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Categorified algebra and quantum mechanics

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Jeffrey Morton

The process some call `categorification' consists of
interpreting set-theoretic structures in mathematics as derived from
category-theoretic structures. Examples include the interpretation of
N as the Burnside rig of the category of finite sets with
product and coproduct, and of N[x] in terms the
category of combinatorial species. This has interesting applications
to quantum mechanics, and in particular the quantum harmonic
oscillator, via Joyal's `combinatorial species', and a new
generalization called `stuff types' described by Baez and Dolan, which
are a special case of Kelly's `clubs'. Operators between stuff
types be represented as rudimentary Feynman diagrams for the
oscillator. In quantum mechanics, we want to represent states in an
algebra over the complex numbers, and also want our Feynman diagrams
to carry more structure than these `stuff operators' can do, and these
turn out to be closely related. We will describe a categorification
of the quantum harmonic oscillator in which the group of `phases' -
that is, U(1), the circle group - plays a special role. We describe
a general notion of `M-stuff types' for any monoid M, and see that
the case M = U(1) provides an interpretation of time evolution in the
combinatorial setting, as well as recovering the usual Feynman rules
for the quantum harmonic oscillator.

2000 MSC:
81P05, 05A15, 18D10, 18B40

*Theory and Applications of Categories,*
Vol. 16, 2006,
No. 29, pp 785-854.

http://www.tac.mta.ca/tac/volumes/16/29/16-29.ps

http://www.tac.mta.ca/tac/volumes/16/29/16-29.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/29/16-29.ps

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