There is a distinct, but closely related meaning of CCR algebra, called the CCR C*-algebra. Let be a real symplectic vector space with nonsingular symplectic form . In the theory of operator algebras the CCR algebra over is the unital C*-algebra generated by elements subject to
These are called the Weyl form of the canonical commutation relations and, in particular, they imply that each is unitary and . It is well known that the CCR algebra is a simple non-separable algebra and is unique up to isomorphism.
for any . The field operators are defined for each as the generator of the one-parameter unitary group on the symmetric Fock space. These are self-adjointunbounded operators, however they formally satisfy
As the assignment is real-linear, so the operators define a CCR algebra over in the sense of Section 1.
The CAR algebra is faithfully represented on by setting
for all and . The fact that these form a C*-algebra is due to the fact that creation and annihilation operators on antisymmetric Fock space are bona-fide bounded operators. Moreover, the field operators satisfy
Let be a real -graded vector space equipped with a nonsingular antisymmetric bilinear superform (i.e. ) such that is real if either or is an even element and imaginary if both of them are odd. The unital *-algebra generated by the elements of subject to the relations
for any two pure elements in is the obvious superalgebra generalization which unifies CCRs with CARs: if all pure elements are even, one obtains a CCR, while if all pure elements are odd, one obtains a CAR.
The graded generalizations of Weyl and Clifford algebras allow the basis-free formulation of the canonical commutation and anticommutation relations in terms of a symplectic and a symmetric non-degenerate bilinear form. In addition the binary elements in this graded Weyl algebra give a basis-free version of the commutation relations of the symplectic and pseudo-orthogonal[clarification needed] Lie algebras.