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TGD as a Generalized Number Theory

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Year 2008



Zero energy ontology, self hierarchy, and the notion of time

In the previous posting I discussed the most recent view about zero energy ontology and p-adicization program. One manner to test the internal consistency of this framework is by formulating the basic notions and problems of TGD inspired quantum theory of consciousness and quantum biology in terms of zero energy ontology. I have discussed these topics already earlier but the more detailed understanding of the role of causal diamonds (CDs) brings many new aspects to the discussion.

In consciousness theory the basic challenges are to understand the asymmetry between positive and negative energies and between two directions of geometric time at the level of conscious experience, the correspondence between experienced and geometric time, and the emergence of the arrow of time. One should also explain why human sensory experience is about a rather narrow time interval of about .1 seconds and why memories are about the interior of much larger CD with time scale of order life time. One should also have a vision about the evolution of consciousness takes place: how quantum leaps leading to an expansion of consciousness take place.

Negative energy signals to geometric past - about which phase conjugate laser light represents an example - provide an attractive tool to realize intentional action as a signal inducing neural activities in the geometric past (this would explain Libet's classical findings), a mechanism of remote metabolism, and the mechanism of declarative memory as communications with the geometric past. One should understand how these signals are realized in zero energy ontology and why their occurrence is so rare.

In the following my intention is to demonstrate that TGD inspired theory of consciousness and quantum TGD proper indeed seem to be in tune and that this process of comparison helps considerably in the attempt to develop the TGD based ontology at the level of details.

1  Causal diamonds as correlates for selves

Quantum jump as a moment of consciousness, self as a sequence of quantum jumps integrating to self, and self hierarchy with sub-selves experienced as mental images, are the basic notion of TGD inspired quantum theory of consciousness. In the most ambitious program self hierarchy reduces to a fractal hierarchy of quantum jumps within quantum jumps.

It is natural to interpret CD:s as correlates of selves. CDs can be interpreted in two manners: as subsets of the generalized imbedding space or as sectors of the world of classical worlds (WCW). Accordingly, selves correspond to CD:s of the generalized imbedding space or sectors of WCW, literally separate interacting quantum Universes. The spiritually oriented reader might speak of Gods. Sub-selves correspond to sub-CD:s geometrically. The contents of consciousness of self is about the interior of the corresponding CD at the level of imbedding space. For sub-selves the wave function for the position of tip of CD brings in the delocalization of sub-WCW.

The fractal hierarchy of CDs within CDs defines the counterpart for the hierarchy of selves: the quantization of the time scale of planned action and memory as T(k) = 2kT0 suggest an interpretation for the fact that we experience octaves as equivalent in music experience.

2. Why sensory experience is about so short time interval?

CD picture implies automatically the 4-D character of conscious experience and memories form part of conscious experience even at elementary particle level: in fact, the secondary p-adic time scale of electron is T=1 seconds defining a fundamental time scale in living matter. The problem is to understand why the sensory experience is about a short time interval of geometric time rather than about the entire personal CD with temporal size of order life-time. The obvious explanation would be that sensory input corresponds to sub-selves (mental images) which correspond to CD:s with T(127) @ .1 s (electrons or their Cooper pairs) at the upper light-like boundary of CD assignable to the self. This requires a strong asymmetry between upper and lower light-like boundaries of CD:s.

  1. The only reasonable manner to explain the situation seems to be that the addition of CD:s within CD:s in the state construction must always glue them to the upper light-like boundary of CD along light-like radial ray from the tip of the past directed light-cone. This conforms with the classical picture according to which classical sensory data arrives from the geometric past with velocity which is at most light velocity.

  2. One must also explain the rare but real occurrence of phase conjugate signals understandable as negative energy signals propagating towards geometric past. The conditions making possible negative energy signals are achieved when the sub-CD is glued to both the past and future directed light-cones at the space-like edge of CD along light-like rays emerging from the edge. This exceptional case gives negative energy signals traveling to the geometric past. The above mentioned basic control mechanism of biology would represent a particular instance of this situation. Negative energy signals as a basic mechanism of intentional action would explain why living matter seems to be so special.

  3. Geometric memories would correspond to the lower boundaries of CD:s and would not be in general sharp because only the sub-CD:s glued to both upper and lower light-cone boundary would be present. A temporal sequence of mental images, say the sequence of digits of a phone number, could corresponds to a sequence of sub-CD:s glued to the upper light-cone boundary.

  4. Sharing of mental images corresponds to a fusion of sub-selves/mental images to single sub-self by quantum entanglement: the space-time correlate for this could be flux tubes connecting space-time sheets associated with sub-selves represented also by space-time sheets inside their CD:s. It could be that these ëpisodal" memories correspond to CD:s at upper light-cone boundary of CD.

On basis of these arguments it seems that the basic conceptual framework of TGD inspired theory of consciousness can be realized in zero energy ontology. Interesting questions relate to how dynamical selves are.

  1. Is self doomed to live inside the same sub-WCW eternally as a lonely god? This question has been already answered: there are interactions between sub-CD:s of given CD, and one can think of selves as quantum superposition of states in CD:s with wave function having as its argument the tips of CD, or rather only the second one since T is assumed to be quantized.

  2. Is there a largest CD in the personal CD hierarchy of self in an absolute sense? Or is the largest CD present only in the sense that the contribution to the contents of consciousness coming from very large CD:s is negligible? Long time scales T correspond to low frequencies and thermal noise might indeed mask these contributions very effectively. Here however the hierarchy of Planck constants and generalization of the imbedding space would come in rescue by allowing dark EEG photons to have energies above thermal energy.

  3. Can selves evolve in the sense that the size of CD increases in quantum leaps so that the corresponding time scale T=2kT0 of memory and planned action increases? Geometrically this kind of leap would mean that CD becomes a sub-CD of a larger CD either at the level of conscious experience or in absolute sense. This leap can occur in two senses: as an increase of the largest p-adic time scale in the personal hierarchy of space-time sheets or as increase of the largest value of Planck constants in the personal dark matter hierarchy. At the level of individual this would mean emergence of increasingly lower frequencies of generalization of EEG and of the levels of dark matter hierarchy with large value of Planck constant.

  4. In 2-D illustration of the leap leading to a higher level of self hierarchy would mean simply the continuation of CD to right or left in the 2-D visualization of CD. Since the preferred M2 is contained in the tangent space of space-time surfaces, and since preferred M2 plays a key role in dark matter hierarchy too, one must ask whether the 2-D illustration might have some deeper truth in it.

3. New view about arrow of time

Perhaps the most fundamental problem related to the notion of time concerns the relationship between experienced time and geometric time. The two notions are definitely different: think only the irreversibility of experienced time and the reversibility of the geometric time and the absence of future of the experienced time. Also the deterministic character of the dynamics in geometric time is in conflict with the notion of free will supported by the direct experience.

In the standard materialistic ontology experienced time and geometric time are identified. In the naivest picture the flow of time is interpreted in terms of the motion of 3-D time=constant surface of space-time towards geometric future without any explanation for why this kind of motion would occur. This identification is plagued by several difficulties. In special relativity the difficulties relate to the impossibility define the notion of simultaneity in a unique manner and the only possible manner to save this notion seems to be the replacement of time=constant 3-surface with past directed light-cone assignable to the world-line of observer. In general relativity additional difficulties are caused by the general coordinate invariance unless one generalizes the picture of special relativity: problems are however caused by the fact that past light-cones make sense only locally. In quantum physics quantum measurement theory leads to a paradoxical situation since the observed localization of the state function reduction to a finite space-time volume is in conflict with the determinism of Schrödinger equation.

TGD forces a new view about the relationship between experienced and geometric time. Although the basic paradox of quantum measurement theory disappears the question about the arrow of geometric time remains.

  1. Selves correspond to CD:s the own sub-WCW:s. These sub-WCW:s and their projections to the imbedding space do not move anywhere. Therefore standard explanation for the arrow of geometric time cannot work. Neither can the experience about flow of time correspond to quantum leaps increasing the size of the largest CD contributing to the conscious experience of self.

  2. The only plausible interpretation is based on quantum classical correspondence and the fact that space-times are 4-surfaces of the imbedding space. If quantum jump corresponds to a shift of quantum superposition of space-time sheets towards geometric past in the first approximation (as quantum classical correspondence suggests), one can indeed understand the arrow of time. Space-time surfaces simply shift backwards with respect to the geometric time of the imbedding space and therefore to the 8-D perceptive field defined by the CD. This creates in the materialistic mind a kind of temporal variant of train illusion. Space-time as 4-surface and macroscopic and macro-temporal quantum coherence are absolutely essential for this interpretation to make sense.

Why this shifting should always take place to the direction of geometric past of the imbedding space? What seems clear is that the asymmetric construction of zero energy states should correlate with the preferred direction. If question is about probabilities, the basic question would be why the probabilities for shifts in the direction of geometric past are higher. Here some alternative attempts to answer this question are discussed.

  1. Cognition and time relate to each other very closely and the required fusion of real physics with various p-adic physics of cognition and intentionality could also have something to do with the asymmetry. Indeed, in the p-adic sectors the transcendental values of p-adic light-cone proper time coordinate correspond to literally infinite values of the real valued light-cone proper time, and one can say that most points of p-adic space-time sheets serving as correlates of thoughts and intentions reside always in the infinite geometric future in the real sense. Therefore cognition and intentionality would break the symmetry between positive and negative energies and geometric past and future, and the breaking of arrow of geometric time could be seen as being induced by intentional action and also due to the basic aspects of cognitive experience.

  2. Zero energy ontology suggests also a possible reason for the asymmetry. Standard quantum mechanics encourages the identification of the space of negative energy states as the dual for the space of positive energy states. There are two kinds of duals. Hilbert space dual is identified as the space of continuous linear functionals from Hilbert space to the coefficient field and is isometrically anti-isomorphic with the Hilbert space. This justifies the bra-ket notation. In the case of vector space the relevant notion is algebraic dual. Algebraic dual can be identified as an infinite direct product of the coefficient field identified as a 1-dimensional vector space. Direct product is defined as the set of functions from an infinite index set I to the disjoint union of infinite number of copies of the coefficient field indexed by I. Infinite-dimensional vector space corresponds to infinite direct sum consisting of functions which are non-vanishing for a finite number of indices only. Hence vector space dual in infinite-dimensional case contains much more states than the vector space and does not have enumerable basis.

    If negative energy states correspond to a subspace of vector space dual containing Hilbert space dual, the number of negative energy states is larger than the number of positive energy states. This asymmetry could correspond to better measurement resolution at the upper light-cone cone boundary so that the state space at lower light-cone boundary would be included via inclusion of HFFs to that associated with the upper light-cone boundary. Geometrically this would mean the possibility to glue to the upper light-cone boundary CD which can be smaller than those associated with the lower one.

  3. The most convincing candidate for an answer comes from consciousness theory. One must understand also why the contents of sensory experience is concentrated around a narrow time interval whereas the time scale of memories and anticipation are much longer. The proposed mechanism is that the resolution of conscious experience is higher at the upper boundary of CD. Since zero energy states correspond to light-like 3-surfaces, this could be a result of self-organization rather than a fundamental physical law.

    1. The key assumption is that CDs have CDs inside CDs and that the vertices of generalized Feynman diagrams are contained within sub-CDs. It is not assumed that CDs are glued to the upper boundary of CD since the arrow of time results from self organization when the distribution of sub-CDs concentrates around the upper boundary of CD. In a category theoretical formulation for generalized Feynman diagrammatics based on this picture is developed.

    2. CDs define the perceptive field for self. Selves are curious about the space-time sheets outside their perceptive field in the geometric future (relative notion) of the imbedding space and perform quantum jumps tending to shift the superposition of the space-time sheets to the direction of geometric past (past defined as the direction of shift!). This creates the illusion that there is a time=snapshot front of consciousness moving to geometric future in fixed background space-time as an analog of train illusion.

    3. The fact that news come from the upper boundary of CD implies that self concentrates its attention to this region and improves the resolutions of sensory experience and quantum measurement here. The sub-CD:s generated in this manner correspond to mental images with contents about this region. As a consequence, the contents of conscious experience, in particular sensory experience, tend to be about the region near the upper boundary.

    4. This mechanism in principle allows the arrow of the geometric time to vary and depend on p-adic length scale and the level of dark matter hierarchy. The occurrence of phase transitions forcing the arrow of geometric time to be same everywhere are however plausible for the reason that the lower and upper boundaries of given CD must possess the same arrow of geometric time.

For details see chapters TGD as a Generalized Number Theory I: p-Adicization Program.



The most recent vision about zero energy ontology and p-adicization

The generalization of the number concept obtained by fusing real and p-adics along rationals and common algbraics is the basic philosophy behind p-adicization. This however requires that it is possible to speak about rational points of the imbedding space and the basic objection against the notion of rational points of imbedding space common to real and various p-adic variants of the imbedding space is the necessity to fix some special coordinates in turn implying the loss of a manifest general coordinate invariance. The isometries of the imbedding space could save the situation provided one can identify some special coordinate system in which isometry group reduces to its discrete subgroup. The loss of the full isometry group could be compensated by assuming that WCW is union over sub-WCW:s obtained by applying isometries on basic sub-WCW with discrete subgroup of isometries.

The combination of zero energy ontology realized in terms of a hierarchy causal diamonds and hierarchy of Planck constants providing a description of dark matter and leading to a generalization of the notion of imbedding space suggests that it is possible to realize this dream. The article TGD: What Might be the First Principles? provides a brief summary about recent state of quantum TGD helping to understand the big picture behind the following considerations.

1. Zero energy ontology briefly

  1. The basic construct in the zero energy ontology is the space CD×CP2, where the causal diamond CD is defined as an intersection of future and past directed light-cones with time-like separation between their tips regarded as points of the underlying universal Minkowski space M4. In zero energy ontology physical states correspond to pairs of positive and negative energy states located at the boundaries of the future and past directed light-cones of a particular CD. CD:s form a fractal hierarchy and one can glue smaller CD:s within larger CD along the upper light-cone boundary along a radial light-like ray: this construction recipe allows to understand the asymmetry between positive and negative energies and why the arrow of experienced time corresponds to the arrow of geometric time and also why the contents of sensory experience is located to so narrow interval of geometric time. One can imagine evolution to occur as quantum leaps in which the size of the largest CD in the hierarchy of personal CD:s increases in such a manner that it becomes sub-CD of a larger CD. p-Adic length scale hypothesis follows if the values of temporal distance T between tips of CD come in powers of 2n. All conserved quantum numbers for zero energy states have vanishing net values. The interpretation of zero energy states in the framework of positive energy ontology is as physical events, say scattering events with positive and negative energy parts of the state interpreted as initial and final states of the event.

  2. In the realization of the hierarchy of Planck constants CD×CP2 is replaced with a Cartesian product of book like structures formed by almost copies of CD:s and CP2:s defined by singular coverings and factors spaces of CD and CP2 with singularities corresponding to intersection M2CD and homologically trivial geodesic sphere S2 of CP2 for which the induced Kähler form vanishes. The coverings and factor spaces of CD:s are glued together along common M2CD. The coverings and factors spaces of CP2 are glued together along common homologically non-trivial geodesic sphere S2. The choice of preferred M2 as subspace of tangent space of X4 at all its points and having interpretation as space of non-physical polarizations, brings M2 into the theory also in different manner. S2 in turn defines a subspace of the much larger space of vacuum extremals as surfaces inside M4×S2.

  3. Configuration space (the world of classical worlds, WCW) decomposes into a union of sub-WCW:s corresponding to different choices of M2 and S2 and also to different choices of the quantization axes of spin and energy and and color isospin and hyper-charge for each choice of this kind. This means breaking down of the isometries to a subgroup. This can be compensated by the fact that the union can be taken over the different choices of this subgroup.

  4. p-Adicization requires a further breakdown to discrete subgroups of the resulting sub-groups of the isometry groups but again a union over sub-WCW:s corresponding to different choices of the discrete subgroup can be assumed. Discretization relates also naturally to the notion of number theoretic braid.

Consider now the critical questions.

  1. Very naively one could think that center of mass wave functions in the union of sectors could give rise to representations of Poincare group. This does not conform with zero energy ontology, where energy-momentum should be assignable to say positive energy part of the state and where these degrees of freedom are expected to be pure gauge degrees of freedom. If zero energy ontology makes sense, then the states in the union over the various copies corresponding to different choices of M2 and S2 would give rise to wave functions having no dynamical meaning. This would bring in nothing new so that one could fix the gauge by choosing preferred M2 and S2 without losing anything. This picture is favored by the interpretation of M2 as the space of longitudinal polarizations.

  2. The crucial question is whether it is really possible to speak about zero energy states for a given sector defined by generalized imbedding space with fixed M2 and S2. Classically this is possible and conserved quantities are well defined. In quantal situation the presence of the lightcone boundaries breaks full Poincare invariance although the infinitesimal version of this invariance is preserved. Note that the basic dynamical objects are 3-D light-like "legs" of the generalized Feynman diagrams.

2. Definition of energy inzero energy ontology

Can one then define the notion of energy for positive and negative energy parts of the state? There are two alternative approaches depending on whether one allows or does not allow wave-functions for the positions of tips of light-cones.

Consider first the naive option for which four momenta are assigned to the wave functions assigned to the tips of CD:s.

  1. The condition that the tips are at time-like distance does not allow separation to a product but only following kind of wave functions

    Ψ = exp(ip�m)Θ(m2) Θ(m0)× Φ(p) , m=m+-m-.

    Here m+ and m- denote the positions of the light-cones and Q denotes step function. F denotes configuration space spinor field in internal degrees of freedom of 3-surface. One can introduce also the decomposition into particles by introducing sub-CD:s glued to the upper light-cone boundary of CD.

  2. The first criticism is that only a local eigen state of 4-momentum operators p = (h/2p) /i is in question everywhere except at boundaries and at the tips of the CD with exact translational invariance broken by the two step functions having a natural classical interpretation. The second criticism is that the quantization of the temporal distance between the tips to T = 2kT0 is in conflict with translational invariance and reduces it to a discrete scaling invariance.

The less naive approach relies of super conformal structures of quantum TGD assumes fixed value of T and therefore allows the crucial quantization condition T=2kT0.

  1. Since light-like 3-surfaces assignable to incoming and outgoing legs of the generalized Feynman diagrams are the basic objects, can hope of having enough translational invariance to define the notion of energy. If translations are restricted to time-like translations acting in the direction of the future (past) then one has local translation invariance of dynamics for classical field equations inside dM4 as a kind of semigroup. Also the M4 translations leading to interior of X4 from the light-like 2-surfaces surfaces act as translations. Classically these restrictions correspond to non-tachyonic momenta defining the allowed directions of translations realizable as particle motions. These two kinds of translations have been assigned to super-canonical conformal symmetries at dM4×CP2 and and super Kac-Moody type conformal symmetries at light-like 3-surfaces. Equivalence Principle in TGD framework states that these two conformal symmetries define a structure completely analogous to a coset representation of conformal algebras so that the four-momenta associated with the two representations are identical .

  2. The condition selecting preferred extremals of Kähler action is induced by a global selection of M2 as a plane belonging to the tangent space of X4 at all its points . The M4 translations of X4 as a whole in general respect the form of this condition in the interior. Furthermore, if M4 translations are restricted to M2, also the condition itself - rather than only its general form - is respected. This observation, the earlier experience with the p-adic mass calculations, and also the treatment of quarks and gluons in QCD encourage to consider the possibility that translational invariance should be restricted to M2 translations so that mass squared, longitudinal momentum and transversal mass squared would be well defined quantum numbers. This would be enough to realize zero energy ontology. Encouragingly, M2 appears also in the generalization of the causal diamond to a book-like structure forced by the realization of the hierarchy of Planck constant at the level of the imbedding space.

  3. That the cm degrees of freedom for CD would be gauge like degrees of freedom sounds strange. The paradoxical feeling disappears as one realizes that this is not the case for sub-CDs, which indeed can have non-trivial correlation functions with either upper or lower tip of the CD playing a role analogous to that of an argument of n-point function in QFT description. One can also say that largest CD in the hierarchy defines infrared cutoff.

3. p-Adic variants of the imbedding space

Consider now the construction of p-adic variants of the imbedding space.

  1. Rational values of p-adic coordinates are non-negative so that light-cone proper time a4,+=(t2-z2-x2-y2) is the unique Lorentz invariant choice for the p-adic time coordinate near the lower tip of CD. For the upper tip the identification of a4 would be a4,-=((t-T)2-z2-x2-y2). In the p-adic context the simultaneous existence of both square roots would pose additional conditions on T. For 2-adic numbers T=2nT0, n 0 (or more generally T=k n0bk 2k), would allow to satisfy these conditions and this would be one additional reason for T=2nT0 implying p-adic length scale hypothesis. The remaining coordinates of CD are naturally hyperbolic cosines and sines of the hyperbolic angle h,4 and cosines and sines of the spherical coordinates q and f.

  2. The existence of the preferred plane M2 of un-physical polarizations would suggest that the 2-D light-cone proper times a2,+ = (t2-z2) a2,- = ((t-T)2-z2) can be also considered. The remaining coordinates would be naturally h,2 and cylindrical coordinates (r,f).

  3. The transcendental values of a4 and a2 are literally infinite as real numbers and could be visualized as points in infinitely distant geometric future so that the arrow of time might be said to emerge number theoretically. For M2 option p-adic transcendental values of r are infinite as real numbers so that also spatial infinity could be said to emerge p-adically.

  4. The selection of the preferred quantization axes of energy and angular momentum unique apart from a Lorentz transformation of M2 would have purely number theoretic meaning in both cases. One must allow a union over sub-WCWs labeled by points of SO(1,1). This suggests a deep connection between number theory, quantum theory, quantum measurement theory, and even quantum theory of mathematical consciousness.

  5. In the case of CP2 there are three real coordinate patches involved . The compactness of CP2 allows to use cosines and sines of the preferred angle variable for a given coordinate patch.

    ξ1= tan(u)× cos(Θ/2)× exp(i(Ψ+Φ)/2) ,

    ξ2= tan(u)× sin(Θ/2)× exp(i(Ψ-Φ)/2).

    The ranges of the variables u,Q, F,Y are [0,p/2],[0,p],[0,4p],[0,2p] respectively. Note that u has naturally only the positive values in the allowed range. S2 corresponds to the values F = Y = 0 of the angle coordinates.

  6. The rational values of the (hyperbolic) cosine and sine correspond to Pythagorean triangles having sides of integer length and thus satisfying m2 = n2+r2 (m2=n2-r2). These conditions are equivalent and allow the well-known explicit solution . One can construct a p-adic completion for the set of Pythagorean triangles by allowing p-adic integers which are infinite as real integers as solutions of the conditions m2=r2s2. These angles correspond to genuinely p-adic directions having no real counterpart. Hence one obtains p-adic continuum also in the angle degrees of freedom. Algebraic extensions of the p-adic numbers bringing in cosines and sines of the angles p/n lead to a hierarchy increasingly refined algebraic extensions of the generalized imbedding space. Since the different sectors of WCW directly correspond to correlates of selves this means direct correlation with the evolution of the mathematical consciousness. Trigonometric identities allow to construct points which in the real context correspond to sums and differences of angles.

  7. Negative rational values of the cosines and sines correspond as p-adic integers to infinite real numbers and it seems that one use several coordinate patches obtained as copies of the octant (x 0,y 0,z 0,). An analogous picture applies in CP2 degrees of freedom.

  8. The expression of the metric tensor and spinor connection of the imbedding in the proposed coordinates makes sense as a p-adic numbers in the algebraic extension considered. The induction of the metric and spinor connection and curvature makes sense provided that the gradients of coordinates with respect to the internal coordinates of the space-time surface belong to the extensions. The most natural choice of the space-time coordinates is as subset of imbedding space-coordinates in a given coordinate patch. If the remaining imbedding space coordinates can be chosen to be rational functions of these preferred coordinates with coefficients in the algebraic extension of p-adic numbers considered for the preferred extremals of Kähler action, then also the gradients satisfy this condition. This is highly non-trivial condition on the extremals and if it works might fix completely the space of exact solutions of field equations. Space-time surfaces are also conjectured to be hyper-quaternionic , this condition might relate to the simultaneous hyper-quaternionicity and Kähler extremal property. Note also that this picture would provide a partial explanation for the decomposition of the imbedding space to sectors dictated also by quantum measurement theory and hierarchy of Planck constants.

4. p-Adic variants for the sectors of WCW

One can also wonder about the most general definition of the p-adic variants of the sectors of the world of classical worlds.

  1. The restriction of the surfaces in question to be expressible in terms of rational functions with coefficients which are rational numbers of belong to algebraic extension of rationals means that the world of classical worlds can be regarded as a a discrete set and there would be no difference between real and p-adic worlds of classical worlds: a rather unexpected conclusion.

  2. One can of course whether one should perform completion also for WCWs. In real context this would mean completion of the rational number valued coefficients of a rational function to arbitrary real coefficients and perhaps also allowance of Taylor and Laurent series as limits of rational functions. In the p-adic case the integers defining rational could be allowed to become p-adic transcendentals infinite as real numbers. Also now also Laurent series could be considered.

  3. In this picture there would be close analogy between the structure of generalized imbedding space and WCW. Different WCW:s could be said to intersect in the space formed by rational functions with coefficients in algebraic extension of rationals just real and p-adic variants of the imbedding space intersect along rational points. In the spirit of algebraic completion one might hope that the expressions for the various physical quantities, say the value of Kähler action, Kähler function, or at least the exponent of Kähler function (at least for the maxima of Kähler function) could be defined by analytic continuation of their values from these sub-WCW to various number fields. The matrix elements for p-adic-to-real phase transitions of zero energy states interpreted as intentional actions could be calculated in the intersection of real and p-adic WCW:s by interpreting everything as real.

For details see chapters TGD as a Generalized Number Theory I: p-Adicization Program.



Prime Hilbert spaces and infinite primes

Kea told in her blog about a result of quantum information science which seems to provide an additional reason why for p-adic physics.

Suppose that one has N-dimensional Hilbert space which allows N+1 mutually unbiased basis. This means that the moduli squared for the inner product of any two states belonging to different basis equals to 1/N. If one knows all transition amplitudes from a given state to all states of all N+1 mutually unbiased basis, one can fully reconstruct the state. For N=pn dimensional N+1 unbiased basis can be found and the article of Durt gives an explicit construction of these basis by applying the properties of finite fields. Thus state spaces with pn elements - which indeed emerge naturally in p-adic framework - would be optimal for quantum tomography. For instance, the discretization of one-dimensional line with length of pn units would give rise to pn-D Hilbert space of wave functions.

The observation motivates the introduction of prime Hilbert space as as a Hilbert space possessing dimension which is prime and it would seem that this kind of number theoretical structure for the category of Hilbert spaces is natural from the point of view of quantum information theory. One might ask whether the tensor product of mutually unbiased bases in the general case could be constructed as a tensor product for the bases for prime power factors. This can be done but since the bases cannot have common elements the number of unbiased basis obtained in this manner is equal to M+1, where M is the smallest prime power factor of N. It is not known whether additional unbiased bases exists.

1. Hierarchy of prime Hilbert spaces characterized by infinite primes

The notion of prime Hilbert space provides a new interpretation for infinite primes, which are in 1-1 correspondence with the states of a supersymmetric arithmetic QFT. The earlier interpretation was that the hierarchy of infinite primes corresponds to a hierarchy of quantum states. Infinite primes could also label a hierarchy of infinite-D prime Hilbert spaces with product and sum for infinite primes representing unfaitfully tensor product and direct sum.

  1. At the lowest level of hierarchy one could interpret infinite primes as homomorphisms of Hilbert spaces to generalized integers (tensor product and direct sum mapped to product and sum) obtained as direct sum of infinite-D Hilbert space and finite-D Hilbert space. (In)finite-D Hilbert space is (in)finite tensor product of prime power factors. The map of N-dimensional Hilbert space to the set of N-orthogonal states resulting in state function reduction maps it to N-element set and integer N. Hence one can interpret the homomorphism as giving rise to a kind of shadow on the wall of Plato's cave projecting (shadow quite literally!) the Hilbert space to generalized integer representing the shadow. In category theoretical setting one could perhaps see generalize integers as shadows of the hierarchy of Hilbert spaces.

  2. The interpretation as a decomposition of the universe to a subsystem plus environment does not seem to work since in this case one would have tensor product. Perhaps the decomposition could be to degrees of freedom to those which are above and below measurement resolution. Perhaps one should try to interpret physically the process of transferring degrees of freedom from tensor product to direct sum.

  3. The construction of these Hilbert spaces would reduce to that of infinite primes. The analog of the fermionic sea would be infinite-D Hilbert space which is tensor product of all prime Hilbert spaces Hp with given prime factor appearing only once in the tensor product. One can "add n bosons" to this state by replacing of any tensor factor Hp with its n+1:th tensor power. One can "add fermions" to this state by deleting some prime factors Hp from the tensor product and adding their tensor product as a finite-direct summand. One can also "add n bosons" to this factor.

  4. At the next level of hierarchy one would form infinite tensor product of all infinite-D prime Hilbert spaces obtained in this manner and repeat the construction. This can be continued ad infinitum and the construction corresponds to abstraction hierarchy or a hierarchy of statements about statements or a hierarchy of n:th order logics. Or a hierarchy of space-time sheets of many-sheeted space-time. Or a hierarchy of particles in which certain many-particle states at the previous level of hierarchy become particles at the new level (bosons and fermions). There are many interpretations.

  5. Note that at the lowest level this construction can be applied also to Riemann Zeta function. ζ would represent fermionic vacuum and the addition of fermions would correspond to a removal of a product of corresponding factors ζp from ζ and addition of them to the resulting truncated ζ function. The addition of bosons would correspond to multiplication by a power of appropriate ζp. At zeros of ζ the modified zeta functions reduce to their fermionic parts. The analog of ζ function at the next level of hierarchy would be product of all these modified ζ functions and probably fails to exist as a smooth function since the product would typically converge to either zero or infinity.

2. Hilbert spaces assignable to infinite integers and rationals make also sense

  1. Also infinite integers make sense since one can form tensor products and direct sums of infinite primes and of corresponding Hilbert spaces. Also infinite rationals exist and this raises the question what kind of state spaces inverses of infinite integers mean.

  2. Zero energy ontology suggests that infinite integers correspond to positive energy states and their inverses to negative energy states. Zero energy states would be always infinite rationals with real norm which equals to real unit.

  3. The existence of these units would give for a given real number an infinite rich number theoretic anatomy so that single space-time point might be able to represent quantum states of the entire universe in its anatomy (number theoretical Brahman=Atman).

    Also the world of classical worlds (light-like 3-surfaces of the imbedding space) might be imbeddable to this anatomy so that basically one would have just space-time surfaces in 8-D space and configuration space would have representation in terms of space-time based on generalized notion of number. Note that infinitesimals around a given number would be replaced with infinite number of number-theoretically non-equivalent real units multiplying it.

3. Should one generalize the notion of von Neumann algebra?

Especially interesting are the implications of the notion of prime Hilbert space concerning the notion of von Neumann algebra -in particular the notion of hyper-finite factors of type II1 playing a key role in TGD framework. Does the prime decomposition bring in additional structure? Hyper-finite factors of type II1 are canonically represented as infinite tensor power of 2×2 matrix algebra having a representation as infinite-dimensional fermionic Fock oscillator algebra and allowing a natural interpretation in terms of spinors for the world of classical worlds having a representation as infinite-dimensional fermionic Fock space.

Infinite primes would correspond to something different: a tensor product of all p×p matrix algebras from which some factors are deleted and added back as direct summands. Besides this some factors are replaced with their tensor powers.

Should one refine the notion of von Neumann algebra so that one can distinguish between these algebras as physically non-equivalent? Is the full algebra tensor product of this kind of generalized hyper-finite factor and hyper-finite factor of type II1 corresponding to the vibrational degrees of freedom of 3-surface and fermionic degrees of freedom? Could p-adic length scale hypothesis - stating that the physically favored primes are near powers of 2 - relate somehow to the naturality of the inclusions of generalized von Neumann algebras to HFF of type II1?

For background see that chapter TGD as a Generalized Number Theory III: Infinite Primes.



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