Irresponsible press releases

Peter Woit has already commented the newest string theory related press release and classified it as the "most outrageously misleading string theory hype of 2007". I cannot but agree.

Physicists Develop Test For “String Theory” is the title of one of the press releases. What one would expect a test for string theory to be? Even educated layman would guess that something which distinguishes string theory from its competitors so that if the string model passes the test, one can say that string model is favored experimentally over its competitors, say quantum field theory models and loop quantum gravity. Educated layman is of course correct.

The press release tells however something totally different for a layman using all trikcs of the language of lawyer. Two excerpts from the part meant for laymen serve as examples.

Until now, experimental verification has not been possible; but researchers at the University of California, Carnegie Mellon University, and the University of Texas are planning a definitive test with the future launch of the Large Hadron Collider…

... scientists have come up with a definitive test that could prove or disprove string theory. The project is described as…

The professors behind the release talk between the lines something different since they must have leave loophole open for accusing the journalists for inaccuracies. The irritated reactions of colleagues are certainly to be expected.

"The beauty of our test is the simplicity of its assumptions," explained Grinstein. "The canonical forms of string theory include three mathematical assumptions — Lorentz invariance (the laws of physics are the same for all uniformly moving observers), analyticity (a smoothness criteria for the scattering of high-energy particles after a collision) and unitarity (all probabilities always add up to one). Our test sets bounds on these assumptions.

The point is that these assumption are common to both field theories and string models!

"If the test does not find what the theory predicts about W boson scattering," he added, "it would be evidence that one of string theory's key mathematical assumptions is violated. In other words, string theory — as articulated in its current form — would be proven impossible."

"If the bounds are satisfied, we would still not know that string theory is correct," Distler said. "But if the bounds are violated, we would know that string theory, as it is currently understood, could not be correct. At the very least, the theory would have to be reshaped in a highly nontrivial way."

This says between the lines that the test is not actually a test of string theory in the sense described in the press release!

For me a test of string theory which does not test string theory is not a test of string theory. Is the situation in string theory really so bad that this kind of pathetic maneuvers are made in attempt to improve the public image of the field? I predict that the effect will not be the desired when it becomes clear that they cheated once again. Trust is the most valuable social capital and physics community as a whole loses it if it tolerates physicists who behave as second rate lawyers.

Glashow, Cohen, and Poincare

The latest New Scientist contains an article about the Very Special Relativity proposed by nobelist Sheldon Glashow and Andrew Cohen. Glashow and Cohen propose that instead of Poincare group, call it P, some subgroup of P might be physically more relevant than the whole P. To not lose four-momentum one must assume that this group is obtained as a semi-direct product of some subgroup of Lorentz group with translations. The smallest subgroup, call it L₂, is a 2-dimensional Abelian group generated by K_x+J_y and K_y-J_x. Here K refers to Lorentz boosts and J to rotations. This group leaves invariant light-like momentum in z direction. By adding J acting in L₂ like rotations in plane, one obtains L₃, the maximal subgroup leaving invariant light-like momentum in z direction. By adding also K_z one obtains the scalings of light-like momentum or equivalently, the isotropy group L₄ of a light-like ray.

The reasons why Glashow and Cohen regard these groups so interesting are following.

1. All kinematical tests of Lorentz invariance are consistent with the reduction of Lorentz invariance to these symmetries.

2. The representations of L₃ are one-dimensional in both massive and massless case (the latter is familiar from massless representations of Poincare group where particle states are characterized by helicity). The mass is invariant only under the smaller group. This might allow to have left-handed massive neutrinos as well as massive fermions with spin dependent mass.

3. The requirement of CP invariance extends all these reduced symmetry groups to the full Poincare group. The observed very small breaking of CP symmetry might correlate with a small breaking of Lorentz symmetry. Matter antimatter asymmetry might relate to the reduced Lorentz invariance.

The idea is highly interesting from TGD point of view. The groups L₃ and L₄ indeed play a very prominent role in TGD.

1. The full Lorentz invariance is obtained in TGD only at the level of the entire configuration space which is union over sub-configuration spaces associated with future and past light-cones (space-time sheets inside future or past light-cone). These sub-configuration spaces decompose further into a union of sub-sub-configuration spaces for which a choice of quantization axes of spin reflects itself at the level of generalized geometry of the imbedding space (quantum classical correspondence requires that the choice of quantization axes has imbedding space and space-time correlates). The construction of the geometry for these sub-worlds of classical worlds reduces to light-cone boundary so that the little group L₃ leaving a given point of light-cone boundary invariant is in a special role in TGD framework.

2. The selection of a preferred light-like momentum direction at light-cone boundary corresponds to the selection of quantization axis for angular momentum playing a key role in TGD view about hierarchy of Planck constants associated with a hierarchy of Jones inclusions implying a breaking of Lorentz invariance induced by the selection of quantization axis. The number theoretic vision about quantum TGD implies a selection of two preferred axes corresponding to time-like and space-like direction corresponding to real and preferred imaginary unit for hyper-octonions. In both cases L₄ emerges naturally.

3. The TGD based identification of Kac-Moody symmetries as local isometries of the imbedding space acting on 3-D light-like orbits of partonic 2-surfaces involves a selection of a preferred light-like direction and thus the selection of L₄.
4. Also the so called massless extremals representing a precisely targeted propagation of patterns of classical gauge fields with light velocity along typically cylindrical tubes without a change in the shape involve L₄. A very general solution ansatz to classical field equations involves a local decomposition of M⁴ to longitudinal and transversal spaces and selection of a light-like direction.

5. Zero energy ontology is fundamental for the interpretation of quantum TGD and could give rise to a spontaneous CP breaking in the sense that for zero energy states positive energy part of the state could correspond to matter whereas negative energy part would correspond to antimatter identified as the analog of phase conjugate laser beams possessing negative energy and propagating towards geometric future. Negative energy part of the state is usually interpreted as a final state of the particle reaction whose detection in TGD framework corresponds to a detection of a zero energy state. S-matrix represents in this framework time like entanglement between positive and negative energy parts of the state: this makes sense only in the quantum theory based on hyper-finite factors of type II₁ since infinite-dimensional unit matrix (SS⁺=Id) has unit trace for them.

   Phase conjugate matter could be regarded as a generalization of phase conjugate laser beams. CP breaking would occur for each space-time sheet separately and the antimatter created in laboratory would reside on space-time sheets different from those usually carrying ordinary matter. The reduction of the Lorentz group to the little group would be a necessary prerequisite for this kind of CP breaking. The arguments of N-point functions in TGD framework indeed correspond to the tips of future and past light-cones depending on whether they represent incoming or outgoing particles.

6. The parton model of hadrons assumes a preferred longitudinal direction of momentum and mass squared decomposes naturally to longitudinal and transversal mass squared. Also p-adic mass calculations rely heavily on this picture and thermodynamics mass squared might be regarded as a longitudinal mass squared. In TGD framework right handed covariantly constant neutrino generates a super-symmetry in CP₂ degrees of freedom and it might be better to regard left-handed neutrino mass as a longitudinal mass.

This list justifies my own hunch that Glashow and Cohen might have discovered something very important. This is in a strong contrast with 23 years of super-string over-hype, the only outcome of which seems to be the weird belief that physics has reached the situation where no predictions are possible because the theory decided to be the only possible one predicts nothing, and would once again demonstrate that a real progress in physics requires careful analysis of is really known and what is just ad hoc beliefs transformed to dogmas.

About the correspondence between infinite primes, points of world of classical worlds, and configuration space spinor fields

The idea that configuration space CH of 3-surfaces, "the world of classical worlds", could be realized in terms of number theoretic anatomies of single space-time point using the real units formed from infinite rationals, is very attractive.

The correspondence of CH points with infinite primes and thus with infinite number of real units determined by them realizing Platonia at single space-time point, can be understood if one assume that the points of CH correspond to infinite rationals via their mapping to hyper-octonion real-analytic rational functions conjectured to define foliations of HO to hyper-quaternionic 4-surfaces inducing corresponding foliations of H.

The correspondence of CH spinors with the real units identified as infinite rationals with varying number theoretical anatomies is not so obvious. It is good to approach the problem by making questions.

1. How the points of CH and CH spinors at given point of CH correspond to various real units? Configuration space Hamiltonians and their super-counterparts characterize modes of configuration space spinor fields rather than only spinors. Does this mean that only ground states of super-conformal representations, which are expected to correspond elementary particles, correspond to configuration space spinors and are coded by infinite primes?

2. How do CH spinor fields (as opposed to CH spinors) correspond to infinite rationals? Configuration space spinor fields are generated by elements of super-conformal algebra from ground states. Should one code the matrix elements of the operators between ground states and creating zero energy states in terms of time-like entanglement between ground states represented by real units and assigned to the preferred points of H characterizing the tips of future and past light-cones and having also interpretation as arguments of n-point functions?

The argument represented in detail in TGD as a Generalized Number Theory III: Infinite Primes is in a nutshell following.

1. CH itself and CH spinors are by super-symmetry characterized by ground states of super-conformal representations and can be mapped to infinite rationals defining real units U_k multiplying the eight preferred H coordinates h^k whereas configuration space spinor fields correspond to discrete analogs of Schrödinger amplitudes in the space whose points have U_k as coordinates. The 8-units correspond to ground states for an 8-fold tensor power of a fundamental super-conformal representation or to a product of representations of this kind.

2. General states are coded by quantum entangled states defined as entangled states of positive and negative energy ground states with entanglement coefficients defined by the product of operators creating positive and negative energy states represented by the units. Normal ordering prescription makes the mapping unique.

3. The condition that various symmetries have number theoretical correlates leads to rather detailed view about the map of ground states to real units. As a matter fact one ends up with a detailed view about number theoretical realization of fundamental symmetries of standard model.

4. It seems that quantal generalization of the fundamental associativity and commutativity conditions might be needed in the sense that quantum states are superpositions over all possible associations associated with a given hyper-octonionic prime. Only infinite integers identifiable as many particle states would reduced to infinite rational integers mappable to rational functions of hyper-octonionic coordinate with rational coefficients. Infinite primes could be genuinely hyper-quaternionic. This would imply automatically color confinement but would allow colored partons.

For more details see the chapter TGD as a Generalized Number Theory III: Infinite Primes. of "TGD as a Generalized Number Theory".

Updated vision about infinite primes

I have updated the chapter about infinite primes so that it conforms with the recent general view about number theoretic aspects of quantum TGD. A lot of obsoletia have been thrown away and new insights have emerged.

1. In particular, the identification of the mapping of infinite primes to space-time surfaces is fixed by associativity condition so that it only yields 4-D surfaces rather than a hierarchy of 4n-D surfaces of 8n-D imbedding spaces. This observation was actually trivial but had escaped my attention.

2. What is especially fascinating is that configuration space and configuration space spinor fields might be represented in terms of the number theoretical anatomy of imbedding space points. Configuration space spinor fields associated with a given sub-configuration space labelled by a preferred point of imbedding space (this includes tip of lightcone) would be analogs of ordinary wave functions defined in the space of points which are identical in the real sense. One can say that physics in a well-defined sense reduces to space-time level after all.

I attach below the abstract of the revised chapter TGD as a Generalized Number Theory III: Infinite Primes.

Infinite primes are besides p-adicization and the representation of space-time surface as a hyper-quaternionic sub-manifold of hyper-octonionic space, basic pillars of the vision about TGD as a generalized number theory and will be discussed in the third part of the multi-chapter devoted to the attempt to articulate this vision as clearly as possible.

1. Why infinite primes are unavoidable

Suppose that 3-surfaces could be characterized by p-adic primes characterizing their effective p-adic topology. p-Adic unitarity implies that each quantum jump involves unitarity evolution U followed by a quantum jump. Simple arguments show that the p-adic prime characterizing the 3-surface representing the entire universe increases in a statistical sense. This leads to a peculiar paradox: if the number of quantum jumps already occurred is infinite, this prime is most naturally infinite. On the other hand, if one assumes that only finite number of quantum jumps have occurred, one encounters the problem of understanding why the initial quantum history was what it was. Furthermore, since the size of the 3-surface representing the entire Universe is infinite, p-adic length scale hypothesis suggest also that the p-adic prime associated with the entire universe is infinite.

These arguments motivate the attempt to construct a theory of infinite primes and to extend quantum TGD so that also infinite primes are possible. Rather surprisingly, one can construct what might be called generating infinite primes by repeating a procedure analogous to a quantization of a super symmetric quantum field theory. At given level of hierarchy one can identify the decomposition of space-time surface to p-adic regions with the corresponding decomposition of the infinite prime to primes at a lower level of infinity: at the basic level are finite primes for which one cannot find any formula.

2. Two views about the role of infinite primes and physics in TGD Universe

Two different views about how infinite primes, integers, and rationals might be relevant in TGD Universe have emerged.

a) The first view is based on the idea that infinite primes characterize quantum states of the entire Universe. 8-D hyper-octonions make this correspondence very concrete since 8-D hyper-octonions have interpretation as 8-momenta. By quantum-classical correspondence also the decomposition of space-time surfaces to p-adic space-time sheets should be coded by infinite hyper-octonionic primes. Infinite primes could even have a representation as hyper-quaternionic 4-surfaces of 8-D hyper-octonionic imbedding space.

b) The second view is based on the idea that infinitely structured space-time points define space-time correlates of mathematical cognition. The mathematical analog of Brahman=Atman identity would however suggest that both views deserve to be taken seriously.

3. Infinite primes and infinite hierarchy of second quantizations

The discovery of infinite primes suggested strongly the possibility to reduce physics to number theory. The construction of infinite primes can be regarded as a repeated second quantization of a super-symmetric arithmetic quantum field theory. Later it became clear that the process generalizes so that it applies in the case of quaternionic and octonionic primes and their hyper counterparts. This hierarchy of second quantizations means enormous generalization of physics to what might be regarded a physical counterpart for a hierarchy of abstractions about abstractions about.... The ordinary second quantized quantum physics corresponds only to the lowest level infinite primes. This hierarchy can be identified with the corresponding hierarchy of space-time sheets of the many-sheeted space-time.

One can even try to understand the quantum numbers of physical particles in terms of infinite primes. In particular, the hyper-quaternionic primes correspond four-momenta and mass squared is prime valued for them. The properties of 8-D hyper-octonionic primes motivate the attempt to identify the quantum numbers associated with CP₂ degrees of freedom in terms of these primes. The representations of color group SU(3) are indeed labelled by two integers and the states inside given representation by color hyper-charge and iso-spin.

4. Infinite primes as a bridge between quantum and classical

An important stimulus came from the observation stimulated by algebraic number theory. Infinite primes can be mapped to polynomial primes and this observation allows to identify completely generally the spectrum of infinite primes whereas hitherto it was possible to construct explicitly only what might be called generating infinite primes.

This in turn led to the idea that it might be possible represent infinite primes (integers) geometrically as surfaces defined by the polynomials associated with infinite primes (integers).

Obviously, infinite primes would serve as a bridge between Fock-space descriptions and geometric descriptions of physics: quantum and classical. Geometric objects could be seen as concrete representations of infinite numbers providing amplification of infinitesimals to macroscopic deformations of space-time surface. We see the infinitesimals as concrete geometric shapes!

5. Various equivalent characterizations of space-times as surfaces

One can imagine several number-theoretic characterizations of the space-time surface.

1. The approach based on octonions and quaternions suggests that space-time surfaces correspond to associative, or equivalently, hyper-quaternionic surfaces of hyper-octonionic imbedding space HO. Also co-associative, or equivalently, co-hyper-quaternionic surfaces are possible. These foliations can be mapped in a natural manner to the foliations of H=M^4\times CP_2 by space-time surfaces which are identified as preferred extremals of the Kähler action (absolute minima or maxima for regions of space-time surface in which action density has definite sign). These views are consistent if hyper-quaternionic space-time surfaces correspond to so called Kähler calibrations.

2. Hyper-octonion real-analytic surfaces define foliations of the imbedding space to hyper-quaternionic 4-surfaces and their duals to co-hyper-quaternionic 4-surfaces representing space-time surfaces.

3. Rational infinite primes can be mapped to rational functions of n arguments. For hyper-octonionic arguments non-associativity makes these functions poorly defined unless one assumes that arguments are related by hyper-octonion real-analytic maps so that only single independent variable remains. These hyper-octonion real-analytic functions define foliations of HO to space-time surfaces if b) holds true.

The challenge of optimist is to prove that these characterizations are equivalent.

6. The representation of infinite primes as 4-surfaces

The difficulties caused by the Euclidian metric signature of the number theoretical norm forced to give up the idea that space-time surfaces could be regarded as quaternionic sub-manifolds of octonionic space, and to introduce complexified octonions and quaternions resulting by extending quaternionic and octonionic algebra by adding imaginary units multiplied with √{-1. This spoils the number field property but the notion of prime is not lost. The sub-space of hyper-quaternions resp.-octonions is obtained from the algebra of ordinary quaternions and octonions by multiplying the imaginary part with √-1. The transition is the number theoretical counterpart for the transition from Riemannian to pseudo-Riemannian geometry performed already in Special Relativity.

The commutative √-1 relates naturally to the algebraic extension of rationals generalized to an algebraic extension of rational quaternions and octonions and conforms with the vision about how quantum TGD could emerge from infinite dimensional Clifford algebra identifiable as a hyper-finite factor of type II₁.

The notions of hyper-quaternionic and octonionic manifold make sense but it is implausible that H=M⁴× CP₂ could be endowed with a hyper-octonionic manifold structure. Indeed, space-time surfaces are assumed to be hyper-quaternionic or co-hyper-quaternionic 4-surfaces of 8-dimensional Minkowski space M⁸ identifiable as the hyper-octonionic space HO. Since the hyper-quaternionic sub-spaces of HO with a fixed complex structure are labelled by CP₂, each (co)-hyper-quaternionic four-surface of HO defines a 4-surface of M⁴× CP₂. One can say that the number-theoretic analog of spontaneous compactification occurs.

Any hyper-octonion analytic function HO--> HO defines a function g: HO--> SU(3) acting as the group of octonion automorphisms leaving a selected imaginary unit invariant, and g in turn defines a foliation of HO and H=M⁴× CP₂ by space-time surfaces. The selection can be local which means that G₂ appears as a local gauge group.

Since the notion of prime makes sense for the complexified octonions, it makes sense also for the hyper-octonions. It is possible to assign to infinite prime of this kind a hyper-octonion analytic polynomial P: HO--> HO and hence also a foliation of HO and H=M⁴× CP₂ by 4-surfaces. Therefore space-time surface could be seen as a geometric counterpart of a Fock state. The assignment is not unique but determined only up to an element of the local octonionic automorphism group G₂ acting in HO and fixing the local choices of the preferred imaginary unit of the hyper-octonionic tangent plane. In fact, a map HO--> S⁶ characterizes the choice since SO(6) acts effectively as a local gauge group.

The construction generalizes to all levels of the hierarchy of infinite primes if one poses the associativity requirement implying that hyper-octonionic variables are related by hyper-octonion real-analytic maps, and produces also representations for integers and rationals associated with hyper-octonionic numbers as space-time surfaces. By the effective 2-dimensionality naturally associated with infinite primes represented by real polynomials 4-surfaces are determined by data given at partonic 2-surfaces defined by the intersections of 3-D and 7-D light-like causal determinants. In particular, the notions of genus and degree serve as classifiers of the algebraic geometry of the 4-surfaces. The great dream is of course to prove that this construction yields the solutions to the absolute minimization of Kähler action.

7. Generalization of ordinary number fields: infinite primes and cognition

Both fermions and p-adic space-time sheets are identified as correlates of cognition in TGD Universe. The attempt to relate these two identifications leads to a rather concrete model for how bosonic generators of super-algebras correspond to either real or p-adic space-time sheets (actions and intentions) and fermionic generators to pairs of real space-time sheets and their p-adic variants obtained by algebraic continuation (note the analogy with fermion hole pairs).

The introduction of infinite primes, integers, and rationals leads also to a generalization of real numbers since an infinite algebra of real units defined by finite ratios of infinite rationals multiplied by ordinary rationals which are their inverses becomes possible. These units are not units in the p-adic sense and have a finite p-adic norm which can be differ from one. This construction generalizes also to the case of hyper- quaternions and -octonions although non-commutativity and in case of octonions also non-associativity pose technical problems to which the reduction to ordinary rational is simplest cure which would however allow interpretation as decomposition of infinite prime to hyper-octonionic lower level constituents. Obviously this approach differs from the standard introduction of infinitesimals in the sense that sum is replaced by multiplication meaning that the set of real units becomes infinitely degenerate.

Infinite primes form an infinite hierarchy so that the points of space-time and imbedding space can be seen as infinitely structured and able to represent all imaginable algebraic structures. Certainly counter-intuitively, single space-time point is even capable of representing the quantum state of the entire physical Universe in its structure. For instance, in the real sense surfaces in the space of units correspond to the same real number 1, and single point, which is structure-less in the real sense could represent arbitrarily high-dimensional spaces as unions of real units.

One might argue that for the real physics this structure is completely invisible and is relevant only for the physics of cognition. On the other hand, one can consider the possibility of mapping the configuration space and configuration space spinor fields to the number theoretical anatomies of a single point of imbedding space so that the structure of this point would code for the world of classical worlds and for the quantum states of the Universe. Quantum jumps would induce changes of configuration space spinor fields interpreted as wave functions in the set of number theoretical anatomies of single point of imbedding space in the ordinary sense of the word, and evolution would reduce to the evolution of the structure of a typical space-time point in the system. Physics would reduce to space-time level but in a generalized sense. Universe would be an algebraic hologram, and there is an obvious connection both with Brahman=Atman identity of Eastern philosophies and Leibniz's notion of monad.

For more details see the revised chapters TGD as a Generalized Number Theory III:Infinite Primes and Infinite Primes and Consciousness.

Galois groups, Jones inclusions, and infinite primes

Langlands program is an attempt to unify mathematics using the idea that all zeta functions and corresponding theta functions could emerge as automorphic functions giving rise to finite-dimensional representations for Galois groups (Galois group is defined as a group of automorphisms of the extension of field F leaving invariant the elements of F). The basic example corresponds to rationals and their extensions. Finite fields G(p,k) and their extensions G(p,nk) represents another example. The largest extension of rationals corresponds to algebraic numbers (algebraically closed set). Although this non-Abelian group is huge and does not exist in the usual sense of the word its finite-dimensional representations in groups GL(n,Z) make sense.

For instance, Edward Witten is working with the idea that geometric variant of Langlands duality could correspond to the dualities discovered in string model framework and be understood in terms of topological version of four-dimensional N=4 super-symmetric YM theory (arXiv:hep-th/060451). In particular, Witten assigns surface operators to the 2-D surfaces of 4-D space-time. This brings unavoidably in mind partonic 2-surfaces and TGD as N=4 super-conformal almost topological QFT.

This observation stimulates some ideas about the role of zeta functions in TGD if one takes the vision about physics as a generalized number theory seriously. In particular, the notion of infinite prime suggests a manner to realize the modular functions as representations of Galois groups. Infinite primes might also provide a new perspective to the concrete realization of Langlands program.

1. Galois groups, Jones inclusions, and quantum measurement theory

The Galois representations appearing in Langlands program could have a concrete physical/cognitive meaning.

1. The Galois groups associated with the extensions of rationals have a natural action on partonic 2-surfaces represented by algebraic equations. Their action would reduce to permutations of roots of the polynomial equations defining the points with a fixed projection to the above mentioned geodesic sphere S² of CP₂ or δ M⁴₊. This makes possible to define modes of induced spinor fields transforming under representations of Galois groups. Galois groups would also have a natural action on configuration space-spinor fields. One can also speak about configuration space spinors invariant under Galois group.

2. Galois groups could be assigned to Jones inclusions having an interpretation in terms of a finite measurement resolution in the sense that the discrete group defining the inclusion leaves invariant the operators generating excitations which are not detectable.

3. The physical interpretation of the finite resolution represented by Galois group would be based on the analogy with particle physics. The field extension K/F implies that the primes (more precisely, prime ideals) of F decompose into products of primes (prime ideals) of K. Physically this corresponds to the decomposition of particle into more elementary constituents, say hadrons into quarks in the improved resolution implied by the extension F→ K. The interpretation in terms of cognitive resolution would be that the primes associated with the higher extensions of rationals are not cognizable: in other words, the observed states are singlets under corresponding Galois groups: one has algebraic/cognitive counterpart of color confinement.

4. For instance, the system labelled by an ordinary p-adic prime could decompose to a system which is a composite of Gaussian primes. Interestingly, the biologically highly interesting p-adic length scale range 10 nm-5 μm contains as many as four Gaussian Mersennes (M_k=(1+i)^k-1, k=151,157,163,167), which suggests that the emergence of living matter means an improved cognitive resolution.

2.Galois groups and infinite primes

The notion of infinite prime suggests a manner to realize the modular functions as representations of Galois groups. Infinite primes might also provide a new perspective to the concrete realization of Langlands program.

1. The discrete Galois groups associated with various extensions of rationals and involved with modular functions which are in one-one correspondence with zeta functions via Mellin transform defined as ∑ x_n n^-s→ ∑ x_nzⁿ. Various Galois groups would have a natural action in the space of infinite primes having interpretation as Fock states and more general bound states of an arithmetic quantum field theory.

2. The number theoretic anatomy of space-time points due to the possibility to define infinite number of number theoretically non-equivalent real units using infinite rationals (see this) allows the imbedding space points themselves to code holographically various things. Galois groups would have a natural action in the space of real units and thus on the number theoretical anatomy of a point of imbedding space.

3. Since the repeated second quantization of the super-symmetric arithmetic quantum field theory defined by infinite primes gives rise to a huge space of quantum states, the conjecture that the number theoretic anatomy of imbedding space point allows to represent configuration space (the world of classical worlds associated with the light-cone of a given point of H) and configuration space spinor fields emerges naturally (see this).

4. Since Galois groups G are associated with inclusions of number fields to their extensions, this inclusion could correspond at quantum level to a generalized Jones inclusion N subset M such that G acts as automorphisms of M and leaves invariant the elements of N. This might be possible if one allows the replacement of complex numbers as coefficient fields of hyper-finite factors of type II₁ with various algebraic extensions of rationals. Quantum measurement theory with a finite measurement resolution defined by Jones inclusion N subset M (see this) could thus have also a purely number theoretic meaning provided it is possible to define a non-trivial action of various Galois groups on configuration space spinor fields via the imbedding of the configuration space spinors to the space of infinite integers and rationals (analogous to the imbedding of space-time surface to imbedding space).

This picture allows to develop rather fascinating ideas about mathematical structures and their relationship to physical world. For instance, the functional form of a map between two sets the points of the domain and target rather than only its value could be coded in a holographic manner by using the number theoretic anatomy of the points. Modular functions giving rise to generalized zeta functions would emerge in especially natural manner in this framework. Configuration space spinor fields would allow a physical realization of the holographic representations of various maps as quantum states.

For more details see the end of the chapter Construction of Quantum Theory: Symmetries of "Towards S-Matrix" and the article Topological Geometrodynamics: an Overall View. See also the article Could local zeta functions take the role of Riemann Zeta in TGD framework?.

Could local zeta functions take the role of Riemann Zeta in TGD framework?

The recent view about TGD (for a summary see this, this, and this) leads to some conjectures about Riemann Zeta.

1. Non-trivial zeros should be algebraic numbers.

2. The building blocks in the product decomposition of ζ should be algebraic numbers for non-trivial zeros of zeta.

3. The values of zeta for their combinations with positive imaginary part with positive integer coefficients should be algebraic numbers.

These conjectures are motivated by the findings that Riemann Zeta seems to be associated with critical systems and by the fact that non-trivial zeros of zeta are analogous to complex conformal weights. The necessity to make such a strong conjectures, in particular the third conjecture, is an unsatisfactory feature of the theory and one could ask how to modify this picture. Also a clear physical interpretation of Riemann zeta is lacking.

1. Local zeta functions and Weil conjectures

Riemann Zeta is not the only zeta (see this and this). There is entire zoo of zeta functions and the natural question is whether some other zeta sharing the basic properties of Riemann zeta having zeros at critical line could be more appropriate in TGD framework.

The so called local zeta functions analogous to the factors ζ_p(s)= 1/(1-p^-s) of Riemann Zeta can be used to code algebraic data about say numbers about solutions of algebraic equations reduced to finite fields. The local zeta functions appearing in Weil's conjectures associated with finite fields G(p,k) and thus to single prime. The extensions G(p,nk) of this finite field are considered. These local zeta functions code the number for the points of algebraic variety for given value of n. Weil's conjectures also state that if X is a mod p reduction of non-singular complex projective variety then the degree for the polynomial multiplying the product ζ(s)×ζ(s-1) equals to Betti number. Betti number is 2 times genus in 2-D case.

It has been proven that the zetas of Weil are associated with single prime p, they satisfy functional equation, their zeros are at critical lines, and rather remarkably, they are rational functions of p^-s. For instance, for elliptic curves zeros are at critical line.

The general form for the local zeta is ζ(s)= exp(G(s)), where G= ∑ g_np^-ns, g_n=N_n/n, codes for the numbers N_n of points of algebraic variety for n^th extension of finite field F with nk elements assuming that F has k=p^r elements. This transformation resembles the relationship Z= exp(F) between partition function and free energy Z= exp(F) in thermodynamics.

The exponential form is motivated by the possibility to factorize the zeta function into a product of zeta functions. Note also that in the situation when N_n approaches constant N_∞, the division of N_n by n gives essentially 1/(1-N_∞p^-s) and one obtains the factor of Riemann Zeta at a shifted argument s-log^p(N_∞). The local zeta associated with Riemann Zeta corresponds to N_n=1.

2. Local zeta functions and TGD

The local zetas are associated with single prime p, they satisfy functional equation, their zeros lie at the critical lines, and they are rational functions of p^-s. These features are highly desirable from the TGD point of view.

2.1 Why local zeta functions are natural in TGD framework?

In TGD framework modified Dirac equation assigns to a partonic 2-surface a p-adic prime p and inverse of the zeta defines local conformal weight (see this). The intersection of the real and corresponding p-adic parton 2-surface is the set containing the points that one is interested in. Hence local zeta sharing the basic properties of Riemann zeta is highly desirable and natural. In particular, if the local zeta is a rational function then the inverse images of rational points of the geodesic sphere are algebraic numbers. Of course, one might consider a stronger constraint that the inverse image is rational. Note that one must still require that p^-s as well as s are algebraic numbers for the zeros of the local zeta (the first two conditions listed in the beginning) if one wants the number theoretical universality.

Since the modified Dirac operator assigns to a given partonic 2-surface a p-adic prime p (see this), one can ask whether the inverse ζ_p^-1(z) of some kind of local zeta directly coding data about partonic 2-surface could define the generalized eigenvalues of the modified Dirac operator and radial super-canonical conformal weights so that the conjectures about Riemann Zeta would not be needed at all.

The eigenvalues of the modified Dirac operator (see this) would in a holographic manner code for information about partonic 2-surface. This kind of algebraic geometric data are absolutely relevant for TGD since U-matrix and probably also S-matrix must be formulated in terms of the data related to the intersection of real and partonic 2-surfaces (number theoretic braids) obeying same algebraic equations and consisting of algebraic points in the appropriate algebraic extension of p-adic numbers. Note that the hierarchy of algebraic extensions of p-adic number fields would give rise to a hierarchy of zetas so that the algebraic extension used would directly reflect itself in the eigenvalue spectrum of the modified Dirac operator and super-canonical conformal weights. This is highly desirable but not achieved if one uses Riemann Zeta.

One must of course leave open the possibility that for real-real transitions the inverse of the zeta defined as a product of the local zetas (very much analogous to Riemann Zeta) defines the conformal weights. This kind of picture would conform with the idea about real physics as a kind of adele formed from p-adic physics.

2.2 Finite field hierarchy is not natural in TGD context

That local zeta functions are assigned with a hierarchy of finite field extensions do not look natural in TGD context. The reason is that these extensions are regarded as abstract extensions of G(p,k) as opposed to a large number of algebraic extensions isomorphic with finite fields as abstract number fields and induced from the extensions of p-adic number fields. Sub-field property is clearly highly relevant in TGD framework just as the sub-manifold property is crucial for geometrizing also other interactions than gravitation in TGD framework.

The O(pⁿ) hierarchy for the p-adic cutoffs would naturally replace the hierarchy of finite fields. This hierarchy is quite different from the hierarchy of finite fields since one expects that the number of solutions becomes constant at the limit of large n and also at the limit of large p so that powers in the function G coding for the numbers of solutions of algebraic equations as function of n should not increase but approach constant N_∞. The possibility to factorize exp(G) to a product exp(G₀)exp(G_∞) would mean a reduction to a product of a rational function and factor(s) ζ_p(s)=1/(1-p^{-s₁}) associated with Riemann Zeta with argument s shifted to s₁=s-log_p(N_∞).

2.3 What data local zetas could code?

The next question is what data the local zeta functions could code.

1. It is not at clear whether it is useful to code global data such as the numbers of points of partonic 2-surface modulo pⁿ. The notion of number theoretic braid occurring in the proposed approach to S-matrix (see this) suggests that the zeta at an algebraic point z of the geodesic sphere S² of CP₂ or of light-cone boundary should code purely local data such as the numbers N_n of points which project to z as function of p-adic cutoff pⁿ. In the generic case this number would be finite for non-vacuum extremals with 2-D S² projection. The n^th coefficient g_n=N_n/n of the function G_p would code the number N_n of these points in the approximation O(pⁿ⁺¹)=0 for the algebraic equations defining the p-adic counterpart of the partonic 2-surface.

2. In a region of partonic 2-surface where the numbers N_n of these points remain constant, ζ(s) would have constant functional form and therefore the information in this discrete set of algebraic points would allow to deduce deduce information about the numbers N_n. Both the algebraic points and generalized eigenvalues would carry the algebraic information.

3. A rather fascinating self referentiality would result: the generalized eigen values of the modified Dirac operator expressible in terms of inverse of zeta would code data for a sequence of approximations for the p-adic variant of the partonic 2-surface. This would be natural since second quantized induced spinor fields are correlates for logical thought in TGD inspired theory of consciousness. Even more, the data would be given at points ζ(s), s a rational value of a super-canonical conformal weight or a value of generalized eigenvalue of modified Dirac operator (which is essentially function s= ζ_p^-1(z) at geodesic sphere of CP₂ or of light-cone boundary).

For more details see the end of the chapter Construction of Quantum Theory: Symmetries of "Towards S-Matrix" and the article Topological Geometrodynamics: an Overall View. See also the article Could local zeta functions take the role of Riemann Zeta in TGD framework?.