Note that all of this can easily be generalized to the case ofdspatial di-
mensions, fordfinite, with the Heisenberg group nowH 2 d+1and the Stone-von
Neumann theorem still true. In the case of an infinite number of degrees of
freedom, which is the case of interest in quantum field theory, the Stone-von
Neumann theorem no longer holds and one has an infinity of inequivalent irre-
ducible representations, leading to quite different phenomena. For more on this
topic see chapter 39.
It is also important to note that the Stone-von Neumann theorem is for-
mulated for Heisenberg group representations, not for Heisenberg Lie algebra
representations. For infinite dimensional representations in cases like this, there
are representations of the Lie algebra that are “non-integrable”: they aren’t
the derivatives of Lie group representations. For such non-integrable represen-
tations of the Heisenberg Lie algebra (i.e., operators satisfying the Heisenberg
commutation relations) there are counter-examples to the analog of the Stone
von-Neumann theorem. It is only for integrable representations that the theo-
rem holds and one has a unique sort of irreducible representation.
13.4 For further reading
For a lot more detail about the mathematics of the Heisenberg group, its Lie
algebra and the Schr ̈odinger representation, see [8], [51], [26] and [94]. An ex-
cellent historical overview of the Stone-von Neumann theorem [74] by Jonathan
Rosenberg is well worth reading. For not just a proof of Stone-von Neumann,
but some motivation, see the discussion in chapter 14 of [41].