Representation Theory
The representation theory of the Heisenberg group is fairly simple – later generalized by Mackey theory – and was the motivation for its introduction in quantum physics, as discussed below.
The key result is the Stone–von Neumann theorem, which, informally stated, says that (with certain technical assumptions) every representation of the Heisenberg group H2n+1 is equivalent to the position operators and momentum operators on Rn. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension 2n.
More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation.
Further, as the Heisenberg group is a semidirect product, its representation theory can be studied in terms of ergodic theory, via ergodic actions of the group, as in the work of George Mackey.
Read more about this topic: Heisenberg Group
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