CCR algebra

Summary

The CCR algebra of a symplectic vector space $(V, \sigma)$ is a C*-algebra generated by unitary elements $W(v)$, $v \in V$. These satisfy the Weyl form of the canonical commutation relations. It is the algebra of observables of a bosonic system with phase space $V$, such as a free quantum field. The Weyl form uses bounded unitaries instead of the unbounded position and momentum operators, so that C*-algebra methods apply. For finite-dimensional $V$ the Stone–von Neumann theorem says that its regular irreducible representations are all equivalent, while for infinite-dimensional $V$ there are many inequivalent ones.