In quantum field theory, if V is a real vector space equipped with a nonsingular real antisymmetric bilinear form (,) (i.e. a symplectic vector space), the unital *-algebra generated by elements of V subject to the relations

fg - gf = i(f,g)

f*=f

for any f, g in V is called the canonical commutation relations (CCR) algebra. The uniqueness of the representations of this algebra when V is finite dimensional is discussed in the Stone-von Neumann theorem.

There is also a corresponding unital C*-algebra, often referred to as the Weyl form of the algebra, generated by e^if subject to

e^ife^ig = e ^{- i(f,g)}e^ige^if

(e^if)*=e^-if

for real numbers c₁, c₂.

If V is equipped with a nonsingular real symmetric bilinear form (,) instead, the unital *-algebra generated by the elements of V subject to the relations

fg + gf = (f,g)

f*=f

for any f, g in V is called the canonical anticommutation relations (CAR) algebra.

If V is a real Z₂-graded vector space equipped with a nonsingular antisymmetric bilinear superform (,) (i.e. (g,f)=-(-1)^|f||g|(g,f) ) such that (f,g) is real if either f or g is an even element and imaginary if both of them are odd, the unital *-algebra generated by the elements of V subject to the relations

fg - ( - 1) ^{| f | | g |} gf = i(f,g)

f*=f, g*=g

for any two pure elements f, g in V is the obvious super generalization which unifies CCRs with CARs.

CCR/CAR algebras only describe free fields, thanks to Haag's theorem.

See also

• canonical commutation relation
• Stone-von Neumann theorem
• Bose-Einstein statistics
• Fermi-Dirac statistics
• Heisenberg group
• Bogoliubov transformation