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

$$\begin{array}{}\text{(1)}& a(f_1)a(f_2)-a(f_2)a(f_1)=\end{array}$$

$$=a^{\star }(f_1)a^{\star }(f_2)-a^{\star }(f_2)a^{\star }(f_1)=0,$$

$$a(f_1)a^{\star }(f_2)-a^{\star }(f_2)a(f_1)=(f_1,f_2)E,f_1,f_2\in H,$$

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

$$\begin{array}{}\text{(2)}& a(f_1)a(f_2)+a(f_2)a(f_1)=\end{array}$$

$$=a^{\star }(f_1)a^{\star }(f_2)+a^{\star }(f_2)a^{\star }(f_1)=0,$$

$$a(f_1)a^{\star }(f_2)+a^{\star }(f_2)a(f_1)=(f_1,f_2)E,f_1,f_2\in H,$$

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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