# CCR and CAR algebras

CCR and CAR algebras

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

:$fg-gf=i\left(f,g\right)$: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 eif subject to

:$e^\left\{ic_1 f\right\}e^\left\{ic_2 f\right\}=e^\left\{i\left(c_1+c_2\right) f\right\}$:$e^\left\{if\right\}e^\left\{ig\right\}=e^\left\{-i\left(f,g\right)\right\}e^\left\{ig\right\}e^\left\{if\right\}$:(eif)*=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

:$fg+gf=\left(f,g\right)$: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

:$fg-\left(-1\right)^gf=i\left(f,g\right)$:f*=f, g*=g

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

ee also

* canonical commutation relation
* Stone-von Neumann theorem
* Bose-Einstein statistics
* Fermi-Dirac statistics
* Heisenberg group
* Weyl algebra
* Bogoliubov transformation
* (−1)F

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