CCR and CAR algebras

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.
CCR and CAR as *-algebras.
Let formula_1 be a real vector space equipped with a nonsingular real antisymmetric bilinear form formula_2 (i.e. a symplectic vector space). The unital *-algebra generated by elements of formula_1 subject to the relations
for any formula_6 in formula_1 is called the canonical commutation relations (CCR) algebra. The uniqueness of the representations of this algebra when formula_1 is finite dimensional is discussed in the Stone–von Neumann theorem.
If formula_1 is equipped with a nonsingular real symmetric bilinear form formula_2 instead, the unital *-algebra generated by the elements of formula_1 subject to the relations
for any formula_6 in formula_1 is called the canonical anticommutation relations (CAR) algebra.
The C*-algebra of CCR.
There is a distinct, but closely related meaning of CCR algebra, called the CCR C*-algebra. Let formula_16 be a real symplectic vector space with nonsingular symplectic form formula_2. In the theory of operator algebras, the CCR algebra over formula_16 is the unital C*-algebra generated by elements formula_19 subject to
These are called the Weyl form of the canonical commutation relations and, in particular, they imply that each formula_22 is unitary and formula_23. It is well known that the CCR algebra is a simple non-separable algebra and is unique up to isomorphism.
When formula_16 is a Hilbert space and formula_2 is given by the imaginary part of the inner-product, the CCR algebra is faithfully represented on the symmetric Fock space over formula_16 by setting
for any formula_28. The field operators formula_29 are defined for each formula_30 as the generator of the one-parameter unitary group formula_31 on the symmetric Fock space. These are self-adjoint unbounded operators, however they formally satisfy
As the assignment formula_33 is real-linear, so the operators formula_29 define a CCR algebra over formula_35 in the sense of Section 1.
The C*-algebra of CAR.
Let formula_16 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 formula_37 subject to the relations
for any formula_28, formula_43.
When formula_16 is separable the CAR algebra is an AF algebra and in the special case formula_16 is infinite dimensional it is often written as formula_46.
Let formula_47 be the antisymmetric Fock space over formula_16 and let formula_49 be the orthogonal projection onto antisymmetric vectors:
The CAR algebra is faithfully represented on formula_47 by setting
for all formula_53 and formula_54. 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 formula_55 satisfy
giving the relationship with Section 1.
Superalgebra generalization.
Let formula_1 be a real formula_58-graded vector space equipped with a nonsingular antisymmetric bilinear superform formula_2 (i.e. formula_60 ) such that formula_61 is real if either formula_62 or formula_63 is an even element and imaginary if both of them are odd. The unital *-algebra generated by the elements of formula_1 subject to the relations
for any two pure elements formula_6 in formula_1 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.