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.
In mathematics, the abstract structure of the CCR and CAR algebras, over any field, not just the complex numbers, is studied by the name of Weyl and Clifford algebras, where many significant results have accrued. One of these is that 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 indefinite orthogonal Lie algebras.