- CCR and CAR algebras
In

quantum field theory , if V is a realvector space equipped with anonsingular realantisymmetric bilinear form (,) (i.e. asymplectic vector space ), theunital *-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 finitedimension al is discussed in theStone-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:$e^\{ic\_1\; f\}e^\{ic\_2\; f\}=e^\{i(c\_1+c\_2)\; f\}$:$e^\{if\}e^\{ig\}=e^\{-i(f,g)\}e^\{ig\}e^\{if\}$:(e

^{if})*=e^{-if}for real numbers c

_{1}, c_{2}.If V is equipped with a

nonsingular realsymmetric 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

**Z**_{2}-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:$fg-(-1)^gf=i(f,g)$:f*=f, g*=g

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

**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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