CCR and CAR algebras


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

CCR and CAR as *-algebras

Let be a real vector space equipped with a nonsingular real antisymmetric bilinear form . The unital *-algebra generated by elements of subject to the relations
for any in is called the canonical commutation relations algebra. The uniqueness of the representations of this algebra when is finite dimensional is discussed in the Stone–von Neumann theorem.
If is equipped with a nonsingular real symmetric bilinear form instead, the unital *-algebra generated by the elements of subject to the relations
for any in is called the canonical anticommutation relations algebra.

The C*-algebra of CCR

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.
When is a Hilbert space and is given by the imaginary part of the inner-product, the CCR algebra is faithfully represented on the symmetric Fock space over by setting
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-adjoint unbounded 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 C*-algebra of CAR

Let be a Hilbert space. In the theory of operator algebras the CAR algebra is the unique C*-completion of the complex unital *-algebra generated by elements subject to the relations
for any,.
When is separable the CAR algebra is an AF algebra and in the special case is infinite dimensional it is often written as.
Let be the antisymmetric Fock space over and let be the orthogonal projection onto antisymmetric vectors:
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
giving the relationship with Section 1.

Superalgebra generalization

Let be a real -graded vector space equipped with a nonsingular antisymmetric bilinear superform 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.