UDC 530.145.1

MATHEMATICAL PHYSICS

A. I. OKSAK

ON THE NORMAL FORM OF OPERATORS IN FOCK SPACE

(Presented by Academician N. N. Bogolyubov on 3 VI 1968)

In the paper certain classes of unbounded operators in the space of second quantization are indicated which can be reduced to normal form. The possibility of such a reduction is closely connected with the domains of definition of the operators and of their adjoints. We shall confine ourselves to the study of operators in Bose space; the generalization to the fermion case presents no difficulties.

1. Definition 1. Let \(C\) be a linear operator with domain of definition \(D(C)\) in a Hilbert space \(H\) and range \(R(C)\) in a Hilbert space \(\mathscr H\); let \(H'\) and \(\mathscr H'\) be (closed) subspaces in \(H\) and \(\mathscr H\), respectively; and let \(P\) and \(\mathscr P\) be projection operators: \(H'=PH\), \(\mathscr H'=\mathscr P\mathscr H\). By the part of the operator \(C\) belonging to the subspaces \(H'\) and \(\mathscr H'\) we shall mean an operator \(X\) with \(D(X)\subset H'\), \(R(X)\subset \mathscr H'\), such that

\[ D(X)=D(C)\cap H', \]

\[ Xf=\mathscr P Cf\equiv \mathscr P C P f \quad \text{for } \forall f\in D(X). \]

Let us note that a part \(X\) of a closed, densely defined operator \(C\), generally speaking, need not admit a closure. However, if the part \(Y\) of the operator \(C^*\), belonging to the subspaces \(\mathscr H'\) and \(H'\), has a dense domain of definition \(D(Y)\subset \mathscr H'\), then the part \(X\) of the operator \(C\), belonging to the subspaces \(H'\) and \(\mathscr H'\), admits a closure, and \(\overline X\subset Y^*\). If \(D(X)\) is dense in \(H'\), then \(Y\) admits a closure and \(\overline Y\subset X^*\).

1. Let \(D_i\) be an arbitrary linear manifold in the Hilbert space \(H_i\) \((i=1,\ldots,n)\); \(L(D_1,\ldots,D_n)\) will henceforth denote the linear span of vectors of the form \(f_1\otimes\cdots\otimes f_n\) with arbitrary \(f_i\in D_i\), in the tensor product \(H_1\otimes\cdots\otimes H_n\) (see \((^1)\)).

Definition 2 \((^2)\). The tensor product \(A_1\otimes\cdots\otimes A_n\) of closed (or closable), densely defined (i.e., with dense domains of definition in the corresponding spaces) operators \(A_i: D(A_i)\subset H_i,\ R(A_i)\subset \mathscr H_i,\ i=1,\ldots,n\), is the closure of the operator \(A\), defined on \(D(A)=L(D(A_1),\ldots,D(A_n))\subset H_1\otimes\cdots\otimes H_n\) with values in \(\mathscr H_1\otimes\cdots\otimes\mathscr H_n\) by the formula:

\[ A(f_1\otimes\cdots\otimes f_n)=(A_1f_1)\otimes\cdots\otimes(A_nf_n). \]

Lemma 1. Let \(A\) and \(B\) be closed operators with dense domains of definition in one and the same Hilbert space \(D(A)\subset H\), \(D(B)\subset H\), and with ranges \(R(A)\subset \mathscr H_1\), \(R(B)\subset \mathscr H_2\), where \(D(A)\supset D(B)\); let \(S\) be a closed densely defined operator, \(D(S)\subset H'\), \(R(S)\subset \mathscr H'\). Then \(D(S\otimes A)\supset D(S\otimes T)\), where \(T=\sqrt{1+B^*B}\) is the positive square root of the self-adjoint operator \(1+B^*B\). If, in addition, one of the requirements is satisfied: (a) \(S\) is bounded, (b) \(B\) has a bounded inverse, then

\[ D(S\otimes A)\supset D(S\otimes B). \tag{1} \]

From Lemma 1 there follows the well-known fact: the tensor product of bounded operators \(S\) and \(A\) is a bounded operator; it is enough in (1) to put \(B=0\): \(D(S\otimes A)\supset D(S\otimes 0)=H'\otimes H\); see \((^2)\), Theorem 2.

1. Denote by \(H^{\otimes n}\) the \(n\)-th tensor power of the space \(H\) (i.e., the tensor product of \(n\) copies of the space \(H\)); \(H^{\vee n}\) is the \(n\)-th symmetric power of \(H\)—this is the subspace in \(H^{\otimes n}\) that is the range of the symmetrization operator \(Q_n\) in \(H^{\otimes n}\) (see (3), Appendix A). Below, by symbols of the type \(A^{(m,n)}\) we shall denote closed (or closable) operators with dense domains of definition \(D(A^{(m,n)}) \subset H^{\vee n}\) and ranges \(R(A^{(m,n)}) \subset H^{\vee m}\). As follows from Sec. 2, the expression \(A^{(m,n)} \otimes B^{(m',n')}\) defines a closed operator with dense domain of definition
   \(D(A^{(m,n)} \otimes B^{(m',n')}) \subset H^{\vee n} \otimes H^{\vee n'}\) and with
   \(R(A^{(m,n)} \otimes B^{(m',n')}) \subset H^{\vee m} \otimes H^{\vee m'}\). It is obvious that \(H^{\vee(n+n')}\) and \(H^{\vee(m+m')}\) are (closed) subspaces, respectively, in \(H^{\vee n} \otimes H^{\vee n'}\) and \(H^{\vee m} \otimes H^{\vee m'}\); therefore we can give the following definition (see also Sec. 1):

Definition 3. The symmetric (tensor) product \(A^{(m,n)} \vee B^{(m',n')}\) of the operators \(A^{(m,n)}\) and \(B^{(m',n')}\) is the part of the operator \(A^{(m,n)} \otimes B^{(m',n')}\) belonging to the subspaces \(H^{\vee(n+n')}\) and \(H^{\vee(m+m')}\).

From Lemma 1 it follows immediately

Theorem 1. Let \(A^{(m,n)}\), \(B^{(m',n)}\), \(S^{(k,l)}\) be closed operators with dense domains of definition respectively in \(H^{\vee n}\), \(H^{\vee n}\), \(H^{\vee l}\) and with values respectively in \(H^{\vee m}\), \(H^{\vee m'}\), \(H^{\vee k}\), and suppose that \(D(A^{(m,n)}) \supset D(B^{(m',n)})\). Assume additionally that \(S^{(k,l)}\) is bounded. Then
\[ D(S^{(k,l)} \vee A^{(m,n)}) \supset D(S^{(k,l)} \vee B^{(m',n)}). \]

It is obvious that the theorem remains valid if, instead of the closed operator \(A^{(m,n)}\), one takes the closable operator \(A^{(m,n)}\).

Lemma 2. If the operators \(X = A^{(m,n)} \vee B^{(m',n')}\) and \(Y = A^{(m,n)*} \vee B^{(m',n')*}\) have dense domains of definition respectively in \(H^{\vee(n+n')}\) and \(H^{\vee(m+m')}\), then they admit closures, with \(\overline{X} \subset Y^*\), \(\overline{Y} \subset X^*\).

1. We turn to the consideration of operators in Fock space. \(H\) is the Hilbert space of one-particle states, \(H^{(n)} = H^{\vee n}\) is the \(n\)-th symmetric power of \(H\);
   \[ \mathscr{H}=\bigoplus_{n=0}^{\infty} H^{(n)} \]
   is the Fock space of Bose particles. Let \(M\) be a self-adjoint operator in \(\mathscr{H}\) with domain of definition \(D(M)\), commuting with the projectors onto the subspaces \(H^{(n)} \subset \mathscr{H}\); \(M^{(n,n)}\) is the part of the operator \(M\) lying in \(H^{(n)}\) with domain of definition
   \[ G^{(n)} = D(M^{(n,n)}). \tag{2} \]

We impose on the operator \(M\) the following essential condition: for any \(s,k=0,1,\ldots\)
\[ D(I^{(s,s)} \vee M^{(k,k)}) \supset D(M^{(s+k,s+k)}) = G^{(s+k)}, \tag{3} \]
where \(I^{(s,s)} \vee M^{(k,k)}\) is the symmetric product of the identity operator \(I^{(s,s)}\) in the space \(H^{(s)}\) with the operator \(M^{(k,k)}\) (see Definition 3).

Starting from an operator \(M\) satisfying condition (3), we construct a class of operators in \(\mathscr{H}\) that expand into a series in creation and annihilation operators. To this end we introduce the space \(\Theta = \Theta_M\) of coefficient functions—this is the set of matrices \(K = [K^{(m,n)}]_{m,n=0,1,\ldots}\), whose elements are operators satisfying the condition
\[ D(K^{(m,n)}) \supset G^{(n)}, \qquad D(K^{(m,n)*}) \supset G^{(m)} \tag{4} \]
(from (4) it follows that \(K^{(m,n)}\) admit closures).

By Theorem 1, from (4), (3) it follows that
\[ D(I^{(s,s)} \vee K^{(m,n)}) \supset G^{(s+n)}, \qquad D(I^{(s,s)} \vee K^{(m,n)*}) \supset G^{(s+m)}, \]
whence, with the aid of Lemma 2, we establish that \(I^{(s,s)} \vee K^{(m,n)}\) admits a closure and
\[ D(I^{(s,s)} \vee K^{(m,n)}) \supset G^{(s+n)}, \qquad D((I^{(s,s)} \vee K^{(m,n)})^*) \supset G^{(s+m)}. \tag{5} \]

We shall denote by \(\mathcal D\) the set of finite (i.e., with a finite number of particles) vectors from \(D(M)\):

\[ \mathcal D=\{\Phi:\ \Phi\in D(M),\ E_n\Phi=0\ \text{for } n>N(\Phi)\}. \tag{6} \]

To each operator in Fock space such that

\[ D(C)\supset\mathcal D,\qquad D(C^*)\supset\mathcal D, \tag{7} \]

there corresponds a matrix \(K\in\Theta\) with elements:

\(K^{(m,n)}\) is the part of the operator \((m!n!)^{-1/2}\cdot C\) belonging to the subspaces \(H^{(n)}\) and \(H^{(m)}\).
\[ \tag{8} \]

Conversely, in general this is false (a matrix \(K\in\Theta\) may correspond to no operator in \(\mathcal H\)).

We shall call a matrix \(K\in\Theta\) Jacobian if, for some \(r\), \(K^{(m,n)}=0\) for \(|m-n|>r\), and finite if only a finite number of elements \(K^{(m,n)}\) are nonzero.

Lemma 3. To every Jacobian (in particular, finite) matrix \(K\in\Theta\) there corresponds (admitting closure) an operator \(C\) satisfying (7), (8).

The inclusions (5) make it possible, for each complex number \(\lambda\), to define a transformation of \(\Theta\) into itself

\[ K\to I_{(\lambda)}*K, \tag{9} \]

where the right-hand side of (9) is a matrix with elements—operators defined on \(G^{(n)}\) by the relation:

\[ (I_{(\lambda)}*K)^{(m,n)} = \sum_{s=0}^{\min(m,n)} \frac{\lambda^s}{s!}\,I^{(s,s)}\vee K^{(m-s,n-s)}. \tag{10} \]

(For \(\lambda=0\), \(I_{(0)}*K\) is defined to be equal to \(K\).)

The transformations (9) have inverses, as follows from the identity

\[ I_{(\lambda)}*[I_{(\mu)}*K]=I_{(\lambda+\mu)}*K. \tag{11} \]

1. For any Jacobian matrix \(A\in\Theta\), \(I_{(1)}*A\) is a Jacobian matrix from \(\Theta\). By Lemma 3 it is associated with an operator \(B\) in \(\mathcal H\), satisfying (7) and the condition:

the part \(B^{(m,n)}\) of the operator \((m!n!)^{-1/2}\cdot B\), belonging to the subspaces \(H^{(n)}\) and \(H^{(m)}\), is equal to

\[ (I_{(1)}*A)^{(m,n)} = \sum_{s=0}^{\min(m,n)} \frac{1}{s!}\,I^{(s,s)}\vee A^{(m-s,n-s)}. \tag{12} \]

Definition 4. The expansion

\[ A\{a^+,a^-\}= \sum_{m,n}(a^+)^m\cdot A^{(m,n)}\cdot(a^-)^n \tag{13} \]

for a Jacobian matrix \(A\in\Theta\) will be called the operator \(B\) corresponding to the Jacobian matrix \(I_{(1)}*A\) and, therefore, defined by the equality (12). (We note that essentially in this form the concept of normal form was used by H. Araki for the study of bounded operators \((^4,\S6)\).)

We generalize the concept of the expansion (13) to the case of an arbitrary (non-Jacobian) matrix \(A\in\Theta\). In this case, by the infinite sum on the right-hand side of (13) one naturally understands the limit

\[ B=\lim_{r\to\infty}\sum_{|m-n|\le r}(a^+)^m\cdot A^{(m,n)}\cdot(a^-)^n, \tag{14} \]

if it exists on the set of finite vectors \(\mathcal D\subset D(M)\).

With the aid of identity (11) one establishes

Theorem 2. Let \(C\) be an operator in the Fock space \(\mathcal H\) satisfying condition (7); let \(K=\|K^{(m,n)}\|\in\Theta\) be the corresponding matrix (8). Then, for the matrix \(A=I_{(-1)}*K\), expression (14) defines an operator \(B\) which is the restriction of the operator \(C\) to the domain \(\mathcal D\):

\[ B=C|_{\mathcal D}. \tag{15} \]

This theorem on reducing operators to normal form for a bounded operator \(C\) was established by F. A. Berezin \({}^{5,6}\) (see also \({}^{4}\), § 6). The question arises whether the expansion (13) for an infinite matrix \(A\) can be understood as the limit of repeated sums:

\[ B=\sum_{m=0}^{\infty}\left(\sum_{n=0}^{\infty}(a^+)^m\cdot A^{(m,n)}\cdot(a^-)^n\right). \tag{16} \]

It turns out that, even for a bounded operator \(C\), the repeated series (16), generally speaking, converges neither in the strong nor in the weak sense (the corresponding example is given in the supplement to \({}^{7}\)); the expansion (16) for an infinite matrix \(A\) can be understood in a certain superweak sense, see \({}^{4,6}\)).

As an application we consider the out- and in-fields (corresponding to particles of positive mass) of quantum field theory with the condition of asymptotic completeness \({}^{8}\). They are defined on the common domain \(D_i(M)\)—the domain of definition of the energy operator \(M=P_0\), which satisfies the basic condition (3). This makes it possible to conclude that the out-fields expand in a series in terms of in-fields.

Remark. Let the matrix \(A\in\Theta\) have the only nonzero element \(K^{(1,0)}\) (respectively \(K^{(0,1)}\)); then to it there corresponds a vector \(f\in H\) such that the operator \(K^{(1,0)}\), taking the vacuum subspace \(H^{(0)}\) into the one-particle subspace \(H^{(1)}\), has the form \(H^{(0)}\ni c\to K^{(1,0)}c=cf\in H^{(1)}\) (respectively, \(K^{(0,1)}\) has the form \(H^{(1)}\ni F\to K^{(0,1)}F=(f,F)\in H^{(0)}\)). For such a matrix \(A\), expression (13) defines the creation operator \(a^+(f)\) (respectively, the annihilation operator \(a^-[f]\))—this is an operator-valued linear (respectively antilinear) functional on \(f\in H\), where \(a^-[f]\subset(a^+(f))^*\), \([a^-[f],a^+(g)]\subset(f,g)\). If \(\mathcal J\) is a certain involuting operator in \(H\), then instead of \(a^-[f]\) one introduces \({}^{6}\) the operator-valued linear functional \(a^-(f)=a^-[\mathcal J f]\).

The author expresses deep gratitude to M. K. Polivanov for numerous discussions in the course of the work, and also to V. S. Vladimirov, I. T. Todorov, and S. S. Khoruzhy for valuable comments.

Moscow Institute of Physics and Technology

Received
16 IV 1968

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