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[1] and quantum field theory.
be a real vector space equipped with a nonsingular real antisymmetric bilinear form
The unital *-algebra generated by elements of
is called the canonical commutation relations (CCR) algebra.
The uniqueness of the representations of this algebra when
is finite dimensional is discussed in the Stone–von Neumann theorem.
is equipped with a nonsingular real symmetric bilinear form
instead, the unital *-algebra generated by the elements of
is called the canonical anticommutation relations (CAR) algebra.
There is a distinct, but closely related meaning of CCR algebra, called the CCR C*-algebra.
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
It is well known that the CCR algebra is a simple (unless the sympletic form is degenerate) non-separable algebra and is unique up to isomorphism.
is given by the imaginary part of the inner-product, the CCR algebra is faithfully represented on the symmetric Fock space over
as the generator of the one-parameter unitary group
These are self-adjoint unbounded operators, however they formally satisfy As the assignment
define a CCR algebra over
In the theory of operator algebras the CAR algebra is the unique C*-completion of the complex unital *-algebra generated by elements
be the orthogonal projection onto antisymmetric vectors: The CAR algebra is faithfully represented on
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.
satisfy giving the relationship with Section 1.
-graded vector space equipped with a nonsingular antisymmetric bilinear superform
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
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.