Arcadian Functor

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Marni D. Sheppeard

Friday, October 31, 2008

M Theory Lesson 235

Once we have selected two different bases in the vector space $\mathbb{C}^{n}$, there is a discrete Fourier transform map between them, represented by an $n \times n$ matrix. For example, when $n = 3$, the Fourier arrows form a triangle. In general, they form an $n$-simplex.

Recall that a simplicial set is a functor from the collection of ordinals $N$, where each $N$ is drawn as an $N+1$-simplex, into the category Set. In other words, the functor first picks out $n$ sets to put at the vertices of each $n$-simplex, for all $n$. We would like to replace the ordinals by MUB simplices. This is natural, because at a fixed $n$ the vertices are labelled by q-numbers of dimension $n$. For example, the MUB triangle of $2 \times 2$ Pauli operators replaces the set $\{ 0,1,2 \}$ of the classical ordinal 2.

This construction uses numbers belonging to finite dimensional vector spaces, rather than sets. But in M Theory it is OK to think of the category of vector spaces as a quantum analogue of Set, and in quantum physics, why would we wish to calculate using classical numbers?

1 Comments:

Blogger Matti Pitkanen said...

Dear Kea,

the root of unity property makes possible the realization of number theoretical universality (fusing p-adic and real physics together).

As I told, the structure constants of the symplectic fusion algebras come as products of roots of unity: C_klm= U_kU_lU_m.
N-D symplectic algebras form an algebra themselves. One obtains a new N-D algebra from two N-D algebras by multiplying the elements one N-D algebra permuted in arbitary manner. Phi_(3,i)= phi_(1,i)phi_(2,P(i)).
One can also identify prime algebras corresponding to N=prime and speak about factor decomposition of fusion algebras.

For the geometric representations of the symplectic algebras these phases have interpretation as analogies for phase factors associate with plane waves of various kinds: momentum eigenstate, spin eigen state, eigenstate of electroweak or color hypercharge and isospin. The geometric representations of fusion algebras are analogous to generalizations of spin networks in which edges carry also momentum and other quantum numbers.

This means that finite measurement resolution for these quantum numbers replaces masses, spin, etc.. by corresponding "plane waves" and conservation laws hold true only within measurement resolution state that the products of these plane wave factors are conserved in particle reactions.

The fact that roots of unity are in question means number theoretical universality so that the discretized n-point functions allow interpretation also as numbers of p-adic number fields (allowing appropriate algebraic extensions). In the case of momentum discretization means replacement of particle in box with particle in discretized box. In case of spin it means just quantization of spin so that quantization can be also understood purely number theoretically. One implication of fusion algebra is quantization of Kahler-magnetic fluxes for symplectic triangles.

The recent-I hope more or less final view as far error corrections are considered- about the realization of construction Feynman graphs in terms of fusion algebras can be found from my blog

October 31, 2008 4:24 PM  

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