# Commutation and anti-commutation relationships, representation of

representation of commutation and anti-commutation relations

A linear weakly-continuous mapping $f \rightarrow a _ {f}$, $f \in L$, from a pre-Hilbert space $L$ into a set of operators acting in some Hilbert space $H$ such that either the commutation relations

$$\tag{1 } a _ {f _ {1} } a _ {f _ {2} } ^ {*} - a _ {f _ {2} } ^ {*} a _ {f _ {1} } = \ ( f _ {1} , f _ {2} ) E ,\ \ a _ {f _ {1} } a _ {f _ {2} } - a _ {f _ {2} } a _ {f _ {1} } = 0 ,$$

or the anti-commutation relations

$$\tag{2 } a _ {f _ {1} } a _ {f _ {2} } ^ {*} + a _ {f _ {2} } ^ {*} a _ {f _ {1} } = \ ( f _ {1} , f _ {2} ) E ,\ \ a _ {f _ {1} } a _ {f _ {2} } + a _ {f _ {2} } a _ {f _ {1} } = 0$$

hold, where $a _ {f} ^ {*}$, $f \in L$, is the adjoint of the operator $a _ {f}$ in $H$, $E$ is the identity operator in $H$ and $( \cdot , \cdot )$ is the scalar product in $L$.

In the case when $L$ is finite-dimensional, all irreducible representations both of relation (1) and of (2) are unitarily equivalent. In the case of an infinite-dimensional space there are infinitely many distinct (not unitarily equivalent) irreducible representations of (1) and (2); for complete separable $L$ they are described in .

Operators $a _ {f}$, $f \in L$, satisfying (1) and (2) form the basis of the so-called second quantization formalism (where $a _ {f}$ is usually called the annihilation operator of a particle in state $f \in L$ and $a _ {f} ^ {*}$ is the creation operator of this particle), often used in the study of quantum physical systems with a large number of degrees of freedom. However in second quantization one uses mainly the so-called Fock [Fok] representation of the commutation and anti-commutation relations; these are irreducible representations with as index space $L$ a separable Hilbert space, while in the space $H$ there exists a so-called vacuum vector that is annihilated by all operators $a _ {f}$, $\sqrt f \in L$.

How to Cite This Entry:
Commutation and anti-commutation relationships, representation of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commutation_and_anti-commutation_relationships,_representation_of&oldid=53373
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article