Canonical Commutation Relation - Representations

Representations

The group H₃(R) generated by exponentiation of the Lie Algebra specified by these commutation relations, =iħ, is called the Heisenberg group.

According to the standard mathematical formulation of quantum mechanics, quantum observables such as x and p should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded—try taking the trace of both sides of the relations.

The canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) unitary operators exp(itx) and exp(isp) which do admit finite-dimensional representations. The resulting braiding relations for these are the so-called Weyl relations, exp(itx) exp(isp) = exp(−iħst) exp(isp) exp(itx). The corresponding group commutator is then exp(itx) exp(isp) exp(−itx) exp(−isp) = exp(−iħst). The uniqueness of the canonical commutation relations between position and momentum is then guaranteed by the Stone-von Neumann theorem.