## Suitable references for the the Stone-von Neumann Theorem

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Hi all,

I am working on a mathematical physics project now and I need to understand the Stone-von Neumann Theorem properly. Wikipedia says that it is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators'.

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If you are familiar with the representation theory, I will recommend  Kirillov : Elements of the theory of representations Mackey : The Theory of Unitary Group Representations  I haven't looked, but suspect that  Mackey : Unitary group representations in physics, probability, and number theory will have some enlightening comments. A toy model of this is looking at representation theory of $SL(2,\mathbb{R})$  on functions annihilated by the laplace beltrami operator. Any books on spectral theory of automorphic form will talk about it.  – isildur May 10 2011 at 1:13

## 4 Answers

von Neumann's original proof is beautiful and quite simple. It deduces the Stone-von Neumann theorem from the Plancherel theorem. This proof can be found in Folland's book "Harmonic analysis in phase space". I have also tried to give an exposition of this proof, along with a proof of a generalization of this theorem due to Mackey, in my paper "An Easy Proof of the Stone-von Neumann-Mackey Theorem" (Expositiones Mathematicae, 24(1):110-118, 2011, also available on the arXiv).

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An excellent reference explaining the history and significance of the Stone von Neumann theorem is Jonathan Rosenberg's

"A Selective History of the Stone-von Neumann Theorem"

available at

http://www-users.math.umd.edu/~jmr/StoneVNart.pdf

Note that one doesn't actually have uniqueness of representations of the position and momentum operators (since they're unbounded). The theorem applies to their exponentiated versions, which give the Heisenberg (mathematician's name) or Weyl (physicist's name) group.

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I first learned about it from the first chapter of Lion and Vergne's book "The Weil representation, Maslov index and theta series", which is really much more down to earth than its title might suggest.

EDIT : It maybe should be emphasized that I am not a representation theorist, and when I read the above book I knew far less about the subject than I do now (which isn't a lot). This seems to me to be a positive thing -- I found the book pretty easy going despite not knowing all that much when I looked at it.

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I would also recommend M. Reed and B. Simon, Methods of modern mathematical physics, vol I, chapter VIII (and maybe vol II, chapter X). The theorem itself is VIII.14 with Corollary after that. The reference is useful also due to counterexample, demonstrating that the idea about “one-to-one correspondence” used in Wikipedia and some other places is not very good one.

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I might be biased, but this is the RIGHT reference. At least for the analysts under us (or better functional analysts or spectral theorists).  – Helge May 17 2011 at 16:04
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