What's new in
TGD as a Generalized Number Theory
Note: Newest contributions are at the top!
Zeros of Riemann Zeta as conformal weights, braids, Jones inclusions, and number theoretical universality of quantum TGD
Quantum TGD relies on a heuristic number theoretical vision lacking a rigorous justification and I have made considerable efforts to reduce this picture to as few basic unproven assumptions as possible. In the following I want briefly summarize some recent progress made in this respect.
1. Geometry of the world of classical worlds as the basic context
The number theoretic conjectures has been inspired by the construction of the geometry of the configuration space consisting of 3-surfaces of M4× CP2, the "world of classical worlds". Hamiltonians defined at δM4+/-× CP2 are basic elements of super-canonical algebra acting as isometries of the geometry of the "world of classical worlds". These Hamiltonians factorize naturally into products of functions of M4 radial coordinate rM which corresponds to a lightlike direction of lightcone boundary δM4+/- and functions of coordinates of rM constant sphere and CP2 coordinates. The assumption has been that the functions in question are powers of form (rM/r0)Δ where Δ has a natural interpretation as a radial conformal conformal weight.
2. List of conjectures
Quite a thick cloud of conjectures surrounds the construction of configuration space geometry and of quantum TGD.
3. The unifying hypothesis
The most recent progress in TGD is based on the finding that these separate hypothesis can be unified to single assumption. The radial conformal weights Δ are not constants but functions of CP2 coordinate expressible as
where ξ1 and ξ2 are the complex coordinates of CP2 transforming linearly under subgroup U(2) of SU(3). The choice of this coordinate system is not completely unique and relates to the choice of directions of color isospin and hyper charge. This choice has a correlates at space-time and configuration space level in accordance with the idea that also quantum measurement theory has geometric correlates in TGD framework. This hypothesis obviously generalizes the earlier assumption which states that Δ is constant and a linear combination of zeros of Zeta.
A couple of comments are in order.
Somehow it is obvious that infinite primes (see this) must have some very deep role to play in quantum TGD and TGD inspired theory of consciousness. What this role precisely is has remained an enigma although I have considered several detailed interpretations (see the link above).
In the following an interpretation allowing to unify the views about fermionic Fock states as a representation of Boolean cognition and p-adic space-time sheets as correlates of cognition is discussed. Very briefly, real and p-adic partonic 3-surfaces serve as space-time correlates for the bosonic super algebra generators, and pairs of real partonic 3-surfaces and their algebraically continued p-adic variants as space-time correlates for the fermionic super generators. Intentions/actions are represented by p-adic/real bosonic partons and cognitions by pairs of real partons and their p-adic variants and the geometric form of Fermi statistics guarantees the stability of cognitions against intentional action.
1. Infinite primes very briefly
Infinite primes have a decomposition to infinite and finite parts allowing an interpretation as a many-particle state of a super-symmetric arithmetic quantum field theory for which fermions and bosons are labelled by primes. There is actually an infinite hierarchy for which infinite primes of a given level define the building blocks of the infinite primes of the next level. One can map infinite primes to polynomials and these polynomials in turn could define space-time surfaces or at least light-like partonic 3-surfaces appearing as solutions of Chern-Simons action so that the classical dynamics would not pose too strong constraints.
The simplest infinite primes at the lowest level are of form mBX/sF + nBsF, X=∏i pi (product of all finite primes). mB, nB, and sF are defined as mB= ∏ipimi, nB= ∏iqini, and sF= ∏iqi, mB and nB have no common prime factors. The simplest interpretation is that X represents Dirac sea with all states filled and X/sF + sF represents a state obtained by creating holes in the Dirac sea. The integers mB and nB characterize the occupation numbers of bosons in modes labelled by pi and qi and sF= ∏iqi characterizes the non-vanishing occupation numbers of fermions.
The simplest infinite primes at all levels of the hierarchy have this form. The notion of infinite prime generalizes to hyper-quaternionic and even hyper-octonionic context and one can consider the possibility that the quaternionic components represent some quantum numbers at least in the sense that one can map these quantum numbers to the quaternionic primes.
The obvious question is whether configuration space degrees of freedom and configuration space spinor (Fock state) of the quantum state could somehow correspond to the bosonic and fermionic parts of the hyper-quaternionic generalization of the infinite prime as proposed here. That hyper-quaternionic (or possibly hyper-octonionic) primes would define as such the quantum numbers of fermionic super generators does not make sense. It is however possible to have a map from the quantum numbers labelling super-generators to the finite primes. One must also remember that the infinite primes considered are only the simplest ones at the given level of the hierarchy and that the number of levels is infinite.
2. Precise space-time correlates of cognition and intention
The best manner to end up with the proposal about how p-adic cognitive representations relate bosonic representations of intentions and actions and to fermionic cognitive representations is through the following arguments.
The discreteness of the intersection of the real space-time sheet and its p-adic variant obtained by algebraic continuation would be a completely universal phenomenon associated with all fermionic states. This suggests that also real-to-real S-matrix elements involve instead of an integral a sum with the arguments of an n-point function running over all possible combinations of the points in the intersection. S-matrix elements would have a universal form which does not depend on the number field at all and the algebraic continuation of the real S-matrix to its p-adic counterpart would trivialize. Note that also fermionic statistics favors strongly discretization unless one allows Dirac delta functions.
The chapters Fusion of p-Adic and Real Variants of Quantum TGD to a More General Theory and TGD as a Generalized Number Theory III: Infinite Primes contain this piece of text too.
The generalization of the number concept obtained roughly by glueing reals and various p-adic numbers and their algebraic extensions together along common rationals and possibly also common algebraics is the starting point of TGD vision about fusion of real physics and various p-adic physics. I call the process of assigning to real physics p-adic physics p-adicization and the heuristic idea is that it corresponds to an algebraic continuation. I realized that the recent progress in the understanding of the formulation of quantum TGD at parton level leads also to a considerable progress in p-adicization.The following text gives a brief summary about the most recent view about what p-adicization might be. This view might be characterized as minimalism and would involve geometrization of only the reduced configuration space consisting of the maxima of Kähler function.
1. p-Adicization at the level of space-time
The minimum amount of p-adicization correspond to the p-adicization for the maxima of the Kähler function. The basic question is whether the equations characterizing real space-time sheet make sense also p-adically. Suppose that TGD indeed reduces to almost topological theory defined by Chern-Simons action for the light-like 3-surfaces interpreted as orbits of partonic 2-surfaces (see this, this, and this). If this is the case, then the starting point here would be the algebraic equations defining light-like partonic 3-surfaces via the condition that the determinant of the induced metric vanishes. If the coordinate functions appearing in the determinant are algebraic functions with algebraic coefficients, p-adicization should make sense. This of course, means the assumption of some preferred coordinates and the construction of solutions of equations leads naturally to such coordinates (see this).
If the corresponding 4-dimensional real space-time sheet is expressible as a hyper-quaternionic surface of hyper-octonionic variant of the imbedding space as number-theoretic vision suggests, it might be possible to construct also the p-adic variant of the space-time sheet by algebraic continuation in the case that the functions appearing in the definition of the space-time sheet are algebraic.
2. p-Adicization of second quantized induced spinor fields
Induction procedure makes it possible to geometrize the concept of a classical gauge fields and also of the spinor fields with internal quantum numbers. In the case of the electro-weak gauge fields induction means the projection of the H-spinor connection to a spinor connection on the space-time surface.
In the most recent formulation induced spinor fields appear only at the 3-dimensional light-like partonic 3-surfaces and the solutions of the modified Dirac equation can be written explicitly (see this, this, and this) as simple algebraic functions involving powers of the preferred coordinate variables very much like various operators in conformal theory can be expressed as Laurent series in powers of a complex variable z with operator valued coefficients. This means that the continuation of the second quantized induced spinor fields to various p-adic number fields is a straightforward procedure. The second quantization of these induced spinor fields as free fields is needed to construct configuration space geometry and anti-commutation relation between spinor fields are fixed from the requirement that configuration space gamma matrices correspond to super-canonical generators.
3. Should one p-adicize at the level of configuration space?
If Duistermaat-Heckman theorem holds true in TGD context, one could express configuration space functional integral in terms of exactly calculable Gaussian integrals around the maxima of the Kähler function defining what might be called reduced configuration space CHred. The huge super-conformal symmetries raise the hope that the rest of S-matrix elements could be deduced using group theoretical considerations so that everything would become algebraic. If this optimistic scenario is realized, the p-adicization of CHred might be enough to p-adicize all operations needed to construct the p-adic variant of S-matrix.
The optimal situation would be that S-matrix elements reduce to algebraic numbers for rational valued incoming momenta and that p-adicization trivializes in the sense that it corresponds only to different interpretations for the imbedding space coordinates (interpretation as real or p-adic numbers) appearing in the equations defining the 4-surfaces. For instance, space-time coordinates would correspond to preferred imbedding space coordinates and the remaining imbedding space coordinates could be rational functions of the latter with algebraic coefficients. Algebraic points in a given extension of rationals would thus be common to real and p-adic surfaces. It could also happen that there are no or very few common algebraic points. For instance, Fermat's theorem says that the surface xn+yn=zn has no rational points for n>2..
This picture is probably too simple. The intuitive expectation is that ordinary S-matrix elements are proportional to a factor which in the real case involves an integration over the arguments of an n-point function of a conformal theory defined at a partonic 2-surface. For p-adic-real transitions the integration should reduce to a sum over the common rational or algebraic points of the p-adic and real surface. Same applies to p1→ p2 type transitions.
If this picture is correct, the p-adicization of the configuration space would mean p-adicization of CHred consisting of the maxima of the Kähler function with respect to both fiber degrees of freedom and zero modes acting effectively as control parameters of the quantum dynamics. If CHred is a discrete subset of CH ultrametric topology induced from finite-p p-adic norm is indeed natural for it. 'Discrete set in CH' need not mean a discrete set in the usual sense and the reduced configuration space could be even finite-dimensional continuum. Finite-p p-adicization as a cognitive model would suggest that p-adicization in given point of CHred is possible for all p-adic primes associated with the corresponding space-time surface (maximum of Kähler function) and represents a particular cognitive representation about CHred.
A basic technical problem is, whether the integral defining the Kähler action appearing in the exponent of Kähler function exists p-adically. Here the hypothesis that the exponent of the Kähler function is identifiable as a Dirac determinant of the modified Dirac operator defined at the light-like partonic 3-surfaces suggests a solution to the problem. By restricting the generalized eigen values of the modified Dirac operator to an appropriate algebraic extension of rationals one could obtain an algebraic number existing both in the real and p-adic sense if the number of the contributing eigenvalues is finite. The resulting hierarchy of algebraic extensions of Rp would have interpretation as a cognitive hierarchy. If the maxima of Kähler function assignable to the functional integral are such that the number of eigenvalues in a given algebraic extension is finite this hypothesis works.
If Duistermaat-Heckman theorem generalizes, the p-adicization of the entire configuration space would be un-necessary and it certainly does not look a good idea in the light of preceding considerations.
Does the quantization of Planck constant transform integer quantum Hall effect to fractional quantum Hall effect?
The TGD based model for topological quantum computation inspired the idea that Planck constant might be dynamical and quantized. The work of Nottale (astro-ph/0310036) gave a strong boost to concrete development of the idea and it took year and half to end up with a proposal about how basic quantum TGD could allow quantization Planck constant associated with M4 and CP2 degrees of freedom such that the scaling factor of the metric in M4 degrees of freedom corresponds to the scaling of hbar in CP2 degrees of freedom and vice versa (see the new chapter Does TGD Predict the Spectrum of Planck constants?). The dynamical character of the scaling factors of M4 and CP2 metrics makes sense if space-time and imbedding space, and in fact the entire quantum TGD, emerge from a local version of an infinite-dimensional Clifford algebra existing only in dimension D=8.
The predicted scaling factors of Planck constant correspond to the integers n defining the quantum phases q=exp(iπ/n) characterizing Jones inclusions. A more precise characterization of Jones inclusion is in terms of group
Gb subset of SU(2) subset of SU(3)
in CP2 degrees of freedom and
in M4 degrees of freedom. In quantum group phase space-time surfaces have exact symmetry such that to a given point of M4 corresponds an entire Gb orbit of CP2 points and vice versa. Thus space-time sheet becomes N(Ga) fold covering of CP2 and N(Gb)-fold covering of M4. This allows an elegant topological interpretation for the fractionization of quantum numbers. The integer n corresponds to the order of maximal cyclic subgroup of G.
In the scaling hbar0→ n× hbar0 of M4 Planck constant fine structure constant would scale as
α= (e2/(4πhbar c)→ α/n ,
and the formula for Hall conductance would transform to
σH =να → (ν/n)× α .
Fractional quantum Hall effect would be integer quantum Hall effect but with scaled down α. The apparent fractional filling fraction ν= m/n would directly code the quantum phase q=exp(iπ/n) in the case that m obtains all possible values. A complete classification for possible phase transitions yielding fractional quantum Hall effect in terms of finite subgroups G subset of SU(2) subset of SU(3) given by ADE diagrams would emerge (An, D2n, E6 and E8 are possible). What would be also nice that CP2 would make itself directly manifest at the level of condensed matter physics.
For more details see the chapter Topological Quantum Computation in TGD Universe, and the chapters Was von Neumann Right After All? and Does TGD predict the Spectrum of Planck Constants? of "TGD: an Overview".
The number theoretic vision about physics (this, this, and this) relies on the idea that physics or, rather what we can know about it, is basically rational number based. Also a generalization of the notion of number is involved. Very roughly, real numbers and various algebraic extensions of p-adic number fields are glued together along common rationals to form a book like structures.
One interpretation would be that space-time surfaces, the induced spinors at space-time surfaces, configuration space spinor fields, S-matrix, etc..., can be obtained by algebraically continuing their values in a discrete subset of rational variant of the geometric structure considered to appropriate completion of rationals (real or p-adic). The existence of the algebraic continuation poses very strong additional constraints on physics but has not provided any practical means to solve quantum TGD.
This view leads to a very powerful iterative method of constructing global solutions of classical field equations from local data and at the same time gives justification for the notion of p-adic fractality, which has provided very successful approach not only to elementary particle physics but also physics at longer scales. The basic idea is that mere p-adic continuity and smoothness imply fractal long range correlations between rational points which are very close p-adically but far from each other in the real sense and vice versa.
1. The emergence of a rational cutoff
For a given p-adic continuation only a subset of rational points is acceptable since the simultaneous requirements of real and p-adic continuity can be satisfied only if one introduces ultraviolet cutoff length scale. This means that the distances between subset of rational points fixing the dynamics of the quantities involved are above some cutoff length scale, which is expected to depend on the p-adic number field Rp as well as a particular solution of field equations. The continued quantities coincide only in this subset of rationals but not in shorter length scales.
The presence of the rational cutoff implies that the dynamics at short scales becomes effectively discrete. Reality is however not discrete: discreteness and rationality only characterize the inherent limitations of our knowledge about reality. This conforms with the fact that our numerical calculations are always discrete and involve finite set of points.
The intersection points of various p-adic continuations with real space-time surface should code for all actual information that a particular p-adic physics can give about real physics in classical sense. There are reasons to believe that real space-time sheets are in the general case characterized by integers n decomposing into products of powers of primes pi. One can expect that for pi-adic continuations the sets of intersection points are especially large and that these p-adic space-time surfaces can be said to provide a good discrete cognitive mimicry of the real space-time surface.
Adelic formula represents real number as product of inverse of its p-adic norms. This raises the hope that taken together these intersections could allow to determine the real surface and thus classical physics to a high degree. This idea generalizes to quantum context too.
The actual construction of the algebraic continuation from a subset of rational points is of course something which cannot be done in practice and this is not even necessary since much more elegant approach is possible.
2. Hierarchy of algebraic physics
One of the basic hypothesis of quantum TGD is that it is possible to define exponent of Kähler action in terms of fermionic determinants associated with the modified Dirac operator derivable from a Dirac action related super-symmetrically to the Kähler action.
If this is true, a very elegant manner to define hierarchy of physics in various algebraic extensions of rational numbers and p-adic numbers becomes possible. The observation is that the continuation to various p-adic numbers fields and their extensions for the fermionic determinant can be simply done by allowing only the eigenvalues which belong to the extension of rationals involved and solve field equations for the resulting Kähler function. Hence a hierarchy of fermionic determinants results. The value of the dynamical Planck constant characterizes in this approach the scale factor of the M4 metric in various number theoretical variants of the imbedding space H=M4× CP2 glued together along subsets of rational points of H. The values of hbar are determined from the requirement of quantum criticality meaning that Kähler coupling strength is analogous to critical temperature.
In this approach there is no need to restrict the imbedding space points to the algebraic extension of rationals and to try to formulate the counterparts of field equations in these discrete imbedding spaces.
3. p-Adic short range physics codes for long range real physics and vice versa
One should be able to construct global solutions of field equations numerically or by engineering them from the large repertoire of known exact solutions. This challenge looks formidable since the field equations are extremely non-linear and the failure of the strict non-determinism seems to make even in principle the construction of global solutions impossible as a boundary value problem or initial value problem.
The hope is that short distance physics might somehow code for long distance physics. If this kind of coding is possible at all, p-adicity should be crucial for achieving it. This suggests that one must articulate the question more precisely by characterizing what we mean with the phrases "short distance" and "long distance". The notion of short distance in p-adic physics is completely different from that in real physics, where rationals very close to each other can be arbitrary far away in the real sense, and vice versa. Could it be that in the statement "Short length scale physics codes for long length scale physics" the attribute "short"/"long" could refer to p-adic/real norm, real/p-adic norm, or both depending on the situation?
The point is that rational imbedding space points very near to each other in the real sense are in general at arbitrarily large distances in p-adic sense and vice versa. This observation leads to an elegant method of constructing solutions of field equations.
Some comments about the construction are in order.
4. p-Adic length scale hypothesis
Approximate p1-adicity implies also approximate p2-adicity of the space-time surface for primes p≈ p1k. p-Adic length scale hypothesis indeed states that primes p≈ 2k are favored and this might be due to simultaneous p≈ 2k- and 2-adicity. The long range fractal correlations in real space-time implied by 2-adicity would indeed resemble those implied by p≈ 2k and both p≈ 2k-adic and 2-adic space-time sheets have larger number of common points with the real space-time sheet.
If the scaling factor λ of hbar appearing in the dark matter hierarchy is in good approximation λ=211 also dark matter hierarchy comes into play in a resonant manner and dark space-time sheets at various levels of the hierarchy tend to have many intersection points with each other.
5. Does cognition automatically solve real field equations in long length scales?
In TGD inspired theory of consciousness p-adic space-time sheets are identified as space-time correlates of cognition. Therefore our thoughts would have literally infinite size in the real topology if p-adics and reals correspond to each other via common rationals.
The cognitive solution of field equations in very small p-adic region would solve field equations in real sense in a discrete point set in very long real length scales. This would allow to understand why the notions of Universe and infinity are a natural part of our conscious experience although our sensory input is about an infinitesimally small region in the scale of universe.
The idea about Universe performing mimicry at all possible levels is one of the basic ideas of TGD inspired theory of consciousness. Universe could indeed understand and represent the long length scale real dynamics using local p-adic physics. The challenge would be to make quantum jumps generating p-adic surfaces having large number of common points with the real space-time surface. We are used to call this activity theorizing and the progress of science towards smaller real length scales means progress towards longer length scales in p-adic sense. Also real physics can represent p-adic physics: written language and computer represent examples of this mimicry.
For more details see the chapter TGD as a Generalized Number Theory I: p-Adicization Program.