Saturday, July 16, 2022

McKay Correspondence from Quantum Arithmetics Replacing Sum and Product with Direct Sum and Tensor Product?

I have already earlier worked with two questions inspired by TGD.
  1. The ideas related to topological quantum computation suggests that it might make sense to replace quantum states with representations of the Galois group or even the coefficient space of state space with a quantum analog of a number field with tensor product and direct sum replacing the multiplication and sum.

    Could one generalize arithmetics by replacing sum and product with direct sum ⊕ and tensor product ⊗ and consider group representations as analogs of numbers? Or could one replace the roots labelling states with representations? Or could even the coefficient field for state space be replaced with the representations? Could one speak about quantum variants of state spaces?

    Could this give a kind of quantum arithmetics or even quantum number theory and possibly also a new kind of quantum analog of group theory. If the direct sums are mapped to ordinary sums in quantum-classical correspondence, this map could make sense under some natural conditions.

  2. McKay graphs (quivers) have irreducible representations as nodes and characterize the tensor product rules for the irreps of finite groups. How general is the McKay correspondence relating these graphs to the Dynkin diagrams of ADE type affine algebras? Could it generalize from finite subgroups of SL(k,C), k=2,3,4 to those of SL(n,C) at least. Is there a deep connection between finite subgroups of SL(n,C), and affine algebras. Could number theory or its quantum counterpart provide insights to the problem?
In the TGD framework M8-H duality relates number theoretic and differential geometric views about physics: could it provide some understanding of this mystery? The proposal is that for cognitive representations associated with extended Dynkin diagrams (EEDs), Galois group Gal acts as Weyl group on McKay diagrams defined by irreps of the isotropy group GalI of given root of a polynomial which is monic polynomial but with roots replaced with direct sums of irreps of GalI. This could work for p-adic number fields and finite fields. One also ends up with a more detailed view about the connection between the hierarchies of inclusion of Galois groups associated with functional composites of polynomials and hierarchies of inclusions of hyperfinite factors of type II1 assignable to the representation of super-symplectic algebra.

See the article McKay Correspondence from Quantum Arithmetics Replacing Sum and Product with Direct Sum and Tensor Product? or the chapter with the same title.

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

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