Canonical commutation relation algebra

In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role in quantum statistical mechanics and quantum field theory.

Let ${\displaystyle V}$ be a real vector space equipped with a nonsingular real antisymmetric bilinear form ${\displaystyle (\cdot ,\cdot )}$ (i.e. a symplectic vector space). The unital *-algebra generated by elements of ${\displaystyle V}$ subject to the relations

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