# Category Theory and Quantum Mechanics

I am wondering if particle interactions in quantum theory can be modeled as a morphism between $2$ categories. My reasoning is that since the states of particles are modeled as vectors in a Hilbert space, given two Hilbert spaces, call them $A$ and $B$, could the interaction be described a morphism $f$ between $A$ and $B$. Suppose the category $C$ is given by $f:A\to B$ Where $A$ is the Hilbert space containing the states of the $2$ particles $\psi_1$ and $\psi_2$ and $B$ is the Hilbert space containing the states of the particles $\phi_1$ and $\phi_2$ after the interaction. So can particle interactions be understood in a category theoretic way, specifically the category of Hilbert Spaces.

-

## 1 Answer

There is an excellent expository article on this by John Baez:

http://arxiv.org/abs/quant-ph/0404040

There is another article by Baez and Lauda on the more general topic of n-categories and their role in physics which is worth checking out:

http://arxiv.org/abs/0908.2469

Of course, for details you can check the references therein, but here are a few of note (in my opinion):

http://arxiv.org/abs/0808.1023

http://arxiv.org/abs/math/0601458

http://arxiv.org/abs/0706.0711

Also, as quantum mechanics can be viewed as a 1-dimensional quantum field theory, the categorical approach to QFT might be of interest to you. In that regard, these references might be of interest:

http://arxiv.org/abs/q-alg/9503002

http://www-math.mit.edu/~lurie/papers/cobordism.pdf

One last thing to go with the theme of (higher) categorical physics: a few months back Urs Schreiber posted this article, which claims to have solved Hilbert's sixth problem (i.e. axiomatizing physics) by using the language of cohesive $\infty$-topoi. It's a massive article that I've only read through a small portion of, but it definitely seems interesting.

-
Thanks :) exactly what I'm looking for. – user118822 Jan 5 '14 at 1:15
Don't mention it. And, actually, now that I think of it a related idea that I find very interesting in this paper (the thesis of one of Baez's students): arxiv.org/abs/1106.4068. The basic idea is that classical mechanics is described mathematically using symplectic geometry, and so in the 60's a quantization procedure was developed, and in this paper the same thing is developed for the field theory analogue of symplectic geometry, multisymplectic geometry. This approach uses a lot of (n-)categorical language for describing quantum theories, so that is why I mention it. – Ralph Mellish Jan 5 '14 at 1:23