Cool Cats
Bob Coecke (an FQXi awardee) is one of few category theorists working close to the foundations of quantum mechanics (although the numbers are increasing at an alarming rate). Today at The Cafe he provided a link to his new paper on the categorical structure of Spekkens' toy quantum mechanics, namely $\dagger$ symmetric monoidal with basis structure (just think of pictures of little boxes and ribbons, which are easy to move around). Mutually unbiased bases are central to the discussion.
In particular, the paper discusses the category of finite sets and relations, in which the two element set acts as a system of observables, capable of modelling much of the behaviour of a qubit, just as in M theory. It will be interesting to extend this observation to a richer arithmetic setting. These days, similarities between sets and Hilbert spaces always invoke the notion of a field with one element, over which a set is just a vector space. And the natural groups to study in association with this field are the braid groups, especially at roots of unity.
Oh wait. Gee, that's what we're using to unify particle mass triplets and mixing matrices. But Dr Mottle tells us that this is completely idiotic and cannot possibly work, so maybe we should just go outside and make snowmen.
In particular, the paper discusses the category of finite sets and relations, in which the two element set acts as a system of observables, capable of modelling much of the behaviour of a qubit, just as in M theory. It will be interesting to extend this observation to a richer arithmetic setting. These days, similarities between sets and Hilbert spaces always invoke the notion of a field with one element, over which a set is just a vector space. And the natural groups to study in association with this field are the braid groups, especially at roots of unity.
Oh wait. Gee, that's what we're using to unify particle mass triplets and mixing matrices. But Dr Mottle tells us that this is completely idiotic and cannot possibly work, so maybe we should just go outside and make snowmen.
posted by Kea | 8:36 AM
2 Comments:
Wow, the "Quantum diaries" post you refer to ran to 124 comments! Finding boundaries for a Higgs mass is exciting news nonetheless.
I looked the Higgs debate in Tommaso's blog. What surprises me was the dogmatic belief of Lubos that coupling constant evolution implies that the low energy physics must be something chaotic and unpredictable. Why should this be the case? Why not ask whether we might be missing something?
The formulation of p-adic coupling coupling constant evolution has demonstrated just the opposite. The basic underlying reason is that p-adic physics at short distances corresponds to real physics at very long distances: the mere continuity and smoothness in p-adic sense give extremely powerful constraints on real physics at long scales. There are also powerful number theoretic existence conditions involved: consider only the generalization of Boltzman weight to integer power of p quantizing p-adic temperatures to T=1/n appearing in mass calculations relying on p-adic thermodynamics for mass squared represented as scaling generator L_0.
Therefore the two (or actually very many) notions of nearness (p-adic for various primes and real) change the situation completely. Low energy physics ceases to be the dust bin containing the dirty things. Simple universal formulas based on p-adic fractality emerge. For instance, discretization of coupling constant evolution to half octaves in length scale and octaves in time scale brings in a hierarchy of mass scales coming as half octaves and p-adic primes near powers of 2 are strongly favored. Masses are precisely predictable, etc... The most important applications are in biology and one of the basic predictions is direct connection with biology and elementary particle physics via assignment of .1 second time scale to electron in zero energy ontology: alpha rhythm defines indeed a fundamental time scale in biology.
One particular implication related also to the problem of Higgs mass is that elementary particles can appear in several mass scales differing by a power of sqrt(2). Quarks do so in TGD based model for hadron masses. This explains also why neutrinos seem to appear in several mass scales. Also Higgs could appear in two mass scales as experiments giving two values of mass differing by a factor of 8 suggest: this I have discussed somewhere in my blog already earlier. The average of these masses would be around 170 GeV and is now excluded so that this picture gets further empirical support. Perhaps it is time to start thinking basics again instead of just taking averages!
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