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Physics as a Generalized Number Theory

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Year 2010



A Possible Explanation of Shnoll Effect

I have already earlier mentioned the work of Russian scientist Shnoll concerning random fluctuations. This work spans four decades and has finally started to gain recognition also in west. By a good luck I found from web an article of Shnoll about strange regularities of what should be random fluctuations. Then Dainis Zeps provided me with a whole collection of similar articles! Thank you Dainis!

The findings of Shnoll led within few days to a considerable progress in the understanding of the relation between p-adic and real probability concepts, the relationship between p-adic physics and quantum groups emerging naturally in TGD based view about finite measurement resolution, the relationship of the hierarchy of Planck constants (in particular the gigantic gravitational Planck constant assignable to the space-time sheets mediating gravitation) and small-p p-adicity, and also with the understanding of the experimental implications of many-sheetedness of space-time in concrete measurement situations in which the measurement apparatus also means non-trivial topology of the space-time.

The most important conclusion is that basic vision about TGD Universe seems manifest itself directly in quantum fluctuations due to quantum coherence in astrophysical scales in practically all kinds of experiments- even in the distributions of financial variables! Needless to tell how far reaching the implications are for quantum gravity. This is one of the biggest surprises of my un-paid professional life which outshines even the repeated surprises caused by the incredibly unintelligent response of most colleagues to my work! I glue below an abstract of a brand new preprint titled A possible Explanation of Shnoll Effect. I attach below the abstract.

Shnoll and collaborators have discovered strange repeating patterns of random fluctuations of physical observables such as the number n of nuclear decays in a given time interval. Periodically occurring peaks for the distribution of the number N(n) of measurements producing n events in a series of measurements as a function of n is observed instead of a single peak. The positions of the peaks are not random and the patterns depend on position and time varying periodically in time scales possibly assignable to Earth-Sun and Earth-Moon gravitational interaction.

These observations suggest a modification of the expected probability distributions but it is very difficult to imagine any physical mechanism in the standard physics framework. Rather, a universal deformation of predicted probability distributions would be in question requiring something analogous to the transition from classical physics to quantum physics.

The hint about the nature of the modification comes from the TGD inspired quantum measurement theory proposing a description of the notion of finite measurement resolution in terms of inclusions of so called hyper-finite factors of type II1 (HFFs) and closely related quantum groups. Also p-adic physics -another key element of TGD- is expected to be involved. A modification of a given probability distribution P(n| λi) for a positive integer valued variable n characterized by rational-valued parameters λi is obtained by replacing n and the integers characterizing λi with so called quantum integers depending on the quantum phase qm=exp(i2π/m). Quantum integer nq must be defined as the product of quantum counterparts pq of the primes p appearing in the prime decomposition of n. One has pq= sin(2π p/m)/sin(2π/m) for p ≠ P and pq=P for p=P. m must satisfy m≥ 3, m≠ p, and m≠ 2p.

The quantum counterparts of positive integers can be negative. Therefore quantum distribution is defined first as p-adic valued distribution and then mapped by so called canonical identification I to a real distribution by the map taking p-adic -1 to P and powers Pn to P-n and other quantum primes to themselves and requiring that the mean value of n is for distribution and its quantum variant. The map I satisfies I(∑ Pn)=∑ I(Pn). The resulting distribution has peaks located periodically with periods coming as powers of P. Also periodicities with peaks corresponding to n=n+n-, n+q>0 with fixed n-q< 0.

The periodic dependence of the distributions would be most naturally assignable to the gravitational interaction of Earth with Sun and Moon and therefore to the periodic variation of Earth-Sun and Earth-Moon distances. The TGD inspired proposal is that the p-dic prime P and integer m characterizing the quantum distribution are determined by a process analogous to a state function reduction and their most probably values depend on the deviation of the distance R through the formulas Δ p/p≈ kpΔ R/R and Δ m/m≈ kmΔ R/R. The p-adic primes assignable to elementary particles are very large unlike the primes which could characterize the empirical distributions. The hierarchy of Planck constants allows the gravitational Planck constant assignable to the space-time sheets mediating gravitational interactions to have gigantic values and this allows p-adicity with small values of the p-adic prime P.

For detals see the new chapter A Possible Explanation of Shnoll Effect.



Considerable progress in generalized Feynman diagrammatics

The following is expanded and somewhat edited response in Kea's blog. For reasons that should become obvious the response deserves to be published also here although I have done this implicitly via links to pdf files in earlier postings. My sincere hope is that at least single really intelligent reader might realize what is is involved;-). This might be enough.

I have been working with twistor program inspired ideas in TGD framework for a couple of years. The basic conceptual elements are following.

  1. The notion of generalized Feyman diagram defined by replacing lines of ordinary Feynman diagram with light-like 3-surfaces (elementary particle sized wormhole contacts with throats carrying quantum numbers) and vertices identified as their 2-D ends - I call them partonic 2-surfaces. Speaking somewhat loosely, generalized Feynman diagrams plus background space-time sheets define the "world of classical worlds" (WCW).

  2. Zero energy ontology (ZEO) and causal diamonds (intersections of future and past directed lightcones). The crucial observation is that in ZEO it is possible to identify off mass shell particles as pairs of on mass shell particles at throats of wormhole contact since both positive and negative signs of energy are possible. The propagator defined by modified Dirac action does not diverge (except for incoming lines) although the fermions at throats are on mass shell. In other words, the generalized eigenvalue of the modified Dirac operator containing a term linear in momentum is non-vanishing and propagator reduces to G=i/λγ , where γ is modified gamma matrix in the direction of stringy coordinate. This means opening of the black box of off mass shell particle-something which for some reason has not occurred to anyone fighting with the divergences of QFTs.

  3. Representation of 8-D gamma matrices in terms of octonionic units and 2-D sigma matrices. Modified gamma matrices at space-time surfaces are quaternionic/associative and allow a genuine matrix representation. As a matter fact, TGD and WCW can be formulated as study of associative local sub-algebras of the local Clifford algebra of 8-D imbedding space parameterized by quaternionic space-time surfaces. Central conjecture is that quaternionic 4-surfaces correspond to preferred extremals of Kähler action identified as critical ones (second variation of Kähler action vanishes for infinite number of deformations defining super-conformal algebra) and allow a slicing to string worldsheets parametrized by points of partonic 2-surfaces.

  4. Number theoretic universality requiring the existence of Feynman amplitudes in all number fields when one allows suitable algebraic extensions: roots of unity are certainly required in order to realize plane waves. Also imbedding space, partonic 2-surfaces and WCW must exist in all number fields and their extensions. These constraints are enormously powerful and the attempts to realize this vision have dominated quantum TGD for last 20 years.

  5. As far as twistors are considered, the first key element is the reduction of the octonionic twistor structure to quaternionic one at space-time surfaces and giving effectively 4-D spinor and twistor structure for quaternionic surfaces.

Quite recently quite a dramatic progress took place in this approach. It was just the simple observation -I should have made if for already half year ago!- that on mass shell property puts enormously strong kinematic restrictions on the loop integrations. With mild restrictions on the number of parallel fermion lines appearing in vertices (there can be several since fermionic oscillator operator algebra defining SUSY algebra generates the parton states)- all loops are manifestly finite and if particles has always mass -say small p-adic thermal mass also in case of massless particles and due to IR cutoff due to the presence largest CD- the number of diagrams is finite. Unitarity reduces to Cutkosky rules automatically satisfied as in the case of ordinary Feynman diagrams.

This is about momentum space aspects of Feynman diagrams but not yet about the functional (not path-) integral over small deformations of the partonic 2-surfaces. It took some time to see that also the functional integrals over WCW can be carried out at general level both in real and p-adic context.

  1. The p-adic generalization of Fourier analysis allows to algebraize integration- the horrible looking technical challenge of p-adic physics- for symmetric spaces for functions allowing the analog of discrete Fourier decomposion. Symmetric space property is indeed essential also for the existence of Kähler geometry for infinite-D spaces as was learned already from the case of loop spaces. Plane waves and exponential functions expressible as roots of unity and powers of p multiplied by the direct analogs of corresponding exponent functions are the basic building bricks and key functions in harmonic analysis in symmetric spaces. The physically unavoidable finite measurement resolution corresponds to algebraically unavoidable finite algebraic dimension of algebraic extension of p-adics (at least some roots of unity are needed). The cutoff in roots of unity is very reminiscent to that occurring for the representations of quantum groups and is certainly very closely related to these as also to the inclusions of hyper-finite factors of type II1 defining the finite measurement resolution.

  2. WCW geometrization reduces to that for a single line of the generalized Feynman diagram defining the basic building brick for WCW. Kähler function decomposes to a sum of "kinetic" terms associated with its ends and interaction term associated with the line itself. p-Adicization boils down to the condition that Kähler function, matrix elements of Kähler form, WCW Hamiltonians and their super counterparts, are rational functions of complex WCW coordinates just as they are for those symmetric spaces that I know of. This allows straightforward continuation to p-adic context. Incredibly simple!
  3. As far as diagrams are considered, everything is manifestly finite as the general arguments (non-locality of Kähler function as functional of 3-surface) developed two decades ago indeed allow to expect. General conditions on the holomorphy properties of the generalized eigenvalues λ of the modified Dirac operator can be deduced from the conditions that propagator decomposes to a sum of products of harmonics associated with the ends of the line and that similar decomposition takes place for exponent of Kähler action identified as Dirac determinant. This guarantees that the convolutions of propagators and vertices give rise to products of harmonic functions which can be Glebsch-Gordanized to harmonics and only the singlet contributes to the WCW integral in given vertex. The still unproven central conjecture is that Dirac determinant equals the exponent of Kähler function.

Ironically, twistors which stimulated all these development do not seem to be absolutely necessary in this approach although they are of course possible. Situation changes if one does not assumes small p-adically thermal mass due to the presence of massless particles and one must sum infinite number of diagrams. Here a potential problem is whether the infinite sum respects the algebraic extension in question.

For a more detailed representation of generalized Feynman diagrammatics see the last section of the pdf article Weak form of electric-magnetic duality, electroweak massivation, and color confinement. For Feynman diagrams and WCW integration see the article How to define generalized Feynman diagrams? summarizing the basic formulas. See also the chapter Physics as a Generalized Number Theory I: p-Adicization Program.



How to perform WCW integrations in generalized Feynman diagrams?

The formidable looking challenge of quantum TGD is to calculate the M-matrix elements defined by the generalized Feynman diagrams. Zero energy ontology (ZEO) has provided profound understanding about how generalized Feynman diagrams differ from the ordinary ones. The most dramatic prediction is that loop momenta correspond to on mass shall momenta for the two throats of the wormhole contact defining virtual particles: the energies of the energies of on mass shell throats can have both signs in ZEO. This predicts finiteness of Feynman diagrams in the fermionic sector. Even more: the number of Feynman diagrams for a given process is finite if also massless particles receive a small mass by p-adic thermodynamics. The mass would be due to IR cutoff provided by the largest CD (causal diamond) involved.

The basic challenges are following.

  1. One should perform the functional integral over world of classical worlds (WCW) for fixed values of on mass shell momenta appearing in internal lines. After this one must perform integral or summation over loop momenta.

  2. One must achieve this also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

The realization that p-adic integrals could be defined if the manifold is symmetric space as the world of classical world (WCW) is proposed to be raises the hope that the WCW integration for Feynman amplitudes could be carried at the general level using Fourier analysis for symmetric spaces. Even more, the possibility to define p-adic intergrals for symmetric spaces suggests that the theory could allow elegant p-adicization. This indeed seems to be the case. It seems that the dream of transforming TGD to a practical calculational machinery does not look non-realistic at all.

I do not bother to type more but give a link to a short article summarizing the basic formulas. For more background see also the article Weak form of electric-magnetic duality, electroweak massivation, and color confinement and the chapter Physics as a Generalized Number Theory I: p-Adicization Program.



Weak form of electric-magnetic duality, particle concept, and Feynman diagrammatics

The notion of electric magnetic duality emerged already two decades ago in attempts to formulate the Kähler geometric of world of classical worlds. Quite recently a considerable step of progress took place in the understanding of this notion. Every new idea must be of course taken with a grain of salt but the good sign is that this concept leads to an identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic flux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement. The fascinating prediction is that the stringy view about elementary particles should become visible at LHC energies.

Zero energy ontology in turn inspires the idea that virtual particles correspond to pairs of on mass shell states assignable to the opposite throats of wormhole contacts: in TGD framework the propagators do not diverge although particles are on mass shell in standard sense. This assumption leads to powerful constraints on the generalized Feynman diagrams giving excellent hopes about the finiteness of loops. Finiteness has been obvious on basis of general arguments but has been very difficult to demonstrate convincingly in the fermionic sector of the theory. In fact, there are good arguments supporting that only a finite number of diagrams contributes to a given reaction: something inspired by the vision about algebraic physics (infinite sums lead out of the algebraic extension used). The reason is that the on mass shell conditions on states at wormhole throats reduce the phase space dramatically, and already in the case of four-vertex loops leave only a discrete set of points under consideration. This implies also finiteness. This wisdom can be combined with the new stringy view about particles to build a very concrete stringy view about generalized Feynman diagrams.

The coutcome of the opening of the black box of virtual particle -an idea forced by the twistorial approach and made possible by zero energy ontology- is something which I dare to regard as a fulfillment of 32 year old dream.

For a more detailed representation of weak electric-magnetic duality see the last section of the pdf article Weak form of electric-magnetic duality, electroweak massivation, and color confinement. For Feynman diagrams and WCW integration see the article How to define generalized Feynman diagrams? summarizing the basic formulas. See also the chapter Physics as a Generalized Number Theory I: p-Adicization Program.



How infinite primes could correspond to quantum states and space-time surfaces?

I became conscious of infinite primes for almost 15 years ago. These numbers were the first mathematical fruit of TGD inspired theory of consciousness and define one of the most unpractical looking aspects of quantum TGD.

Their construction is however structurally similar to a repeated second quantization of an arithmetic super-symmetry quantum field theory with states labeled by primes. An attractive identification of the hierarchy is in terms of the many-sheeted space-time. Also the abstraction hierarchy of conscious thought and hierarchy of n:th order logics naturally correspond to this infinite hierarchy. We ourselves are at rather lowest level of this hierarchy. Propositional logic and first order logic at best and usually no logic at all;-)

By generalizing from rational primes to hyper-octonionic primes one has good hopes about a direct connection with physics. The reason is that the automorphism group of octonions respecting a preferred imaginary unit is SU(3)subset G2 and physically corresponds to color group in the formulation of the number theoretical compactification stating equivalence of the formulations of TGD based on the identification of imbedding space with 8-dimensional hyperquaternions M8 and M4× CP2. The components of hyper-octonion behave like two color singlets and triplet and antitriplet. For a given hyper-octonionic prime there exists a discrete subgroup of SU(3) respecting the prime property and generating a set of primes at octonionic 6-sphere. For a given prime one can realize a finite number of color multiplets in this discrete space. The components in the hyper-complex subspace M2 remaining invariant under SU(3) can be identified as components of momentum in this subspace. M2 is needed for massless particles the preferred extremals of Kähler action assign this space to each point of space-time surface as space non-physical polarizations.

There are two kinds of infinite primes differing only by the sign of the "small" part of the infinite prime and for second kind of primes one can consider the action of SU(2) subgroup of SU(3) and corresponding discrete subgroups of SU(2) respecting prime property (note that this suggests a direct connection with the Jones inclusions of hyper-finite factors of type II1!). These representations give rise to two SU(2) multiplets and their orbital excitations identifiable as deformations of the partonic 2-surface. Four components of hyper-octonion remain invariant under SU(2) and have interpretation as momentum in M2 and electroweak charges. Therefore a pair of these primes characterizes standard model quantum numbers of particle if discrete wave functions in the space of primes are allowed. For color singlet particles single prime is enough. At the level of infinite primes one obtains extremely rich structure and it is possible to map the states of quantum TGD to these number theoretical states. Only the genus of partonic 2-surface responsible for family replication phenomenon fails to find an obvious interpretation in this picture.

The completely unexpected by-product is a prediction for the spectrum of quantum states and quantum numbers including masses so that infinite primes and rationals are not so unpractical as one might think! This prediction is really incredible since it applies to the entire hierarchy of second quantizations in which many particle states of previous level become particles of the new level (corresponding physically to space-time sheets condensed to a larger space-time sheet or causal diamonds inside larger causal diamond CD).

In zero energy ontology positive and negative energy states correspond to infinite integers and their inverses respectively and their ratio to a hyper-octonionic unit. The wave functions in this space induced from those for finite hyper-octonionic primes define the quantum states of the sub-Universe defined by given CD and sub-CDs. These phases can be assigned to any point of the 8-dimensional imbedding space M8 interpreted as hyper-octonions so that number theoretic Brahman=Atman identity or algebraic holography is realized! These incredibly beautiful infinite primes are both highly spiritual and highly practical just as a real spiritual person experienced directly Brahman=Atman state is;-). A fascinating possibility is that even M-matrix- which is nothing but a characterization of zero energy state- could find an elegant formulation as entanglement coefficients associated with the pair of the integer and inverse integer characterizing the positive and negative energy states.

A fascinating possibility is that even M-matrix- which is nothing but a characterization of zero energy state- could find an elegant formulation as entanglement coefficients associated with the pair of the integer and inverse integer characterizing the positive and negative energy states.

  1. The great vision is that associativity and commutativity conditions fix the number theoretical quantum dynamics completely. Quantum associativity states that the wave functions in the space of infinite primes, integers, and rationals are invariant under associations of finite hyper-octonionic primes (A(BC) and (AB)C are the basic associations), physics requires associativity only apart from a phase factor. The condition of commutativity poses a more familiar condition implying that permutations induce only a phase factor which is +/- 1 for boson and fermion statistics and a more general phase for quantum group statistics for the anyonic phases, which correspond to nonstandard values of Planck constant in TGD framework. These symmetries induce time-like entanglement for zero energy stats and perhaps non-trivial enough M-matrix.

  2. One must also remember that besides the infinite primes defining the counterparts of free Fock states of supersymmetric QFT, also infinite primes analogous to bound states are predicted. The analogy with polynomial primes illustrates what is involved. In the space of polynomials with integer coefficients polynomials of degree one correspond free single particle states and one can form free many particle states as their products. Higher degree polynomials with algebraic roots correspond to bound states being not decomposable to a product of polynomials of first degree in the field of rationals. Could also positive and negative energy parts of zero energy states form a analog of bound state giving rise to highly non-trivial M-matrix?

Also a rigorous interpretation of complexified octonions emerges in zero energy ontology.

  1. The two tips of causal diamond CD define two preferred points of M4. The fixing of quantization axes of color fixes in CP2 also a point at both light-like boundaries of CD. The moduli space for CDs is therefore M4× CP2 × M4++× CP2 and its M8 counterpart is obtained by replacing CP2 with E4 so that a space which correspond locally to complexified octonions is the outcome. p-Adic length scale hypothesis suggests very strongly a quantization of the second factor to a set of hyperboloids with light-cone proper time come as powers of 2. For other values of Planck constant rational multiples of these are obtained. This suggests quantization also for hyperboloids and CP2.

  2. An attractive hypothesis is that infinite-primes determine the discretization as Ga subset SU(2)subset SU(3) and Gb subset SU(3) orbits of the points of hyperboloid and CP2. The interpretation would be in terms of cosmology. The Robertson Walker space-time would be replaced with this discrete space meaning in particular that cosmic time identified as Minkowski proper time is quantized in powers of two. One prediction is quantization of cosmic redshift resulting from quantization of Lorentz boosts and has been indeed observed and extremely difficult to understand in standard cosmology. We would observe infinite primes directly!

I do not bother to type more. Interested reader can read the brief pdf file explaining all this in detail or read the chapter Physics as Generalized Number Theory III: Infinite Primes.



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