Trying to fuse the basic mathematical ideas of quantum TGD to a single coherent whole

Abstract

The theoretical framework behind TGD involves several different strands and the goal is to unify them to a single coherent whole. TGD involves number theoretic and geometric visions about physics and $M^8-H$ duality, analogous to Langlands duality, is proposed to unify them. Also quantum classical correspondence (QCC) is a central aspect of TGD. One should understand both the $M^8-H$ duality and QCC at the level of detail.

The following mathematical notions are expected to be of relevance for this goal.

\begin{enumerate}

\item Von Neumann algebras, call them $M$, in particular hyperfinite factors of type $II_1$ (HFFs), are in a central role. Both the geometric and number theoretic side, QCC could mathematically correspond to the relationship between $M$ and its commutant $M'$.

For instance, symplectic transformations leave induced K\"ahler form invariant and various fluxes of K\"ahler form are symplectic invariants and correspond to classical physics commuting with quantum physics coded by the super symplectic algebra (SSA). On the number theoretic side, the Galois invariants assignable to the polynomials determining space-time surfaces are analogous classical invariants.

\item The generalization of ordinary arithmetics to quantum arithmetics obtained by replacing $+$ and $\times$ with $\oplus$ and $\otimes$ allows us to replace the notions of finite and p-adic number fields with their quantum variants. The same applies to various algebras.

\item Number theoretic vision leads to adelic physics involving a fusion of various p-adic physics and real physics and to hierarchies of extensions of rationals involving hierarchies of Galois groups involving inclusions of normal subgroups. The notion of adele can be generalized by replacing various p-adic number fields with the p-adic representations of various algebras.

\item The physical interpretation of the notion of infinite prime has remained elusive although a formal interpretation in terms of a repeated quantization of a supersymmetric arithmetic QFT is highly suggestive. One can also generalize infinite primes to their quantum variants. The proposal is that the hierarchy of infinite primes generalizes the notion of adele.

\end{enumerate}


The formulation of physics as K\"ahler geometry of the "world of classical worlds" (WCW) involves f 3 kinds of algebras $A$; supersymplectic isometries $SSA$ acting on $\delta M^4_+\times CP_2$, affine algebras $Aff$ acting on light-like partonic orbits, and isometries $I$ of light-cone boundary $\delta M^4_+$, allowing hierarchies $A_n$.


The braided Galois group algebras at the number theory side and algebras $\{A_n\}$ at the geometric side define excellent candidates for inclusion hierarchies of HFFs. $M^8-H$ duality suggests that $n$ corresponds to the degree $n$of the polynomial $P$ defining space-time surface and that the $n$ roots of $P$ correspond to $n$ braid strands at $H$ side. Braided Galois group would act in $A_n$ and hierarchies of Galois groups would induce hierarchies of inclusions of HFFs. The ramified primes of $P$ would correspond to physically preferred p-adic primes in the adelic structure formed by p-adic variants of $A_n$ with $+$ and $\times$ replaced with $\oplus$ and $\otimes$.