Permutation relationships

permutation relations

Rules for permuting the product of two creation or annihilation operators. That is, for the annihilation operators $ \{ {a( f ) } : {f \in H } \} $ and the adjoint creation operators $ \{ {a ^ \star ( f ) } : {f \in H } \} $, where $ H $ is some Hilbert space, acting in the symmetric Fock space $ F( H) $ over $ H $, these relationships take the form

$$ \tag{1 } a( f _ {1} ) a ( f _ {2} ) - a( f _ {2} ) a( f _ {1} ) = $$

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

$$ a( f _ {1} ) a ^ \star ( f _ {2} ) - a ^ \star ( f _ {2} ) a( f _ {1} ) = ( f _ {1} , f _ {2} ) E ,\ f _ {1} , f _ {2} \in H, $$

where $ ( \cdot , \cdot ) $ is the inner product in $ H $ and $ E $ is the identity operator acting in $ F( H) $. The relations (1) are also called the commutation relations. In the case of an anti-symmetric Fock space, the creation and annihilation operators permute in accordance with the rules

$$ \tag{2 } a( f _ {1} ) a( f _ {2} ) + a( f _ {2} ) a( f _ {1} ) = $$

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

$$ a( f _ {1} ) a ^ \star ( f _ {2} ) + a ^ \star ( f _ {2} ) a( f _ {1} ) = ( f _ {1} , f _ {2} ) E ,\ f _ {1} , f _ {2} \in H, $$

which are called the anti-commutation relations.

In the case of an infinite-dimensional space $ H $, besides the creation and annihilation operators acting in Fock spaces over $ H $ there exist other irreducible representations not equivalent to them for the commutation and anti-commutation relations, i.e. other families of operators acting in some Hilbert space and satisfying the permutation rules (1) or (2) [1], . In the case of a finite-dimensional Hilbert space $ H $, all the irreducible representations of (1) or (2) are unitarily equivalent.

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The abbreviations CCR and CAR, which stand for canonical commutation relations and canonical anti-commutation relations are often used for relations (1) and (2). One also speaks of CCR algebras and CAR algebras.

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