Annihilation operators

A family of closed linear operators $\{a(f):f\in H\}$, where $H$ is some Hilbert space, acting on a Fock space constructed from $H$( i.e. on the symmetrization ${\mathrm{\Gamma }}^s(H)$ or anti-symmetrization ${\mathrm{\Gamma }}^a(H)$ of the space of tensors over $H$) such that on the vector $(f_1\otimes \cdots \otimes f_n{)}_{\alpha }\in {\mathrm{\Gamma }}^{\alpha }(H)$, $\alpha =s,a$, consisting of the symmetrized $(\alpha =s)$ or anti-symmetrized $(\alpha =a)$ tensor product of a sequence of elements $f_1\dots f_n\in H$, $n=1,2\dots$ in $H$, they are given by the formulas:

$$\begin{array}{}\text{(1)}& a(f)(f_1\otimes \cdots \otimes f_n{)}_s=\end{array}$$

$$=\sum _{i=1}^n(f,f_i)(f_1\otimes \cdots \otimes f_{i-1}\otimes f_{i+1}\otimes \cdots \otimes f_n{)}_s$$

in the symmetric case, and

$$\begin{array}{}\text{(2)}& a(f)(f_1\otimes \cdots \otimes f_n{)}_a=\end{array}$$

$$=\sum _{i=1}^n(-1{)}^{i-1}(f,f_i)(f_i\otimes \cdots \otimes f_{i-1}\otimes f_{i+1}\otimes \cdots \otimes f_n{)}_a$$

in the anti-symmetric case; the empty vector $\mathrm{\Omega }\in {\mathrm{\Gamma }}^{\alpha }(H)$, $\alpha =s,a$( i.e. the unit vector in the subspace of constants in ${\mathrm{\Gamma }}^{\alpha }(H)$) is mapped to zero by $a(f)$. In these formulas $(\cdot ,\cdot )$ is the inner product in $H$. The operators $\{a^{\ast }(f):f\in H\}$ dual to the operators $a(f)$ are called creation operators; their action on the vectors $(f_1\otimes \cdots \otimes f_n{)}_{\alpha }$, $\alpha =s,a$, $n=1,2\dots$ is given by the formulas

$$\begin{array}{}\text{(3)}& a^{\ast }(f)(f_1\otimes \cdots \otimes f_n{)}_{\alpha }=(f\otimes f_1\otimes \cdots \otimes f_n{)}_{\alpha }\end{array}$$

and

$$a^{\ast }(f)\mathrm{\Omega }=f.$$

As a consequence of these definitions, for each $n>0$ the subspace ${\mathrm{\Gamma }}_n^{\alpha }(H{)}_{\alpha }^{\otimes n}$, $\alpha =s,a$, the symmetrized or anti-symmetrized $n$- th tensor power of $H$, is mapped by $a(f)$ into ${\mathrm{\Gamma }}_{n-1}^{\alpha }$ and by $a^{\ast }(f)$ into ${\mathrm{\Gamma }}_{n+1}^{\alpha }(H)$.

In quantum physics, the Fock space ${\mathrm{\Gamma }}^{\alpha }(H)$, $\alpha =s,a$, is interpreted as the state space of a system consisting of an arbitrary (finite) number of identical quantum particles, the space $H$ is the state space of a single particle, the subspace ${\mathrm{\Gamma }}_n^{\alpha }(H)$ corresponds to the states of the system with $n$ particles, i.e. states in which there are just $n$ particles. A state with $n$ particles is mapped by $a(f)$ to a state with $n-1$ particles ( "annihilation" of a particle), and by $a^{\ast }(f)$ to a state with $n+1$ particles ( "creation" of a particle).

The operators $a(f)$ and $a^{\ast }(f)$ form irreducible families of operators satisfying the following permutation relations: In the symmetric case (the commutation relations)

$$\begin{array}{}\text{(4)}& a(f_1)a(f_2)-a(f_2)a(f_1)=\end{array}$$

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

$$a^{\ast }(f_2)a(f_1)-a(f_1)a^{\ast }(f_2)=-(f_1,f_2)E;$$

and in the anti-symmetric case (the anti-commutation relations)

$$\begin{array}{}\text{(5)}& a(f_1)a(f_2)+a(f_2)a(f_1)=\end{array}$$

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

$$a(f_1)a^{\ast }(f_2)+a^{\ast }(f_2)a(f_1)=(f_1,f_2)E,$$

where $E$ is the identity operator in ${\mathrm{\Gamma }}^s(H)$ or ${\mathrm{\Gamma }}^a(H)$. Besides the families of operators $a(f)$ and $a^{\ast }(f)$, $f\in H$, described here, there exist in the case of an infinite-dimensional space $H$ also other irreducible representations of the commutation and anti-commutation relations (4) and (5), not equivalent to those given above. Sometimes they are also called creation and annihilation operators. In the case of a finite-dimensional space $H$, all irreducible representations of the commutation or anti-commutation relations are unitarily equivalent.

The operators $\{a(f),a^{\ast }(f):f\in H\}$ are in many connections convenient "generators" in the set of all linear operators acting in the space ${\mathrm{\Gamma }}^{\alpha }(H)$, $\alpha =s,a$, and the representation of such operators as the sum of arbitrary creation and annihilation operators (the normal form of an operator) is very useful in applications. The connection with this formalism bears the name method of second quantization, cf. [1].

In the particular, but for applications important, case in which $H=L_2({\mathbf{R}}^{\nu },d^{\nu }x)$, $\nu =1,2,...$( or in a more general case $H=L_2(M,Q)$, where $(M,Q)$ is a measure space), the family of operators $\{a(f),a^{\ast }(f):f\in L_2({\mathbf{R}}^{\nu },d^{\nu }x)\}$ defines two operator-valued generalized functions $a(x)$ and $a^{\ast }(x)$ such that

$$a(f)=\underset{{\mathbf{R}}^{\nu }}{\int }a(x)f(x)d^{\nu }x,a^{\ast }(f)=\underset{{\mathbf{R}}^{\nu }}{\int }{a}^{\ast }(x)\bar{f}(x)d^{\nu }x.$$

The introduction of $a(x)$ and $a^{\ast }(x)$ turns out to be convenient for the formalism of second quantization (e.g. it allows one directly to consider operators of the form

$$\underset{({\mathbf{R}}^{\nu }{)}^{m+n}}{\int }K(x_1\dots x_n;y_1\dots y_m)a^{\ast }(x_1)\dots a^{\ast }(x_n)\times$$

$$\times a(y_1)\dots a(y_n)d^{\nu }{x}_1\dots d^{\nu }{x}_n{d}^{\nu }{y}_1\dots d^{\nu }{y}_m,$$

$$n,m=1,2\dots$$

where $K(x_1\dots x_n;y_1\dots y_m)$ is a certain "sufficiently-good" function), without having to recourse to their decomposition as a series in the monomials

$$a^{\ast }(f_1)\dots a^{\ast }(f_n)a(g_1)\dots a(g_m),$$

where

$$f_1\dots f_n,g_1\dots g_m\in L_2({\mathbf{R}}^{\nu },d^{\nu }x).$$

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