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. 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. Let V {displaystyle V} be a real vector space equipped with a nonsingular real antisymmetric bilinear form ( ⋅ , ⋅ ) {displaystyle (cdot ,cdot )} (i.e. a symplectic vector space). The unital *-algebra generated by elements of V {displaystyle V} subject to the relations for any f ,   g {displaystyle f,~g} in V {displaystyle V} is called the canonical commutation relations (CCR) algebra. The uniqueness of the representations of this algebra when V {displaystyle V} is finite dimensional is discussed in the Stone–von Neumann theorem. If V {displaystyle V} is equipped with a nonsingular real symmetric bilinear form ( ⋅ , ⋅ ) {displaystyle (cdot ,cdot )} instead, the unital *-algebra generated by the elements of V {displaystyle V} subject to the relations for any f ,   g {displaystyle f,~g} in V {displaystyle V} is called the canonical anticommutation relations (CAR) algebra. There is a distinct, but closely related meaning of CCR algebra, called the CCR C*-algebra. Let H {displaystyle H} be a real symplectic vector space with nonsingular symplectic form ( ⋅ , ⋅ ) {displaystyle (cdot ,cdot )} . In the theory of operator algebras, the CCR algebra over H {displaystyle H} is the unital C*-algebra generated by elements { W ( f ) :   f ∈ H } {displaystyle {W(f):~fin H}} subject to These are called the Weyl form of the canonical commutation relations and, in particular, they imply that each W ( f ) {displaystyle W(f)} is unitary and W ( 0 ) = 1 {displaystyle W(0)=1} . It is well known that the CCR algebra is a simple non-separable algebra and is unique up to isomorphism. When H {displaystyle H} is a Hilbert space and ( ⋅ , ⋅ ) {displaystyle (cdot ,cdot )} is given by the imaginary part of the inner-product, the CCR algebra is faithfully represented on the symmetric Fock space over H {displaystyle H} by setting for any f , g ∈ H {displaystyle f,gin H} . The field operators B ( f ) {displaystyle B(f)} are defined for each f ∈ H {displaystyle fin H} as the generator of the one-parameter unitary group ( W ( t f ) ) t ∈ R {displaystyle (W(tf))_{tin mathbb {R} }} on the symmetric Fock space. These are self-adjoint unbounded operators, however they formally satisfy