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Physics as a Generalized Number Theory
Note: Newest contributions are at the top!
Riemann Hypothess (RH) states that the non-trivial (critical) zeros of zeta lie at critical line s=1/2. It would be interesting to know how many physical justifications for why this should be the case has been proposed during years. Probably this number is finite, but very large it certainly is. In Zero Energy Ontology (ZEO) forming one of the cornerstones of the ontology of quantum TGD, the following justification emerges naturally. I represented it in the answer to previous posting, but there was stupid error in the answer so that I represent the corrected argument here.
The understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During years I have made several attempts to understand coupling evolution.
See the new chapter Does Riemann Zeta Code for Generic Coupling Constant Evolution? or the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?.
Number Theoretic Universality (NTU) in the strongest form says that all numbers involved at "basic level" (whatever this means!) of adelic TGD are products of roots of unity and of power of a root of e defining finite-dimensional extensions of p-adic numbers (ep is ordinary p-adic number). This is extremely powerful physics inspired conjecture with a wide range of possible mathematical applications.
Super-symplectic conformal weights and Riemann zeta
The existence of WCW geometry highly nontrivial already in the case of loop spaces. Maximal group of isometries required and is infinite-dimensional. Super-symplectic algebra is excellent candidate for the isometry algebra. There is also extended conformal algebra associated with δ CD. These algebras have fractal structure. Conformal weights for isomorphic subalgebra n-multiples of those for entire algebra. Infinite hierarchy labelled by integer n>0. Generating conformal weights could be poles of fermionic zeta ζF. This demands n>0. Infinite number of generators with different non-vanishing conformal weight with other quantum numbers fixed. For ordinary conformal algebras there are only finite number of generating elements (n=1).
If the radial conformal weights for the generators of g consist of poles of ζF, the situation changes. ζF is suggested by the observation that fermions are the only fundamental particles in TGD.
Are the zeros of Riemann zeta number theoretically universal?
Dyson's comment about Fourier transform of Riemann Zeta is very interesting concerning NTU for Riemann zeta.
where u(p) is a root of unity that is ya(p)=2π (m(a)+r(p))/log(p) and forming a subset of a lattice with a lattice constant y0=2π/log(p), which itself need not be a zero.
In terms of stationary phase approximation the zeros ya(p) associated with p would have constant stationary phase whereas for ya(pi≠ p)) the phase piya(pi) would fail to be stationary. The phase eixy would be non-stationary also for x≠ log(pk) as function of y.
What this speculative picture from the point of view of TGD?
The number theoretic vision has begun to show its power. The basic hierarchies of quantum TGD would reduce to a hierarchy of algebraic extensions of rationals and the parameters - such as the degrees of the irreducible polynomials characterizing the extension and the set of ramified primes - would characterize quantum criticality and the physics of dark matter as large heff phases. The identification of preferred p-adic primes as remified primes of the extension and generalization of p-adic length scale hypothesis as prediction of NMP are basic victories of this vision (see this and this).
By strong form of holography the parameters characterizing string world sheets and partonic 2-surfaces serve as WCW coordinates. By various conformal invariances, one expects that the parameters correspond to conformal moduli, which means a huge simplification of quantum TGD since the mathematical apparatus of superstring theories becomes available and number theoretical vision can be realized. Scattering amplitudes can be constructed for a given algebraic extension and continued to various number fields by continuing the parameters which are conformal moduli and group invariants characterizing incoming particles.
There are many un-answered and even un-asked questions.
1. Some basic notions
Some basic facts about extensions are in order. I emphasize that I am not a specialist.
1.1. Basic facts
The algebraic extensions of rationals are determined by roots of polynomials. Polynomials be decomposed to products of irreducible polynomials, which by definition do not contain factors which are polynomials with rational coefficients. These polynomials are characterized by their degree n, which is the most important parameter characterizing the algebraic extension.
One can assign to the extension primes and integers - or more precisely, prime and integer ideals. Integer ideals correspond to roots of monic polynomials Pn(x)=xn+..a0 in the extension with integer coefficients. Clearly, for n=0 (trivial extension) one obtains ordinary integers. Primes as such are not a useful concept since roots of unity are possible and primes which differ by a multiplication by a root of unity are equivalent. It is better to speak about prime ideals rather than primes.
Rational prime p can be decomposed to product of powers of primes of extension and if some power is higher than one, the prime is said to be ramified and the exponent is called ramification index. Eisenstein's criterion states that any polynomial Pn(x)= anxn+an-1xn-1+...a1x+ a0 for which the coefficients ai, i<n are divisible by p and a0 is not divisible by p2 allows p as a maximally ramified prime. mThe corresponding prime ideal is n:th power of the prime ideal of the extensions (roughly n:th root of p). This allows to construct endless variety of algebraic extensions having given primes as ramified primes.
Ramification is analogous to criticality. When the gradient potential function V(x) depending on parameters has multiple roots, the potential function becomes proportional a higher power of x-x0. The appearance of power is analogous to appearance of higher power of prime of extension in ramification. This gives rise to cusp catastrophe. In fact, ramification is expected to be number theoretical correlate for the quantum criticality in TGD framework. What this precisely means at the level of space-time surfaces, is the question.
1.2 Galois group as symmetry group of algebraic physics
I have proposed long time ago that Galois group acts as fundamental symmetry group of quantum TGD and even made clumsy attempt to make this idea more precise in terms of the notion of number theoretic braid. It seems that this notion is too primitive: the action of Galois group must be realized at more abstract level and WCW provides this level.
First some facts (I am not a number theory professional, as the professional reader might have already noticed!).
Galois group acts in the space of integers or prime ideals of the algebraic extension of rationals and it is also physically attractive to consider the orbits defined by ideals as preferred geometric structures. If the numbers of the extension serve as parameters characterizing string world sheets and partonic 2-surfaces, then the ideals would naturally define subsets of the parameter space in which Galois group would act.
The action of Galois group would leave the space-time surface invariant if the sheets co-incide at ends but permute the sheets. Of course, the space-time sheets permuted by Galois group need not co-incide at ends. In this case the action need not be gauge action and one could have non-trivial representations of the Galois group. In Langlands correspondence these representation relate to the representations of Lie group and something similar might take place in TGD as I have indeed proposed.
Remark: Strong form of holography supports also the vision about quaternionic generalization of conformal invariance implying that the adelic space-time surface can be constructed from the data associated with functions of two complex variables, which in turn reduce to functions of single variable.
If this picture is correct, it is possible to talk about quantum amplitudes in the space defined by the numbers of extension and restrict the consideration to prime ideals or more general integer ideals.
2. How new degrees of freedom emerge for ramified primes?
How the new discrete degrees of freedom appear for ramified primes?
3. About the physical interpretation of the parameters characterizing algebraic extension of rationals in TGD framework
It seems that Galois group is naturally associated with the hierarchy heff/h=n of effective Planck constants defined by the hierarchy of quantum criticalities. n would naturally define the maximal order for the element of Galois group. The analog of singular covering with that of z1/n would suggest that Galois group is very closely related to the conformal symmetries and its action induces permutations of the sheets of the covering of space-time surface.
Without any additional assumptions the values of n and ramified primes are completely independent so that the conjecture that the magnetic flux tube connecting the wormhole contacts associated with elementary particles would not correspond to very large n having the p-adic prime p characterizing particle as factor (p=M127=2127-1 for electron). This would not induce any catastrophic changes.
TGD based physics could however change the situation and reduce number theoretical degrees of freedom: the intuitive hypothesis that p divides n might hold true after all.
The argument would explain the existence of preferred p-adic primes. It does not yet explain p-adic length scale hypothesis stating that p-adic primes near powers of 2 are favored. A possible generalization of this hypothesis is that primes near powers of prime are favored. There indeed exists evidence for the realization of 3-adic time scale hierarchies in living matter (see this) and in music both 2-adicity and 3-adicity could be present, this is discussed in TGD inspired theory of music harmony and genetic code (see this).
The weak form of NMP might come in rescue here.
A long-standing question has been the origin of preferred p-adic primes characterizing elementary particles. I have proposed several explanations and the most convincing hitherto is related to the algebraic extensions of rationals and p-adic numbers selecting naturally preferred primes as those which are ramified for the extension in question.
See the chapter Unified Number Theoretic Vision .
The twistor approach to TGD require that the expression of light-likeness of M4 momenta in terms of twistors generalizes to 8-D case. The light-likeness condition for twistors states that the 2× 2 matrix representing M4 momentum annihilates a 2-spinor defining the second half of the twistor. The determinant of the matrix reduces to momentum squared and its vanishing implies the light-likeness. This should be generalized to a situation in one has M4 and CP2 twistor, which are not light-like separately but light-likeness in 8-D sense holds true (allowing massive particles in M4 sense and thus generalization of twistor approach for massive particles).
The case of M8=M4× E4
M8-H duality suggests that it might be useful to consider first the twistorialiation of 8-D light-likeness first the simpler case of M8 for which CP2 corresponds to E4. It turns out that octonionic representation of gamma matrices provide the most promising formulation.
In order to obtain quadratic dispersion relation, one must have 2× 2 matrix unless the determinant for the 4× 4 matrix reduces to the square of the generalized light-likeness condition.
The case of M8=M4× CP2
What about twistorialization in the case of M4× CP2? The introduction of wave functions in the twistor space of CP2 seems to be enough to generalize Witten's construction to TGD framework and that algebraic variant of twistors might be needed only to realize quantum classical correspondence. It should correspond to tangent space counterpart of the induced twistor structure of space-time surface, which should reduce effectively to 4-D one by quaternionicity of the space-time surface.
To sum up, the generalization of the notion of twistor to 8-D context allows description of massive particles using twistors but requires that octonionic Dirac equation is introduced. If one requires that octonionic and ordinary description of Dirac equation are equivalent, the description is possible only at surfaces having at most 1-D CP2 projection - geodesic circle for the most stringent option. The boundaries of string world sheets are such surfaces and also string world sheets themselves if they have 1-D CP2 projection, which must be geodesic circle if also induce gauge potentials are required to vanish. In spirit with M8-H duality, string boundaries give rise to classical M8 twistorizalization analogous to the standard M4 twistorialization and generalize 4-momentum to massless 8-momentum whereas imbedding space spinor harmonics give description in terms of four-momentum and color quantum numbers. One has SO(4)-SU(3) duality: a wave function in the space of 8-momenta corresponds to SO(4) description of hadrons at low energies as opposed to that for quarks at high energies in terms of color. The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle (EP) in the most concrete form: gravitational four-momentum equals to inertial four-momentum. EP for internal quantum numbers is clearly more delicate. In twistorialization also helicity is brought and for CP2 degrees of freedom M8 helicity means that electroweak spin is described in terms of helicity.
Biologists have a principle known as "ontogeny recapitulates phylogeny" (ORP) stating that the morphogenesis of the individual reflects evolution of the species. The principle seems to be realized also in theoretical physics - at least in TGD Universe. ORP would now say that the evolution of theoretical physics via the emergence of increasingly complex notion of particle reflects the structure physics itself. Point like particles are really there as points at partonic 2-surfaces carrying fermion number: their 1-D orbits correspond to the boundaries of string world sheets; 2-D hyper-complex string world sheets in flat space (M4× S1) are there and carry induced spinors; also complex (or co-complex) partonic 2-surfaces (Euclidian string world sheets) and carry particle numbers; 3-D space-like surfaces at the ends of causal diamonds (CDs) and the 3-D light-like orbits of partonic 2-surfaces are there; 4-D space-time surfaces are there as quaternionic or co-quaternionic sub-manifolds of 8-D octonionic imbedding space: there the hierarchy ends since there are no higher-dimensional classical number fields. ORP would thus also realize evolution of mathematics at the level of physics.
The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle in the most concrete form: gravitational four-momentum equals to inertial four-momentum.
See the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts or the article Classical part of the twistor story.