A further insight to adelic physics comes from the possible physical interpretation of the L-functions appearing also in Langlands program (see this. The most important L-function would be generalization of Riemann zeta to extension of rationals. I have proposed several roles for ζ, which would be the simplest L-function assignable to rational primes, and for its zeros.
Suppose for a moment that more precise formulations of these physics inspired conjectures hold true and even that their generalization for extensions K/Q of rationals holds true. This would solve quite a portion of adelic physics! Not surprisingly, the generalization of zeta function was proposed already by Dedekind (see this).
- Riemann zeta itself could be identifiable as an analog of partition function for a system with energies given by logarithms of prime. In ZEO this function could be regarded as complex square root of thermodynamical partition function in accordance with the interpretation of quantum theory as complex square root of thermodynamics.
- The zeros of zeta could define the conformal weights for the generators of super-symplectic algebra so that the number of generators would be infinite. The rough idea - certainly not correct as such except at the limit of infinitely large CD - is that corresponding functions would correspond to functions of radial light-like coordinate rM of light-cone boundary (boundary of causal diamond) of form (rM/r0)sn, sn=1/2+iy, sn would be radial conformal weight. Periodic boundary conditions for CD do not allow all possible zeros as conformal weights so that for given CD only finite subset corresponds to generators of supersymplectic algebra. Conformal confinement would hold true in the sense that the sum sn for physical states would be integer. Roots and their conjugates should appear as pairs in physical states.
- On basis of numerical evidence Dyson (see this) has conjectured that the Fourier transform for the set formed by zeros of zeta consists of primes so that one could regard zeros as one-dimensional quasi-crystal. This hypothesis makes sense if the zeros of zeta decompose into disjoint sets such that each set corresponds to its own prime (and its powers) and one has piy= Um/n=exp(i2π m/n) (see the appendix of this). This hypothesis is motivated by number theoretical universality.
- I have considered the possibility (see this) that the inverse of the electro-weak U(1) coupling constant for a gauge field assignable to the Kähler form of CP2 corresponds to poles of the fermionic zeta ζF(s)= ζ(s)/ζ(2s) coming from sn/2 (denominator) and pole at s=1 (numerator) zeros of zeta assignable to rational primes. Here one can consider scaling of argument of ζF(s). More general coupling constant evolutions could correspond to ζF(m(s)) where m(s)= (as+b)/(cs+d) is Möbius transformation performed for the argument mapping upper complex plane to itself so that a,b,c,d are real and also rational by number theoretical universality.
From Wikipedia one learns that for Galois extensions L/K the ratio ζL/ζK is so called Artin L-function of the regular representation (group algebra) of Galois group factorizing in terms of irreps of Gal(L/K) is holomorphic (no poles!) so that ζL must have also the zeros of ζK. This holds in the special case K=Q. Therefore extension of rationals can only bring new zeros but no new poles!
- The definition of Dedekind zeta function ζK relies on the product representation and analytic continuation allows to deduce ζK elsewhere. One has a product over prime ideals of K/Q of rationals with the factors 1/(1-p-s) associated with the ordinary primes in Riemann zeta replaced with the factors X(P) =1/(1-NK/Q(P)-s), where P is prime for the integers O(K) of extension and NK/Q(P) is the norm of P in the extension. In the region s>1 where the product converges, ζK is non-vanishing and s=1 is a pole of ζK. The functional identifies of ζ hold true for ζK as well. Riemann hypothesis is generalized for ζK.
- It is possible to interpret ζK in terms of a physical picture. By the general results (see this) one NK/Q(P)= pr, r>0 integer. One can deduce for r a general expression. This implies that one can arrange in ζK all primes P for which the norm is power or given p in the same group. The prime ideals p of ordinary integers decompose to products of prime ideals P of the extension: one has p= ∏r=1g Prer, where er is so called ramification index. One can say that each factor of ζ decomposes to a product of factors associated with corresponding primes P with norm a power of p. In the language of physics, the particle state represented by p decomposes in an improved resolution to a product of many-particle states consisting of er particles in states Pr, very much like hadron decomposes to quarks.
The norms of NK/Q(Pr) = pdr satisfy the condition ∑r=1g dr er= n. Mathematician would say that the prime ideals of Q modulo p decompose in n-dimensional extension K to products of prime power ideals Prer and that Pr corresponds to a finite field G(p,dr) with algebraic dimension dr. The formula ∑r=1g dr er = n reflects the fact the dimension n of extension is same independent of p even when one has g<n and ramification occurs.
Physicist would say that the number of degrees of freedom is n and is preserved although one has only g<n different particle types with er particles having dr internal degrees of freedom. The factor replacing 1/(1-p-s) for the general prime p is given by ∏r=1g 1/(1-p-erdrs).
- There are only finite number of ramified primes p having er>1 for some r and they correspond to primes dividing the so called discriminant D of the irreducible polynomial P defining the extension. D mod p obviously vanishes if D is divisible by p. For second order polynomials P=x2+bx+c equals to the familiar D=b2-4c and in this case the two roots indeed co-incide. For quadratic extensions with D= b2-4c>0 the ramified primes divide D.
Remark: Resultant R(P,Q) and discriminant D(P)= R(P,dP/dx) are elegant tools used by number theorists to study extensions of rationals defined by irreducible polynomials. From Wikipedia articles one finds an elegant proof for the facts that R(P,Q) is proportional to the product of differences of the roots of P and Q, and D to the product of squares for the differences of distinct roots. R(P,Q)=0 tells that two polynomials have a common root. D mod p=0 tells that polynomial and its derivative have a common root so that there is a degenerate root modulo p and the prime is indeed ramified. For modulo p reduction of P the vanishing of D(P) mod p follows if D is divisible by p. There exist clearly only a finite number of primes of this kind.
Most primes are unramified. If one has maximum number n of factors in the decomposition and er=1, maximum splitting of p occurs. The factor 1/(1-p-s) is replaced with its n:th power 1/(1-p-s)n. The geometric interpretation is that space-time sheet is replaced with n-fold covering and each sheet gives one factor in the power. It is also possible to have a situation in which no splitting occurs and one as er=1 for one prime Pr=p. The factor is in this case equal to 1/(1-p-ns).
See the article p-Adization and adelic physics or the chapter Philosophy of adelic physics.
- This result is quite far reaching if one accepts the hypothesis about super-symplectic conformal weights as zeros of ζK and the conjecture about coupling constant evolution. In the case of ζF,K this means new poles meaning new conformal weights due to increased complexity and a modification of the conjecture for the coupling constant evolution due to new primes in extension. The outcome looks physically sensible.
- Quadratic field Q(m1/2) serves as example. Quite generally, the factorization of rational primes to the primes of extension corresponds to the factorization of the minimal polynomial for the generating element θ for the integers of extension and one has p= Piei, where ei is ramification index. The norm of p factorizes to the produce of norms of Piei.
Rational prime can either remain prime in which case x2-m does not factorize mod p, split when x2-m factorizes mod P, or ramify when it divides the discriminant of x2-m = 4m. From this it is clear that for unramfied primes the factors in ζ are replaced by either 1/(1-p-s)2 or 1/(1-p-2s)= 1/(1-p-s)(1+p-s). For a finite number of unramified primes one can have something different.
For Gaussian primes with m=-1 one has er=1 for p mod 4=3 and er=2 for p=~mod~4=1. zK therefore decomposes into two factors corresponding to primes p ~mod~4=3 and p mod 4=1. One can extract out Riemann zeta and the remaining factor
∏p mod 4=3 1/(1-p-s) × ∏p mod 4=1 1/(1+p-s)
should be holomorphic and without poles but having possibly additional zeros at critical line. That ζK should possess also the poles of ζ as poles looks therefore highly non-trivial.