MATHEMATICS

S. S. Sannikov

ON REPRESENTATIONS OF A CONTINUOUS GROUP BY UNBOUNDED OPERATORS

(Presented by Academician A. I. Mal’cev on 9 XI 1966)

1. In connection with the introduction into quantum theory of complex angular momenta \( (^{1}) \), in \( (^{2}) \) linear representations of the rotation group \(O_3\) were constructed, corresponding to arbitrary values of the angular momentum and spin*. In contrast to the well-known finite-dimensional representations, to which the classical definition of a representation of a compact continuous Lie group \( (^{4,5}) \), based on the concept of a bounded operator, leads, the representations constructed are infinite-dimensional, generally speaking nonunitary and multivalued. Such representations are given by unbounded operators in a certain linear topological space and are connected with a generalization of the concept of a function on a group to a very broad class of singular functions.

In this note elements are given of the theory of representations of a continuous Lie group by unbounded operators, and one example of such a representation is analyzed in the case of the rotation group \(O_3\) (a compact group).

1. Let \(G\) be a Lie group; let \(\mathscr L\) be a certain linear topological space on which unbounded operators \(T\) act; \(D_T\) and \(R_T\) are the domains of definition and ranges of the operator \(T\) in \(\mathscr L\) (\(D_T\) need not coincide with all of \(\mathscr L\)).

Definition 1. A mapping \(g \to T(g)\), \(g \in G\), where \(T(g)\) are unbounded operators, will be called a representation of the group \(G\) in \(\mathscr L\), if \(\mathscr L \supset D_{T(g)}\) for every \(T(g)\), and on the domain of definition the operators \(T(g)\) satisfy all group axioms, in particular
\[ T(g_1 g_2)=T(g_1)T(g_2). \]

We shall consider only such mappings for which, for any \(g \in G\), each set \(D_{T(g)}\) is everywhere dense in \(\mathscr L\), and moreover \(R_{T(g_1)} \cap D_{T(g_2)}\)** for any \(g_1, g_2 \in G\), and for any countable number of sets \(D_{T(g)}\), the intersection \(\bigcap D_{T(g)}\) is dense in \(\mathscr L\). Here, by a representation is understood, generally speaking, a projective representation
\[ T(g_1)T(g_2)=\eta(g_1,g_2)T(g_1g_2), \qquad T(e)=1; \tag{1} \]
\(\eta(g_1,g_2)\) is a numerical function***.

Definition 2. The elements \(g_1, g_2 \in G\) will be considered close if \(T(g_1)f\) is close to \(T(g_2)f\) in the topology of \(\mathscr L\) for \(f \in D_{T(g_1)} \cap D_{T(g_2)}\). Thus a new topology is specified on \(G\)****.

Definition 3. The representation \(g \to T(g)\) is irreducible in \(\mathscr L\) if \(\mathscr L\) contains no closed subspaces that include the domain of definition of every \(T(g)\).

* In \( (^{3}) \) these representations were used to describe a certain type of unstable state in quantum mechanics.

** Here \(\displaystyle \bigcup_{g\in G} D_{T(g)} = 0\).

*** The important question of the existence of a covering of \(G\), of which the exact mapping is \(T(g)\), is discussed in a concrete example of a representation of the group \(O_3\) (see below).

**** The introduction of a topology with respect to a given function or class of functions goes back to J. von Neumann \( (^{6}) \).

Such an \(\mathscr L\) is constructed as follows. Suppose that \(f \in \mathscr L\). Let, for example, \(f\) be one of the elements of the canonical basis of a representation of the Lie algebra of the group \(G\).* A dense set in \(\mathscr L\) will be the linear span of the orbit \(T(g)f\). It is formed by finite sums of the form
\[ \sum_i A_i T(g_i)f, \]
where \(A_i\) are complex numbers. At the same time certain \(g_0 \in \Delta_f \subset G\) (the set \(\Delta_f\) does not form a subgroup) must be excluded. Indeed, \(T(g)\) are unbounded operators; consequently, there will be such \(g_0\) for which \(f \notin D_{T(g_0)}\).

Introduce in \(\mathscr L\) the weakest locally convex separable topology defined by the family of seminorms \(P_\varphi(f)=|f(\varphi)|\), \(f \in \mathscr L\), \(\varphi \in \mathscr L'\) (the \(P_\varphi(f)\) do not take infinite values), where the linear functionals \(f(\varphi)\) have the property that
\[ f(T(g)\varphi)=(T^{-1}(g)f)(\varphi), \]
even if \(T(g)\varphi \notin \mathscr L'\). A sequence \(f_n \in \mathscr L\) converges to \(f\) if \(P_\varphi(f_n)\to P_\varphi(f)\) for all \(\varphi \in \mathscr L'\).

Definition 4. Two representations \(T_1(g)\) and \(T_2(g)\) in \(\mathscr L_1\) and \(\mathscr L_2\) shall be called equivalent if there exists an operator \(A\), mapping a dense set in \(\mathscr L_1\) onto a dense set in \(\mathscr L_2\), and an operator \(B\), mapping a dense set in \(\mathscr L_2'\) onto a dense set in \(\mathscr L_1'\), such that for elements of the dense set
\[ (Af_1)(\varphi_2)=f_1(B\varphi_2). \]

Obviously, all representations defined by unbounded operators are infinite-dimensional.

3. Example. As an example, consider one representation of the rotation group \(O_3\) to which the problem of extracting the square root of a spinor leads \((^7)\). In the case of spinors of the second rank, extraction of the root leads to the algebra \(A(a_1,a_2)\), generated by the operators of the (complex) coordinate and momentum of quantum mechanics \((^{8,9})\):
\[ a_1=z,\qquad a_2=-\frac{d}{dz},\qquad [a_1,a_2]=1, \]
where \(z\) is a complex variable. In \(A\) there are given linear canonical transformations
\[ a_i \to a_i' = u_i^k a_k \]
with unitary unimodular matrices
\[ u= \begin{pmatrix} \alpha & \beta\\ -\bar\beta & \bar\alpha \end{pmatrix}, \qquad |\alpha|^2+|\beta|^2=1, \]
forming the group \(SU(2)\).** For us the following auxiliary result is important.

Lemma \((^7)\). The transformations \(a_i \to a_i'\) can be represented as inner automorphisms in \(A\)
\[ u_i^k a_k = T(u)a_iT^{-1}(u), \tag{2} \]
locally isomorphic to the group \(O_3\).

Corollary. Formula (2) defines a projective representation of the group \(O_3\) of the form (1).

Construct a linear representation with operators \(T(u)\) in a certain class of entire analytic functions of the complex variable \(z\). Let \(f(z)\in D_{T(u)}\). We define the action of \(T(u)\) on \(f(z)\) by the formula
\[ T(u)f(z)=T(u)f(z)T^{-1}(u)(T(u)\cdot 1) =f\bigl(T(u)zT^{-1}(u)\bigr)\chi(z;u) \]
\[ = f\left(\alpha z-\bar\beta\,\frac{d}{dz}\right)\chi(z;u), \tag{3} \]
where \(\chi(z;u)=T(u)\cdot 1\) and \(\chi(z;e)=1\). The group condition gives an equation for \(\chi(z;u)\)
\[ T(u_2)\chi(z;u_1)=\chi(z;u_2u_1), \tag{4} \]

\[ \overline{\phantom{MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}} \]

* The existence of such representations is guaranteed by the requirements imposed on \(D_{T(g)}\).

** This consideration also extends to the general case of unimodular transformations with matrices
\[ v= \begin{pmatrix} \alpha & \beta\\ \gamma & \delta \end{pmatrix}, \qquad \alpha\delta-\beta\gamma=1. \]
As a result, we arrive at a representation by unbounded operators of the Lorentz group \(\mathscr L_4\) (a noncompact group).

where the parameters of the transformation \(T(u_2u_1)\) are expressed in terms of the parameters of the transformations \(T(u_1)\) and \(T(u_2)\) by the formulas known for the rotation group.

A solution of (4) is

\[ \varkappa(z;u)=\frac{1}{\sqrt{\alpha}}\exp\left(\frac{1}{2}z^2\frac{\beta}{\alpha}\right). \tag{5} \]

Consider the class \(\mathscr L\) of entire analytic functions of order \(\rho\leq 2\) and type \(0\leq K<\infty\), even with respect to the substitution \(z\mapsto -z\). Introduce in \(\mathscr L\) the weakest of the locally convex separable topologies defined by the family of seminorms \(P_\varphi(f)=|[f,\varphi]|\) with respect to the scalar product

\[ [f,\varphi]=\int f(z)\overline{(I\varphi(z))}\,d\mu(z) \tag{6} \]

with measure \((^{9,10})\)

\[ d\mu(z)=\pi^{-1}\exp(-z\overline z)\,dz,\qquad dz=dx\,dy. \tag{7} \]

In (6), \(I\varphi(z)=\varphi(iz)\), \(f\in\mathscr L\), and \(\varphi\in\mathscr L'\), where \(\mathscr L'\) is the class of entire analytic functions of order \(\rho<2\) and type \(0\leq K<\infty\).

Consider the system of functions \(e^{\frac12\sigma z^2}\), where \(\sigma\) is a complex variable. These functions form a complete system in \(\mathscr L\). Indeed, any function \(f(z)\in\mathscr L\) can be represented in the form

\[ f(z)=\int e^{\frac12\sigma z^2}\,\tilde f(2\sigma)\,d\mu(\sigma), \tag{8} \]

where the integration is over the whole complex plane \((\sigma)\) with measure (7); moreover, we have \(\tilde f(z^2)=f(z)\). The inverse of formula (8) has the form (choosing the sheet on which \((\sqrt{\sigma})^2=\sigma\))

\[ \tilde f(2\sigma)=\int \operatorname{ch}(\overline z\sqrt{2\sigma})f(z)\,d\mu(z). \tag{9} \]

Since

\[ \int \operatorname{ch}(\overline z\sqrt{2\sigma})e^{\frac12\overline\sigma w^2}\,d\mu(\sigma)=\operatorname{ch}(\overline z w),\qquad \int e^{\frac12\overline\sigma z^2}\operatorname{ch}(\overline z\sqrt{2\tau})\,d\mu(z)=e^{\overline\sigma\tau}, \tag{10} \]

and for any entire analytic functions \(f(z)=f(-z)\) and \(\tilde f(\sigma)\) we have

\[ \int \operatorname{ch}(z\overline w)f(w)\,d\mu(w)=f(z),\qquad \int e^{\sigma\overline\tau}\tilde f(\tau)\,d\mu(\tau)=\tilde f(\sigma), \]

this proves the completeness of the system of functions \(e^{\frac12\sigma z^2}\) in \(\mathscr L\). As follows from (9), (10), the systems of functions \(e^{\frac12 z^2\overline\sigma}\), \(\operatorname{ch}(\sqrt{2\sigma}\overline z)\) have biorthogonality properties.

Next we have

\[ T(u)e^{\frac12 z^2\sigma}=\eta(u,\sigma)e^{\frac12 z^2\sigma(u)}, \tag{11} \]

i.e., under the action of \(T(u)\) the functions \(e^{\frac12 z^2\sigma}\) pass into expressions of the same form. Computations give

\[ \eta(u,\sigma)=\frac{1}{\sqrt{-\beta\sigma+\alpha}},\qquad \sigma(u)=\frac{\alpha\sigma+\beta}{-\overline\beta\sigma+\overline\alpha},\qquad \sigma\equiv\sigma(e). \tag{12} \]

By the definition of the class \(\mathscr L\), values \(\sigma(u)=\infty\) must be excluded. Therefore, for fixed \(u\), the domain of definition of the operator \(T(u)\) is not the whole system of functions complete in \(\mathscr L\), but only those functions of the form (11) for which \(\sigma\ne \overline\alpha/\overline\beta\). Consequently, \(T(u)\) are unbounded operators.*

* \(\mathscr L\) can be completed by adjoining to it “ideal” elements to which \(\sigma=\infty\) corresponds. Formulas (11), (12) show how to extend \(T(u)\) to such elements. As a result we obtain a space \([\mathscr L]\) invariant with respect to the representation \(T(u)\).

It is not difficult to verify that \(\mathscr L\) contains the domain of definition of each \(T(u)\) and contains no subspaces possessing the same property. Thus, the following holds.

Theorem. The mapping \(u \to T(u)\) (3), as a linear irreducible representation of the group \(SU(2)\), is realized by unbounded operators in the class of entire analytic functions of order \(\rho \leqslant 2\) and type \(0<K<\infty\).

We shall call the following functions on the group the matrix elements of the representation:

\[ D_{\tau,\bar\sigma}(u) = \int \operatorname{ch}\!\left(\sqrt{2\tau \bar z}\right) \left(T(u)e^{\frac12 z\bar\sigma}\right)d\mu(z) = \eta(u,\bar\sigma)e^{\bar\tau\sigma(u)}. \tag{13} \]

These are singular functions on the group. As formulas (11), (12), (13) show, the representation under consideration is two-valued on the group \(SU(2)\) and four-valued on the group \(O_3\). Indeed, in the Euler parametrization

\[ \alpha=\cos\frac{\vartheta}{2}\,e^{-i(\varphi_1+\varphi_2)/2},\quad \beta=-i\sin\frac{\vartheta}{2}\,e^{i(\varphi_1-\varphi_2)/2}. \]

The functions (13) are single-valued on the covering space in which the Euler angles \(\varphi_1,\varphi_2,\vartheta\) take values in the region \(0\leqslant \varphi_1,\varphi_2<8\pi,\ 0\leqslant \vartheta\leqslant \pi\). The group space of such a covering is a certain three-dimensional manifold in an infinite-dimensional space.

In conclusion, we note that with respect to a complete system in \(\mathscr L\) and to the matrix elements, the approach considered here differs from the canonical one \((^2)\), which makes substantial use of infinitesimal methods.

Physical-Technical Institute
Academy of Sciences of the Ukrainian SSR

Received
3 XI 1966

CITED LITERATURE

\(^{1}\) T. Regge, Nuove Cim., 14, 951 (1959).
\(^{2}\) C. C. Sannikov, Yadern. Fiz., 2, 570 (1965).
\(^{3}\) S. S. Sannikov, Phys. Lett., 19, 216 (1965).
\(^{4}\) H. Weyl, Math. Zs., 23, 271; 24, 328 (1925).
\(^{5}\) B. L. van der Waerden, Math. Zs., 36, 780 (1932).
\(^{6}\) J. von Neumann, Trans. Am. Math. Soc., 36, 445 (1934).
\(^{7}\) C. C. Sannikov, DAN, 172, No. 1 (1967).
\(^{8}\) V. Fock, Zs. Phys., 49, 339 (1928).
\(^{9}\) V. Bargmann, Comm. Pure and Appl. Math., 14, 187 (1961).
\(^{10}\) C. C. Sannikov, ZhETF, 49, 1913 (1965).