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Physics as a Generalized Number Theory

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Year 2011



How quantum arithmetics affects basic TGD and TGD inspired view about life and consciousness?

The vision about real and p-adic physics as completions of rational physics or physics associated with extensions of rational numbers is central element of number theoretical universality. The physics in the extensions of rationals are assigned with the interaction of real and p-adic worlds.

  1. At the level of the world of classical worlds (WCW) the points in the intersection of real and p-adic worlds are 2-surfaces defined by equations making sense both in real and p-adic sense. Rational functions with polynomials having rational (or algebraic coefficients in some extension of rationals) would define the partonic 2-surface. One can of course consider more stringent formulations obtained by replacing 2-surface with certain 3-surfaces or even by 4-surfaces.

  2. At the space-time level the intersection of real and p-adic worlds corresponds to rational points common to real partonic 2-surface obeying same equations (the simplest assumption). This conforms with the vision that finite measurement resolution implies discretization at the level of partonic 2-surfaces and replaces light-like 3-surfaces and space-like 3-surfaces at the ends of causal diamonds with braids so that almost topological QFT is the outcome.

How does the replacement of rationals with quantum rationals modify quantum TGD and the TGD inspired vision about quantum biology and consciousness?

What happens to p-adic mass calculations and quantum TGD?

The basic assumption behind the p-adic mass calculations and all applications is that one can assign to a given partonic 2-surface (or even light-like 3-surface) a preferred p-adic prime (or possibly several primes).

The replacement of rationals with quantum rationals in p-adic mass calculations implies effects, which are extremely small since the difference between rationals and quantum rationals is extremely small due to the fact that the primes assignable to elementary particles are so large (M127=2127-1 for electron). The predictions of p-adic mass calculations remains almost as such in excellent accuracy. The bonus is the uniqueness of the canonical identification making the theory unique.

The problem of the original p-adic mass calculations is that the number of common rationals (plus possible algebraics in some extension of rationals) is same for all primes p. What is the additional criterion selecting the preferred prime assigned to the elementary particle?

Could the preferred prime correspond to the maximization of number theoretic negentropy for a quantum state involved and therefore for the partonic 2-surface by quantum classical correspondence? The solution ansatz for the modified Dirac equation indeed allows this assignment (see this): could this provide the first principle selecting the preferred p-adic prime? Here the replacement of rationals with quantum rationals improves the situation dramatically.

  1. Quantum rationals are characterized by a quantum phase q=exp(i2π/p) and thus by prime p (in the most general but not so plausible case by an integer n). The set of points shared by real and p-adic partonic 2-surfaces would be discrete also now but consist of points in the algebraic extension defined by the quantum phase q=exp(i2π/p).

  2. What is of crucial importance is that the number of common quantum rational points of partonic 2-surface and its p-adic counterpart would depend on the p-adic prime p. For some primes p would be large and in accordance with the original intuition this suggests that the interaction between p-adic and real partonic 2-surface is stronger. This kind of prime is the natural candidate for the p-adic prime defining effective p-adic topology assignable to the partonic 2-surface and elementary particle. Quantum rationals would thus bring in the preferred prime and perhaps at the deepest possible level that one can imagine.

What happens to TGD inspired theory of consciousness and quantum biology?

The vision about rationals as common to reals and p-adics is central for TGD inspired theory of consciousness and the applications of TGD in biology.

  1. One can say that life resides in the intersection of real and p-adic worlds. The basic motivation comes from the observation that number theoretical entanglement entropy can have negative values and has minimum for a unique prime (see this). Negative entanglement entropy has a natural interpretation as a genuine information and this leads to a modification of Negentropy Maximization Principle (NMP) allowing quantum jumps generating negentropic entanglement. This tendency is something completely new: NMP for ordinary entanglement entropy would force always a state function reduction leading to unentangled states and the increase of ensemble entropy.

    What happens at the level of ensemble in TGD Universe is an interesting question. The pessimistic view is that the generation of negentropic entanglement is accompanied by entropic entanglement somewhere else guaranteeing that second law still holds true. Living matter would be bound to pollute its environment if the pessimistic view is correct. I cannot decide whether this is so: this seems like deciding whether Riemann hypothesis is true or not or perhaps unprovable.

  2. Replacing rationals with quantum rationals however modifies somewhat the overall vision about what life is. It would be quantum rationals which would be common to real and p-adic variants of the partonic 2-surface. Also now an algebraic extension of rationals would be in question so that the proposal would be only more specific. The notion of number theoretic entropy still makes sense so that the basic vision about quantum biology survives the modification.

  3. The large number of common points for some prime would mean that the quantum jump transforming p-adic partonic 2-surface to its real counterpart would take place with a large probability. Using the language of TGD inspired theory of consciousness one would say that the intentional powers are strong for the conscious entity involved. This applies also to the reverse transition generating a cognitive representation if p-adic-real duality induced by the canonical identification is true. This conclusion seems to apply even in the case of elementary particles. Could even elementary particles cognize and intend in some primitive sense? Intriguingly, the secondary p-adic time scale associated with electron defining the size of corresponding CD is .1 seconds defining the fundamental 10 Hz bio-rhythm. Just an accident or something very deep: a direct connection between elementary particle level and biology perhaps?

For details and background see the new chapter Quantum Arithmetics and the Relationship between Real and p-Adic Physics.



Quantum Arithmetics and the Relationship between Real and p-Adic Physics

p-Adic physics involves two only partially understood questions.

  1. Is there a duality between real and p-adic physics? What is its precice mathematic formulation? In particular, what is the concrete map p-adic physics in long scales (in real sense) to real physics in short scales? Can one find a rigorous mathematical formulation of canonical identification induced by the map p→ 1/p in pinary expansion of p-adic number such that it is both continuous and respects symmetries.

  2. What is the origin of the p-adic length scale hypothesis suggesting that primes near power of two are physically preferred? Why Mersenne primes are especially important?

A possible answer to these questions relies on the following ideas inspired by the model of Shnoll effect. The first piece of the puzzle is the notion of quantum arithmetics formulated in non-rigorous manner already in the model of Shnoll effect.

  1. Quantum arithmetics is induced by the map of primes to quantum primes by the standard formula. Quantum integer is obtained by mapping the primes in the prime decomposition of integer to quantum primes. Quantum sum is induced by the ordinary sum by requiring that also sum commutes with the quantization.

  2. The construction is especially interesting if the integer defining the quantum phase is prime. One can introduce the notion of quantum rational defined as series in powers of the preferred prime defining quantum phase. The coefficients of the series are quantum rationals for which neither numerator and denominator is divisible by the preferred prime.

  3. p-Adic--real duality can be identified as the analog of canonical identification induced by the map p→ 1/p in the pinary expansion of quantum rational. This maps maps p-adic and real physics to each other and real long distances to short ones and vice versa. This map is especially interesting as a map defining cognitive representations.

Quantum arithmetics inspires the notion of quantum matrix group as counterpart of quantum group for which matrix elements are ordinary numbers. Quantum classical correspondence and the notion of finite measurement resolution realized at classical level in terms of discretization suggest that these two views about quantum groups are closely related. The preferred prime p defining the quantum matrix group is identified as p-adic prime and canonical identification p→ 1/p is group homomorphism so that symmetries are respected.

  1. The quantum counterparts of special linear groups SL(n,F) exists always. For the covering group SL(2,C) of SO(3,1) this is the case so that 4-dimensional Minkowski space is in a very special position. For orthogonal, unitary, and orthogonal groups the quantum counterpart exists only if quantum arithmetics is characterized by a prime rather than general integer and when the number of powers of p for the generating elements of the quantum matrix group satisfies an upper bound characterizing the matrix group.

  2. For the quantum counterparts of SO(3) (SU(2)/ SU(3)) the orthogonality conditions state that at least some multiples of the prime characterizing quantum arithmetics is sum of three (four/six) squares. For SO(3) this condition is strongest and satisfied for all integers, which are not of form n= 22r(8k+7)). The number r3(n) of representations as sum of squares is known and r3(n) is invariant under the scalings n→ 22rn. This means scaling by 2 for the integers appearing in the square sum representation.

  3. r3(n) is proportional to the so called class number function h(-n) telling how many non-equivalent decompositions algebraic integers have in the quadratic algebraic extension generated by (-n)1/2.

The findings about quantum SO(3) suggest a possible explanation for p-adic length scale hypothesis and preferred p-adic primes.

  1. The basic idea is that the quantum matrix group which is discrete is very large for preferred p-adic primes. If cognitive representations correspond to the representations of quantum matrix group, the representational capacity of cognitive representations is high and this kind of primes are survivors‍ in the algebraic evolution leading to algebraic extensions with increasing dimension.

  2. The preferred primes correspond to a large value of r3(n). It is enough that some of their multiples do so (the 22r multiples of these do so automatically). Indeed, for Mersenne primes and integers one has r3(n)=0, which was in conflict with the original expectations. For integers n=2Mm however r3(n) is a local maximum at least for the small integers studied numerically.

  3. The requirement that the notion of quantum integer applies also to algebraic integers in quadratic extensions of rationals requires that the preferred primes (p-adic primes) satisfy p=8k+7. Quite generally, for the integers n=22r(8k+7) not representable as sum of three integers the decomposition of ordinary integers to algebraic primes in the quadratic extensions defined by (-n)1/2 is unique. Therefore also the corresponding quantum algebraic integers are unique for preferred ordinary prime if it is prime also in the algebraic extension. If this were not the case two different decompositions of one and same integer would be mapped to different quantum integers. Therefore the generalization of quantum arithmetics defined by any preferred ordinary prime, which does not split to a product of algebraic primes, is well-defined for p=22r(8k+7).

  4. This argument was for quadratic extensions but also more complex extensions defined by higher polynomials exist. The allowed extensions should allow unique decomposition of integers to algebraic primes. The prime defining the quantum arithmetics should not decompose to algebraic primes. If the algebraic evolution leadis to algebraic extensions of increasing dimension it gradually selects preferred primes as survivors.

For details and background see the new chapter Quantum Arithmetics and the Relationship between Real and p-Adic Physics.



Could TGD be an integrable theory?

During years evidence supporting the idea that TGD could be an integrable theory in some sense has accumulated. The challenge is to show that various ideas about what integrability means form pieces of a bigger coherent picture. Of course, some of the ideas are doomed to be only partially correct or simply wrong. Since it is not possible to know beforehand what ideas are wrong and what are right the situation is very much like in experimental physics and it is easy to claim (and has been and will be claimed) that all this argumentation is useless speculation. This is the price that must be paid for the luxury of genuine thinking.

Integrable theories allow to solve nonlinear classical dynamics in terms of scattering data for a linear system. In TGD framework this translates to quantum classical correspondence. The solutions of modified Dirac equation define the scattering data. The conjecture is that octonionic real-analyticity with space-time surfaces identified as surfaces for which the imaginary part of the biquaternion representing the octonion vanishes solves the field equations. This conjecture generalizes the conformal invariance to its octonionic analog. If this conjecture is correct, the scattering data should define a real analytic function whose octonionic extension defines the space-time surface as a surface for which its imaginary part in the representation as bi-quaternion vanishes. There are excellent hopes about this thanks to the reduction of the modified Dirac equation to geometric optics.

For details and background the reader can consult to the article An attempt to understand preferred extremals of Kähler action and to the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts.



p-Adic fractality, canonical identification, and symmetries

The original motivation for the canonical identification

I: ∑ xnpn →∑ xnp-n,

and its variants - in particular the variant mapping real rationals with the defining integers below a pinary cutoff to p-adic rationals - was that it defines a continuous map from p-adics to reals and produces beautiful p-adic fractals as a map from reals to p-adics by canonical identification followed by a p-adically smooth map in turn followed by the inverse of the canonical identification.

The first drawback was that the map does not commute with symmetries. Second drawback was that the standard canonical identification from reals to p-adics with finite pinary cutoff is two-valued for finite integers. The canonical real images of these transcendentals are also transcendentals. These are however countable whereas p-adic algebraics and transcendentals having by definition a non-periodic pinary expansion are uncountable. Therefore the map from reals to p-adics is single valued for almost all p-adic numbers.

On the other hand, p-adic rationals form a dense set of p-adic numbers and define "almost all" for the purposes of numerics! Which argument is heavier? The direct identification of reals and p-adics via common rationals commutes with symmetries in an approximation defined by the pinary cutoff an is used in the canonical identification with pinary cutoff mapping rationals to rationals.

Symmetries are of extreme importance in physics. Is it possible to imagine the action of say Poincare transformations commuting with the canonical identification in the sets of p-adic and real transcendentals? This might be the case.

  1. Wick rotation is routinely used in quantum field theory to define Minkowskian momentum integrals. One Wick rotates Minkowski space to Euclidian space, performs the integrals, and returns to Minkowskian regime by using the inverse of Wick rotation. The generalization to the p-adic context is highly suggestive. One could map the real Minkowski space to its p-adic counterpart, perform Poincare transformation there, and return back to the real Minkowski space using the inverse of the rational canonical identification.

  2. For p-adic transdendentals one would a formal automorph of Poincare group as IPI-1 and this Poincare group could seen as a fractal counterpart of the ordinary Poincare group. Mathematician would regard I as the analog of intertwining operator, which is linear map between Hilbert spaces. This variant of Poincare symmetry would be exact in the transcendental realm since canonical identification is continuous. For rationals this symmetry would fail.

  3. For rationals which are constructed as ratios of small enough integers, the rational Poincare symmetry with group elements involving rationals constructed from small enough integers would be an exact symmetry. For both options the use of preferred coordinates, most naturally linear Minkowski coordinates would be essential since canonical identification does not commute with general coordinate transformations.

  4. Which of these Poincare symmetries corresponds to the physical Poincare symmetry? The above argument does not make it easy to answer the question. One can however circumvent it. Maybe one could distinguish between rational and transcendental regime in the sense that Poincare group and other symmetries would be realized in different manner in these regimes?

Note that the analog of Wick rotation could be used also to define p-adic integrals by mapping the p-adic integration region to real one by some variant of canonical identification continuously, performing the integral in the real context, and mapping the outcome of the integral to p-adic number by canonical identification. Again preferred coordinates are essential and in TGD framework such coordinates are provided by symmetries. This would allow a numerical treatment of the p-adic integral but the map of the resulting rational to p-adic number would be two valued. The difference between the images would be determined by the numerical accuracy when p-adic expansions are used. This method would be a numerical analog of the analytic definition of p-adic integrals by analytic continuation from the intersection of real and p-adic worlds defined by rational values of parameters appearing in the expressions of integrals.

For details and background see the chapter p-Adic numbers and generalization of number concept.



Could octonion analyticity solve the field equations of quantum TGD?

There are pressing motivations for understanding the preferred extremals of Kähler action. For instance, the conformal invariance of string models naturally generalizes to 4-D invariance defined by quantum Yangian of quantum affine algebra (Kac-Moody type algebra) characterized by two complex coordinates and therefore explaining naturally the effective 2-dimensionality (see this). The problem is however how to assign a complex coordinate with the string world sheet having Minkowskian signature of metric. One can hope that the understanding of preferred extremals could allow to identify two preferred complex coordinates whose existence is also suggested by number theoretical vision giving preferred role for the rational points of partonic 2-surfaces in preferred coordinates. The best one could hope is a general solution of field equations in accordance with the hints that TGD is integrable quantum theory.

A lot is is known about properties of preferred extremals and just by trying to integrate all this understanding, one might gain new visions. The problem is that all these arguments are heuristic and rely heavily on physical intuition. The following considerations relate to the space-time regions having Minkowskian signature of the induced metric. The attempt to generalize the construction also to Euclidian regions could be very rewarding. Only a humble attempt to combine various ideas to a more coherent picture is in question.

The core observations and visions are following.

  1. Hamilton-Jacobi coordinates for M4 (discussed in this chapter) define natural preferred coordinates for Minkowskian space-time sheet and might allow to identify string world sheets for X4 as those for M4. Hamilton-Jacobi coordinates consist of light-like coordinate m and its dual defining local 2-plane M2⊂ M4 and complex transversal complex coordinates (w,w*) for a plane E2x orthogonal to M2x at each point of M4. Clearly, hyper-complex analyticity and complex analyticity are in question.

  2. Space-time sheets allow a slicing by string world sheets (partonic 2-surfaces) labelled by partonic 2-surfaces (string world sheets).

  3. The quaternionic planes of octonion space containing preferred hyper-complex plane are labelled by CP2, which might be called CP2mod (see this). The identification CP2=CP2mod motivates the notion of M8--M4× CP2 duality (see this). It also inspires a concrete solution ansatz assuming the equivalence of two different identifications of the quaternionic tangent space of the space-time sheet and implying that string world sheets can be regarded as strings in the 6-D coset space G2/SU(3). The group G2 of octonion automorphisms has already earlier appeared in TGD framework.

  4. The duality between partonic 2-surfaces and string world sheets in turn suggests that the CP2=CP2mod conditions reduce to string model for partonic 2-surfaces in CP2=SU(3)/U(2). String model in both cases could mean just hypercomplex/complex analyticity for the coordinates of the coset space as functions of hyper-complex/complex coordinate of string world sheet/partonic 2-surface.

The considerations of this section lead to a revival of an old very ambitious and very romantic number theoretic idea, which I already thought to be dead.

  1. To begin with express octonions in the form o=q1+Iq2, where qi is quaternion and I is an octonionic imaginary unit in the complement of fixed a quaternionic sub-space of octonions. Map the preferred coordinates of H=M4× CP2 to octonionic coordinate, form an arbitrary octonion analytic function having expansion with real Taylor or Laurent coefficients to avoid problems due to non-commutativity and non-associativity. Map the outcome to a point of H to get a map H→ H. This procedure is nothing but a generalization of Wick rotation to get an 8-D generalization of analytic map.

  2. Identify the preferred extremals of Kähler action as surfaces obtained by requiring the vanishing of the imaginary part of an octonion analytic function. Partonic 2-surfaces and string world sheets would correspond to commutative sub-manifolds of the space-time surface and of imbedding space and would emerge naturally. The ends of braid strands at partonic 2-surface would naturally correspond to the poles of the octonion analytic functions. This would mean a huge generalization of conformal invariance of string models to octonionic conformal invariance and an exact solution of the field equations of TGD and presumably of quantum TGD itself.

For background see the chapter TGD as Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts and the article An attempt to understand preferred extremals of Kauml;hler action



How to understand transcendental numbers in terms of infinite integers?

Santeri Satama made in my blog a very interesting question about transcendental numbers. The reformulation of Santeri's question could be "How can one know that given number defined as a limit of rational number is genuinely algebraic or transcendental?". I answered to the question and since it inspired a long sequence of speculations during my morning walk on the sands of Tullinniemi I decided to expand my hasty answer to a blog posting.

The basic outcome was the proposal that by bringing TGD based view about infinity based on infinite primes, integers, and rationals one could regard transcendental numbers as algebraic numbers by allowing genuinely infinite numbers in their definition.

  1. In the definition of any transcendental as a limit of algebraic number (root of a polynomial and rational in special case) in which integer n approaches infinity one can replace n with any infinite integer. The transcendental would be an algebraic number in this generalized sense. Among other things this might allow polynomials with degree given by infinite integer if they have finite number of terms. Also mathematics would be generalized number theory, not only physics!

  2. Each infinite integer would give a different variant of the transcendental: these variants would have different number theoretic anatomies but with respect to real norm they would be identical.

  3. This would extend further the generalization of number concept obtained by allowing all infinite rationals which reduce to units in real sense and would further enrich the infinitely rich number theoretic anatomy of real point and also of space-time point. Space-time point would be the Platonia. One could call this number theoretic Brahman=Atman identity or algebraic holography.

1. How can one know that the real number is transcendental?

The difficulty of telling whether given real number defined as a limit of algebraic number boils down to the fact that there is no numerical method for telling whether this kind of number is rational, algebraic, or transcendental. This limitation of numerics would be also a restriction of cognition if p-adic view about it is correct. One can ask several questions. What about infinite-P p-adic numbers: if they make sense could it be possible to cognize also transcendentally? What can we conclude from the very fact that we cognize transcendentals? Transcendentality can be proven for some transcendentals such as π. How this is possible? What distinguishes "knowably transcendentals" like π and e from those, which are able to hide their real number theoretic identity?

  1. Certainly for "knowably transcendentals" there must exist some process revealing their transcendental character. How π and e are proven to be transcendental? What in our mathematical cognition makes this possible? First of all one starts from the definitions of these numbers. e can be defined as the limit of the rational number (1+1/n)n and 2π could be defined as the limit for the length of the circumference of a regular n-side polygon and is a limit of an algebraic number since Pythagoras law is involved in calculating the length of the side. The process of proving "knowable transcendentality" would be a demonstration that these numbers cannot be solutions of any polynomial equation.

  2. Squaring of circle is not possible because π is transcendental. When I search Wikipedia for squaring of circle I find a link to Weierstrass theorem allowing to prove that π and e are transcendentals. In the formulation of Baker this theorem states the following: If α1,...,αn are distinct algebraic numbers then the numbers eα1,...,eαn are linearly independent over algebraic numbers and therefore transcendentals. One says that the extension Q(eα1,...,eαn) of rationals has transcendence degree n over Q. This is something extremely deep and unfortunately I do not know what is the gist of the proof. In any case the proof defines a procedure of demonstrating "knowable transcendentality" for these numbers. The number of these transcendentals is huge but countable and therefore vanishingly small as compared to the uncountable cardinality of all transcendentals.

  3. This theorem allows to prove that π and e are transcendentals. Suppose on the contrary that π is algebraic number. Then also iπ would be algebraic and the previous theorem would imply that e=-1 is transcendental. This is of course a contradiction. Theorem also implies that e is transcendental. But how do we know that e=-1 holds true? Euler deduced this from the connection between exponential and trigonometric functions understood in terms of complex analysis and related number theory. Clearly, rational functions and exponential function and its inverse -logarithm- continued to complex plane are crucial for defining e and π and proving also e=-1. Exponent function and logarithm appear everywhere in mathematics: in group theory for instance. All these considerations suggest that "knowably transcendental" is a very special mathematical property and deserves a careful analysis.

2. Exponentiation and formation of set of subsets as transcendence

What is so special in exponentiation? Why it sends algebraic numbers to "knowably transcendentals". One could try to understand this in terms of exponentiation which for natural numbers has also an interpretation in terms of power set just as product has interpretation in terms of Cartesian product.

  1. In Cantor's approach to the notion of infinite ordinals exponentiation is involved besides sum and product. All three binary operations - sum, product, exponent are expressed set theoretically. Product and sum are "algebraic" operations. Exponentiation is "non-algebraic" binary operation defined in terms of power set (set of subsets). For m and n definining the cardinalities of sets X and Y, mn defines the cardinality of the set YX defining the number of functions assigning to each point of Y a point of X. When X is two-element set (bits 0 and 1) the power set is just the set of all subsets of Y which bit 1 assigned to the subset and 0 with its complement. If X has more than two elements one can speak of decompositions of Y to subsets colored with different colors- one color for each point of X.

  2. The formation of the power set (or of its analog for the number of colors larger than 2) means going to the next level of abstraction: considering instead of set the set of subsets or studying the set of functions from the set. In the case of Boolean algebras this means formation of statements about statements. This could be regarded as the set theoretic view about transcendence.

  3. What is interesting that 2-adic integers would label the elements of the power set of integers (all possible subsets would be allowed, for finite subsets one would obtain just natural numbers) and p-adic numbers the elements in the set formed by coloring integers with p colors. One could thus say that p-adic numbers correspond naturally to the notion of cognition based on power sets and their finite field generalizations.

  4. But can one naively transcend the set theoretic exponent function for natural numbers to that defined in complex plane? Could the "knowably transcendental" property of numbers like e and π reduce to the transcendence in this set theoretic sense? It is difficult to tell since this notion of power applies only to integers m,n rather than to a pair of transcendentals e,π. Concretization of e in terms of sets seems impossible: it is very difficult to imagine what sets with cardinality e and π could be.

3. Infinite primes and transcendence

TGD suggests also a different identification of transcendence not expressible as formation of a power set or its generalizations.

  1. The notion of infinite primes replaces the set theoretic notion of infinity with purely number theoretic one.

    1. The mathematical motivation could be the need to avoid problems like Russell's antinomy. In Cantorian world a given ordinal is identified as the ordered set of all ordinals smaller than it and the set of all ordinals would define an ordinal larger than every ordinal and at the same time member of all ordinals.

    2. The physical motivation for infinite primes is that their construction corresponds to a repeated second quantization of an arithmetic supersymmetric quantum field theory such that the many particle states of the previous level become elementary particles of the new level. At the lowest level finite primes label fermionic and bosonic states. Besides free many-particle states also bound states are obtained and correspond at the first level of the hierarchy to genuinely algebraic roots of irreducible polynomials.

    3. The allowance of infinite rationals which as real numbers reduce to real units implies that the points of real axes have infinitely rich number theoretical anatomy. Space-time point would become the Platonia. One could speak of number theoretic Brahman=Atman identity or algebraic holography. The great vision is that the World of Classical Worlds has a mathematical representation in terms of the number theoretical anatomy of space-time point.

  2. Transcendence in purely number theoretic sense could mean a transition to a higher level in the hierarchy of infinite primes. The scale of new infinity defined as the product of all prime at the previous level of hierarchy would be infinitely larger than the previous one. Quantization would correspond to abstraction and transcendence.

This idea inspires some questions.

  1. Could infinite integers allow the reduction of transcendentals to algebraic numbers when understood in general enough sense. Could real algebraic numbers be reduced to infinite rationals with finite real values (for complex algebraic numbers this is certainly not the case)? If so, then all real numbers would be rationals identified as ratios of possibly infinite integers and having finite value as real numbers? This turns out to be too strong a statement. The statement that all real numbers can be represented as finite or infinite algebraic numbers looks however sensible and would reduce mathematics to generalized number theory by reducing limiting procedure involved with the transition from rationals to reals to algebraic transcendence. This applies also to p-adic numbers.

  2. p-Adic cognition for finite values of prime p does not explain why we have the notions of π and e and more generally, that of transcendental number. Could the replacement of finite-p p-adic number fields with infinite-P p-adic number fields allow us to understand our own mathematical cognition? Could the infinite-P p-adic number fields or at least integers and corresponding space-time sheets make possible mathematical cognition able to deduce analytic formulas in which transcendentals and transcendental functions appear making it possible to leave the extremely restricted realm of numerics and enter the realm of mathematics? Lie group theory would represent a basic example of this transcendental aspect of cognition. Maybe this framework might allow to understand why we can have the notion of transcendental number!

4. Identification of real transcendentals as infinite algebraic numbers with finite value as real numbers

The following observations suggests that it could be possible to reduce transcendentals to generalized algebraic numbers in the framework provided by infinite primes. This would mean that not only physics but also mathematics (or at least "physical mathematics") could be seen as generalized number theory.

  1. In the definition of any transcendental as an n→ ∞ limit of algebraic number (root of a polynomial and rational in special case), one can replace n with any infinite integer if n appears as an argument of a function having well defined value at this limit. If n appears as the number of summands or factors of product, the replacement does not make sense. For instance, an algebraic number could be defined as a limit of Taylor series by solving the polynomial equation defining it. The replacement of the upper limit of the series with infinite integer does not however make sense. Only transcendentals (and possibly also some algebraic numbers) allowing a representation as n→ ∞ limit with n appearing as argument of expression involving a finite number of terms can have representation as infinite algebraic number. The rule would be simple.

    Transcendentals or algebraic numbers allowing an identification as infinite algebraic number must correspond to a term of a sequence with a fixed number of terms rather than sum of series or infinite product.

  2. Each infinite integer gives a different variant of the transcendental: these variants would have different number theoretic anatomies but with respect to the real norm they would be identical.

  3. The heuristic guess is that any genuine algebraic number has an expression as Taylor series obtained by writing the solution of the polynomial equation as Tarylor expansion. If so, algebraic numbers must be introduced in the standard manner and do not allow a representation as infinite rationals. Only transcendentals would allow a representation as infinite rationals or infinite algebraic numbers. The infinite variety of representation in terms of infinite integers would enormously expand the number theoretical anatomy of the real point. Do all transcendentals allow an expression containing a finite number of terms and N appearing as argument? Or is this the defining property of only "knowably transcendentals"?

One can consider some examples to illustrate the situation.

  1. The transcendental π could be defined as πN=-iN(eiπ/N-1), where eiπ/N is N:th root of unity for infinite integer N and as a real number real unit. In real sense the limit however gives π. There are of course very many definitions of π as limits of algebraic numbers and each gives rise to infinite variety of number theoretic anatomies of π.

  2. One can also consider the roots exp(i2π n/N) of the algebraic equation xN=1 for infinite integer N. One might define the roots as limits of Taylor series for the exponent function but it does not make sense to define the limit when the cutoff for the Taylor series approaches some infinite integer. These roots would have similar multiplicative structure as finite roots of unity with pn:th roots with p running over primes defining the generating roots. The presence of Nth roots of unity f or infinite N would further enrich the infinitely rich number theoretic anatomy of real point and therefore of space-time points.

  3. There would be infinite variety of Neper numbers identified as eN=(1+1/N)N, N any infinite integer. Their number theoretic anatomies would be different but as real numbers they would be identical.

To conclude, the talk about infinite primes might sound weird in the ears of a layman but mathematicians do not lose their peace of mind when they here the word "infinity". The notion of infinity is relative. For instance, infinite integers are completely finite in p-adic sense. One can also imagine completely "real-worldish" realizations for infinite integers (say as states of repeatedly second quantized arithmetic quantum field theory and this realization might provide completely new insights about how to undestand bound states in ordinary QFT).

For details and background see the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?.



About the structure of the Yangian algebra

The attempt to understand Langlands conjecture in TGD framework (see this) led to a completely unexpected progress in the understanding of the Yangian symmetry expected to be the basic symmetry of quantum TGD (see this) and the following vision suggesting how conformal field theory could be generalized to four-dimensional context is a fruit of this work.

The structure of the Yangian algebra is quite intricate and in order to minimize confusion easily caused by my own restricted mathematical skills it is best to try to build a physical interpretation for what Yangian really is and leave the details for the mathematicians.

  1. The first thing to notice is that Yangian and quantum affine algebra are two different quantum deformations of a given Lie algebra. Both rely on the notion of R-matrix inducing a swap of braid strands. R-matrix represents the projective representations of the permutation group for braid strands and possible in 2-dimensional case due to the non-commutativity of the first homotopy group for 2-dimensional spaces with punctures. The R-matrix Rq(u,v) depends on complex parameter q and two complex coordinates u,v. In integrable quantum field theories in M2 the coordinates u,v are real numbers having identification as exponentials representing Lorenz boosts. In 2-D integrable conformal field theory the coordinates u,v have interpretation as complex phases representing points of a circle. The assumption that the coordinate parameters are complex numbers is the safest one.

  2. For Yangian the R-matrix is rational whereas for quantum affine algebra it is trigonometric. For the Yangian of a linear group quantum deformation parameter can be taken to be equal to one by a suitable rescaling of the generators labelled by integer by a power of the complex quantum deformation parameter q. I do not know whether this true in the general case. For the quantum affine algebra this is not possible and in TGD framework the most interesting values of the deformation parameter correspond to roots of unity.

Slicing of space-time sheets to partonic 2-surfaces and string world sheets

The proposals is t the preferred extremals of Kähler action are involved in an essential manner the slicing of the space-time sheets by partonic 2-surfaces and string world sheets. Also an analogous slicing of Minkowski space is assumed and there are infinite number of this kind of slicings defining what I have called Hamilton-Jaboci coordinates (see this). What is really involved is far from clear. For instance, I do not really understand whether the slicings of the space-time surfaces are purely dynamical or induced by special coordinatizations of the space-time sheets using projections to special kind of sub-manifolds of the imbedding space, or are these two type of slicings equivalent by the very property of being a preferred extremal. Therefore I can represent only what I think I understand about the situation.

  1. What is needed is the slicing of space-time sheets by partonic 2-surfaces and string world sheets. The existence of this slicing is assumed for the preferred extremals of Kähler action (see this). Physically the slicing corresponds to an integrable decomposition of the tangent space of space-time surface to 2-D space representing non-physical polarizations and 2-D space representing physical polarizations and has also number theoretical meaning.

  2. In zero energy ontology the complex coordinate parameters appearing in the generalized conformal fields should correspond to coordinates of the imbedding space serving also as local coordinates of the space-time surface. Problems seem to be caused by the fact that for string world sheets hyper-complex coordinate is more natural than complex coordinate. Pair of hyper-complex and complex coordinate emerge naturally as Hamilton-Jacobi coordinates for Minkowski space encountered in the attempts to understand the construction of the preferred extremals of Kähler action.

    Also the condition that the flow lines of conserved isometry currents define global coordinates lead to the to the analog of Hamilton-Jacobi coordinates for space-time sheets (see this). The physical interpretation is in terms of local polarization plane and momentum plane defined by local light-like direction. What is so nice that these coordinates are highly unique and determined dynamically.

  3. Is it really necessary to use two complex coordinates in the definition of Yangian-affine conformal fields? Why not to use hyper-complex coordinate for string world sheets? Since the inverse of hyper-complex number does not exist when the hyper-complex number is light-like, hyper-complex coordinate should appear in the expansions for the Yangian generalization of conformal field as positive powers only. Intriguingly, the Yangian algebra is "one half" of the affine algebra so that only positive powers appear in the expansion. Maybe the hyper-complex expansion works and forces Yangian-affine instead of doubly affine structure. The appearance of only positive conformal weights in Yangian sector could also relate to the fact that also in conformal theories this restriction must be made.

  4. It seems indeed essential that the space-time coordinates used can be regarded as imbedding space coordinates which can be fixed to a high degree by symmetries: otherwise problems with general coordinate invariance and with number theoretical universality would be encountered.

  5. The slicing by partonic 2-surfaces could (but need not) be induced by the slicing of CD by parallel translates of either upper or lower boundary of CD in time direction in the rest frame of CD (time coordinate varying in the direction of the line connecting the tips of CD). These slicings are not global. Upper and lower boundaries of CD would definitely define analogs of different coordinate patches.

Physical interpretation of the Yangian of quantum affine algebra

What the Yangian of quantum affine algebra or more generally, its super counterpart could mean in TGD framework? The key idea is that this algebra would define a generalization of super conformal algebras of super conformal field theories as well as the generalization of super Virasoro algebra. Optimist could hope that the constructions associated with conformal algebras generalize: this includes the representation theory of super conformal and super Virasoro algebras, coset construction, and vertex operator construction in terms of free fields. One could also hope that the classification of extended conformal theories defined in this manner might be possible.

  1. The Yangian of a quantum affine algebra is in question. The heuristic idea is that the two R-matrices - trigonometric and rational- are assignable to the swaps defined by space-like braidings associated with the braids at 3-D space-like ends of space-time sheets at light-like boundaries of CD and time like braidings associated with the braids at 3-D light-like surfaces connecting partonic 2-surfaces at opposite light-like boundaries of CD. Electric-magnetic duality and S-duality implied by the strong form of General Coordinate Invariance should be closely related to the presence of two R-matrices. The first guess is that rational R-matrix is assignable with the time-like braidings and trigonometric R-matrix with the space-like braidings. Here one must or course be very cautious.

  2. The representation of the collection of Galois groups associated with infinite primes in terms of braided symplectic flows for braid of braids of .... braids implies that there is a hierarchy of swaps: swaps can also exchange braids of ...braids. This would suggest that at the lowest level of the braiding hierarchy the R-matrix associated with a Kac-Moody algebra permutes two braid strands which decompose to braids. There would be two different braided variants of Galois groups.

  3. The Yangian of the affine Kac-Moody algebra could be seen as a 4-D generalization of the 2-D Kac-Moody algebra- that is a local algebra having representation as a power series of complex coordinates defined by the projections of the point of the space-time sheet to geodesic spheres of light-cone boundary and geodesic sphere of CP2.

  4. For the Yangian the generators would correspond to polynomials of the complex coordinate of string world sheet and for quantum affine algebra to Laurent series for the complex coordinate of partonic 2-surface. What the restriction to polynomials means is not quite clear. Witten sees Yangian as one half of Kac-Moody algebra containing only the generators having n≥ 0. This might mean that the positivity of conformal weight for physical states essential for the construction of the representations of Virasoro algebra would be replaced with automatic positivity of the conformal weight assignable to the Yangian coordinate.

  5. Also Virasoro algebra should be replaced with the Yangian of Virasoro algebra or its quantum counterpart. This construction should generalize also to Super Virasoro algebra. A generalization of conformal field theory to a theory defined at 4-D space-time surfaces using two preferred complex coordinates made possible by surface property is highly suggestive. The generalization of conformal field theory in question would have two complex coordinates and conformal invariance associated with both of them. This would therefore reduce the situation to effectively 2-dimensional one rather than 3-dimensional: this would be nothing but the effective 2-dimensionality of quantum TGD implied by the strong form of General Coordinate Invariance.

  6. This picture conforms with what the generalization of D=4 N=4 SYM by replacing point like particles with partonic 2-surfaces would suggest: Yangian is replaced with Yangian of quantum affine algebra rather than quantum group. Note that it is the finite measurement resolution alone which brings in the quantum parameters q1 and q2. The finite measurement resolution might be relevant for the elimination of IR divergences.

How to construct the Yangian of quantum affine algebra?

The next step is to try to understand the construction of the Yangian of quantum affine algebra.

  1. One starts with a given Lie group G. It could be the group of isometries of the imbedding space or subgroup of it or even the symplectic group of the light-like boundary of CD× CP2 and thus infinite-dimensional. It could be also the Lie group defining finite measurement resolution with the dimension of Cartan algebra determined by the number of braid strands.

  2. The next step is to construct the affine algebra (Kac-Moody type algebra with central extension). For the group defining the measurement resolution the scalar fields assigned with the ends of braid strands could define the Cartan algebra of Kac-Moody type algebra of this group. The ordered exponentials of these generators would define the charged generators of the affine algebra.

    For the imbedding space isometries and symplectic transformations the algebra would be obtained by localizing with respect to the internal coordinates of the partonic 2-surface. Note that also a localization with respect to the light-like coordinate of light-cone boundary or light-like orbit of partonic 2-surface is possible and is strongly suggested by the effective 2-dimensionality of light-like 3-surfaces allowing extension of conformal algebra by the dependence on second real coordinate. This second coordinate should obviously correspond to the restriction of second complex coordinate to light-like 3-surface. If the space-time sheets allow slicing by partonic 2-surfaces and string world sheets this localization is possible for all 2-D partonic slices of space-time surface.

  3. The next step is quantum deformation to quantum affine algebra with trigonometric R-matrix Rq1(u,v) associated with space-like braidings along space-like 3-surfaces along the ends of CD. u and v could correspond to the values of a preferred complex coordinate of the geodesic sphere of light-cone boundary defined by rotational symmetry. It choice would fix a preferred quantization axes for spin.

  4. The last step is the construction of Yangian using rational R-matrix Rq2(u,v). In this case the braiding is along the light-like orbit between ends of CD. u and v would correspond to the complex coordinates of the geodesic sphere of CP2. Now the preferred complex coordinate would fix the quantization axis of color isospin.

These arguments are of course heuristic and do not satisfy any criteria of mathematical rigor and the details could of course change under closer scrutinity. The whole point is in the attempt to understand the situation physically in all its generality. The most important outcome is the conjecture that the incredibly powerful mathematical apparatus of 2-dimensional conformal field theories might have a generalization to four-dimensional context based on Yangians of quantum affined algebras. This might explain the miracles of both twistor approach and string approach.

How 4-D generalization of conformal invariance relates to strong form of general coordinate invariance?

The basic objections that one can rise to the extension of conformal field theory to 4-D context come from the successes of p-adic mass calculations. p-Adic thermodynamics relies heavily on the properties of partition functions for super-conformal representations. What happens when one replaces affine algebra with (quantum) Yangian of affine algebra? Ordinary Yangian involves the original algebra and its dual and from these higher multilocal generators are constructed. In the recent case the obvious interpretation for this would be that one has Kac-Moody type algebra with expansion with respect to complex coordinate w for partonic 2-surfaces and its dual algebra with expansion with respect to hyper-complex coordinate of string world sheet.

p-Adic mass calculations suggest that the use of either algebra is enough to construct single particle states. Or more precisely, local generators are enough. I have indeed proposed that the multilocal generators are relevant for the construction of bound states. Also the strong form of general coordinate invariance implying strong form of holography, effective 2-dimensionality, electric-magnetic duality and S-duality suggest the same. If one could construct the states representing elementary particles solely in terms of either algebra, there would be no danger that the results of p-adic mass calculations are lost. Note that also the necessity to restrict the conformal weights of conformal representations to be non-negative would have nice interpretation in terms of the duality.

For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article Langlands conjectures in TGD framework.



Is quantal Boolean reverse engineering possible?

The quantal version of Boolean algebra means that the basic logical functions have quantum inverses. The inverse of C=A ∧ B represents the quantum superposition of all pairs A and B for which A∧ B=C hols true. Same is true for ∨. How could these additional quantum logical functions with no classical counterparts extend the capacities of logician?

What comes in mind is logical reverse engineering. Consider the standard problem solving situation repeatedly encountered by my hero Hercule Poirot. Someone has been murdered. Who could have done it? Who did it? Actually scientists who want to explain instead of just applying the method to get additional items to the CVC, meet this kind of problem repeatedly. One has something which looks like an experimental anomaly and one has to explain it. Is this anomaly genuine or is it due to a systematic error in the information processing? Could the interpretation of data be somehow wrong? Is the model behind experiments based on existing theory really correct or has something very delicate been neglected? If a genuine anomaly is in question (someone has been really murdered- this is always obvious in the tales about the deeds of Hercule Poirot since the mere presence of Hercule guarantees the murder unless it has been already done) one encounters what might be called Poirot problem in honor of my hero. As a matter fact, from the point of view of Boolean algebra, one has the same reverse Boolean engineering problem irrespective of whether it was a genuine anomaly or not.

This brings in my mind the enormously simplified problem. The logical statement C is found to be true. Which pairs A,B could have implied C as C=A∧ B (or A∨ B). Of course, much more complex situations can be considered where C corresponds to some logical function C=f(A1,A2,...,An). Quantum Poirot could use quantum computer able to realize the co-gates for gates AND and OR (essentially time reversals) and write a quantum computer program solving the problem by constructing the Boolean co-function of Boolean function f.

What would happen in TGD Universe obeying zero energy ontology is following.

  1. The statement C is represented as as positive energy part of zero energy state (analogous to initial state of physical event) and A1,..An is represented as one state in the quantum superposition of final states representing various value combinations for A1,...,An. Zero energy states (rather than only their evolution) represets the arrow of time. The M-matrix characterizing time-like entanglement between positive and negative energy states generalizes generalizes S-matrix. S-matrix is such that initial states have well defined particle numbers and other quantum numbers whereas final states do not. They are analogous to the outcomes of quantum measurement in particle physics.

  2. Negentropy Maximization Principle maximizing the information contents of conscious experience (sic!) forces state function reduction to one particular A1,...,An and one particular value combination consistent with C is found in each state function reduction. At the ensemble level one obtains probabilities for various outcomes and the most probable combination might represent the most plausible candidate for the murderer in quantum Poirot problem. Also in particle physics one can only speak about plausibility of the explanation and this leads to the endless n sigma talk. Note that it is absolutely essential that state function reduction occurs. Ironically, quantum problem solving causes dissipation at the level of ensemble but the ensemble probabilities carry actually information! Second law of thermodynamics tells us that Nature is a pathological problem solver- just like my hero!

  3. In TGD framework basic logical binary operations have a representation at the level of Boolean algebra realized in terms of fermionic oscillator operators. They have also space-time correlates realized topologically. ∧ has a representation as the analog of three-vertex of Feynman graph for partonic 2-surfaces: partonic 2-surfaces are glued along the ends to form outgoing partonic 2-surface. ∨ has a representation as the analog of stringy trouser vertex in which partonic surfaces fuse together. Here TGD differs from string models in a profound manner.

To conclude, I am a Boolean dilettante and know practically nothing about what quantum computer theorists have done- in particular I do not know whether they have considered quantum inverse gages. My feeling is that only the gates with bits replaced with qubits are considered: very natural when one thinks in terms of Boolean logic. If this is really the case, quantal co-AND and co-OR having no classical counterparts would bring a totally new aspect to quantum computation in solving problems in which one cannot do without (quantum) Poirot and his little gray (quantum) brain cells.

For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?.



TGD and Physical Mathematics

I discussed in What's New the TGD based vision about Langlands program. I have actually written several blog postings (and What's News) related to the relationship between TGD and physical mathematics during this year (see this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, and this). I try also to become (and remain!) conscious about possible sources of inconsistencies to see what might go wrong.

I see the attempt to understand the relation between Langlands program and TGD as a part of a bigger project the goal of which is to relate TGD to physical mathematics. The basic motivations come from the mathematical challenges of TGD and from the almost-belief that the beautiful mathematical structures of the contemporary physical mathematics must be realized in Nature somehow.

The notion of infinite prime is becoming more and more important concept of quantum TGD and also a common denominator. The infinite-dimensional symplectic group acting as the isometry group of WCW geometry and symplectic flows seems to be another common denominator. Zero energy ontology together with the notion of causal diamond is also a central concept. A further common denominator seems to be the notion of finite measurement resolution allowing discretization. Strings and super-symmetry so beautiful notions that it is difficult to imagine physics without them although super string theory has turned out to be a disappointment in this respect. In the following I mention just some examples of problems that I have discussed during this year.

Infinite primes are certainly something genuinely TGD inspired and it is reasonable to consider their possible role in physical mathematics.

  1. The set theoretic view about the fundamentals of mathematics is inspired by classical physics. Cantor's view about infinite ordinals relies on set theoretic representation of ordinals and is plagued by difficulties (say Russel's paradox) (see this). Infinite primes provide an alternative to Cantor's view about infinity based on divisility alone and allowing to avoid these problems. Infinite primes are obtained by a repeated second quantization of an arithmetic quantum field theory and can be seen as a notion inspired by quantum physics. The conjecture is that quantum states in TGD Universe can be labelled by infinite primes and that standard model symmetries can be understood in terms of octonionic infinite primes defined in appropriate manner.

    The replacement of ordinals with infinite primes would mean a modification of the fundamentals of physical mathematics. The physicists's view about the notion set is also much more restricted than the set theoretic view. Subsets are typically manifolds or even algebraic varieties and they allow description in terms of partial differential equations or algebraic equations.

    Boolean algebra is the quintessence of mathematical logic and TGD suggests that quantum Boolean algebra should replace Boolean algebra (see this). The representation would be in terms of fermionic Fock states and in zero energy ontology fermionic parts of the state would define Boolean states of form A→ B. This notion might be useful for understanding the physical correlates of Boolean cognition and might also provide insights about fundamentals of physical mathematics itself. Boolean cognition must have space-time correlates and this leads to a space-time description of logical OR resp. AND as a generalization of trouser diagram of string models resp. fusion along ends of partonic 2-surfaces generalizing the 3-vertex of Feyman diagrammatics. These diagrams would give rise to fundamental logic gates.

  2. Infinite primes can be represented using polynomials of several variables with rational coefficients (see this). One can solve the zeros of these polynomials iteratively. At each step one can identify a finite Galois group permuting the roots of the polynomial (algebraic function in general). The resulting Galois groups can be arranged into a hierarchy of Galois groups and the natural idea is that the Galois groups at the upper levels act as homomorphisms of Galois groups at lower levels. A generalization of homology and cohomology theories to their non-Abelian counterparts emerges (see this): the square of the boundary operation yields unit element in normal homology but now an element in commutator group so that abelianization yields ordinary homology. The proposal is that the roots are represented as punctures of the partonic 2-surfaces and that braids represent symplectic flows representing the braided counterpars of the Galois groups. Braids of braids of.... braids structrure of braids is inherited from the hierarchical structure of infinite primes.

    That braided Galois groups would have a representation as symplectic flows is exactly what physics as generalized number theory vision suggests and is applied also to understand Langlands conjectures. Langlands program would be modified in TGD framework to the study of the complexes of Galois groups associated with infinite primes and integers and have direct physical meaning.

The notion of finite measurement resolution realized at quantum level as inclusions of hyper-finite factors and at space-time level in terms of braids replacing the orbits of partonic 2-surfaces - is also a purely TGD inspired notion and gives good hopes about calculable theory.

  1. The notion of finite measurement resolution leads to a rational discretization needed by both the number theoretic and geometric Langlands conjecture. The simplest manner to understand the discretization is in terms of extrema of Chern-Simons action if they correspond to "rational" surfaces. The guess that the rational surfaces are dense in the WCW just as rationals are dense in various number fields is probably quite too optimistic physically. Algebraic partonic 2-surfaces containin typically finite number of rational points having interpretation in terms of finite measurement resolution. Same might apply to algebraic surfaces as points of WCW in given quantum state.

  2. The charged generators of the Kac-Moody algebra associated with the Lie group G defining measurement resolution correspond to tachyonic momenta in free field reprsesentation using ordered exponentials. This raises unpleasant question. One should have also a realization for the coset construction in which Kac-Moody variant of the symplectic group of δ M4+/- and Kac-Moody algebra of isometry group of H assignable to the light-like 3-surfaces (isometries at the level of WCW resp. H) define a coset representation. Equivalence Principle generalizes to the condition that the actions of corresponding super Virasoro algebras are identical. Now the momenta are however non-tachyonic.

    How these Kac-Moody type algebras relate? From p-adic mass calculations it is clear that the ground states of super-conformal representations have tachyonic conformal weights. Does this mean that the ground states can be organized into representations of the Kac-Moody algebra representing finite measurement resolution? Or are the two Kac-Moody algebra like structures completely independent. This would mean that the positions of punctures cannot correspond to the H-coordinates appearing as arguments of sympletic and Kac-Moody algebra giving rise to Equivalence Principle. The fact that the groups associated with algebras are different would allow this.

TGD is a generalization of string models obtained by replacing strings with 3-surfaces. Therefore it is not surprising that stringy structures should appear also in TGD Universe and the strong form of general coordinate invariance indeed implies this. As a matter fact, string like objects appear also in various applications of TGD: consider only the notions of cosmic string (see this) and nuclear string (see this). Magnetic flux tubes central in TGD inspired quantum biology making possible topological quantum computation (see this) represent a further example.

  1. What distinguishes TGD approach from Witten's approach is that twisted SUSY is replaced by string model like theory with strings moving in the moduli space for partonic 2-surfaces or string world sheets related by electric-magnetic duality. Higgs bundle is replaced with the moduli space for punctured partonic 2-surfaces and its electric dual for string world sheets. The new element is the possibility of trouser vertices and generalization of 3-vertex if Feynman diagrams having interpretation in terms of quantum Boolean algebra.

  2. Stringy view means that all topologies of partonic 2-surfaces are allowed and that also quantum superpositions of different topologies are allowed. The restriction to single topology and fixed moduli would mean sigma model. Stringy picture requires quantum superposition of different moduli and genera and this is what one expects on physical grounds. The model for CKM mixing indeed assumes that CKM mixing results from different topological mixings for U and D type quarks (see this) and leads to the notion of elementary particle vacuum functional identifiable as a particular automorphic form (see this).

  3. The twisted variant of N=4 SUSY appears as TQFT in many mathematical applications proposed by Witten and is replaced in TGD framework by the stringy picture. Supersymmetry would naturally correspond to the fermionic oscillator operator algebra assignable to the partonic 2-surfaces or string world sheet and SUSY would be broken.

When I look what I have written about various topics during this year I find that symplectic invariance and symplectic flows appear repeatedly.

  1. Khovanov homology provides very general knot invariants. I rephrased Witten's formulation about Khovanov homology as TQFT in TGD framework here. Witten's TQFT is obtained by twisting a 4-dimensional N=4 SYM. This approach generalizes the original 3-D Chern-Simons approach of Witten. Witten applies twisted 4-D N=4 SYM also to geometric Langlands program and to Floer homology.

    TGD is an almost topological QFT so that the natural expectation is that it yields as a side product knot invariants, invariants for braiding of knots, and perhaps even invariants for 2-knots: here the dimension D=4 for space-time surface is crucial. One outcome is a generalization of the notion of Wilson loop to its 2-D variant defined by string world sheetw and a unique identification of string world sheet for a given space-time surface. The duality between the descriptions based on string world sheets and partonic 2-surfaces is central. I have not yet discussed the implications of the conjectures inpired by Langlands program for the TGD inspired view about knots.

  2. Floer homology generalizes the usual Morse theory and is one of the applications of topological QFTs discussed by Witten using twisted SYM. One studies symplectic flows and the basic objects are what might regarded as string world sheets referred to as pseudo-holomorphic surfaces. It is now wonder that also here TGD as almost topological QFT view leads to a generalization of the QFT vision about Floer homology (see this). The new result from TGD point of view was the realization that the naivest possible interpretation for Kähler action for a preferred extremal is correct. The contribution to Kähler action from Minkowskian regions of space-time surface is imaginary and has identification as Morse function whereas Euclidian regions give the real contribution having interpretation as Kähler function. Both contributions reduce to 3-D Chern-Simons terms and under certain additional assumptions only the wormhole throats at which the signature of the induced metric changes from Minkowskian to Euclidian contribute.

  3. Gromov-Witten invariants are closely related to Floer homology and their definition involves quantum cohomology in which the notion of intersection for two varieties is more general taking into account "quantum fuzzines". The stringy trouser vertex represent the basic diagram: the incoming string world sheets intersect because they can fuse to single string world sheet. Amazingy, this is just that OR in quantum Boolean algebra suggested by TGD. Another diagram would be AND responsible for genuine particle reactions in TGD framework. There would be a direct connection with quantum Boolean algebra.

Number theoretical universality is one of the corner stones of the vision about physics as generalized number theory. One might perhaps say that a similar vision has guided Grothendieck and his followers.

  1. The realization of this vision involves several challenges. One of them is definition of p-adic integration. At least integration in the sense of comology is needed and one might also hope that numerical approach to integration exists. It came as a surprise to me that something very similar to number theoretical universality has inspired also mathematicians and that there exist refined theories inspired by the notion of motive introduced by Groethendieck to define universal cohomology applying in all number fields. One application and also motivation for taking motives very seriously is notivic integration which has found applications in in physics as a manner to calculate twistor space integrals defining scattering amplitudes in twistor approach to N=4 SUSY. The essence of motivic integral is that integral is an algebraic operation rather than defined by a measure. One ends up with notions like scissor group and integration as processing of symbols. This is of course in spirit with number theoretical approach where integral as measure is replaced with algebraic operation. The problem is that numerics made possible by measure seems to be lost.

  2. The TGD inspired proposal for the definition of p-adic integral relies on number theoretical universality reducing the integral essentially to integral in the rational intersection of real and p-adic worlds. An essential role is played at the level of WCW by the decomposition of WCW to a union of symmetric spaces allowing to define what the p-adic variant of WCW is. Also this would conform with the vision that infinite-dimensional geometric existence is unique just from the requirement that it exists. One can consider also the possibility of having p-adic variant of numerical integration (see this).

Twistor approach has led to the emergence of motives to physics and twistor approach is also what gives hopes that some day quantum TGD could be formulated in terms of explicit Feynman rules or their twistorial generalization (see this and this).

  1. The Yangian symmetry discovered first in integrable quantum theories is responsible for the success fo the twistorial approach. What distinguishes Yangian symmetry from standard symmetries is that the generators of Lie algebra are multilocal. Yangian symmetry is generalized in TGD framework since point like particles are replaced by partonic 2-surfaces meaning that Lie group is replaced with Kac-Moody group or its generalization. Finite measurement resolution however replaces them with discrete set of points definining braid strands so that a close connection with twistor approach and ordinary Yangian symmetry is suggestive in finite measurement resolution. Also the fact that Yangian symmetry relates closely to topological string models supports the expectation that the proposed stringy view about quantum TGD could allow to formulate twistorial approach to TGD.

  2. The vision about finite measurement resolution represented in terms of effective Kac-Moody algebra defined by a group with dimension of Cartan algebra given by the number of braid strands must be consistent with the twistorial picture based on Yangians and this requires extension to Yangian algebra. In Yangian picture one cannot speak about single partonic 2-surface alone and the same is true about the TGD based generalization of Langlands probram. Collections of two-surfaces and possibly also string world sheets are always involved. Multilocality is also required by the basic properties of quantum states in zero energy ontology.

  3. The Kac-Moody group extended to Yangian and defining finite measurement resolution would naturally correspond to the gauge group of N=4 SUSY and braid points to the arguments of N-point functions. The new element would be representation of massive particles as bound states of massles particles giving hopes about cancellation of IR divergences and about exact Yangian symmetry. Second new element would be that virtual particles correspond to wormholes for which throats are massless but can have different momenta and opposite signs of energies. This implies that absence of UV divergences and gives hopes that the number of Feynman diagrams is effectively finite and that there is simple expression of twistorial diagrams in terms of Feynman diagrams (see this).

For details and background the reader can consult either to the chapter Langlands Program and TGD of "Physics as Generalized Number Theory" or to the articles Langlands Conjectures in TGD Framework, How infinite primes relate to other views about mathematical infinity?, Motives and Infinite primes, and Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2-knots?. Also the previous blog postings during this year give a view about the development of ideas.



Langlands Conjectures in TGD Framework

During last years I have done work in attempt to relate TGD to the new developments in mathematics. The evolution of ideas has been especially fast during last year and I have reported about these developments in various postings. The latest articles are How infinite primes relate to other views about mathematical infinity? and Motives and Infinite Primes.

What makes me happy is that TGD is not only receiving experimental support from LHC and other particle accelerators but also providing profound insights inspiring mathematical conjectures. What is also highly satisfying that the physically motivated visions such as the need for number theoretic universality are guiding the development of modern mathematics. The notion of motives introduced by Grothendieck is a good example about number theoretical universality and relates to the need to define integral- at least in cohomological sense- for all number fields: very concrete challenge also in TGD framework.

Langlands program is one of the hot areas of what might be called physical mathematics. The above mentioned number theoretical universality is one of the guiding lines in this approach. The program relies on very general conjectures about a connection between number theory and harmonic analysis relating the representations of Galois groups with the representations of certain kinds of Lie groups to each other. Langlands conjecture has many forms and it is indeed a conjecture and many of them are inprecise since the notions involved are not sharply defined.

Peter Woit noticed that Ed Frenkel had given a talk with rather interesting title "What do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?"? I listened the talk and found it very inspiring. The talk provides bird's eye of view about some basic aspects of Langlands program using the language understood by physicist. Also the ideas concerting the connection between Langlands duality and electric-magnetic duality generalized to S-duality in the context of non-Abelian gauge theories and string theory context and developed by Witten and Kapustin and followers are summarized. In this context D=4 and twisted version of N=4 SYM familiar from twistor program and defining a topological QFT appears.

For some years ago I made my first attempt to understand what Langlands program is about and tried to relate it to TGD framework (see this). At that time I did not really understand the motivations for many of the mathematical structures introduced. In particular, I did not really understand the motivations for introducing the gigantic Galois group of algebraic numbers regarded as algebraic extension of rationals.

  1. Why not restrict the consideration to finite Galois groups or their braided counterparts (as I indeed effectively did in my first approach)? At that time I concentrated on the question what enormous Galois group of algebraic numbers regarded as algebraic extension of rationals could mean, and proposed that it could be identified as a symmetric group consisting of permutations of infinitely many objects. The definition of this group is however far from trivial. Should one allow as generators of the group only the permutations affecting only finite number of objects or permutations of even infinite number of objects?

    The analogous situation for the sequences of binary digits would lead to a countable set of sequence of binary digits forming a discrete set of finite integers in real sense or to 2-adic integers forming a 2-adic continuum. Something similar could be expected now. The physical constraints coming the condition that the elements of symmetric group allow lifting to braidings suggested that the permutations permuting infinitely many objects should be periodic meaning that the infinite braid decomposes to an infinite number of identical N-braids and braiding is same for all of them. The p-adic analog would be p-adic integers, which correspond to rationals having periodical expansion in powers of p. Braids would be therefore like pinary digits. I regarded this choice as the most realistic one at that time. I failed to realize the possibility of having analogs of p-adic integers by general permutations. In any case, this observation makes clear that the unrestricted Galois group is analogous to a Lie group in topology analogous to p-adic topology rather than to discrete group. Neither did I realize that the Galois groups could be finite and be associated with some other field than rationals, say a Galois group associated with the field of polynomials of n-variable with rational coefficients and with its completion with coefficients replaced by algebraic numbers.

  2. The ring of adeles can be seen as a Cartesian product of non-vanishing real numbers R× with the infinite Cartesian product ∏ Zp having as factors p-adic integers Zp for all values of prime p. Rational adeles are obtained by replacing R with rationals Q and requiring that multiplication of rational by integers is equivalent with multiplication of any Zp with rational. Finite number of factors in Zp can correspond to Qp: this is required in to have finite adelic norm defined as the product of p-adic norms. This definition implicitly regards rationals as common to all number fields involved. At the first encounter with adeles I did not realize that this definition is in spirit with the basic vision of TGD.

    The motivation for the introduction of adele is that one can elegantly combine the algebraic groups assignable to rationals (or their extensions) and all p-adic number fields or even more general function fields such as polynomials with some number of argument at the same time as a Cartesian product of these groups as well as to finite fields. This is indeed needed if one wants to realize number theoretic universality which is basic vision behind physics as generalized number theory vision. This approach obviously means enormous economy of thought irrespective of whether one takes adeles seriously as a physicist.

The talk of Frenkel inspired me to look again for Langlands program in TGD framework taking into the account of various developments that have occured in TGD during these years. I realized again that ideas develop unconsciously during the years and that many questions which remained unanswered for some years ago had found obvious answers. Instead of writing a 10 page posting I attach the abstract of pdf article "Langlands conjectures in TGD framework" at my homepage.

The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following.

  1. Zero energy ontology and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M4 and of CP2). This notion leads to the notion of partonic 2-surfaces at the light-like boundaries of CD and to the notion of string world sheet.

  2. Electric-magnetic duality realized in terms of string world sheets and partonic 2-surfaces. The group G and its Langlands dual LG would correspond to the time-like and space-like braidings. Duality predicts that the moduli space of string world sheets is very closely related to that for the partonic 2-surfaces. The strong form of 4-D general coordinate invariance implying electric-magnetic duality and S-duality as well as strong form of holography indeed predicts that the collection of string world sheets is fixed once the collection of partonic 2-surfaces at light-like boundaries of CD and its sub-CDs is known.

  3. The proposal is that finite measurement resolution is realized in terms of inclusions of hyperfinite factors of type II1 at quantum level and represented in terms of confining effective gauge group (see this). This effective gauge group could be some associate of G: gauge group, Kac-Moody group or its quantum counterpart, or so called twisted quantum Yangian strongly suggested by twistor considerations ("symmetry group" hitherto). At space-time level the finite measurement resolution would be represented in terms of braids at space-time level which come in two varieties correspond to braids assignable to space-like surfaces at the two light-like boundaries of CD and with light-like 3-surfaces at which the signature of the induced metric changes and which are identified as orbits of partonic 2-surfaces connecting the future and past boundaries of CDs.

    There are several steps leading from G to its twisted quantum Yangian. The first step replaces point like particles with partonic 2-surfaces: this brings in Kac-Moody character. The second step brings in finite measurement resolution meaning that Kac-Moody type algebra is replaced with its quantum version. The third step brings in zero energy ontology: one cannot treat single partonic surface or string world sheet as independent unit: always the collection of partonic 2-surfaces and corresponding string worlds sheets defines the geometric structure so that multilocality and therefore quantum Yangian algebra with multilocal generators is unavoidable.

    In finite measurement resolution geometric Langlands duality and number theoretic Langlands duality are very closely related since partonic 2-surface is effectively replaced with the punctures representing the ends of braid strands and the orbit of this set under a discrete subgroup of G defines effectively a collection of "rational" 2-surfaces. The number of the "rational" surfaces in geometric Langlands conjecture replaces the number of rational points of partonic 2-surface in its number theoretic variant. The ability to compute both these numbers is very relevant for quantum TGD.

  4. The natural identification of the associate of G is as quantum Yangian of Kac-Moody type group associated with Minkowskian open string model assignable to string world sheet representing a string moving in the moduli space of partonic 2-surface. The dual group corresponds to Euclidian string model with partonic 2-surface representing string orbit in the moduli space of the string world sheets. The Kac-Moody algebra assigned with simply laced G is obtained using the standard tachyonic free field representation obtained as ordered exponentials of Cartan algebra generators identified as transversal parts of M4 coordinates for the braid strands.

  5. Langlands duality involves besides harmonic analysis side also the number theoretic side. Galois groups (collections of them) defined by infinite primes and integers having representation as symplectic flows defining braidings. I have earlier proposed that the hierarchy of these Galois groups define what might be regarded as a non-commutative homology and cohomology. Also the effective symmetry group has this kind of representation which explains why the representations of these two kinds of groups are so intimately related. This relationship could be seen as a generalization of the MacKay correspondence between finite subgroups of SU(2) and simply laced Lie groups.

  6. The symplectic group of the light-cone boundary acting as isometries of the WCW geometry allowing to represent projectively both Galois groups and effective symmetry groups as symplectic flows so that the non-commutative cohomology would have braided representation. This leads to braided counterparts for both Galois group and effective symetry group.

  7. The moduli space for Higgs bundle playing central role in the approach of Witten and Kapustin to geometric Landlands program is in TGD framework replaced with the conformal moduli space for partonic 2-surfaces. It is not however possible to speak about Higgs field although moduli defined the analog of Higgs vacuum expectation value. Note that in TGD Universe the most natural assumption is that all Higgs like states are "eaten" by gauge bosons so that also photon and gluons become massive. This mechanism would be very general and mean that massless representations of Poincare group organize to massive ones via the formation of bound states. It might be however possible to see the contribution of p-adic thermodynamics depending on genus as analogous to Higgs contribution since the conformal moduli are analogous to vacuum expectation of Higgs field.

For details and background the reader can consult either to the chapter Langlands Program and TGD or to the article Langlands Conjectures in TGD Framework.



Quantum Boolean algebra instead of Boolean algebra?

Mathematical logic relies on the notion of Boolean algebra, which has a well-known representation as the algebra of sets which in turn has in algebraic geometry a representation in terms of algebraic varieties. This is not however attractive at space-time level since the dimension of the algebraic variety is different for the intersection resp. union representing AND resp. OR so that only only finite number of ANDs can appear in the Boolean function. TGD inspired interpretation of the fermionic sector of the theory in terms of Boolean algebra inspires more concrete ideas about the the realization of Boolean algebra at both quantum level and classical space-time level and also suggests a geometric realization of the basic logical functions respecting the dimension of the representative objects.

  1. In TGD framework WCW spinors correspond to fermionic Fock states and an attractive interpretation for the basis of fermionic Fock states is as Boolean algebra. In zero energy ontology one consider pairs of positive and negative energy states and zero energy states could be seen as physical correlates for statements A→ B or A↔ B with individual state pairs in the quantum superposition representing various instances of the rule A→ B or A↔ B. The breaking of time reversal invariance means that either the positive or negative energy part of the state (but not both) can correspond to a state with precisely definine number of particles with precisely defining quantum numbers such as four-momentum. At the second end one has scattered state which is a superposition of this kind of many-particle states. This would suggest that A→ B is the correct interpretation.

  2. In quantum group theory the notion of co-algebra is very natural and the binary algebraic operations of co-algebra are in a well-defined sense time reversals of those of algebra. Hence there is a great temptation to generalize Boolean algebra to include its co-algebra so that one might speak about quantum Boolean algebra. The vertices of generalized Feynman diagrams represent two topological binary operations for partonic two surfaces and there is a strong temptation to interpret them as representations for the operations of Boolean algebra and its co-algebra.

    1. The first vertex corresponds to the analog of a stringy trouser diagram in which partonic 2-surface decays to two and the reversal of this representing fusion of partonic 2-surfaces. In TGD framework this diagram does not represent classically particle decay or fusion but the propagation of particle along two paths after the decay or the reversal of this process. The Boolean analog would be logical OR (A∨B) or set theoretical union A∪B resp. its co-operation. The partonic two surfaces would represent the arguments (resp. co-arguments) A and B.

    2. Second one corresponds to the analog of 3-vertex for Feynman diagram: the three 3-D "lines" of generalized Feynman diagram meet at the partonic 2-surface. This vertex (co-vertex) is the analog of Boolean AND (A∧B) or intersection A∩B of two sets resp. its co-operation.

    3. I have already earlier ended up with the proposal that only three-vertices appear as fundamental vertices in quantum TGD (see this). The interpretation of generalized Feynman diagrams as a representation of quantum Boolean algebra would give a deeper meaning for this proposal.

    These vertices could therefore have interpretation as a space-time representation for operations of Boolean algebra and its co-algebra so that the space-time surfaces could serve as classical correlates for the generalized Boolean functions defined by generalized Feynman diagrams and expressible in terms of basic operations of the quantum Boolean algebra. For this representation the dimension of the variety representing the value of Boolean function at classical level is the same as as the dimension of arguments: that is two. Hence this representation is not equivalent with the representation provided by algebraic geometry for which the dimension of the geometric variety representing A∧ B and A∨ B in general differs from that for A and B. If one however restricts the algebra to that assignable to braid strands, statements would correspond to points at partonic level, so that one would have discrete sets and the set theoretic representation of quantum Boolean algebra could make sense. Discrete sets are indeed the only possibility since otherwise the dimension of intersection and union are different if algebraic varieties are in question.

  3. The breaking of time reversal invariance is accompanied by a generation of entropy and loss of information. The interpretation at the level of quantum Boolean algebra would be following. The Boolean function and and OR assign to two statements a single statement: this means a gain of information and at the level of physics this is indeed the case since entropy is reduced in the process reducing the number of particles. The occurrence of co-operations of AND and OR corresponds to particle decays and uncertainty about the path along which particle travels (dispersion of wave packet) and therefore loss of information.

    1. The "most logical" interpretation for the situation is in conflict with the identification of the arrow of logic implication with the arrow of time: the direction of Boolean implication arrow and the arrow of geometric time would be opposite so that final state could be said to imply the initial state. The arrow of time would weaken logical equivalence to implication arrow.

    2. If one naively identifies the arrows of logical implication and geometric time so that initial state can be said to imply the final state, second law implies that logic becomes fuzzy. Second law would weak logical equivalence to statistical implication arrow.

    3. The natural question is whether just the presence of both algebra and co-algebra operations causing a loss of information in generalized Feynman diagrams could lead to what might be called fuzzy Boolean functions expressing the presence of entropic element appears at the level of Boolean cognition.

  4. This picture requires a duality between Boolean algebra and its co-algebra and this duality would naturally correspond to time reversal. Skeptic can argue that there is no guarantee about the existence of the extended algebra analogous to Drinfeld double that would unify Boolean algebra and its dual. Only the physical intuition suggests its existence.

These observations suggest that generalized Feynman diagrams and their space-time counterparts could have a precise interpretation in quantum Boolean algebra and that one should perhaps consider the extension of the mathematical logic to quantum logic. Alternatively, one could argue that quantum Boolean algebra is more like a model for what mathematical cognition could be in the real world.

For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?.



How infinite primes relate to other views about mathematical infinity?

Infinite primes is a purely TGD inspired notion. The notion of infinity is number theoretical and infinite primes have well defined divisibility properties. One can partially order them by the real norm. p-Adic norms of infinite primes are well defined and finite. The construction of infinite primes is a hierarchical procedure structurally equivalent to a repeated second quantization of a supersymmetric arithmetic quantum field theory. At the lowest level bosons and fermions are labelled by ordinary primes. At the next level one obtains free Fock states plus states having interpretation as bound many particle states. The many particle states of a given level become the single particle states of the next level and one can repeat the construction ad infinitum. The analogy with quantum theory is intriguing and I have proposed that the quantum states in TGD Universe correspond to octonionic generalizations of infinite primes.

It is interesting to compare infinite primes (and integers) to the Cantorian view about infinite ordinals and cardinals. The basic problems of Cantor's approach relate to the axiom of choice, continuum hypothesis, and Russell's antinomy: all these problems are due to the definition of ordinals as sets. In TGD framework infinite primes, integers, and rationals are defined purely algebraically so that these problems are avoided. It is not surprising that these approaches are not equivalent. For instance, sum and product for Cantorian ordinals are not commutative unlike for infinite integers defined in terms of infinite primes.

Set theory defines the foundations of modern mathematics. Set theory relies strongly on classical physics, and the obvious question is whether one should reconsider the foundations of mathematics in light of quantum physics. Is set theory really the correct approach to axiomatization?

  1. Quantum view about consciousness and cognition leads to a proposal that p-adic physics serves as a correlate for cognition. Together with the notion of infinite primes this suggests that number theory should play a key role in the axiomatics.

  2. Algebraic geometry allows algebraization of the set theory and this kind of approach suggests itself strongly in physics inspired approach to the foundations of mathematics. This means powerful limitations on the notion of set.

  3. Finite measurement resolution and finite resolution of cognition could have implications also for the foundations of mathematics and relate directly to the fact that all numerical approaches reduce to an approximation using rationals with a cutoff on the number of binary digits.

  4. The TGD inspired vision about consciousness implies evolution by quantum jumps meaning that also evolution of mathematics so that no fixed system of axioms can ever catch all the mathematical truths for the simple reason that mathematicians themselves evolve with mathematics.

It is interesting to discuss the possible impact of these observations on the foundations of physical mathematics assuming that one accepts the TGD inspired view about infinity, about the notion of number, and the restrictions on the notion of set suggested by classical TGD.

I do not bother to type all the text here. For details the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?.



How could one calculate p-adic integrals numerically?

Riemann sum gives the simplest numerical approach to the calculation of real integrals. Also p-adic integrals should allow a numerical approach and very probably such approaches already exist and "motivic integration" presumably is the proper word to google. The attempts of an average physicist to dig out this kind of wisdom from the vastness of mathematical literature however lead to a depression and deep feeling of inferiority. The only manner to avoid the painful question "To whom should I blame for ever imagining that I could become a real mathematical physicist some day?" is a humble attempt to extrapolate real common sense to p-adic realm. One must believe that the almost trivial Riemann integral must have an almost trivial p-adic generalization although this looks far from obvious.

1. A proposal for p-adic numerical integration

The physical picture provided by quantum TGD gives strong constraints on the notion of p-adic integral.

  1. The most important integrals should be over partonic 2-surfaces. Also p-adic variants of 3-surfaces and 4-surfaces can be considered. The p-adic variant of Kähler action would be an especially interesting integral and reduces to Chern-Simons terms over 3-surfaces for preferred extremals. One should use this definition also in the p-adic context since the reduction of a total divergence to boundary term is not expected to take place in numerical approach if one begins from a 4-dimensional Kähler action since in p-adic context topological boundaries do not exist. The reduction to Chern-Simons term means also a reduction to cohomology and p-adic cohomology indeed exists.

    At the first step one could restrict the consideration to algebraic varieties - in other words zero loci for a set of polynomials Pi(x) at the boundary of causal diamond consisting of pieces of δ M4+/-× CP2. 5 equations are needed. The simplest integral would be the p-adic volume of the partonic 2-surface.

  2. The numerics must somehow rely on the p-adic topology meaning that very large powers pn are very small in p-adic sense. In the p-adic context Riemann sum makes no sense since the sum never has p-adic norm larger than the maximum p-adic norm for summands so that the limit would give just zero. Finite measurement resolution suggests that the analog for the limit Δ x→ 0 is pinary cutoff O(pn)=0, n→ ∞, for the function f to be integrated. In the spirit of algebraic geometry one must asume at least power series expansion if not even the representability as a polynomial or rational function with rational or p-adic coefficients.

  3. Number theoretic approach suggests that the calculation of the volume vol(V) of a p-adic algebraic variety V as integral should reduce to the counting of numbers for the solutions for the equations fi(x)=0 defining the variety. Together with the finite pinary cutoff this would mean counting of numbers for the solutions of equations fi(x) mod pn=0 . The p-adic volume Vol(V,n) of the variety in the measurement resolution O(pn)=0 would be simply the number of p-adic solutions to the equations fi(x) mod pn=0. Although this number is expected to become infinite as a real number at the limit n→ ∞, its p-adic norm is never larger than one. In the case that the limit is a well-defined as p-adic integer, one can say that the variety has a well-defined p-adic valued volume at the limit of infinite measurement resolution. The volume Vol(V,n) could behave like npn and exist as a well defined p-adic number only if np is divisible by p.

  4. The generalization of the formula for the volume to an integral of a function over the volume is straightforward. Let f be the function to be integrated. One considers solutions to the conditions f(x) =y, where y is p-adic number in resolution O(pn)=0, and therefore has only a finite number of values. The condition f(x)-y=0 defines a codimension 1 sub-variety Vy of the original variety and the integral is defined as the weighted sum ∑y y × vol(Vy), where y denotes the point in the finite set of allowed values of f(x) so that calculation reduces to the calculation of volumes also now.

2. General coordinate invariance

From the point of view of physics general coordinate invariance of the volume integral and more general integrals is of utmost importance.

  1. The general coordinate invariance with respect to the internal coordinates of surface is achieved by using a subset of imbedding space-cooordinates as preferred coordinates for the surface. This is of also required if one works in algebraic geometric setting. In the case of projective spaces and similar standard imbedding spaces of algebraic varieties natural preferred coordinates exist. In TGD framework the isometries of M4× CP2 define natural preferred coordinate systems.

  2. The question whether the formula can give rise to a something proportional to the volume in the induced metric in the intersection of real and rational worlds interesting. One could argue that one must include the square root of the determinant of the induced metric to the definition of volume in preferred coordinates but this might not be necessary. In fact, p-adic integration is genuine summation whereas the determinant of metric corresponds density of volume and need not make no sense in p-adic context. Could the fact that the preferred coordinates transform in simple manner under isometries of the imbedding space (linearly under maximal subgoup) alone guarantee that the information about the imbedding space metric is conveyed to the formula?

  3. Indeed, since the volume is defined as the number of p-adic points, the proposed formula should be invariant at least under coordinate transformations mediated by bijections of the preferred coordinates expressible in terms of rational functions. In fact, even more general bijections mapping p-adic numbers to p-adic numbers could be allowed since they effectively mean the introduction of new summation indices. Since the determinant of metric changes in coordinate transformations this requires that the metric determinant is not present at all. Thus summation is what allows to achieve the p-adic variant of general coordinate invariance.

  4. This definition of volume and more general integrals amounts to solving the remaining coordinates of imbedding space as (in general) many-valued functions of these coordinates. In the integral those branches contribute to the integral for which the solution is p-adic number or belongs to the extension of p-adic numbers in question. By p-adic continuity the number of p-adic value solutions is locally constant. In the case that one integrates function over the surface one obtains effectively many-valued function of the preferred coordinates and can perform separate integrals over the branches.

3. Numerical iteration procedure

A convenient iteration procedure is based on the representation of integrand f as sum ∑kfk of functions associated with different p-adic valued branches zk=zk(x) for the surface in the coordinates chosen and identified as a subset of preferred imbedding space coordinates. The number of branches zk contributing is by p-adic continuity locally constant.

The function fk - call it g for simplicity - can in turn be decomposed into a sum of piecewise constant functions by introducing first the piecewise constant pinary cutoffs gn(x) obtained in the approximation O(pn+1)=0. One can write g as

g(x)= ∑ hn (x) , h0(x)=g0(x) ,

hn=gn(x)-gn-1(x) for n>0 .

Note that hn(x) is of form gn(x)= an(x)pn, an(x) ∈ {0,p-1} so that the representation for integral as a sum of integrals for piecewise constant functions hn converge rapidly. The technical problem is the determination of the boundaries of the regions inside which these functions contribute.

The integral reduces to the calculation of the number of points for given value of hn(x) and by the local constancy for the number of p-adic valued roots zk(x) the number of points for N0k≥ 0 pk= N0/(1-p), where N0 is the number of points x with the property that not all points y= x(1+O(p)) represent p-adic points z(x). Hence a finite number of calculational steps is enough to determine completely the contribution of given value to the integral and the only approximation comes from the cutoff in n for hn(x).

4. Number theoretical universality

This picture looks nice but it is far from clear whether the resulting integral is that what physicist wants. It is not clear whether the limit Vol(V,n), n→ ∞, exists or even should exist always.

  1. In TGD Universe a rather natural condition is algebraic universality requiring that the p-adic integral is proportional to a real integral in the intersection of real and p-adic worlds defined by varieties identified as loci of polynomials with integer/rational coefficients. Number theoretical universality would require that the value of the p-adic integral is p-adic rational (or algebraic number for extensions of p-adic numbers) equal to the value of the real integral and in algebraic sense independent of the number field. In the eyes of physicist this condition looks highly non-trivial. For a mathematician it should be extremely easy to show that this condition cannot hold true. If true the equality would represent extremely profound number theoretic truth.

    The basic idea of the motivic approach to integration is to generalize integral formulas so that the same formula applies in any number field: the specialization of the formula to given number field would give the integral in that particular number field. This is of course nothing but number theoretical universality. Note that the existence of this kind of formula requires that in the intersection of the real and p-adic worlds real and p-adic integrals reduce to the same rational or transcendentals (such as log(1+x) and polylogarithms).

  2. If number theoretical universality holds true one can imagine that one just takes the real integral, expresses it as a function of the rational number valued parameters (continuable to real numbers) characterizing the integrand and the variety and algebraically continues this expression to p-adic number fields. This would give the universal formula which can be specified to any number field. But it is not at all clear whether this definition is consistent with the proposed numerical definition.

  3. There is also an intuitive expectation in an apparent conflict with the number theoretic universality. The existence of the limit for a finite number p-adic primes could be interpreted as mathematical realization of the physical intuition suggesting that one can assign to a given partonic 2-surface only a finite number of p-adic primes. Indeed, quantum classical correspondence combined with the p-adic mass calculations suggests that the partonic 2-surfaces assignable to a given elementary particle in the intersection of real and p-adic worlds corresponds to a finite number of p-adic primes somehow coded by the geometry of the partonic 2-surface.

    One way out of the difficulty is that the functions - say polynomials - defining the surface have as coefficients powers of en. For given prime p only the powers of ep exist p-adically so that only the primes p dividing n would be allowed. The transcendenteals of form log(1+px) and their polylogarithmic generalizations resulting from integrals in the intersection of real and p-adic worlds would have the same effect. Second way out of the difficulty would be based on the condition that the functional integral over WCW ("world of classical worlds") converges. There is a good argument stating that the exponent of Kähler action reduces to an exponent of integer n and since all powers of n appear the convergence is achieved only for p-adic primes dividing n.

5. Can number theoretical universality be consistent with the proposed numerical definition of the p-adic integral?

The equivalence of the proposed numerical integral with the algebraic definition of p-adic integral motivated by the algebraic formula in real context expressed in terms of various parameters defining the variety and the integrand and continued to all number fields would be such a number theoretical miracle that it deserves italics around it:

For algebraic surfaces the real volume of the variety equals apart from constant C to the number of p-adic points of the variety in the case that the volume is expressible as p-adic integer.

The proportionality constant C can depend on p-adic number field , and the previous numerical argument suggests that the constant could be simply the factor 1/(1-p) resulting from the sum of p-adic points in p-adic scales so short that the number of the p-adic branches zk(x) is locally constant. This constant is indeed needed: without it the real integrals in the intersection of real and p-adic worlds giving integer valued result I=m would correspond to functions for which the number of p-adic valued points is finite.

The statement generalizes to apply also to the integrals of rational and perhaps even more general functions. The equivalence should be considered in a weak form by allowing the transcendentals contained by the formulas have different meanings in real and p-adic number fields. Already the integrals of rational functions contain this kind of transcendentals.

The basic objection that number of p-adic points without cannot give something proportional to real volume with an appropriate interpretation cannot hold true since real integral contains the determinant of the induced metric. As already noticed the preferred coordinates for the imbedding space are fixed by the isometries of the imbedding space and therefore the information about metric is actually present. For constant function the correspondence holds true and since the recipe for performing of the integral reduce to that for an infinite sum of constant functions, it might be that the miracle indeed happens.

The proposal can be tested in a very simple manner. The simplest possible algebraic variety is unit circle defined by the condition x2+y2=1.

  1. In the real context the circumference is 2π and p-adic transcendental requiring an infinite-dimensional algebraic extension defined in terms of powers of 2π. Does this mean that the number of p-adic points of circle at the limit n→ ∞ for the pinary cutoff O(pn)=0 is ill-defined? Should one define 2π as this integral and say that the motivic integral calculus based on manipulation of formulas reduces the integrals to a combination of p-adically existing numbers and 2π? In motivic integration the outcome of the integration is indeed formula rather than number and only a specialization gives it a value in a particular number field. Does 2π have a specialization to the original p-adic number field or should one introduce it via transcendental extension?

  2. The rational points (x,y)=(k/m,l/m) of the p-adic unit circle would correspond to Pythagorean triangles satisfying k2+l2= m2 with the general solution k= r2-s2, l=2rs, m=r2+s2. Besides this there is an infinite number of p-adic points satisfying the same equation: some of the integers k,l,m would be however infinite as real integers. These points can be solved by starting from O(p)=0 approximation (k,l,m)→ (k,l,m)~ mod~ p== (k0,l0,m0). One must assume that the equations are satisfied only modulo p so that Pythagorean triangles modulo p are the basic objects. Pythagorean triangles can be also degenerate modulo p so that either k0,l0 or evenm0 vanishes. Note that for surfaces xn+yn=zn no non-trivial solutions exists for xn,yn,zn<p for n> 2 and all p-adic points are infinite as real integers.

    The Pythagorean condition would give a constraint between higher powers in the expressions for k,l and m. The challenge would be to calculate the number of this kind of points. If one can choose the integers k-(k mod p) and l-(l mod p) freely and solve m-(m mod p) from the quadratic equations uniquely, the number of points of the unit circle consisting of p-adic integers must be of form N0/(1-p). At the limit n→ ∞ the p-adic length of the unit circle would be in p-adic topology equal to the number of modulo p Pythagorean triangles (r,s) satisfying the condition (r2+s2)2<p. The p-adic counterpart of 2π would be ordinary p-adic number depending on p. This definition of the length of unit circle as number of its modulo p Pythagorean points also Pythagoras would have agreed with since in the Pythagorean world view only rational triangles were accepted.

  3. One can look the situation also directly solving y as y=+/- (1-x2)1/2. The p-adic square root exists always for x=O(pn), n>0. The number of these points x is 2/(1-p) taking into account the minus sign. For x=O(p0) the square root exist for roughly one half of the integers n∈ {0,p-1}. The number of integers (x2)0 is therefore roughly (p-1)/2. The study of p=5 case suggests that the number of integers (1-(x2)0)0∈ {0,p-1} which are squares is about (p-1)/4. Taking into account the +/- sign the number of these points by N0≈ (p-1)/2. In this case the higher O(p) contribution to x is arbitrary and one obtains total contribution N0/(1-p). Altogether one would have (N0+2)/(1-p) so that eliminating the proportionality factor the estimate for the p-adic counterpart of 2π would be (p+3)/2.

  4. One could also try a trick. Express the points of circle as (x,y)=(cos(t),sin(t)) such that t is any p-adic number with norm smaller than one in p-adic case. This unit circle is definitely not the same object as the one defined as algebraic variety in plane. One can however calculate the number of p-adic points at the limit n→ ∞. Besides t=0, all p-adic numbers with norm larger than p-n and smaller than 1 are acceptable and one obtains as a result N(n)= 1+ pn-1, where "1" comes from overall important point t=0. One has N(n)→ 1 in p-adic sense. If t=0 is not allowed the length vanishes p-adically. The circumference of circle in p-adic context would have length equal to 1 in p-adic topology so that no problems would be encountered (numbers exp(i2π/n) would require algebraic extension of p-adic numbers and would not exist as power series).

    The replacement of the coordinates (x,y) with coordinate t does not respect the rules of algebraic geometry since trigonometric functions are not algebraic functions. Should one allow also exponential and trigonometric functions and their inverses besides rational functions and define circle also in terms of these. Note that these functions are exceptional in that corresponding transcendental extensions -say that containing e and its powers- are finite-dimensional?

  5. To make things more complicated, one could allow algebraic extensions of p-adic numbers containing roots Un=exp(i2π/n) of unity. This would affect the count too but give a well-defined answer if one accepts that the points of unit circle correspond to the Pythagorean points multiplied by the roots of unity.

A question inspired by this example is whether the values of p-adic integrals as p-adic numbers could be determined by the few lowest powers of p with higher order contribution giving something proportional to an infinite power of p.

6. p-Adic thermodynamics for measurement resolution?

The proposed definition is rather attractive number theoretically since everything would reduce to the counting of p-adic points of algebraic varieties. The approach generalizes also to algebraic extensions of p-adic numbers. Mathematicians and also physicists love partition functions, and one can indeed assign to the volume integral a partition function as p-adic valued power series in powers Z(t)=∑ vntn with the coefficients vn giving the volume in O(pn)=0 cutoff. One can also define partition functions Zf(t)= ∑ fntn, with fn giving the integral of f in the same approximation.

Could this kind of partition functions have a physical interpretation as averages over physical measurements over different pinary cutoffs? p-Adic temperature can be identified as t = p1/T, T=1/k. For p-adically small temperatures the lowest terms corresponding to the worst measurement resolution dominate. At first this sounds counter-intuitive since usually low temperatures are thought to make possible good measurement resolution. One can however argue that one must excite p-adic short range degrees of freedom to get information about them. These degrees of freedom correspond to the higher pinary digits by p-adic length scale hypothesis and high energies by Uncertainty Principle. Hence high p-adic temperatures are needed. Also measurement resolution would be subject to p-adic thermodynamics rather than being freely fixed by the experimentalist.

For details see the new chapter Motives and Infinite Primes or the article with same title.



What about counterparts of T, S, and U dualities in TGD framework?

The natural question is what could be the TGD counterparts of S-, T- and U-dualities. If one accepts the identification of U-duality as product U=ST and the proposed counterpart of T duality as a strong form of general coordinate invariance discussed in previous posting, it remains to understand the TGD counterpart of S-duality - in other words electric-magnetic duality - relating the theories with gauge couplings g and 1/g. Quantum criticality selects the preferred value of gK: Kähler coupling strength is very near to fine structure constant at electron length scale and can be equal to it. Since there is no coupling constant evolution associated with αK, it does not make sense to say that gK becomes strong and is replaced with its inverse at some point. One should be able to formulate the counterpart of S-duality as an identity following from the weak form of electric-magnetic duality and the reduction of TGD to almost topological QFT. This seems to be the case.

TGD based view about S-duality

The following arguments suggests that in TGD framework S duality is realized for each preferred extremal of Kähler action separately whereas in standard view the duality would be realized only at the level of path integral defining the partition function.

  1. For preferred extremals the interior parts of Kähler action reduces to a boundary term because the term jμAμ vanishes. The weak form of electric-magnetic duality requires that Kähler electric charge is proportional to Kähler magnetic charge, which implies reduction to abelian Chern-Simons term: the Kähler coupling strength does not appear at all in Chern-Simons term. The proportionality constant beween the electric and magnetic parts JE and JB of Kähler form however enters into the dynamics through the boundary conditions stating the weak form of electric-magnetic duality. At the Minkowskian side the proportionality constant must be proportional to gK2 to guarantee a correct value for the unit of Kähler electric charge - equal to that for electric charge in electron length scale- from the assumption that electric charge is proportional to the topologically quantized magnetic charge. It has been assumed that

    JE= αK JB

    holds true at both sides of the wormhole throat but this is an un-necessarily strong assumption at the Euclidian side. In fact, the self-duality of CP2 Kähler form stating

    JE=JB

    favours this boundary condition at the Euclidian side of the wormhole throat. Also the fact that one cannot distinguish between electric and magnetic charges in Euclidian region since all charges are magnetic can be used to argue in favor of this form. The same constraint arises from the condition that the action for CP2 type vacuum extremal has the value required by the argument leading to a prediction for gravitational constant in terms of the square of CP2 radius and αK the effective replacement gK2 → 1 would spoil the argument.

  2. Minkowskian and Euclidian regions should correspond to a strongly/weakly interacting phase in which Kähler magnetic/electric charges provide the proper description. In Euclidian regions associated with CP2 type extremals there is a natural interpretation of interactions between magnetic monopoles associated with the light-like throats: for CP2 type vacuum extremal itself magnetic and electric charges are actually identical and cannot be distinguished from each other. Therefore the duality between strong and weak coupling phases seems to be trivially true in Euclidian regions if one has JB= JE at Euclidian side of the wormhole throat. This is however an un-necessarily strong condition as the following argument shows.

  3. In Minkowskian regions the interaction is via Kähler electric charges and elementary particles have vanishing total Kähler magnetic charge consisting of pairs of Kähler magnetic monopoles so that one has confinement characteristic for strongly interacting phase. Therefore Minkowskian regions naturaly correspond to a weakly interacting phase for Kähler electric charges. One can write the action density at the Minkowskian side of the wormhole throat as

    (JE2-JB2)/αK= αKJB2 - JB2K .

    The exchange JE↔ JB accompanied by gK2→ -1/gK2 leaves the action density invariant. Since only the behavior of the vacuum functional infinitesimally near to the wormhole throat matters by almost topological QFT property, the duality is realized. Note that the argument goes through also in Euclidian regions so that it does not allow to decide which is the correct form of weak form of electric-magnetic duality.

  4. S-duality could correspond geometrically to the duality between partonic 2-surfaces responsible for magnetic fluxes and string worlds sheets responsible for electric fluxes as rotations of Kähler gauge potentials around them and would be very closely related with the counterpart of T-duality implied by the strong form of general coordinate invariance and saying that space-like 3-surfaces at the ends of space-time sheets are equivalent with light-like 3-surfaces connecting them.

The boundary condition JE=JB at the Euclidian side of the wormhole throat inspires the question whether all Euclidian regions could be self-dual so that the density of Kähler action would be just the instanton density. Self-duality follows if the deformation of the metric induced by the deformation of the canonically imbedded CP2 is such that in CP2 coordinates for the Euclidian region the tensor (gαβgμν -gανgμβ)/g1/2 remains invariant. This is certainly the case for CP2 type vacuum extremals since by the light-likeness of M4 projection the metric remains invariant. Also conformal scalings of the induced metric would satisfy this condition. Conformal scaling is not consistent with the degeneracy of the 4-metric at the wormhole throat. Self-duality is indeed an un-necessarily strong condition.

Comparison with standard view about dualities

One can compare the proposed realization of T-, S and U-duality to the more general dualities defined by the modular group SL(2,Z), which in QFT framework can hold true for the path integral over all possible gauge field configurations. In the resent case the dualities hold true for every preferred extremal separately and the functional integral is only over the space-time projections of fixed Kähler form of CP2. Modular invariance for Maxwell action was discussed by E. Verlinde for Maxwell action with θ term for a general 4-D compact manifold with Euclidian signature of metric. In this case one has path integral giving sum over infinite number of extrema characterized by the cohomological equivalence class of the Maxwell field the action exponential to a high degree. Modular invariance is broken for CP2: one obtains invariance only for τ→ τ+2 whereas S induces a phase factor to the path integral.

  1. In the recent case these homology equivalence classes would correspond to homology equivalence classes of holomorphic partonic 2-surfaces associated with the critical points of K\"ahler function with respect to zero modes.

  2. In the case that the Euclidian contribution to the Kähler action is expressible solely in terms of wormhole throat Chern-Simons terms, and one can neglect the measurement interaction terms, the exponent of Kähler action can be expressed in terms of Chern-Simons action density as

    L= τ LC-S ,

    LC-S=J∧ A ,

    τ=1/gK2 +ik/4π , k=1 .

    Here the parameter τ transforms under full SL(2,Z) group as

    τ→ (aτ+b)/(cτ+d) .

    The generators of SL(2,Z) transformations are T: τ → τ+1, S:τ→-1/τ. The imaginary part in the exponents corresponds to Kac-Moody central extension k=1.

    This form corresponds also to the general form of Maxwell action with CP breaking θ term given by

    L= 1/gK2 J∧*J +i(θ/8π2) J∧J , θ=2π .

    Hence the Minkowskian part mimicks the θ term but with a value of θ for which the term does not give rise to CP breaking in the case that the action is full action for CP2 type vacuum extremal so that the phase equals to 2π and phase factor case is trivial. It would seem that the deviation from the full action for CP2 due to the presence of wormhole throats reducing the value of the full Kähler action for CP2 type vacuum extremal gives rise to CP breaking. One can visualize the excluded volume as homologically non-trivial geodesic spheres with some thickness in two transverse dimensions. At the limit of infinitely thin geodesic spheres CP breaking would vanish. The effect is exponentially sensitive to the volume deficit.

CP breaking and ground state degeneracy

Ground state degeneracy due to the possibility of having both signs for Minkowskian contribution to the exponent of vacuum functional provides a general view about the description of CP breaking in TGD framework.

  1. In TGD framework path integral is replaced by inner product involving integral over WCV. The vacuum functional and its conjugate are associated with the states in the inner product so that the phases of vacuum functionals cancel if only one sign for the phase is allowed. Minkowskian contribution would have no physical significance. This of course cannot be the case. The ground state is actually degenerate corresponding to the phase factor and its complex conjugate since kenosqrtg can have two signs in Minkowskian regions. Therefore the inner products between states associated with the two ground states define 2× 2 matrix and non-diagonal elements contain interference terms due to the presence of the phase factor. At the limit of full CP2 type vacuum extremal the two ground states would reduce to each other and the determinant of the matrix would vanish.

  2. A small mixing of the two ground states would give rise to CP breaking and the first principle description of CP breaking in systems like K-Kbar and of CKM matrix should reduce to this mixing. K0 mesons would be CP even and odd states in the first approximation and correspond to the sum and difference of the ground states. Small mixing would be present having exponential sensitivity to the actions of CP2 type extremals representing wormhole throats. This might allow to understand qualitatively why the mixing is about 50 times larger than expected for B0 mesons.

  3. There is a strong temptation to assign the two ground states with two possible arrows of geometric time. At the level of M-matrix the two arrows would correspond to state preparation at either upper or lower boundary of CD. Do long- and shortlived neutral K mesons correspond to almost fifty-fifty orthogonal superpositions for the two arrow of geometric time or almost completely to a fixed arrow of time induced by environment? Is the dominant part of the arrow same for both or is it opposite for long and short-lived neutral measons? Different lifetimes would suggest that the arrow must be the same and apart from small leakage that induced by environment. CP breaking would be induced by the fact that CP is performed only K0 but not for the environment in the construction of states. One can probably imagine also alternative interpretations.
Remark: The proportionality of Minkowskian and Euclidian contributions to the same Chern-Simons term implies that the critical points with respect to zero modes appear for both the phase and modulus of vacuum functional. The Kähler function property does not allow extrema for vacuum functional as a function of complex coordinates of WCW since this would mean Kähler metric with non-Euclidian signature. If this were not the case the stationary values of phase factor and extrema of modulus of the vacuum functional would correspond to different configurations.

For details see the new chapter Motives and Infinite Primes or the article with same title.



K-theory, branes, and TGD

K-theory is an essential part of the motivic cohomology. Unfortunately, this theory is very abstract and the articles written by mathematicians are usually incomprehensible for a physicist. Hence the best manner to learn K-theory is to learn about its physics applications. The most important applications are brane classification in super string models and M-theory. The excellent lectures by Harah Evslin with title What doesn't K-theory classify? make it possible to learn the basic motivations for the classification, what kind of classifications are possible, and what are the failures. Also the Wikipedia article gives a bird's eye of view about the problems. As a by-product one learns something about the basic ideas of K-theory.

In the sequel I will discuss critically the basic assumptions of brane world scenario, sum up my understanding about the problems related to the topological classification of branes and also to the notion itself, ask what goes wrong with branes and demonstrate how the problems are avoided in TGD framework, and conclude with a proposal for a natural generalization of K-theory to include also the division of bundles inspired by the generalization of Feynman diagrammatics in quantum TGD, by zero energy ontology, and by the notion of finite measurement resolution.

Brane world scenario

The brane world scenario looks attractive from the mathematical point of view ine one is able to get accustomed with the idea that basic geometric objects have varying dimensions. Even accepting the varying dimensions, the basic physical assumptions behind this scenario are vulnerable to criticism.

  1. Branes are geometric objects of varying dimension in the 10-/11-dimensional space-time -call it M- of superstring theory/M-theory. In M-theory the fundamental strings are replaced with M-branes, which are 2-D membranes with 3-dimensional orbit having as its magnetic dual 6-D M5-brane. Branes are thought to emerge non-perturbatively from fundamental 2-branes but what this really means is not understood. One has D-p-branes with Dirichlet boundary conditions fixing a p+1-dimensional surface of M as brane orbit: one of the dimensions corresponds to time. Also S-branes localized in time have been proposed.

  2. In the description of the classical limit branes interact with the classical fields of the target space by the generalization of the minimal coupling of charged point-like particle to electromagnetic gauge potential. The coupling is simply the integral of the gauge potential over the world-line - the value of 1-form for the wordline. Point like particle represents 0-brane and in the case of p-brane the generalization is obtained by replacing the gauge potential represented by a 1-from with p+1-form. The exterior derivative of this p+1-form is p+2-form representing the analog of electromagnetic field. Complete dimensional democracy strongly suggests that string world sheets should be regarded as 1-branes.

  3. From TGD point of view the introduction of branes looks a rather ad hoc trick. By generalizing the coupling of electromagnetic gauge potential to the word line of point like particle one could introduce extended objects of various dimensions also in the ordinary 4-D Maxwell theory but they would be always interpreted as idealizations for the carriers of 4- currents. Therefore the crucial step leading to branes involves classical idealization in conflict with Uncertainty Principle and the genuine quantal description in terms of fields coupled to gauge potentials.

    My view is that the most natural interpretation for what is behind branes is in terms of currents in D=10 or D= 11 space-time. In this scheme branes have role only as semi-classical idealizations making sense only above some scale. Both the reduction of string theories to quantum field theories by holography and the dynamical character of the metric of the target space conforms with super-gravity interpretation. Internal consistency requires also the identification of strings as branes so that superstring theories and M-theory would reduce to an idealization to 10-/11-dimensional quantum gravity.

In this framework the brave brane world episode would have been a very useful Odysseia. The possibility to interpret various geometric objects physically has proved to be an extremely powerful tool for building provable conjectures and has produced lots of immensely beautiful mathematics. As a fundamental theory this kind of approach does not look convincing to me.

The basic challenge: classify the conserved brane charges associated with branes

One can of course forget these critical arguments and look whether this general picture works. The first thing that one can do is to classify the branes topologically. I made the same question about 32 years ago in TGD framework: I thought that cobordism for 3-manifolds might give highly interesting topological conservation laws. I was disappointed. The results of Thom's classical article about manifold cobordism demonstrated that there is no hope for really interesting conservation laws. The assumption of Lorentz cobordism meaning the existence of global time-like vector field would make the situation more interesting but this condition looked too strong and I could not see a real justification for it. In generalized Feynman diagrammatics there is no need for this kind of condition.

There are many alternative approaches to the classification problem. One can use homotopy, homology, cohomology and their relative and other variants, topological or algebraic K-theory, twisted K-theory, and variants of K-theory not yet existing but to be proposed within next years. The list is probably endless unless something like motivic cohomology brings in enlightment.

  1. First of all one must decide whether one classifies p-dimensional time=constant sections of p-branes or their p+1-dimensional orbits. Both approaches have been applied although the first one is natural in the standard view about spontaneous compactification. For the first option topological invariants could be seen as conserved charges: homotopy invariants and homological and cohomological characteristics of branes provide this kind of invariants. For the latter option the invariants would be analogous to instanton number characterizing the change of magnetic charge.

  2. Purely topological invariants come first in mind. Homotopy groups of the brane are invariants inherent to the brane (the brane topology can however change). Homological and cohomological characteristics of branes in singular homology characterize the imbedding to the target space. There are also more delicate differential topological invariants such as de Rham cohomology defining invariants analogous to magnetic charges. Dolbeault cohomology emerges naturally for even-dimensional branes with complex structure.

  3. Gauge theories - both abelian and non-Abelian - define a standard approach to the construction of brane charges for the bundle structures assigned with branes. Chern-Simons classes are fundamental invariants of this kind. Also more delicate invariants associated with gauge potentials can be considered. Chern-Simons theory with vanishing field strengths for solutions of field equations provides a basic example about this. For intance, SU(2) Chern-Simons theory provides 3-D topological invariants and knot invariants.

  4. More refined approaches involve K-theory -closely related to motivic cohomology - and its twisted version. The idea is to reduce the classification of branes to the classification of the bundle structures associated with them. This approach has had remarkable successes but has also its short-comings.

The challenge is to find the mathematical classification which suits best the physical intuitions (, which might be fatally wrong as already proposed) but is universal at the same time. This challenge has turned out to be tough. The Ramond-Ramond (RR) p-form fields of type II superstring theory are rather delicate objects and a source of most of the problems. The difficulties emerge also by the presence of Neveu-Schwartz 3-form H =dB defining classical background field.

K-theory has emerged as a good candidate for the classification of branes. It leaves the confines of homology and uses bundle structures associated with branes and classifies these. There are many K-theories. In topological K-theory bundles form an algebraic structure with sum, difference, and multiplication. Sum is simply the direct sum for the fibers of the bundle with common base space. Product reduces to a tensor product for the fibers. The difference of bundles represents a more abstract notion. It is obtained by replacing bundles with pairs in much the same way as rationals can be thought of as pairs of integers with equivalence (m,n)= (km,kn), k integer. Pairs (n,1) representing integers and pairs (1,n) their inverses. In the recent case one replaces multiplication with sum and regards bundle pairs and (E,F) and (E+G,F+G) equivalent. Although the pair as such remains a formal notion, each pair must have also a real world representativs. Therefore the sign for the bundle must have meaning and corresponds to the sign of the charges assigned to the bundle. The charges are analogous to winding of the brane and one can call brane with negative winding antibrane. The interpretation in terms of orientation looks rather natural. Later a TGD inspired concrete interpretation for the bundle sum, difference, product and also division will be proposed.

Problems related to the existence of spinor structure

Many problems in the classification of brane charges relate to the existence of spinor structure. The existence of spinor structure is a problem already in general general relativity since ordinary spinor structure exists only if the second Stiefel-Whitney class of the manifold is non-vanishing: if the third Stiefel-Whitney class vanishes one can introduce so called spinc structure. This kind of problems are encountered already in lattice QCD, where periodic boundary conditions imply non-uniqueness having interpretation in terms of 16 different spinor structures with no obvious physical interpretation. One the strengths of TGD is that the notion of induced spinor structure eliminates all problems of this kind completely. One can therefore find direct support for TGD based notion of spinor structure from the basic inconsistency of QCD lattice calculations!

  1. Freed-Witten anomaly appearing in type II string theories represents one of the problems. Freed and Witten show that in the case of 2-branes for which the generalized gauge potential is 3-form so called spinc structure is needed and exists if the third Stiefel-Whitney class w3 related to second Stiefel Whitney class whose vanishing guarantees the existence of ordinary spin structure (in TGD framework spinc structure for CP2 is absolutely essential for obtaining standard model symmetries).

    It can however happen that w3 is non-vanishing. In this case it is possible to modify the spinc structure if the condition w3+[H]=0 holds true. It can however happen that there is an obstruction for having this structure - in other words w3+[H] does not vanish - known as Freed-Witten anomaly. In this case K-theory classification fails. Witten and Freed argue that physically the wrapping of cycle with non-vanishing w3 + [H] by a Dp-brane requires the presence of D(p-2) brane cancelling the anomaly. If D(p-2) brane ends to anti-Dp in which case charge conservation is lost. If there is not place for it to end one has semi-infinite brane with infinite mass, which is also problematic physically. Witten calls these branes baryons: these physically very dubious objects are not classified by K-theory.

  2. The non-vanishing of w3+[H]=0 forces to generalize K-theory to twisted K-theory. This means a modification of the exterior derivative to get twisted de Rham cohomology and twisted K-theory and the condition of closedness in this cohomology for certain form becomes the condition guaranteeing the existence of the modified spinc structure. D-branes act as sources of these fields and the coupling is completely analogous to that in electrodynamics. In the presence of classical Neveu-Schwartz (NS-NS) 3-form field H associated with the back-ground geometry the field strength Gp+1 = dCp is not gauge invariant anymore. One must replace the exterior derivative with its twisted version to get twisted de Rham cohomology:

    d→ d+ H∧ .

    There is a coupling between p- and p+2-forms together and gauge symmetries must be modified accordingly. The fluxes of twisted field strengths are not quantized but one can return to original p-forms which are quantized. The coupling to external sources also becomes more complicated and in the case of magnetic charges one obtains magnetically charged Dp-branes. Dp-brane serves as a source for D(p-2)- branes.

    This kind of twisted cohomology is known by mathematicians as Deligne cohomology. At the level of homology this means that if branes with dimension of p are presented then also branes with dimension p+2 are there and serve as source of Dp-branes emanating from them or perhaps identifiable as their sub-manifolds. Ordinary homology fails in this kind of situation and the proposal is that so called twisted K-theory could allow to classify the brane charges.

  3. A Lagrangian formulation of brane dynamics based on the notion of p-brane democracy due to Peter Townsend has been developed by various authors.

Ashoke Sen has proposed a grand vision for understanding the brane classification in terms of tachyon condensation in absence of NS-NS field H. The basic observation is that stacks of space-filling D- and anti D-branes are unstable against process called tachyon condensation which however means fusion of p+1-D brane orbits rather than p-dimensional time slicse of branes. These branes are however accompanied by lower-dimensional branes and the decay process cannot destroy these. Therefore the idea arises that suitable stacks of D9 branes and anti-D9-branes could code for all lower-dimensional brane configurations as the end products of the decay process.

This leads to a creation of lower-dimensional branes. All decay products of branes resulting in the decay cascade would be by definition equivalent. The basic step of the decay process is the fusion of D-branes in stack to single brane. In bundle theoretic language one can say that the D-branes and anti-D branes in the stack fuse together to single brane with bundle fiber which is direct sum of the fibers on the stack. This fusion process for the branes of stack would correspond in topological K-theory. The fusion of D-branes and anti-D branes would give rise to nothing since the fibers would have opposite sign. The classification would reduce to that for stacks of D9-branes and anti D9-branes.

Problems with Hodge duality and S-duality

The K-theory classification is plagued by problems all of which need not be only technical.

  1. R-R fields are self dual and since metric is involved with the mapping taking forms to their duals one encounters a problem. Chern characters appearing in K-theory are rational valued but the presence of metric implies that the Chern characters for the duals need not be rational valued. Hence K-theory must be replaced with something less demanding.

    The geometric quantization inspired proposal of Diaconescu, Moore and Witten is based on the polarization using only one half of the forms to get rid of the proboem. This is like thinking the 10-D space-time as phase space and reducing it effectively to 5-D space: this brings strongly in mind the identification of space-time surfaces as hyper-quaternionic (associative) sub-manifolds of imbedding space with octonionic structure and one can ask whether the basic objects also in M-theory should be taken 5-dimensional if this line of thought is taken seriously. An alternative approach uses K-theory to classify the intersections of branes with 9-D space-time slice as has been porposed by Maldacena, Moore and Seiberg.

  2. There another problem related to classification of the brane charges. Witten, Moore and Diaconescu have shown that there are also homology cycles which are unstable against decay and this means that twisted K-theory is inconsistent with the S-duality of type IIB string theory. Also these cycles should be eliminated in an improved classification if one takes charge conservation as the basic condition and an hitherto un-known modification of cohomology theory is needed.

  3. There is also the problem that K-theory for time slices classifies only the R-R field strengths. Also R-R gauge potentials carry information just as ordinary gauge potentials and this information is crucial in Chern-Simons type topological QFTs. K-theory for entire target space classifies D-branes as p+1-dimensional objects but in this case the classification of R-R field strengths is lost.

The existence of non-representable 7-D homology classes for targent space dimension D>9

There is a further nasty problem which destroys the hopes that twisted K-theory could provide a satisfactory classification. Even worse, something might be wrong with the superstring theory itself. The problem is that not all homology classes allow a representation as non-singular manifolds. The first dimension in which this happens is D=10, the dimension of super-string models! Situation is of course the same in M-theory. The existence of the non-representables was demonstrated by Thom - the creator of catastrophe theory and of cobordism theory for manifolds- for a long time ago.

What happens is that there can exist 7-D cycles which allow only singular imbeddings. A good example would be the imbedding of twistor space CP3, whose orbit would have conical singularity for which CP3 would contract to a point at the "moment of big bang". Therefore homological classification not only allows but demands branes which are orbifolds. Should orbifolds be excluded as unphysical? If so then homology gives too many branes and the singular branes must be excluded by replacing the homology with something else. Could twisted K-theory exclude non-representable branes as unstable ones by having non-vanishing w3+[H]? The answer to the question is negative: D6-branes with w3+[H]=0 exist for which K-theory charges can be both vanishing or non-vanishing.

One can argue that non-representability is not a problem in superstring models (M-theory) since spontaneous compactification leads to M× X6 (M× X7). On the other hand, Cartesian product topology is an approximation which is expected to fail in high enough length scale resolution and near big bang so that one could encounter the problem. Most importantly, if M-theory is theory of everything it cannot contain this kind of beauty spots.

What could go wrong with super string theory and how TGD circumvents the problems?

As a proponent of TGD I cannot avoid the temptation to suggest that at least two things could go wrong in the fundamental physical assumptions of superstrings and M-theory.

  1. The basic failure would be the construction of quantum theory starting from semiclassical approximation assuming localization of currents of 10 - or 11-dimensional theory to lower-dimensional sub-manifolds. What should have been a generalization of QFT by replacing pointlike particles with higher-dimensional objects would reduce to an approximation of 10- or 11-dimensional supergravity.

    This argument does not bite in TGD. 4-D space-time surfaces are indeed fundamental objects in TGD as also partonic 2-surfaces and braids. This role emerges purely number theoretically inspiring the conjecture that space-time surfaces are associative sub-manifolds of octonionic imbedding spaces, from the requirement of extended conformal invariance, and from the non-dynamical character of the imbedding space.

  2. The condition that all homology equivalence classes are representable as manifolds excludes all dimensions D> 9 and thus super-strings and M-theory as a physical theory. This would be the case since branes are unavoidable in M-theory as is also the landscape of compactifications. In semiclassical supergravity interpretation this would not be catastrophe but if branes are fundamental objects this shortcoming is serious. If the condition of homological representability is accepted then target space must have dimension D<10 and the arguments sequence leading to D=8 and TGD is rather short. The number theoretical vision provides the mathematical justification for TGD as the unique outcome.

  3. The existence of spin structure is clearly the source of many problems related to R-R form. In TGD framework the induction of spinc structure of the imbedding space resolves all problems associated with sub-manifold spin structures. For some reason the notion of induced spinor structure has not gained attention in super string approach.

  4. Conservative experimental physicist might criticize the emergence of branes of various dimensions as something rather weird. In TGD framework electric-magnetic duality can be understood in terms of general coordinate invariance and holography and branes and their duals have dimension 2, 3, and 4 organize to sub-manifolds of space-time sheets. The TGD counterpart for the fundamental M-2-brane is light-like 3-surface. Its magnetic dual has dimension given by the general formula pdual= D-p-4, where D is the dimension of the target space. In TGD one has D=8 giving pdual= 2. The first interpretation is in terms of self-duality. A more plausible interpretation relies on the identification of the duals of light-like 3-surfaces as spacelike-3-surfaces at the light-like boundaries of CD. General Coordinate Invariance in strong sense implies this duality. For partonic 2-surface one would have p=2 and pdual=3. The identification of the dual would be as space-time surface. The crucial distinction to M-theory would be that branes of different dimension would be sub-manifolds of space-time surface.

  5. For p=0 one would have pdual=4 assigning five-dimensional surface to orbits of point-like particles identifiable most naturally as braid strands. One cannot assign to it any direct physical meaning in TGD framework and gauge invariance for the analogs of brane gauge potentials indeed excludes even-dimensional branes in TGD since corresponding forms are proportional to Kähler gauge potential (so that they would be analogous to odd-dimensional branes allowed by type IIB superstrings).

    4-branes could be however mathematically useful by allowing to define Morse theory for the critical points of the Minkowskian part of Kähler action. While writing this I learned that Witten has proposed a 4-D gauge theory approach with N=4 SUSY to the classification of knots. Witten also ends up with a Morse theory using 5-D space-times in the category-theoretical formulation of the theory. For some time ago I also proposed that TGD as almost topological QFT defines a theory of knots, knot braidings, and of 2-knots in terms of string world sheets. Maybe the 4-branes could be useful for understanding of the extrema of TGD of the Minkowskian part of Kähler action which would take take the same role as Hamiltonian in Floer homology: the extrema of 5-D brane action would connect these extrema.

  6. Light-like 3-surfaces could be seen as the analogs von Neuman branes for which the boundary conditions state that the ends of space-like 3-brane defined by the partonic 2-surfaces move with light-velocity. The interpretation of partonic 2-surfaces as space-like branes at the ends of CD would in turn make them D-branes so that one would have a duality between D-branes and N-brane interpretations. T-duality exchanges von Neumann and Dirichlet boundary conditions so that strong from of general coordinate invariance would correspond to both electric-magnetic and T-duality in TGD framework. Note that T-duality exchanges type IIA and type IIB super-strings with each other.

  7. What about causal diamonds and their 7-D lightlike boundaries? Could one regard the light-like boundaries of CDs as analogs of 6-branes with light-like direction defining time-like direction so that space-time surfaces would be seen as 3-branes connecting them? This brane would not have magnetic dual since the formula for the dimensions of brane and its magnetic dual allows positive brane dimension p only in the range (1,3).

Can one identify the counterparts of R-R and NS-NS fields in TGD?

R-R and NS-NS 3-forms are clearly in fundamental role in M-theory. Since in TGD partonic 2-surfaces define the analogs of fundamental M-2-branes, one can wonder whether these 3-forms could have TGD counterparts.

  1. In TGD framework the 3-forms G3,A =dC2,A defined as the exterior derivatives of the two-forms C2,A identified as products C2,A=HAJ of Hamiltonians HA of δ M4+/-× CP2 with Kähler forms of factors of δ M4+/-× CP2 define an infinite family of closed 3-forms belonging to various irreducible representations of rotation group and color group. One can consider also the algebra generated by products HA A, HAJ, HA A∧ J, HA J∧ J, where A resp. J denotes the Kähler gauge potential resp. Kähler form or either δ M4+/- or CP2. A resp. Also the sum of Kähler potentials resp. forms of δ M4+/- and CP2 can be considered.

  2. One can define the counterparts of the fluxes ∫ Adx as fluxes of HA A over braid strands, HAJ over partonic 2-surfaces and string world sheets, HA A∧ J over 3-surfaces, and HAJ∧ J over space-time sheets. Gauge invariance however suggests that for non-constant Hamiltonians one must exclude the fluxes assigned to odd dimensional surfaces so that only odd-dimensional branes would be allowed. This would exclude 0-branes and the problematic 4-branes. These fluxes should be quantized for the critical values of the Minkowskian contributions and for the maxima with respect to zero modes for the Euclidian contributions to Kähler action. The interpretation would be in terms of Morse function and Kähler function if the proposed conjecture holds true. One could even hope that the charges in Cartan algebra are quantized for all preferred extremals and define charges in these irreducible representations for the isometry algebra of WCW. The quantization of electric fluxes for string world sheets would give rise to the familiar quantization of the rotation ∫ E dl of electric field over a loop in time direction taking place in superconductivity.

  3. Should one interpret these fluxes as the analogs of NS-NS-fluxes or R-R fluxes? The exterior derivatives of the forms G3 vanish which is the analog for the vanishing of magnetic charge densities (it is however possible to have the analogs of homological magnetic charge). The self-duality of Ramond p-forms could be posed formally (Gp= *G8-p) but does not have any implications for p< 4 since the space-time projections vanish in this case identically for p>3. For p=4 the dual of the instanton density J∧ J is proportional to volume form if M4 and is not of topological interest. The approach of Witten eliminating one half of self dual R-R-fluxes would mean that only the above discussed series of fluxes need to be considered so that one would have no troubles with non-rational values of the fluxes nor with the lack of higher dimensional objects assignable to them. An interesting question is whether the fluxes could define some kind of K-theory invariants.

  4. In TGD imbedding space is non-dynamical and there seems to be no counterpart for the NS 3-form field H=dB. The only natural candidate would correspond to Hamiltonian B=J giving H=dB=0. At quantum level this might be understood in terms of bosonic emergence meaning that only Ramond representations for fermions are needed in the theory since bosons correspond to wormhole contacts with fermion and anti-fermions at opposite throats. Therefore twisted cohomology is not needed and there is no need to introduce the analogy of brane democracy and 4-D space-time surfaces containing the analogs of lower-dimensional brains as sub-manifolds are enough. The fluxes of these forms over partonic 2-surfaces and string world sheets defined non-abelian analogs of ordinary gauge fluxes reducing to rotations of vector potentials and suggested be crucial for understanding braidings of knots and 2-knots in TGD framework. Note also that the unique dimension D=4 for space-time makes 4-D space-time surfaces homologically self-dual so that only they are needed.

Could one divide bundles?

TGD differs from string models in one important aspects: stringy diagrams do not have interpretation as analogs of vertices of Feynman diagrams: the stringy decay of partonic 2-surface to two pieces does not represent particle decay but a propagation along different paths for incoming particle. Particle reactions in turn are described by the vertices of generalized Feynman diagrams in which the ends of incoming and outgoing particles meet along partonic 2-surface. This suggests a generalization of K-theory for bundles assignable to the partonic 2-surfaces. It is good to start with a guess for the concrete geometric realization of the sum and product of bundles in TGD framework.

  1. The analogs of string diagrams could represent the analog for direct sum. Difference between bundles could be defined geometrically in terms of trouser vertex A+B→ C. B would by definition represent C-A. Direct sum could make sense for single particle states and have as space-time correlate the conservation of braid strands.

  2. A possible concretization in TGD framework for the tensor product is in terms of the vertices of generalized Feynman diagrams at which incoming light-like 3-D orbits of partons meet along their ends. The tensor product of incoming state spaces defined by fermionic oscillator algebras is naturally formed. Tensor product would have also now as a space-time correlate conservation of braid strands. This does not mean that the number of braid strands is conserved in reactions if also particular exchanges can carry the braid strands of particles coming to the vertex.

Why not define also division of bundles in terms of the division for tensor product? In terms of the 3-vertex for generalized Feynman diagrams A⊗ B=C representing tensor product B would be by definition C/A. Therefore TGD would extend the K-theory algebra by introducing also division as a natural operation necessitated by the presence of the join along ends vertices not present in string theory. I would be surprised if some mathematician would not have published the idea in some exotic journal. Below I represent an argument that this notion could be also applied in the mathematical description of finite measurement resolution in TGD framework using inclusions of hyper-finite factor. Division could make possible a rigorous definition for for non-commutative quantum spaces.

Tensor division could have also other natural applications in TGD framework.

  1. One could assign bundles M+ and M- to the upper and lower light-like boundaries of CD. The bundle M+/M- would be obtained by formally identifying the upper and lower light-like boundaries. More generally, one could assign to the boundaries of CD positive and negative energy parts of WCW spinor fields and corresponding bundle structures in "half WCW". Zero energy states could be seen as sections of the unit bundle just like infinite rationals reducing to real units as real numbers would represent zero energy states.

  2. Finite measurement resolution would encourage tensor division since finite measurement resolution means essentially the loss of information about everything below measurement resolution represented as a tensor product factor. The notion of coset space formed by hyper-finite factor and included factor could be understood in terms of tensor division and give rise to quantum group like space with fractional quantum dimension in the case of Jones inclusions. Finite measurement resolution would therefore define infinite hierarchy of finite dimensional non-commutative spaces characterized by fractional quantum dimension. In this case the notion of tensor product would be somewhat more delicate since complex numbers are effectively replaced by the included algebra whose action creates states not distinguishable from each other. The action of algebra elements to the state |B> in the inner product < A|B> must be equivalent with the action of its hermitian conjugate to the state < A|. Note that zero energy states are in question so that the included algedra generates always modifications of states which keep it as a zero energy state.

For more details see the new chapter Infinite Primes and Motives or the article with same title.



How detailed the quantum classical correspondence can be?

Can the dynamics defined by preferred extremals of Kähler action be dissipative in some sense? The generation of the arrow of time has a nice realization in zero energy ontology as a choice of well-defined particle numbers and other quantum numbers at the "lower" end of CD. By quantum classical correspondence this should have a space-time correlate. Gradient dynamics is a highly phenomenological realization of the dissipative dynamics and one must try to identify a microscopic variant of dissipation in terms of entropy growth of some kind. If the arrow of time and dissipation has space-time correlate, there are hopes about the identification of this kind of correlate.

Quantum classical correspondence has been perhaps the most useful guiding principle in the construction of quantum TGD. What is says that not only quantum numbers but also quantum jump sequences should have space-time correlates: about this the failure of strict determinism of Kähler action gives good hopes. Even the quantum superposition - at least for certain situations - might have space-time correlates.

  1. Measurement interaction term in the modified Dirac action at the upper end of CD indeed defines a coupling to the classical dynamics kenociteallb/Dirac in a very delicate manner. This kind of measurement interaction is indeed basic element of quantum TGD. Also the color and charges and angular momentum associated with the Hamiltonians at point of braids could couple to the dynamics via the boundary conditions.

  2. The braid strand with a given Hamiltonian could obey Hamiltonian equations of motion: this would give rise to a skeleton of space-time defined by braid strands possibly continued to string world sheets and would provided different realization of quantum classical correspondence. Symplectic tringulation suggests by the symplectic QFT proposed to describe physics in zero modes would add to the skeleton edges connecting string ends continued to 2-D sheets in the interior of space-time.

  3. Quantum TGD can be regarded as a square root of thermodynamics in well-defined sense. Could it be possible to couple the Hermitian square root of density matrix appearing in M-matric and characterizing zero energy state thermally to the geometry of space-time sheets by coupling it to the classical dynamical via boundary conditions depending on its eigenvalues? The necessity to choose single eigenvalue spoils the attempt and one obtains only a representation for single measurement outcome. It seems that one can achieve only a representation of the ensemble at space-time level consisting of space-time sheets representing various outcomes of measurement. This ensemble would be realized as ensemble of sub-CDs for a given CD.

  4. One can pose even more ambigious question: could quantum superposition of WCW spinor fields have a space-time correlate in the sense that all space-time surfaces in the superposition would carry information about the superposition itself? Obviously this would mean self-referentiality via quantum-classical feedback.

The following discussion concentrates on possible space-time correlates for the quantum superposition of WCW spinor fields and for the arrow of time.

  1. It seems difficult to imagine space-time correlate for the quantum superposition of final states with varying quantum numbers since these states correspond to quantum superpositions of different space-time surfaces. How could one code information about quantum superposition of space-time surfaces to the space-time surfaces appearing in the superposition? This kind of self-referentiality seems to be necessary if one requires that various quantum numbers characterizing the superposition (say momentum) couple via boundary conditions to the space-time dynamics.

  2. The failure of non-determinism of quantum dynamics is behind dissipation and strict determinism fails for Kähler action. This gives hopes that the dynamics induces also arrow of time. Energy non-conservation is of course excluded and one should be able to identify a measure of entropy and the analog of second law of thermodynamics telling what happens at for preferred extremals when the situation becomes non-deterministic. The vertices of generalized Feynman graphs are natural places were non-determinism emerges as are also sub-CDs. Naive physical intuition would suggest that dissipation means generation of entropy: the vertices would favor decay of particles rather than their spontaneous assembly. The analog of blackhole entropy assignable to partonic 2-surfaces might allow to characterize this quantatively. The symplectic area of partonic 2-surface could be a symplectic invariant of this kind.

  3. Could the mysterious branching of partonic 2-surfaces -obviously analogous to even more mysterious branching of quantum state in many worlds interpretation of quantum mechanics- assigned to the multivalued character of the correspondence between canonical momentum densities and time derivatives of H coordinates allow to understand how the arrow of time is represented at space-time level? Recall that this brancing is what implies the effective hierarchy of Planck constants as integer multiples of its minimal value absolutely crucial for the application of TGD in biology and consciousness and to the understanding of dark matter as large hbar phases

    1. This branching would effectively replace CD with its singular covering with number of branches dependin on space-time region. The relative homology with respect to the upper boundary of CD (so that the branches of the trees would effectively meet there) could define the analog of Floer homology with various paths defined by the orbits of partonic 2-surfaces along lines of generalize Feynman diagram defining the first homology group. Typically tree like structures would be involved with the ends of the tree at the upper boundary of CD effectively identified.

    2. This branching could serve as a representation for the branching of quantum state to a superposition of eigenstates of measured quantum observables. If this is the case, the various branches to which partonic 2-surface decays at partonic 2-surface would more or less relate to quantum superposition of final states in particle reaction. The number of branches would be finite by finite measurement resolution. For a given choice of the arrow of geometric time the partonic surface would not fuse back at the upper end of CD.

    3. Rather paradoxically, the space-time correlate for the dissipation would reduce the dissipation by increasing the effective value of hbar: the interpretation would be however in terms of dark matter identified in terms of large hbar phase. In the same manner dissipation would be accompanied by evolution since the increase of hbar naturally implies formation of macroscopically quantum coherent states. The space-time representation of dissipation would compensate the increase of entropy at the ensemble level.

    4. The geometric representation of quantum superposition might take place only in the intersection of real and p-adic worlds and have interpretation in terms of cognitive representations. In the intersection one can also have a generalization of second law (see this) in which the generation of genuine negentropy in some space-time regions via the build up of cognitive representation compensated by the generation of entropy at other space-time regions. The entropy generating behavior of living matter conforms with this modification of the second law. The negentropy measure in question relies on the replacement of logarithms of probabilities with logarithms of their p-adic norms and works for rational probabilities and also their algebraic variants for finite-dimensional algebraic extensions of rationals.

    5. Each state in the superposition of WCW quantum states would contain this representation as its space-time correlate realizing self-referentiality at quantum level in the intersection of real and p-adic worlds. Also the state function reduced members of ensemble could contain this cognitive representation at space-time level. Essentially quantum memory making possible self-referential linguistic representation of quantum state in terms of space-time geometry and topology would be in question. The formulas written by mathematicians would define similar map from quantum level to the space-time level making possible to "see" one's thoughts.

For more details see the new chapter Infinite Primes and Motives of "Physics as Generalized Number Theory" or the article with same title.



Floer homology and TGD

TGD can be seen as almost topological quantum field theory. This could have served as a motivation for spending most of last months to the attempt to learn some of the mathematics related to various kind of homologies and cohomologies. The decisive stimulus came from the attempt to understand the basic ideas of motivic cohomology. I am not a specialist and do not have any ambition or abilities to become such. My goals is to see whether these ideas could be applied in quantum TGD.

Documentation is the best manner to develop ideas and the learning process has materialized as a new chapter entitled Infinite Primes and Motives of "Physics as Generalized Number Theory". It soon became clear that much of the mathematics needed by TGD has existed for decades and developing all the time. The difficult task is to understand the essentials of this mathematics and translate to the language that I talk and understand. Also generalization is unavoidable. Those who think that new physics can be done by taking math as such are wasting their time.

Among another things I have been learning about various cohomologies and homologies - about quantum cohomology, about Floer homology and topological string theories, about Gromov-Witten invariants,... It would be very naive to think that these notions would work as such in TGD framework. It looks however very plausible that the their generalizations to TGD exist, and could be very useful in the more detailed formulation of quantum TGD. The crucially important notion is finite measurement resolution making everything almost topological and highly number theoretic. In this brain-stormy spirit I have even become a proud father of my own pet homology, which I have christened as braided Galois homology. It is based on the correspondence between infinite primes and polynomials of several variables and is formulated in braided group algebras with braidings realized as symplectic flows and generalizing somewhat the usual notion of homology meaning that the square of boundary operation gives something in commutator group reducing to unit element of ordinary homology only in the factor group obtained by dividing with the commutator group.

Floer homology in its original form replaces Morse function in symplectic manifold M in the loop space LM of M. The loops can be seen as homotopies of Hamiltonians and paths in loops space describe cylinders in M. With an appropriate choice of symplectic action these cylinderes can be regarded as (pseudo-)holomorphic surface completely analogous to string orbits. By combining Floer's theory with Witten's discovery about the connection between the Morse theory and supersymmetry one ends up with topological QFTs as a manner to formulate Floer homology and various variants of this notion- in particular topological QFTs characterizing topology of three-manifolds.

This kind of learning periods are very useful as a rule since they allow to improve bird's eye of view about TGD and its problems. The understanding of both quantum TGD and its classical counterpart is still far from from comprehensive.

For instance, the view about the physical and mathematical roles of Kähler actions for Euclidian and Minkowskian space-time regions is far from clear. Do they provide dual descriptions as suggested or are both needed? Kähler action for preferred extremal in Euclidian regions defines naturally positive definite Kähler function. But can one regard the Kähler action in Minkowskian regions as equivalent definition for Kähler function or should one regard it as imaginary as the presence of square root of metric determinant would suggest? What could be the interpretation in this case? The basic ideas about Floer homology suggest and answer to these questions.

  1. Since quantum fluctuating WCW degrees of freedom correspond to a symmetric space assignable to the symplectic group in TGD framework symplectic geometry is of special interest from TGD point of view. Floer homology is indeed about symplectic geometry as also Gromov-Witten invariants and topological string theories developed for the purpose of calculating these invariants. Hence the question whether Floer homology could have a generalization to TGD framework is highly relevant.

  2. As such Floer homology for M4× CP2 is deadly boring since it reduces to ordinary singular homology. The correspondence between canonical momentum densities of Kähler action and time derivatives of imbedding space coordinates is however one-to-many- and inspires the replacement of the imbedding space with its singular covering with different space-time regions corresponding to different number of sheets for the covering. The effective hierarchy of Planck constants emerges as a result. The homology in WCW could be mapped to the homology of this structure just as the homology of loop space of M is mapped to that of M in Floer theory.

  3. The obvious question is how to generalize Floer homology to TGD framework and the obvious guess is that Kähler action for preferred extremals must take the role of symplectic action for pseudo-holomorphic surfaces which could in fact be replaced with hyper-quaternionic space-time surfaces containing string world sheets whose ends defined braid strands carrying quantum numbers and which intersect partonic 2-surfaces at the future and past light-like boundaries of CDs. This actually suggests an obvious generalization for quantum cohomology based on quantal notion of intersection: partonic surfaces intersect if there exist a string world sheets connecting them. Fuzzy intersection has interpretation in terms of causal dependence: by effective 2-dimensionality this causal dependence is along light-like 3-surfaces and along space-like 3-surfaces at the boundaries of CDs. The notion of quantum intersection is so beautiful that one an almost forgive for the theoricians who have begun to take seriously the idea about branes connected by strings.

  4. The question providing the new insight is simple. Could Kähler function allow to define Morse theory? The answer is negative. Kähler metric must be positive definite so that the Hessian associated with it in quantum fluctuating degrees of freedom must have positive signature: no saddle points are possible in quantum fluctuating degrees of freedom although in zero modes they are allowed. Second counter argument is that quantum Morse theory is based on path integral rather than functional integral.

    How could one circumvent this difficulty? Could Kähler action in Minkowskian regions- naturally imaginary by negative sign of metric determinant- give an imaginary contribution to the vacuum functional and define Morse function so that both Kähler and Morse would find a prominent role in the world order of TGD? Maybe! The presence of Kähler function and Morse function in the vacuum functional would give much more direct connection with the path integral approach and Kähler function would also make path integral well-defined since one integrates only over preferred extremals of Kähler action for which Kähler action reduces to Chern-Simons term coming from Minkowskian region and contribution from Euclidian region (generalized Feynman graph).

Should one assume that the reduction to Chern-Simons terms occurs for the preferred extremals in both Minkowskian and Euclidian regions or only in Minkowskian regions?

  1. All arguments for this have been represented for Minkowskian regions involve local light-like momentum direction which does not make sense in the Euclidian regions. This does not however kill the argument: one can have non-trivial solutions of Laplacian equation in the region of CP2 bounded by wormhole throats: for CP2 itself only covariantly constant right-handed neutrino represents this kind of solution and at the same time supersymmetry. In the general case solutions of Laplacian represent broken super-symmetries and should be in one-one correspondences with the solutions of the modified Dirac equation. The interpretation for the counterparts of momentum and polarization would be in terms of classical representation of color quantum numbers.

    If the reduction occurs in Euclidian regions, it gives in the case of CP2 two 3-D terms corresponding to two 3-D gluing regions for three coordinate patches needed to define coordinates and spinor connection for CP2 so that one would have two Chern-Simons terms. Without any other contributions the first term would be identical with that from Minkowskian region apart from imaginary unit. Second Chern-Simons term would be however independent of this. For wormhole contacts the two terms could be assigned with opposite wormhole throats and would be identical with their Minkowskian cousins from imaginary unit. This looks a little bit strange.

  2. There is however a very delicate issue involved. Quantum classical correspondence requires that the quantum numbers of partonic states must be coded to the space-time geometry, and this is achieved by adding to the action a measurement interaction term which reduces to what is almost a gauge term present only in Chern-Simons-Dirac equation but not at space-time interior. This term would represent a coupling to Poincare quantum numbers at the Minkowskian side and to color and electro-weak quantum numbers at CP2 side. Therefore the net Chern-Simons contributions and would be different.

  3. There is also a very beautiful argument stating that Dirac determinant for Chern-Simons-Dirac action equals to Kähler function, which would be lost if Euclidian regions would not obey holography. The argument obviously generalizes and applies to both Morse and Kähler function.
In any case, it is still too early to give up the possibility that these two parts of Kähler action (real and positive- imaginary) provide dual descriptions as functional integral and path integral: Wick rotations is what comes in mind. Certainly, the rigorous definition of the path integral would be as difficult -should one say hopeless- as in ordinary QFT.

Floer homology and Gromov-Witten invariants provide also other insights about quantum TGD. For more details see the new chapter Infinite Primes and Motives or the article with same title.



Twistors, hyperbolic 3-manifolds, and zero energy ontology

While performing web searches for twistors and motives I have begun to realize that Russian mathematicians have been building the mathematics needed by quantum TGD for decades while realizing the great visions of Grothendieck. Maybe I am also beginning to vaguely grasp something about the connection of Grassmannian twistor approach to the motivic integrals. In the following I make comments about three articles that I found from web.

The latest finding was the article Volumes of hyperbolic manifolds and mixed Tate motives by Goncharov- one of the great Russian mathematicians involved with the drama. The article is about polylogarithms emerging in twistor calculations and their relationship to the volumes of hyperbolic n-manifolds. I do not of course understand anything about the jargon of the article: it is written by a specialist for specialists and I can only try to understand the general notions and the possible meaning of the results from TGD point of view.

Hyperbolic n-manifolds are n-manifolds equipped with complete Riemann metric having constant sectional curvature equal to -1 (with a suitable choice of length unit) and therefore obeying Einstein's equations with cosmological constant. They are obtained as coset spaces on proper-time constant hyperboloids of n+1-dimensional Minkowski space by dividing by the action of discrete subgroup of SO(n,1), whose action defines a lattice like structure on the hyperboloid. What is remarkable is that the volumes of these closed spaces are homotopy invariants in a well-defined sense.

What is even more remarkable that hyperbolic 3-manifolds are completely exceptional in that there are very many of them. The complements of knots and links in 3-sphere are often cusped hyperbolic 3-manifolds (having therefore tori as boundaries). Also Haken manifolds are hyperbolic. Says Wikipedia:

According to Thurston's geometrization conjecture, proved by Perelman, any closed, irreducible, atoroidal 3-manifold with infinite fundamental group is hyperbolic. There is an analogous statement for 3-manifolds with boundary.

Therefore there are very many hyperbolic 3-manifolds.

The geometrization conjecture of Thurston allows to see hyperbolic 3-manifolds in a wider framework. The theorem states that compact 3-manifolds can be decomposed canonically into sub-manifolds that have geometric structures. It was Perelman who sketched the proof of the conjecture. The prime decomposition with respect to connected sum reduces the problem to the classification of prime 3-manifolds and geometrization conjecture states that closed 3-manifold can be cut along tori such that the interior of each piece has a geometric structure with finite volume serving as a topological invariant. There are 8 possible geometric structures in dimension three and they are characterized by the isometry group of the geometry and the isotropy group of point.

Important is also the behavior under Ricci flowtgij= -2Rij: here t is not space-time coordinate but a parameter of homotopy. If I have understood correctly, Ricci flow is a dissipative flow gradually polishing the metric for a particular region of 3-manifold to one of the 8 highly symmetric local metrics defining topological invariants. This conforms with the general vision about dissipation as source of maximal symmetries. For compact n-manifolds the normalized Ricci flow ∂tgij= -2Rij +(2/n)Rgij preserving the volume makes sense. Interestingly, for n=4 the right hand side is Einstein tensor so that the solutions of vacuum Einstein's equations in dimension four are fixed points of normalized Ricci flow. Ricci flow expands the negatively curved regions and contracts the positively curved regions of space-time time. Hyperbolic geometries represent one these 8 geometries and for the Ricci flow is expanding. The outcome is amazingly simple and gives also support for the idea that the preferred extremals of Kähler action could represent maximally symmetries 4-geometries defining topological invariants: the preferred extremals would be maximally symmetric representatives with a given topology or algebraic geometry.

The volume spectrum for hyperbolic 3-manifolds forms a countable set which is however not discrete: the statement that one can assign to them ordinal ωω does not have any obvious meaning for the man of the street;-). What comes into my simple mind is that p-adic integers and more generally, profinite spaces with infinite number of points, might be something similar: one can enumerate them by infinitely long sequences of pinary digits so that they are countable (I do not know whether also infinite p-adic primes must be allowed and whether they could somehow correspond the hierarchy of infinite ordinals). They are totally disconnected in real sense but do not form a discrete set since since can connect any two points by a p-adically continuous curve.

What makes twistor people excited is that the polylogarithms emerging from twistor integrals (see this and this) seem to be expressible in terms of the volumes of hyperbolic manifolds. What fascinates me is that the polylogarithms in question make sense also p-adically and that the moduli spaces for causal diamonds -or rather, for the double light-cones associated with their M4 projections with second tip fixed - are naturally lattices of the 3-dimensional hyperbolic space defined by all positions of the second tip and 3-dimensional hyperbolic spaces are the most interesting ones! In the intersection of the real and p-adic worlds both algebraic universality and finite measurement resolution require number theoretic discretization so that the 3-volume volume could be quantized in discrete manner.

For n=3 the group defining the lattice is a discrete subgroup of the group of SO(3,1) which equals to PSL(2,C) obtained by identifying SL(2,C) matrices with opposite sign. The divisor group defining the lattice and hyperbolic spaces as its lattice cell is therefore a subgroup of PSL(2,Zc), where Zc denotes complex integers. Recall that PSL(2,Zc) acts also in complex plane (and therefore on partonic 2-surfaces) as discrete Möbius transformations whereas PSL(2,Z) correspond to 3-braid group. Reader is perhaps familiar with fractal like orbits of points of plane under iterated Möbius transformations. The lattice cell of this lattice obtained by identifying symmetry related points defines hyperbolic 3-manifolds. Therefore zero energy ontology realizes directly the hyperboliic manifolds whose volumes should somehow represent the poly-logarithms.

The volumes are topological invariants in the sense that homeomorphism does not affect the volume of the space in question if it is given hyperbolic metric. The spectrum of volumes is said to be highly transcendental. In the intersection of real and p-adic worlds only algebraic volumes are possible unless one allows extension by say finite number of roots of e (ep is p-adic number). The p-adic existence of polylogarithms suggests that also p-adic variants of hyperbolic spaces make sense and that one can assign to them volume as topological invariant although the notion of ordinary volume integral is problematic. In fact, hyperbolic spaces are symmetric spaces and the general arguments that I have developed earlier allow to imagine what the p-adic variants of real symmetric spaces could be.

Not surprisingly, also AdS-CFT enthusiasts would like to have similar invariants for for AdS (Minkowskian analog of hyperbolic space) and even dS (Minkowskian analog of sphere). Mitchell Porter gives a link to the talk of Maldacena. The expected non-compactness of these spaces implies infinite volume and this problem should be circumvented somehow.

Maybe the preferred role of hyperbolic spaces over AdS and dS might finally select between TGD and M-theory like approach. This would simplify matters enormously since 10-dimensional holography would reduce to 4-dimensional one and would have a direct connection with physics as we have used to know it. For condensed matter physicists expected to say something interesting about this real world already the complexities of 3-D world represent a tough enough challenge and the formulation of the problems in terms of 10-dimensional blackholes migh be too much;-).

For more details see the new chapter Infinite Primes and Motives or the article with same title.



Motives and twistors in TGD

Motivic cohomology has turned out to pop up in the calculations of the twistorial amplitudes using Grassmannian approach (see this and this). The amplitudes reduce to multiple residue integrals over smooth projective sub-varieties of projective spaces. Therefore they represent the simplest kind of algebraic geometry for which cohomology theory exists.

Also in Grothendieck's vision about motivic cohomology projective spaces are fundamental as spaces to which more general spaces can be mapped in the construction of the cohomology groups (factorization). In the previous posting I gave an abstract of a chapter about motives and TGD explaining a proposal for a non-commutative variant of homology theory based on a hierarchy of Galois groups assigned with the zero locus of polynomial and its restrictions to lover dimension planes obtained by putting variables appearing in it to zero one by one: the basic idea is simple but I would have never discovered it without infinite primes.

The basic problem is to define boundary homomorphism for the hierarchy of Galois groups Gk satisfying the non-abelian generalization of δ2=0 stating that the image under δ2 belongs to the commutator subgroup of Gk-2 and therefore is mapped to zero in abelianization, which means division by commutator sub-group.

  1. The proposal is also that the roots can be represented as points of 2-D surface (partonic 2-surface) and that Galois groups can be lifted to braid groups acting on a braid of braids of .... to which infinite primes can be mapped. Infinite primes at n:th level of hierarchy describe a states of n times quantized arithmetic SUSY for which the many particles states of the previous level take the role of elementary particles.

  2. The basic idea is very physical: the braiding for a braid of braids induces braiding of sub-braids and this is represented as a homomorphism of the Galois group lifted to braid group of the braid to the corresponding groups of sub-braids. This nothing but a representation of symmetries and braiding as a isotopic flow gives excellent hopes about a unique realization of the boundary homomorphism.

  3. This SUSY is physically extremely interesting since irreducible polynomials of degree n> 1 have interpretation as bound states. Therefore bound states, which are the basic problem of perturbative quantum field theory, would have purely number theoretic meaning. As a matter fact, infinite rationals reducing to real units in real sense represent zero energy states in zero energy ontology and it is natural to assign Galois group hierarchies also to the poles of this rational function.

Summarizing, the infinite prime - irreducible polynomial - braid - quantum state connection suggests very deep connections between number theory, algebraic geometry, topological quantum field theories, and super-symmetric quantum field theories. The article Motives and Infinite Primes gives a more detailed discussion.

Defining integration in p-adic context is one of the basic challenges of quantum TGD in which real and various p-adic physics ought to be unified to a larger theory by realizing what I have called number theoretical Universality. Grothendieck's motivic comology can be seen as a program for the realization of integration of forms making sense also in p-adic context. In the following I shall discuss some aspects of the problem in TGD framework. The discussion of course fails to satisfy all standards of mathematical rigor but it relies of extremely deep and general physical principles and my conviction is that good physics is the best guideline for developing good mathematics.

Number theoretic universality, residue integrals, and symplectic symmetry

A key challenge in the realization of the number theoretic universality is the definition of p-adic definite integral. In twistor approach integration reduces to the calculation of multiple residue integrals over closed varieties. These could exist also for p-adic number fields. Even more general integrals identifiable as integrals of forms can be defined in terms of motivic cohomology.

Yangian symmetry (see this and this) is the symmetry behind the successes of twistor Grassmannian approach and has a very natural realization in zero energy ontology (see this). Also the basic prerequisites for twistorialization are satisfied. Even more, it is possible to have massive states as bound states of massless ones and one can circumvent the IR difficulties of massless gauge theories. Even UV divergences are tamed since virtual particles consist of massless wormhole throats without bound state condition on masses. Space-like momentum exchanges correspond to pairs of throats with opposite sign of energy.

Algebraic universality could be realized if the calculation of the scattering amplitudes reduces to multiple residue integrals just as in twistor Grassmannian approach. This is because also p-adic integrals could be defined as residue integrals. For rational functions with rational coefficients field the outcome would be an algebraic number apart from power of 2π, which in p-adic framework is a nuisance unless it is possible to get rid of it by a proper normalization or unless one can accepts the infinite-dimensional transcendental extension defined by 2π. It could also happen that physical predictions do not contain the power of 2π.

Motivic cohomology defines much more general approach allowing to calculate analogs of integrals of forms over closed varieties for arbitrary number fields. In motivic integration - to be discussed below - the basic idea is to replace integrals as real numbers with elements of so called scissor group whose elements are geometric objects. In the recent case one could consider the possibility that (2π)n is interpreted as torus (S1)n regarded as an element of scissor group which is free group formed by formal sums of varieties modulo certain natural relations meaning.

Motivic cohomology allows to realize integrals of forms over cycles also in p-adic context. Symplectic transformations are transformation leaving areas invariant. Symplectic form and its exterior powers define natural volume measures as elements of cohomology and p-adic variant of integrals over closed and even surfaces with boundary might make sense. In TGD framework symplectic transformations indeed define a fundamental symmetry and quantum fluctuating degrees of freedom reduce to a symplectic group assignable to δ M4+/-× CP2 in well-defined sense (see this). One might hope that they could allow to define scissor group with very simple canonical representatives- perhaps even polygons- so that integrals could be defined purely algebraically using elementary area (volume) formulas and allowing continuation to real and p-adic number fields. The basic argument could be that varieties with rational symplectic volumes form a dense set of all varieties involved.

How to define the p-adic variant for the exponent of Kähler action?

The exponent of Kähler function defined by the Kähler action (integral of Maxwell action for induced Kähler form) is central for quantum at least in the real sector of WCW. The question is whether this exponent could have p-adic counterpart and if so, how it should be defined.

In the real context the replacement of the exponent with power of p changes nothing but in the p-adic context the interpretation is affected in a dramatic manner. Physical intuition provided by p-adic thermodynamics (see this) suggests that the exponent of Kähler function is analogous to Bolzmann weight replaced in the p-adic context with non-negative power of p in order to achieve convergence of the series defining the partition function not possible for the exponent function in p-adic context.

  1. The quantization of Kähler function as K= rlog(m/n), where r is integer, m>n is divisible by a positive power of p and n is indivisible by a power of p, implies that the exponent of Kähler function is of form (m/n)r and therefore exists also p-adically. This would guarantee the p-adic existence of the vacuum functional for any prime dividing m and for a given prime p would select a restricted set of p-adic space-time sheets (or partonic 2-surfaces) in the intersection of real and p-adic worlds. It would be possible to assign several p-adic primes to a given space-time sheet (or partonic 2-surface). In elementary particle physics a possible interpretation is that elementary particle can correspond to several p-adic mass scales differing by a power of two (see this). One could also consider a more general quantization of Kähler action as sum K=K1+K2 where K1=rlog(m/n) and K2=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e1/n can be considered.

  2. The natural continuation to p-adic sector would be the replacement of integer coefficient r with a p-adic integer. For p-adic integers not reducing to finite integers the p-adic norm of the vacuum functional would however vanish and their contribution to the transition amplitude vanish unless the number of these space-time sheets increases with an exponential rate making the net contribution proportional to a finite positive power of p. This situation would correspond to a critical situation analogous to that encounted in string models as the temperature approaches Hagedorn temperature and the number states with given energy increases as fast as the Boltzmann weight. Hagedorn temperature is essentially due to the extended nature of particles identified as strings. Therefore this kind of non-perturbative situation might be encountered also now.

  3. Rational numbers m/n with n not divisible by p are also infinite as real integers. They are somewhat problematic. Does it make sense to speak about algebraic extensions of p-adic numbers generated by p1/n and giving n-1 fractional powers of p in the extension or does this extension reduce to something equivalent with the original p-adic number field when one redefines the p-adic norm as |x|p → |vert x|1/n? Physically this kind of extension could have a well defined meaning. If this does not make sense, it seems that one must treat p-adic rationals as infinite real integers so that the exponent would vanish p-adically.

  4. If one wants that Kähler action exists p-adically a transcendental extension of rational numbers allowing all powers of log(p) and log(k), where k<p is primitive p-1:th root of unity in G(p). A weaker condition would be an extension to a ring with containing only log(p) and log(k) but not their powers. That only single k<p is needed is clear from the identity log(kr)=rlog(k), from primitive root property, and from the possibility to expand log(kr+pn), where n is p-adic integer, to powers series with respect to p. If the exponent of Kähler function is the quantity coding for physics and naturally required to be ordinary p-adic number, one could allow log(p) and log(k) to exists only in symbolic sense or in the extension of p-adic numbers to a ring with minimal dimension.

    Remark: One can get rid of the extension by log(p) and log(k) if one accepts the definition of p-adic logarithm as log(x)=log(p-kx/x0) for x=pk(x0+ py), |y|p<1. To me this definition looks somewhat artificial since this function is not strictly speaking the inverse of exponent function but it might have a deeper justification.

  5. What happens in the real sector? The quantization of Kähler action cannot take place for all real surfaces since a discrete value set for Kähler function would mean that WCW metric is not defined. Hence the most natural interpretation is that the quantization takes place only in the intersection of real and p-adic worlds, that is for surfaces which are algebraic surfaces in some sense. What this actually means is not quite clear. Are partonic 2-surfaces and their tangent space data algebraic in some preferred coordinates? Can one find a universal identification for the preferred coordinates- say as subset of imbedding space coordinates selected by isometries?

If this picture inspired by p-adic thermodynamics holds true, p-adic integration at the level of WCW would give analog of partition function with Boltzman weight replaced by a power of p reducing a sum over contributions corresponding to different powers of p with WCW integra.l over space-time sheets with this value of Kähler action defining the analog for the degeneracy of states with a given value of energy. The integral over space-time sheets corresponding to fixed value of Kähler action should allow definition in terms of a symplectic form defined in the p-adic variant of WCW. In finite-dimensional case one could worry about odd dimension of this sub-manifold but in infinite-dimensional case this need not be a problem. Kähler function could defines one particular zero mode of WCW Kähler metric possessing an infinite number of zero modes.

One should also give a meaning to the p-adic integral of Kähler action over space-time surface assumed to be quantized as multiples of log(m/n).

  1. The key observation is that Kähler action for preferred extrememals reduces to 3-D Chern-Simons form by the weak form of electric-magnetic duality. Therefore the reduction to cohomology takes place and the existing p-adic cohomology gives excellent hopes about the existence of the p-adic variant of Kähler action. Therefore the reduction of TGD to almost topological QFT would be an essential aspect of number theoretical universality.

  2. This integral should have a clear meaning also in the intersection of real and p-adic world. Why the integrals in the intersection would be quantized as multiple of log(m/n), m/n divisible by a positive power of p? Could log(m/n) relate to the integral of ∫1p dx/x, which brings in mind ∮ dz/z in residue calculus. Could the integration range [1,m/n] be analogous to the integration range [0,2π]. Both multiples of 2π and logarithms of rationals indeed emerge from definite integrals of rational functions with rational coefficients and allowing rational valued limits and in both cases 1/z is the rational function responsible for this.

  3. log(m/n) would play a role similar to 2π in the approach based on motivic integration where integral has geometric objects as its values. In the case of 2π the value would be circle. In the case of log(m/n) the value could be the arc between the points r=m/n>1 and r=1 with r identified the radial coordinate of light-cone boundary with conformally invariant length measures dr/r. One can also consider the idea that log(m/n) is the hyperbolic angle analogous to 2π so that these two integrals could correspond to hyper-complex and complex residue calculus respectively.

  4. TGD as almost topological QFT means that for preferred extremals the Kähler action reduces to 3-D Chern-Simons action, which is indeed 3-form as cohomology interpretation requires, and one could consider the possibility that the integration giving log(m/n) factor to Kähler action is associated with the integral of Chern-Simons action density in time direction along light-like 3-surface and that the integral over the transversal degrees of freedom could be reduced to the flux of the induced CP2 Kähler form. The logarithmic quantization of the effective distance between the braid end points the in metric defined by modified gamma matrices has been proposed earlier.

Since p-adic objects do not possess boundaries, one could argue that only the integrals over closed varieties make sense. Hence the basic premise of cohomology would fail when one has p-adic integral over braid strand since it does not represent closed curve. The question is whether one could identify the end points of braid in some sense so that one would have a closed curve effectively or alternatively relative cohomology. Periodic boundary conditions is certainly one prerequisite for this kind of identification.

  1. In one of the many cohomologies known as quantum cohomology one indeed assumes that the intersection of varieties is fuzzy in the sense that two surfaces for points are connected by a curve of certain kind known as pseudo-holomorphic curve can be said to intersect at these points.

  2. In the construction of the solutions of the modified Dirac equation one assumes periodic boundary conditions so that in physical sense these points are identified (see this). This assumption actually reduces the locus of solutions of the modified Dirac equation to a union of braids at light-like 3-surfaces so that finite measurement resolution for which discretization defines space-time correlates becomes an inherent property of the dynamics. The coordinate varying along the braid strands is light-like so that the distance in the induced metric vanishes between its end points (unlike the distance in the effective metric defined by the modified gamma matrices): therefore also in metric sense the end points represent intersection point. Also the effective 2-dimensionality means are effectively one and same point.

  3. The effective metric 2-dimensionality of the light-like 2-surfaces implies the counterpart of conformal invariance with the light-like coordinate varying along braid strands so that it might make sense to say that braid strands are pseudo-holomorphic curves. Note also that the end points of a braid along light-like 3-surface are not causally independent: this is why M-matrix in zero energy ontology is non-trivial. Maybe the causal dependence together with periodic boundary conditions, light-likeness, and pseudo-holomorphy could imply a variant of quantum cohomology and justify the p-adic integration over the braid strands.

Motivic integration

While doing web searches related to motivic cohomology I encountered also the notion of motivic measure proposed first by Kontsevich. Motivic integration is a purely algebraic procedure in the sense that assigns to the symbol defining the variety for which one wants to calculate measure. The measure is not real valued but takes values in so called scissor group, which is a free group with group operation defined by a formal sum of varieties subject to relations. Motivic measure is number theoretical universal in the sense that it is independent of number field but can be given a value in particular number field via a homomorphism of motivic group to the number field with respect to sum operation.

Some examples are in order.

  1. A simple example about scissor group is scissor group consisting operations needed in the algorithm transforming plane polygon to a rectangle with unit edge. Polygon is triangulated; triangles are transformed to rectangle using scissors; long rectangles are folded in one half; rectangles are rescaled to give an unit edge (say in horizontal direction); finally the resulting rectangles with unit edge are stacked over each other so that the height of the stack gives the area of the polygon. Polygons which can be transformed to each other using the basic area preserving building bricks of this algorithm are said to be congruent.

    The basic object is the free abelian group of polygons subject to two relations analogous to second homology group. If P is polygon which can be cut to two polygons P1 and P2 one has [P]=[P1]+[P2]. If P and P' are congruent polygons, one has [P]=[P']. For plane polygons the scissor group turns out to be the group of real numbers and the area of polygon is the area of the resulting rectangle. The value of the integral is obtained by mapping the element of scissor group to a real number by group homomorphism.

  2. One can also consider symplectic transformations leaving areas invariant as allowed congruences besides the slicing to pieces as congruences appearing as parts of the algorithm leading to a standard representation. In this framework polygons would be replaced by a much larger space of varieties so that the outcome of the integral is variety and integration means finding a simple representative for this variety using the relations of the scissor group. One might hope that a symplectic transformations singular at the vertices of polygon combined with with scissor transformations could reduce arbitrary area bounded by a curve into polygon.

  3. One can identify also for discrete sets the analog of scissor group. In this case the integral could be simply the number of points. Even more abstractly: one can consider algebraic formulas defining algebraic varieties and define scissor operations defining scissor congruences and scissor group as sums of the formulas modulo scissor relations. This would obviously abstract the analytic calculation algorithm for integral. Integration would mean that transformation of the formula to a formula stating the outcome of the integral. Free group for formulas with disjunction of formulas is the additive operation(see this). Congruence must correspond to equivalence of some kind. For finite fields it could be bijection between solutions of the formulas. The outcome of the integration is the scissor group element associated with the formula defining the variety.

  4. For residue integrals the free group would be generated as formal sums of even-dimensional complex integration contours. Two contours would be equivalent if they can be deformed to each other without going through poles. The standard form of variety consists of arbitrary small circles surrounding the poles of the integrand multiplied by the residues which are algebraic numbers for rational functions. This generalizes to rational functions with both real and p-adic coefficients if one accepts the identification of integral as a variety modulo the described equivalence so that (2π)n corresponds to torus (S1)n. One can replace torus with 2π if one accepts an infinite-dimensional algebraic extension of p-adic numbers by powers of 2π. A weaker condition is that one allows ring containing only the positive powers of 2π.

  5. The Grassmannian twistor approach for two-loop hexagon Wilson loop gives classical polylogarithms Lk(s) (see this). General polylogarithm is defined by obey the recursion formula:

    Lis+1(z)= ∫0zLis(t)dt/t .

    Ordinary logarithm Li1(s) = -log(1-s) exists p-adically and generates a hierarchy containing dilogarithm, trilogarithm, and so on, which each exist p-adically for |x|p < 1 as is easy to see. If one accepts the general definition of p-adic logariths one finds that the entire function series exists p-adically for integer values of s. An interesting question is how strong constraints p-adic existence gives to the thetwistor loop integrals and to the underlying QFT.

  6. The ring having p-adic numbers as coefficients and spanned by transcendentals log(k) and log(p), where k is primitive root of unity in G(p) emerges in the proposed p-adicization of vacuum functional as exponent of Kähler action. The action for the preferred extremals reducing to 3-D Chern-Simons action for space-time surfaces in the intersection of real and p-adic worlds would be expressible p-adically as a linear combination of log(p) and log(k). log(m/n) expressible in this manner p-adically would be the symbolic outcome of p-adic integral ∫ dx/x between rational points. x could be identified as a preferred coordinate along braid strand. A possible identification for x earlier would be as the length in the effective metric defined by modified gamma matrices appearing in the modified Dirac equation (see this).

Infinite rationals and multiple residue integrals as Galois invariants and Galois groups as symmetry groups of quantum physics

In TGD framework one could consider also another kind of cohomological interpretation. The basic structures are braids at light-like 3-surfaces and space-like 3-surfaces at the ends of space-time surfaces. Braids intersects have common ends points at the partonic 2-surfaces at the light-like boundaries of a causal diamond. String world sheets define braid cobordism and in more general case 2-knot (see this)). Strong form of holography with finite measurement resolution would suggest that physics is coded by the data associated with the discrete set of points at partonic 2-surfaces. Cohomological interpretation would in turn would suggest that these points could be identified as intersections of string world sheets and partonic 2-surface defining dual descriptions of physics and would represent intersection form for string world sheets and partonic 2-surfaces.

Infinite rationals define rational functions and one can assign to them residue integrals if the variables xn are interpreted as complex variables. These rational functions could be replaced with a hierarchy of sub-varieties defined by their poles of various dimensions. Just as the zeros allow realization as braids or braids also poles would allow a realization as braids of braids. Hence the n-fold residue integral could have a representation in terms of braids. Given level of the braid hierarchy with n levels would correspond to a level in the hierarchy of complex varieties with decreasing complex dimension.

One can assign also to the poles (zeros of polynomial in the denominator of rational function) Galois group and obtains a hierarchy of Galois groups in this manner. Also the braid representation would exists for these Galois groups and define even cohomology and homology if they do so for the zeros. The intersections of braids with of the partonic 2-surfaces would represent the poles in the preferred coordinates and various residue integrals would have representation in terms of products of complex points of partonic 2-surface in preferred coordinates. The interpretation would be in terms of quantum classical correspondence.

Galois groups transform the poles to each other and one can ask how much information they give about the residue integral. One would expect that the n-fold residue integral as a sum over residues expressible in terms of the poles is invariant under Galois group. This is the case for the simplest integrals in plane with n poles and probably quite generally. Physically the invariance under the hierarchy of Galois group would mean that Galois groups act as the symmetry group of quantum physics. This conforms with the number theoretic vision and one could justify the formula for the residue integral also as a definition motivated by the condition of Galois invariance. Of course, all symmetric functions of roots would be Galois invariants and would be expected to appear in the expressions for scattering amplitudes.

The Galois groups associated with zeros and poles of the infinite rational seem to have a clear physical significance. This can be understood in zero energy ontology if positive (negative) physical states are indeed identifiable as infinite integers and if zero energy states can be mapped to infinite rationals which as real numbers reduce to real units. The positive/negative energy part of the zero energy state would correspond to zeros/poles in this correspondence. An interesting question is how strong correlations the real unit property poses on the two Galois group hierarchies. The asymmetry between positive and negative energy states would have interpretation in terms of the thermodynamic arrow of geometric time (see this) implied by the condition that either positive or negative energy states correspond to state function reduced/prepared states with well defined particle numbers and minimum amount of entanglement.

For more details see the new chapter Infinite Primes and Motives or the article with same title.



Infinite primes and motives

In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it.

Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution.

The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states.

  1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes xn=0, xn=xn-1=0, etc...

  2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group Gk would be analogous to the relative homology group relative to the plane xk-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description.

  3. The algebraic counterpart of boundary map would map the elements of Gk identified as analog of homotopy group to the commutator group [Gk-2,Gk-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra.

  4. That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δikδjk-1jkδik-1 maps to the group algebra of the commutator group [Gk-2,Gk-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration.The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices.

  5. The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M2+/-× CP2 representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of fermion modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry.

  6. A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II1. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial).

This construction would realize thge number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as an infinite-D geometry, and TGD as a generalized number theory visions.

This picture leads also to a proposal how p-adic integrals could be defined in TGD framework.

  1. The calculation of twistorial amplitudes reduces to multi-dimensional residue calculus. Motivic integration gives excellent hopes for the p-adic existence of this calculus and braid representation would give space-time representation for the residue integrals in terms of the braid points representing poles of the integrand: this would conform with quantum classical correspondence. The power of 2π appearing in multiple residue integral is problematic unless it disappears from scattering amplitudes. Otherwise one must allow an extension of p-adic numbers to a ring containing powers of 2π.

  2. Weak form of electric-magnetic duality and the general solution ansatz for preferred extremals reduce the Kähler action defining the Kähler function for WCW to the integral of Chern-Simons 3-form. Hence the reduction to cohomology takes places at space-time level and since p-adic cohomology exists there are excellent hopes about the existence of p-adic variant of Kähler action. The existence of the exponent of Kähler gives additional powerful constraints on the value of the Kähler fuction in the intersection of real and p-adic worlds consisting of algebraic partonic 2-surfaces and allows to guess the general form of the Kähler action in p-adic context.

  3. One also should define p-adic integration for vacuum functional at the level of WCW. p-Adic thermodynamics serves as a guideline leading to the condition that in p-adic sector exponent of Kähler action is of form (m/n)r, where m/n is divisible by a positive power of p-adic prime p. This implies that one has sum over contributions coming as powers of p and the challenge is to calculate the integral for K= constant surfaces using the integration measure defined by an infinite power of Kähler form of WCW reducing the integral to cohomology which should make sense also p-adically. The p-adicization of the WCW integrals has been discussed already earlier using an approach based on harmonic analysis in symmetric spaces and these two approaches should be equivalent. One could also consider a more general quantization of Kähler action as sum K=K1+K2 where K1=rlog(m/n) and K2=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e1/n can be considered.

  4. If the Galois group algebras indeed define a representation for WCW spinor fields in finite measurement resolution, also WCW integration would reduce to summations over the Galois groups involved so that integrals would be well-defined in all number fields.

p-Adic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in one-one correspondence with Boolean algebras and have typically 2-adic topologies. A generalization to p-adic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2n>p>pn-1 statements is encouraged by p-adic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and p-adic space-time physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed.

For details see the new chapter Infinite Primes and Motives.



Finding the roots of polynomials defined by infinite primes

Infinite primes identifiable as analogs of free single particle states and bound many-particle states of a repeatedly second quantized supersymmetric arithmetic quantum field theory correspond at n:th level of the hierarchy to irreducible polynomials in the variable Xn which corresponds to the product of all primes at the previous level of hierarchy. At the first level of hierarchy the roots of this polynomial are ordinary algebraic numbers but at higher levels they correspond to infinite algebraic numbers which are somewhat weird looking creatures. These numbers however exist p-adically for all primes at the previous levels because one one can develop the roots of the polynomial in question as powers series in Xn-1 and this series converges p-adically. This of course requires that infinite-p p-adicity makes sense. Note that all higher terms in series are p-adically infinitesimal at higher levels of the hierarchy. Roots are also infinitesimal in the scale defined Xn. Power series expansion allows to construct the roots explicitly at given level of the hierarchy as the following induction argument demonstrates.

  1. At the first level of the hierarchy the roots of the polynomial of X1 are ordinary algebraic numbers and irreducible polynomials correspond to infinite primes. Induction hypothesis states that the roots can be solved at n:th level of the hierarchy.

  2. At n+1:th level of the hierarchy infinite primes correspond to irreducible polynomials

    Pm(Xn+1)= ∑s=0,...,m ps Xsn+1 .

    The roots R are given by the condition

    Pm(R)=0 .

    The ansatz for a given root R of the polynomial is as a Taylor series in Xn:

    R= ∑ rkXnk ,

    which indeed converges p-adically for all primes of the previous level. Note that R is infinitesimal at n+1:th level. This gives

    Pm(R)=∑s=0,...,m ps (∑ rkXnk)s=0 .

    1. The polynomial contains constant term (zeroth power of Xn+1 given by

      Pm(r0)=∑s=0,...,m pr r0s .

      The vanishing of this term determines the value of r0. Although r0 is infinite number the condition makes sense by induction hypothesis. One can indeed interpret the vanishing condition Pm(r0)=0 as a vanishing of a polynomial at the n:th level of hierarchy having coefficients at n-1:th level and continue the process down to the lowest level of hierarchy obtaining m:th order polynomial at each step. At the lowest level of the hierarchy one obtains just ordinary polynomial equation having finite algebraic numbers as roots.

    2. If one has found the values of r0 one can solve the coefficients rs, s>0 as linear expressions of the coefficients rt, t0.

    3. The naive expectation is that the fundamental theorem of algebra generalizes so that that the number of different roots r0 would be equal to m in the irreducible case. This seems to be the case. Suppose that one has constructed a root R of Pm. One can write Pm(Xn+1) in the form

      Pm(Xn+1)= (Xn+1-R) × Pm-1(Xn+1) ,

      and solve Pm-1 by expanding Pm as Taylor polynomial with respect to Xn+1-R. This is achieved by calculating the derivatives of both sides with respect to Xn+1. The derivatives are completely well-defined since purely algebraic operations are in question. For instance, at the first step one obtains Pm-1(R)=(dPm/dXn+1)(R). The process stops at m:th step so that m roots are obtained.

What is remarkable that the construction of the roots at the first level of the hierarchy forces the introduction of p-adic number fields and that at higher levels also infinite-p p-adic number fields must be introduced. Therefore infinite primes provide a higher level concept implying real and p-adic number fields. If one allows all levels of the hierarchy, a new number Xn must be introduced at each level of the hierarchy. About this number one knows all of its lower level p-adic norms and infinite real norm but cannot say anything more about them. The conjectured correspondence of real units built as ratios of infinite integers and zero energy states however means that these infinite primes would be represented as building blocks of quantum states and that the points of imbedding space would have infinitely complex number theoretical anatomy able to represent zero energy states and perhaps even the world of classical worlds associated with a given causal diamond.

For background see the chapter TGD as a Generalized Number Theory III: Infinite Primes and for the pdf version of the argument the chapter Non-Standard Numbers and TGD.



Non-Standard Numbers and TGD

I had opportunity to read articles of Elemer Rosinger about possible physical applications of non-standard numbers and it was natural to compare these numbers with the generalization of real numbers inspired by the notion of infinite primes. I dediced to attach the commentary as a new chapter to "Physics as a Generalized Number Theory". The abstract gives a rough overall view about the commentary.

The chapter represents a comparison of ultrapower fields (loosely surreals, hyper-reals, long line) and number fields generated by infinite primes having a physical interpretation in Topological Geometrodynamics. Ultrapower fields are discussed in very physicist friendly manner in the articles of Elemer Rosinger and these articles are taken as a convenient starting point. The physical interpretations and principles proposed by Rosinger are considered against the background provided by TGD. The construction of ultrapower fields is associated with physics using the close analogies with gauge theories, gauge invariance, and with the singularities of classical fields. Non-standard numbers are compared with the numbers generated by infinite primes and it is found that the construction of infinite primes, integers, and rationals has a close similarity with construction of the generalized scalars. The construction replaces at the lowest level the index set Λ=N of natural numbers with algebraic numbers A, Frechet filter of N with that of A, and R with unit circle S1 represented as complex numbers of unit magnitude. At higher levels of the hierarchy generalized -possibly infinite and infinitesimal- algebraic numbers emerge. This correspondence maps a given set in the dual of Frechet filter of A to a phase factor characterizing infinite rational algebraically so that correspondence is like representation of algebra. The basic difference between two approaches to infinite numbers is that the counterpart of infinitesimals is infinitude of real units with complex number theoretic anatomy: one might loosely say that these real units are exponentials of infinitesimals.

For details see the new chapter Non-Standard Numbers and TGD.



Generalization of thermodynamics allowing negentropic entanglement and a model for conscious information processing

Costa de Beauregard considers a model for information processing by a computer based on an analogy with Carnot's heat engine (see this). I am grateful for Stephen Paul King for bringing this article to my attention in Time discussion group and also for inspiring discussions which also led to the birth of this section. As such the model Beauregard for computer does not look convincing as a model for what happens in biological information processing.

Combined with TGD based vision about living matter, the model however inspires a model for how conscious information is generated and how the second law of thermodynamics must be modified in TGD framework. The basic formulas of thermodynamics remain as such since the modification means only the replacement S→ S-N, where S is thermodynamical entropy and N the negentropy associated with negentropic entanglement. This allows to circumvent the basic objections against the application of Beauregard's model to living systems. One can also understand why living matter is so effective entropy producer as compared to inanimate matter and also the characteristic decomposition of living systems to highly negentropic and entropic parts as a consequence of generalized second law.

I do not bother to type further and give instead a link to the article Generalization of thermodynamics allowing negentropic entanglement and a model for conscious information processing at my homepage and also to the chapter Negentropy Maximization Principle.



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