F. A. BEREZIN

ON CANONICAL TRANSFORMATIONS IN THE SECOND-QUANTIZATION REPRESENTATION

(Presented by Academician I. G. Petrovskii, January 10, 1963)

1.

Consider a Hilbert space with an involution \(L\). Let \(\hat a(f)\), \(\hat a^{*}(f)\), \(f \in L\), be linear functionals on \(L\) with values in the set of linear operators in the Hilbert space \(\mathcal H\). We shall assume that the operators \(\hat a^{*}(f)\), \(\hat a(f)\) satisfy the usual (fermion or Bose) commutation relations, that they form an irreducible family, and that in the space \(\mathcal H\) there exists a vacuum vector \(\hat\Phi_0\): \(\hat a(f)\hat\Phi_0 = 0\).

Let \(\Phi, \Psi\) be operators in \(L\) having a common everywhere dense domain of definition \(D\), endowed with its own topology and invariant with respect to the involution. We denote by \(\widetilde L\) the space of continuous linear functionals on \(D\).

We shall assume that \(\widetilde L\) contains \(L\) as a dense set. The value of an element \(F \in \widetilde L\) on \(\varphi \in D\) will be denoted in the same way as the scalar product in \(L\): \(F(\varphi) = (F,\varphi^{*})\) (\(*\) is the involution in \(L\)). Define the operators \(\overline\Phi, \overline\Psi\): \(f\overline\Phi = (f^{*}\Phi)^{*}\), \(f\overline\Psi = (f^{*}\Psi)^{*}\), \(f \in L\).

Consider the operator linear functionals in \(D\):

\[ \hat b(f)=\hat a(f\Phi)+\hat a^{*}(f\Psi)+(F,f^{*}),\qquad \hat b^{*}(f)=\hat a(f\overline\Psi)+\hat a^{*}(f\overline\Phi)+(f,F). \tag{1} \]

If transformation (1) is invertible and the operators \(\hat b(f)\), \(\hat b^{*}(f)\) satisfy the same permutation relations as the operators \(\hat a(f)\), \(\hat a^{*}(f)\), then it is called a linear canonical transformation. In the fermion case \(F = 0\).

It is well known that if the space \(L\) is finite-dimensional, then in the space \(\mathcal H\) there exists a unitary operator \(\hat U\) which generates transformation (1):

\[ \hat b(f)=\hat U\hat a(f)\hat U^{-1},\qquad \hat b^{*}(f)=\hat U\hat a^{*}(f)\hat U^{-1}. \tag{2} \]

In the general case this is not always so. We shall agree to call a canonical transformation proper if there exists a unitary operator satisfying condition (2), and improper in the opposite case. It is known (), that in order that the canonical transformation (1) be proper, it is necessary and sufficient that the operator \(\Psi\) be a Hilbert–Schmidt operator and that the functional \(F\) belong to \(L\).

We note that in order that transformation (1) be canonical, definite relations must hold between the operators \(\Phi\) and \(\Psi\). These relations imply that, in the case when the transformation is proper, the operator \(\Phi\) is bounded**. Thus, in this case the domain \(D\) on which the operator functionals \(\hat b(f)\) and \(\hat b^{*}(f)\) are defined coincides with \(L\).

Let \(\mathcal A\) be some canonical transformation given by operators \(\Phi,\Psi\) and a functional \(F\), defined on the domain \(D\). Consider a sequence of canonical transformations \(\mathcal A_n\) having the following properties: 1) all operators \(\Phi_n,\Psi_n\) and functionals \(F_n\) are defined on the domain \(D\); 2) as \(n \to \infty\), \(\|\Phi_n f-\Phi f\|\to0\), \(\|\Psi_n f-\Psi f\|\to0\), where \(f\) is an arbitrary element of \(D\); \(F_n \to F\) in the strong topology of the space \(\widetilde L\).

In this case we shall call the transformation \(\mathcal A\) the limit of the transformations \(\mathcal A_n\).

* In (*) this theorem is proved for homogeneous transformations; however, by the same method a general result can be obtained.

** In the fermion case the operators \(\Phi\) and \(\Psi\) are bounded for any canonical transformation.

Theorem 1. Every linear canonical transformation is the limit of proper linear canonical transformations.

1. Let \(\mathcal A_n\) be a sequence of proper canonical transformations converging to an improper transformation \(\mathcal A\); let \(\hat U_n\) be the unitary operators in \(\mathcal H\) implementing the transformations \(\mathcal A_n\).

Definition. An operator \(\hat A\) in \(\mathcal H\), defined on the domain \(D_{\hat A}\), sustains the improper canonical transformation \(\mathcal A\) if, for every \(n\), the operators \(\hat U_n \hat A \hat U_n^{-1}\) are defined on \(D_{\hat A}\), and for every \(f \in D_{\hat A}\) there exists, in the strong sense, the limit \(\lim_{n \to \infty} \hat U_n \hat A \hat U_n^{-1} f\), and this limit does not depend on the choice of the sequence of canonical transformations \(\mathcal A_n\) converging to \(\mathcal A\).

In this section we shall describe the set of bounded operators that sustain all (linear) canonical transformations in the Fermi case.

Consider the operators
\[ \hat p(f)=\hat a(f)+\hat a^{*}(f), \qquad \hat q(f)=\frac{1}{i}\bigl(\hat a(f)-\hat a^{*}(f)\bigr). \tag{3} \]

For what follows it is convenient, instead of operator functionals, to consider operator generalized functions. In this connection consider a realization of the space \(L\) by means of functions with summable square on a certain set \(M\) endowed with a measure; suppose that, under this realization, the involution in \(L\) becomes complex conjugation.

Define the operator generalized function \(\hat p(x)\) by the equality
\[ \hat p(f)=\int \hat p(x) f(x)\,dx. \]
The operator generalized functions \(\hat q(x)\), \(\hat a(x)\), \(\hat a^{*}(x)\) are defined analogously.

Theorem 2. In order that a bounded operator \(\hat A\) sustain all canonical transformations, it is necessary and sufficient that it be representable in the form of a strongly convergent series
\[ \hat A=\sum \frac{1}{\sqrt{m!n!}}\int K_{mn}(x_1\ldots x_m \mid y_1\ldots y_n)\, \]
\[ {}\times \hat p(x_1)\ldots \hat p(x_m)\hat q(y_1)\ldots \hat q(y_n)\,d^{m}x\,d^{n}y, \tag{4} \]
where \(K_{mn}(x_1\ldots x_m \mid y_1\ldots y_n)\) are square-summable functions, antisymmetric separately in \(x_1\ldots x_m\) and in \(y_1\ldots y_n\), and satisfying the condition
\[ \sum \int \left|K_{mn}(x_1\ldots x_m \mid y_1\ldots y_n)\right|^{2}\,d^{m}x\,d^{n}y < \infty . \]

The set of operators of the form (4) forms a ring, which we shall denote by \(\mathfrak B\). In \(\mathfrak B\) one can introduce a trace according to the formula
\[ \operatorname{sp}_1 \hat A = K_{00}. \]

It is not difficult to verify that, with this, the usual requirement is satisfied: for any \(\hat A_1,\hat A_2 \in \mathfrak B\),
\[ \operatorname{sp}_1 \hat A_1\hat A_2=\operatorname{sp}_1 \hat A_2\hat A_1 . \]

Let us note that if the space \(L\) has dimension \(N<\infty\), then \(\mathfrak B\) is the ring of all matrices of order \(2^N\), and the trace introduced differs from the ordinary one by a factor:
\[ \operatorname{sp}_1 \hat A=\frac{1}{2^N}\operatorname{sp}\hat A . \]

We give the idea of the proof of sufficiency. Introduce in \(\mathfrak B\) the scalar product
\[ (A_1,A_2)=\operatorname{sp}_1(A_1A_2). \]
The completion of \(\mathfrak B\) with respect to this scalar product will be denoted by \(\overline{\mathfrak B}\).

It is not difficult to verify that every canonical transformation \(\mathcal A\) generates a unitary operator in the Hilbert space \(\overline{\mathfrak B}\). We denote this operator by \(U_{\mathcal A}\). It is not difficult to verify, further, that if \(\mathcal A_n \to \mathcal A\) in the sense of the definition given in Section 1, then \(U_{\mathcal A_n}\to U_{\mathcal A}\) in the strong sense.

The topology of the Hilbert space \(\overline{\mathfrak B}\) does not coincide with any natural topology in the space of operators. Consequently, in \(\overline{\mathfrak B}\) there may exist elements that do not correspond to operators in \(\mathcal H\).

Let us formulate, in this connection, a general lemma which completes the proof of sufficiency and is, moreover, of independent interest.

Lemma. Let \(\hat A_n \in \mathfrak B\) be a sequence of operators whose norms are bounded by a common constant \(c\). Then, if \(\hat A_n \to \hat A \in \overline{\mathfrak B}\) in the sense that \((\hat A-\hat A_n,\hat A-\hat A_n)\to 0\), then \(\hat A \in \mathfrak B\), \(\hat A_n \to \hat A\) in the strong sense, and \(\|\hat A\|\le c\).

1. Let us dwell on the connection between the normal form of an operator and its representation in the form (4). Consider the exterior algebras \(\mathfrak G_a\) and \(\mathfrak G_p\) with generators (functions with anticommuting values) \(a(x), a^*(x)\) and \(p(x), q(x)\), respectively:
   \[ \{a(x),a^*(x')\}=\{a(x),a(x')\}=\{a^*(x),a^*(x')\} =\{p(x),p(x')\}=\{p(x),q(x')\}=\{q(x),q(x')\}=0. \]
   To the representation of each operator in normal form and in the form (4) we assign elements of the algebras \(\mathfrak G_a\) and \(\mathfrak G_p\) (functionals):
   \[ A(a^*,a)= \]
   \[ =\sum \int L_{mn}(x_1\ldots x_m\mid y_1\ldots y_n)\, a^*(x_1)\cdots a^*(x_m)a(y_1)\cdots a(y_n)\,d^m x\,d^n y; \tag{5} \]
   \[ \mathfrak A(p,q)= \sum \frac{1}{\sqrt{m!n!}}\int K_{mn}(x_1\ldots x_m\mid y_1\ldots y_n)\times \]
   \[ {}\times p(x_1)\cdots p(x_m)q(y_1)\cdots q(y_n)\,d^m x\,d^n y \tag{6} \]
   (the functions \(K_{mn}\) in (6) are the same as in (4). For the functionals \(A(a^*,a)\), see \((^2)\)).

Using the continual integral over anticommuting variables \((^2)\), the connection between \(A(a^*,a)\) and \(\mathfrak A(p,q)\) can be written in the form*:
\[ \mathfrak A(p,q)=\int \exp\left[-i\int \left(q(x)+\frac{a^*(x)-a(x)}{i\sqrt2}\right) \left(p(x)-\frac{a(x)+a^*(x)}{\sqrt2}\right)\,dx\right]\times \]
\[ {}\times A\left(\frac{a^*}{\sqrt2},\frac{a}{\sqrt2}\right)\prod da^*\,da. \tag{7} \]

We omit the inversion of this formula.

The canonical transformation (1) gives rise to a relation between the operators
\[ \hat p=\hat a+\hat a^*,\quad \hat q=\frac1i(\hat a-\hat a^*) \]
and
\[ \hat p'=\hat b+\hat b^*,\quad \hat q'=\frac1i(\hat b-\hat b^*). \]
Replacing in this relation \(\hat p,\hat q,\hat p',\hat q'\) by \(p,q,p',q'\), we obtain a linear transformation in the algebra \(\mathfrak G_p\):
\[ p'=Ap+Bq,\qquad q'=Cp+Dq, \]
where \(A,B,C,D\) are operators expressible in a definite way through \(\Phi,\Psi\).

It turns out that the functional \({}'\mathfrak A(p,q)\), corresponding to the transformed operator \({}'\hat A\), is expressed through the functional \(\mathfrak A(p,q)\), corresponding to the operator \(\hat A\), by the formula
\[ {}'\mathfrak A(p,q)=\mathfrak A(p',q'), \]
i.e. the transformation of the functionals \(\mathfrak A(p,q)\) reduces to a change of variables**.

We note that not every operator can be written in the form (4). For exam—

* Recall the definition of the integral. In the case when the Grassmann algebra \(\mathfrak G\) has a finite number of generators \(x_1,\ldots,x_N\), the integral is defined as follows: \(\int dx_i=0,\ \int x_i\,dx_i=1\); the symbols \(dx_i\) anticommute with one another and with \(x_k\); a multiple integral is understood as repeated. The continual integral is the limit of \(n\)-fold integrals as \(n\to\infty\).

** In contrast to the functionals \(\mathfrak A(p,q)\), the functionals \(A(a^*,a)\) transform according to rather complicated formulas involving continual integration (see \((^2)\)).

for example, the operator \(\hat A=\int K(x,y)\hat a^*(x)\hat a(y)\,dx\,dy\) is representable in the form (4) if and only if \(K(x,y)\) is the kernel of an operator with an absolutely convergent trace. The same applies to the operator
\[ \hat A=\exp\left\{i\int K(x,y)a^*(x)a(y)\,dx\,dy\right\}. \]
Thus, if \(K(x,y)\) is the kernel of a self-adjoint, but non-nuclear operator, then \(\hat A\) is an example of a bounded operator that does not withstand all canonical transformations.

1. Let us pass to the Bose case. Consider the linear space \(\widetilde L\), consisting of pairs of real functions \((p(x),q(x))\). Suppose that a probability measure \(\mu\) is concentrated in the space \(\widetilde L\)*. To each operator \(\hat A\), written in normal form, we assign a functional on \(\widetilde L\):

\[ \mathfrak A(p,q)=\int \exp\left\{\frac12\int\left[\left(p-i\frac{b+b^*}{\sqrt2}\right)^2+ \left(q+\frac{b^*-b}{\sqrt2}\right)^2\right]dx\right\}\times \]
\[ {}\times A\left(\frac{ib^*}{\sqrt2},\frac{ib}{\sqrt2}\right)\prod db^*\,db, \tag{8} \]

where \(A(a^*,a)\) is the functional corresponding to the normal form of \(\hat A\)**.
We omit the inversion of formula (8).

The continual integral in this formula is understood as the limit of finite-dimensional ones, where in the finite-dimensional approximation

\[ b=\xi+i\eta,\quad b^*=\xi-i\eta,\quad \prod db^*db=\pi^{-n}d^n\xi\,d^n\eta,\quad \prod dp\,dq=(2\pi)^{-n}d^n p\,d^n q. \]

Just as in the Fermi case, the canonical transformation (1) gives rise to a relation between the operators
\(\hat p=\frac1{\sqrt2}(\hat a+\hat a^*)\),
\(\hat q=\frac1{i\sqrt2}(\hat a-\hat a^*)\) and
\(\hat p'=\frac1{\sqrt2}(\hat b+\hat b^*)\),
\(\hat q'=\frac1{i\sqrt2}(\hat b-\hat b^*)\).
Replacing in this relation \(\hat p,\hat q,\hat p',\hat q'\) by \(p,q,p',q'\), we obtain a linear transformation in \(\widetilde L\):

\[ p'(x)=\int\bigl(A(x,y)p(y)+B(x,y)q(y)\bigr)\,dy+f_1(x); \]
\[ q'(x)=\int\bigl(C(x,y)p(y)+D(x,y)q(y)\bigr)\,dy+f_2(x). \tag{9} \]

Theorem 3. Let the operator \(\hat A\) correspond to a functional \(\mathfrak A(p,q)\) that is summable with its square with respect to the measure \(\mu(p,q)\), and is quasi-invariant with respect to the transformation (9). Then the operator \(\hat A\) withstands the canonical transformation (1), and the transformed operator corresponds to the functional \({}'\mathfrak A(p,q)=\mathfrak A(p',q')\).

Thus, the transformation of the functional \(\mathfrak A(p,q)\) reduces to a change of variables.

Received
10 I 1963

References

1. K. O. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields, N. Y., 1953.
2. F. A. Berezin, DAN, 137, No. 2 (1961).
3. R. A. Minlos, Tr. Mosk. matem. obshch., 8, 497 (1959).
4. E. Wigner, Phys. Rev., 40, 749 (1932).

* Such a situation is possible if, for example, \(\widetilde L\) is the space conjugate to a nuclear one (³).

** The functional \(A(a^*,a)\) is defined in the same way as the analogous functional in the Fermi case, with the only difference that \(a(x),a^*(x)\) are ordinary complex-valued functions (²). In the case of a finite number of degrees of freedom, the writing of operators by means of the functions \(\mathfrak A(p,q)\) was first considered by Wigner (⁴).