*resp.*elementary particle level could correspond to the difference between L-functions and polynomials.

This vision inspires the question whether the generalization of the number theoretic view of TGD so that besides rational polynomials (subject to some restrictions) also L-functions, which have a nice physical interpretation if RH holds true for them, can be defined via their roots 4-surfaces in M^{8}_{c} and by M^{8}-H duality 4-surfaces in H. Both conformal confinement (in weak and strong form) and Galois confinement (having also weak and strong form) support the view that L-functions are Langlands duals of the partition functions defining quantum states.

If L functions indeed appear as a generalization of polynomials and define space-time surfaces, there must be a very deep reason for this.

- The key idea of computationalism is that computers can emulate/mimic each other. Universe should be able to emulate itself. Could WCW level and space-time level mimic each other? If this were the case, it could take place via QCC. If so, it should be possible to assign to a quantum state a space-time surface as its classical space-time correlate and vice versa.
- There are several space-time surfaces with a given Galois group but fixing the polynomial P fixes the space-time surface. An interesting possibility is that the observed classical space-time corresponds to superposition of space-time surfaces with the same discretization defined by the extension defined by the polynomial P. If so, the superposition of space-time surfaces would be effectively absent in the measurement resolution used and the quantum world would look classical.
- A given polynomial P fixes the mass shells H
^{3}⊂ M^{4}⊂ M^{8}but does not fix the space-time surface X^{4}completely since the polynomial hypothesis says nothing about the intersections of X^{4}with H^{3}defining 3-surfaces. The associativity hypothesis for the normal space of X^{4}⊂ M^{8}(see this and this) implies holography, which fixes X^{4}to a high degree for a given X^{3}. Holography is not expected to be completely deterministic: this non-determinism is proposed to serve as a correlate for intentionality.If space-time has boundaries, the boundaries X

^{2}of X^{3}⊂ H^{3}could be ends of light-like 3-surfaces X^{3}_{L}(see this). An attractive idea is that they are hyperbolic manifolds or pieces of a tessellation defined by a hyperbolic manifold as the analog of a unit cell (see this). The ends X^{2}of these 3-surfaces at the boundaries of CD would define partonic 2-surfaces.By quantum criticality of the light-like 3-surfaces satisfying det(-g

_{4})=0 (see this), their time evolution is not expected to be completely unique. If the extended conformal invariance of 3-D light-like surfaces is broken to a subgroup with conformal weights, which are multiples of integer n the conformal algebra defines a non-compact group serving as a reductive group allowing extensions of irreps of Galois group to its representations.One can also consider space-time surfaces without boundaries. They would define coverings of M

^{4}and there would be several overlapping projections to H^{3}, which would meet along 2-D surfaces as analogies of boundaries of 3-space. Also in this case, the idea that the X^{3}is a hyperbolic 3-manifold is attractive. - Quantum TGD involves a general mechanism reducing the infinite-D symmetry groups to finite-D groups, which has an interpretation in terms of finite measurement resolution (see this) describable both in terms of inclusions of hyperfinite factors of type II
_{1}and inclusions of extensions of rationals inducing inclusions of cognitive representations. One can also consider an interpretation in terms of symmetry breaking.This reduction means that the conformal weights of the generators of the Lie-algebras of these groups have a cutoff so that radial conformal weight associated with the light-like coordinate of δ M

^{4}_{+}is below a maximal value n_{max}. The generators with conformal weight n>n_{max}and their commutators with the entire algebra would act like a gauge algebra, whereas for n≤ n_{max}they generate genuine symmetries. The alternative interpretation is that the gauge symmetry breaks from n_{max}=0 to n_{max}>0 by transforming to dynamical symmetry.Note that the gauge conditions for the Virasoro algebra and Kac-Moody algebra are assumed to have n

_{max}=0 so that a breaking of conformal invariance would be in question for n_{max}>0. - The natural expectation is that the representation of the Galois group for these space-time surfaces defines representations in various degrees of freedom in terms of the semi-direct products of the Langlands duals
^{L}G^{0}with the Galois group (here^{L}G^{0}denotes the connected component of Langlands dual of G). Semi-direct product means that the Galois group acts on the algebraic group G assignable to algebraic extension by affecting the matrix elements of the group element.There are several candidates for the group G (see this). G could correspond to a conformal cutoff A

_{n}of algebra A, which could be the super symplectic algebra SSA of δ M^{4}× CP_{2}, the infinite-D algebra I of isometries of δ M^{4}_{+}, or the algebra Conf extended conformal symmetries of δ M4_{+}. Also the extended conformal algebra and extended Kac-Moody type algebras of H isometries associated with the light-like partonic orbits can be considered. - One could assign to these representations modular forms interpreted as generalized partition functions, kind of complex square roots of thermodynamic partition functions. Quantum TGD can be indeed formally regarded as a complex square root of thermodynamics. This partition function could define a ground state for a space of zero energy state defined in WCW as a superposition over different light-like 3-surfaces.

- Could the quantum states at WCW level have classical space-time correlates as space-time surfaces, which would be defined by the L-functions associated with the modular forms assignable to finite-D representations of Galois group having a physical interpretation as partition functions?
- Could this give rise to a kind of feedback loop representing increasingly higher abstractions as space-time surfaces. This sequence could continue endlessly. This picture brings in mind the hierarchy of infinite primes (see this).
Many-sheeted space-time would represent a hierarchy of abstractions. The longer the scale of the space-time sheet the higher the level in the hierarchy.

If Riemann hypothesis (RH) is true and the roots of L-functions are algebraic numbers, L-functions are in many aspects like rational polynomials and motivate the idea that, besides rationals polynomials, also L-functions could define space-time surfaces as kinds of higher level classical representations of physics.

One concretization of Langlands program would be the extension of the representations of the Galois group to the polynomials P to the representations of reductive groups appearing naturally in the TGD framework. Elementary particle vacuum functionals are defined as modular invariant forms of Teichmüller parameters. Multiple residue integral is proposed as a manner to obtain L-functions defining space-time surfaces.

One challenge is to construct Riemann zeta and the associated ξ function and the Hadamard product leads to a proposal for the Taylor coefficients c_{k} of ξ(s) as a function of s(s-1). One would have c_{k}= ∑_{i,j}c_{k,ij}e^{i/k}e^{(-1)1/22πj/n}, c_{k,ij}∈ {0,\pm 1}. e^{1/k} is the hyperbolic analogy for a root of unity and defines a finite-D transcendental extension of p-adic numbers and together with n:th roots of unity powers of e^{1/k} define a discrete tessellation of the hyperbolic space H^{2}.

See the article Some New Ideas Related to Langlands Program *viz.* TGD or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

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