Lie groups, algebras, Angular Momentum and Turings generators


One of the things we have been learning about is the “degeneracy-busting” power of symmetry operators. Or, should there be multiple eigenkets (certain states, within a superposition space) that have the same eigenvalue (and the same energy in particular, for the energy operator), we can use lie algebras to bust these correlations.

In crypto, this is what we would want to be doing! Think of it has busting the hidden correlations hiding within the data model, that allow eigenvectors to be considered “same”. We want them all “different”.

image

http://turingarchive.org/viewer/?id=133&title=05a

In the section of his manuscript on permutations concerning with using search processes to find exceptional groups (no annoying cycles, within the group), Turing wishes to establish that the desired group property holds, once the generators are spread across several enigma wheels in a cage, each contributing a (tensored) generator term to a multi-component wave function. And let’s not forget his big picture argument, either: that the normal subgroup (acting as generator) is the SYMMETRY group (or the alternating group). Hmm..

At this point, we think in the terms taught to us by the physicists. That is, our symmetry-group “exists” to help “generate” sequences of symmetry-operations, that bust the degeneracy. And, we recall how a lie algebra – a product of two generator terms (each being a functional symmetry-operation element of the group) has appropriate properties – that don’t interfere with rotational invariance of certain underlying systems –  but still allow certain classes of calculation. In particular, we saw one class of calculation that enabled enumeration through the ordered set of eigenkets of a certain eigenvalue.

An important element of the “algorithm” setting up the lie algebra consisted of calculating each of the new (non-commutative) generators, and showing that the product of any two – be they original or calculated or some linear combination – was closed. That is, it identified another of the generator set.

We can think of this generate-the-generators phase as a way of calculating an instance of the degeneracy-busting algorithm (or custom-operator). Since the net result is a function of the group used in the baseline, we have n of them.

what we notice about the case Turing diagrams, above is that we see a example of two generator-operations (that induce an actions). If we continue the metaphor that the (α0 α1) cycle is the “coin” of the quantum random walk (whose impetus induces actions) and the (α2 α-2) are the eigenvectors of the energy operator, then we note how a geometric “rotation” gives (α-3 α-2) as the new impulsing/clocking mechanism. (Just rotate the largest field-numbering line clockwise a bit). And if you now reflect, the second cycle becomes (α-1 α-5).

This seems just a little unlikely to be concidence!

He seems to be showing, without saying it as usual, that his exceptional group has invariant properties, such that one can rotate and reflect…

I think we should go read 1930s book on lie algebra, and note how “exceptional” was used (and for what purposes in those early gedanken experiments).

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Computer Programmer who often does network administration with focus on security servers. Very strong in Microsoft Azure cloud!
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