## Sunday, July 31, 2022

### Wheels and quantum arithmetics

Gary Ehlenberg gave a link to a Wikipedia article telling of Wheel theory (this) and said that he now has a name for what he has been working with. I am sure that this kind of adventure is a wonderful mathematical experience.

I looked at the link and realized that it might be very relevant to the TGD inspired idea about quantum arithmetics (see this).

I understood that Wheel structure is special in the sense that division by zero is well defined and multiplication by zero gives a non-vanishing result. The wheel of fractions, discussed in the Wikipedia article as an example of wheel structure, brings into mind a generalization of arithmetics and perhaps even of number theory to its quantum counterpart obtained by replacing + and - with direct sum ⊕ and tensor product &otimes: for irreps of finite groups with trivial representation as multiplicative unit: Galois group is the natural group in TGD framework.

One could also define polynomial equations for the extension of integers (multiples of identity representation) by irreps and solve their roots.

This might allow us to understand the mysterious McKay correspondence. McKay graph codes for the tensor product structure for irreps of a finite group, now Galois group. For subgroups of SL(n,C), the graphs are extended Dynkin diagrams for affine ADE groups.

Could wheel structure provide a more rigorous generalization of the notion of the additive and multiplicative inverses of the representation somehow to build quantum counterparts of rationals, algebraic numbers and p-adics and their extensions?

1. One way to achieve this is to restrict consideration to the quantum analogs of finite fields G(p,n): + and x would be replaced with ⊕ and ⊗ obtained as extensions by the irreps of the Galois group in TGD picture. There would be quantum-classical correspondence between roots of quantum polynomials and ordinary monic polynomials.
2. The notion of rational as a pair of integers (now representations) would provide at least a formal solution of the problem, and one could define non-negative rationals.

p-Adically one can also define quite concretely the inverse for a representation of form R=1 ⊕ O(p), where O(p) is proportional to p (p-fold direct sum) of a representation, as a geometric series.

3. Negative integers and rationals pose a problem for ordinary integers and rationals: it is difficult to imagine what direct sum of -n irreps could mean.

The definition of the negative of representation could work in the case of p-adic integers: -1 = (p-1)⊗ (1 ⊕ p*1 ⊕ p^2*1 ⊕...) would be generalized by replacing 1 with trivial representation. Infinite direct sum would be obtained but it would converge rapidly in p-adic topology.

4. Could 1/pn make sense in the Wheel structure so that one would obtain the analog of a p-adic number field? The definition of rationals as pairs might allow this since only non-negative powers of p need to be considered. p would represent zero but multiplication by p would give a non-vanishing result.
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.