linking permutation group swapping with global phase changes, and quantum mechanics multi-particle analysis


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see minute 19

Susskind makes some more connections for us – THIS time between the two halves of the Turing manuscript on permutations, lie algebras, and norm contraction by normalizer subgroups generating operator-norms, operator-kets, and super-operators. Specifically, WE can now link up the “Swapping” nature of typical permutation groups that underlies the discussion in the first half of the paper with the theory probably implied by the discussion in the second half. Susskind’s elaboration of the relationship between swapping of particles in a multi-particle generalization of his topic.

 

Of course, permutation groups are prototypical programs – for 2-particle swapping instructions. Given an array of group elements, any two can be swapped according to the cycle “program”; and sequences of cycles form products (of cycles) that give one a prototypical programming language. As Turing well knew.

What we now figure, due to Susskind, is that such programs are having an effect in phase space. While on the one hand global phase makes no difference, on the other hand it introduces notions of degeneracy – in which multiple descriptions of the same thing exist. Much as in phase (change) signal modulation, sequences of phase changes act as the physical layer bearer for communication. The sequences of cycles are formulated according to particular algebras – imposing constraints on choices of “cycle programs” – that create orthogonal relationships between the alternative descriptions. Or, as we saw on re-reading a Forney paper on coset-codes yesterday, one so uses binary convolutional encoders to choose coset translates of a signal constellation out of which to engineer reliable coding channels.

We know that Turing was arguing, in his method for searching out non-exceptional subgroups of wheel wirings, that the solution  – as as product of cycles – would itself exhibit only certain swaps within a given cycle or between the product terms. There were two fundamental cycles within the geometry, automorphic translates of which formed variants of the “program” for walking the geometry (now with elements re-labelled), while still  fashioning a discrete hilbert space. The automorphism group and the semi-direct group theory constrained the swapping of elements labels within “programming cycles” that then described how to generate ‘leaf-node swapping’ that would create the p-adic distances he needed for his (small-l 2) inner product space.

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