MATHEMATICS

D. P. ZHELOBENKO

DESCRIPTION OF A CERTAIN CLASS OF REPRESENTATIONS OF THE LORENTZ GROUP

(Presented by Academician N. N. Bogolyubov, 1 IV 1958)

Irreducible linear representations of the Lorentz group have been completely described in ((^1)). However, since the Lorentz group is noncompact, it has representations that do not decompose into irreducible ones. Therefore there arises the problem of describing all (up to equivalence) linear representations, for example all bounded representations in Banach spaces. In the present note a partial solution of this problem is given, namely, a certain, quite broad, class of representations, called below the class (K_0), is described. In the exposition we use the notation and results of ((^1)).

Below we shall have in mind representations (a \to T_a) of the group (\mathfrak A) in reflexive Banach spaces satisfying the conditions: 1) the operators (T_a) are bounded for every fixed (a); 2) ((T_a \xi, \varphi)) is continuous in (a) for all (\xi \in R,\ \varphi \in R').

1. Canonical model. Let (R_0) be a complex reflexive Banach space and (\varepsilon \to A_\varepsilon) some representation of the subgroup (E), given in (R_0) (i.e. a one-parameter group of operators in this space). Further, let (m) be an integer. From these data (i.e. from (R_0), (A_\varepsilon), and (m)) we define a representation of the group (\mathfrak A) in the following way.

The space of the representation is (R = L_m[R_0])—the set of all measurable vector-functions (f(u)), (u \in \mathfrak u), with values in (R_0), satisfying the conditions

[
f(\gamma u)=e^{im\omega}f(u)\quad \text{for } \gamma=
\begin{pmatrix}
e^{-i\omega} & 0\
0 & e^{i\omega}
\end{pmatrix};
\qquad
\int |f(u)|^2\,du<\infty .
]

The operations in (L_m[R_0]) are defined in the usual way; the norm

[
|f|=\left(\int |f(u)|^2\,du\right)^{1/2}.
]

The space conjugate to (L_m[R_0]) is (L_m[R_0']). Define in (L_m[R_0]) the representation (a \to T_a) of the group (\mathfrak A) in the following way:

[
T_a f(u)=\varepsilon_{22}^{\prime -2} A_{\varepsilon'} f(u\bar a)
\quad \text{for } ua=\varepsilon'\zeta(u\bar a),\ \varepsilon' \in E,\ \zeta \in Z,\ u\bar a \in \mathfrak u .
\tag{1}
]

The lowest weight (k_0) (the least of the weights (k) for which (\mathfrak M_j^k \ne (0))) is (k_0=|m|/2). In the conjugate representation the role of (A_\varepsilon) is played by the operator

[
\hat A_\varepsilon = A_{\varepsilon^{-1}}'
]

(the prime denotes the conjugate operator).

All representations (S_{m,\rho}) described in ((^1)) are contained in this model (when (R_0) is one-dimensional and (A_\varepsilon=\varepsilon_{22}^{2\alpha}(\varepsilon)=\varepsilon_{22}^{i\rho})).

1. Structure of the group ring. We formulate a preliminary result, on the basis of which the class of representations under study will be singled out.

The well-known analogue of the Plancherel formula ((^{2})) maps the space (L_2(\mathfrak A)) onto the Hilbert space (H), consisting of infinite matrices*

[
\lambda=\lambda(\chi)=\left| \lambda^{k_1k_2}{j_1j_2}(\chi)\right|,\qquad
\sum\sum_m \int}
\left|\lambda^{k_1k_2}_{j_1j_2}(m,\rho)\right|^2(m^2+\rho^2)\,d\rho<\infty,
]

with scalar product

[
(\lambda,\mu)=
\sum_{\substack{k_1,k_2\ j_1,j_2}}\sum_m \int
\lambda^{k_1k_2}{j_1j_2}(m,\rho)\,
\overline{\mu^{k_1k_2}\,(m^2+\rho^2)\,d\rho .}(m,\rho)
]

The matrix index is (\binom{k}{j}), (k=0,1,2,\ldots,\ |j|\leq k); the elements (\lambda^{k_1k_2}_{j_1j_2}(\chi)) are functions of the character (\chi=(m,\rho)); they differ from zero only when (|m|/2\leq \min(k_1,k_2)).

Under this mapping, the image of the ring (X\subset L_2(\mathfrak A)) is the ring (\Lambda={\lambda}) with ordinary matrix multiplication and Hermitian conjugation as involution. In order to construct representations of the ring (\Lambda), it is necessary to know the class of functions (\lambda^{k_1k_2}_{j_1j_2}(\chi)) as functions of the character (\chi).

Theorem 1. The ring (\Lambda) consists of all matrices (\lambda\in H) whose elements (\lambda^{k_1k_2}_{j_1j_2}(\chi)) satisfy the conditions:

1) (\lambda^{k_1k_2}{j_1j_2}(\chi)=\lambda^{k_1k_2}(m,\rho)) are analytically continuable to the whole complex plane (\rho=\rho_1+i\rho_2), where they are entire functions of integral order of growth equal to one;**

[
\varkappa^{k_1}{m\rho}\varkappa^{k_2}}\lambda^{k_1k_2{j_1j_2}(m,\rho)
=
\varkappa^{k_1}}\varkappa^{k_2{m\rho}\lambda^{k_1k_2}(-m,-\rho);
\tag{2}
]

[
\varkappa^{k_1}{m,il}\varkappa^{k_2}}\lambda^{k_1k_2{j_1j_2}(m,il)
=
\varkappa^{k_1}}\varkappa^{k_2{-l,-im}\lambda^{k_1k_2}(l,im),
]

3)

[
m,l=-k,-k+1,\ldots,k,\qquad k=\min(k_1,k_2);
\tag{3a}
]

[
\lambda^{k_1k_2}_{j_1j_2}(m,il)=0,\quad \text{if }
\begin{cases}
\text{either } k_2<|l/2|\leq k_1,\quad l>0,\
\text{or } k_1<|l/2|\leq k_2,\quad l<0,
\end{cases}
\tag{3b}
]

where it is denoted

[
\varkappa^k_{m\rho}=
\prod_{\mu=|m/2|}^{k}
\frac{2\mu+i\rho}{\sqrt{4\mu^2+\rho^2}} .
]

For elements (x\in X^k_j), only the function (\lambda^{kk}_{jj}(m,\rho)=\lambda(m,\rho)) differs from zero, and conditions (2), (3) in this case take the form

[
\lambda(m,\rho)=\lambda(-m,-\rho);\qquad
\lambda(m,il)=\lambda(l,im),\quad |m|\leq 2k,\quad |l|\leq 2k .
]

* We have in mind the kernels (K(u_1,u_2;m,\rho)) in the basis of functions (f^k_j(u)=\sqrt{2k+1}\,c^k_{\nu j}(u)) with (\nu=m/2).

** That is,

[
\int_{-\infty+i\rho_2}^{\infty+i\rho_2}
\left|P(\rho_1+i\rho_2)\lambda(m,\rho_1+i\rho_2)\right|^2\,d\rho_1
\leq C_1e^{C|\rho_2|},
]

where (P(\rho)) is an arbitrary polynomial in (\rho), and the constants (C_1) and (C) are their own for each function (\lambda) and each polynomial (P).

If the representation $\vartheta$ is reducible and, in one of the invariant subspaces, induces the representation $\gamma$, we shall say that $\gamma$ is contained in $\vartheta$. As is known (${}^2$), the representations $S_{m,\rho}$ are reducible if $\rho^2=-(|m|+2n)^2$, $n=1,2,\ldots$.

The relations (3) have the following consequence:

I. A reducible representation $S_{m,\rho}$ contains only one irreducible representation: a spinor representation, if $\operatorname{Im}\rho>0$, and an infinite-dimensional one (namely $S_{i\rho,im}$), if $\operatorname{Im}\rho<0$.

The irreducible (infinite-dimensional) representations $S_{m,il}$, $l=0,\pm1,\pm2,\ldots$, $m=|l|+1, |l|+2,\ldots$, will be called special.

Definition of the class $K_0$. A representation of the group $\mathfrak A$ belongs to the class $K_0$ if neither it nor its conjugate contains any special or spinor representations.

1. Main result.

Theorem 2. Every representation of the class $K_0$ decomposes into a discrete direct sum of representations, each of which is equivalent to one of the representations of the canonical model; moreover, in this model the operators $A_\varepsilon$ and $A'\varepsilon$ have no eigenvalues $\varepsilon^n$, where $n$ is an integer of the same parity as $m$, $|n|\ne |m|$.

Idea of the proof. A representation of the group $\mathfrak A$ is called normal (M. A. Naimark) if the linear span of vectors of the form $\xi=T_x\eta$, where $\eta$ runs through $\mathfrak M_{k_0}^{k_0}$ and $x$ runs through the whole ring $X$, is dense in $R$, and the analogous set of vectors is dense in $R'$. All irreducible representations are normal.

M. A. Naimark proved the following proposition:

II (equivalence criterion). Two normal representations of the ring $X$ with the same lowest weight $k_0$ are equivalent if and only if the representations of the ring $X_{k_0}^{k_0}$ induced in the subspaces $\mathfrak M_{k_0}^{k_0}$ are equivalent.

Thus, the study of representations of the commutative ring $X_{k_0}^{k_0}$ leads to a description of all normal representations.

III. Representations given by the model (1) are normal only in the case when the operators $A_\varepsilon$ and $A'{\varepsilon^{-1}}$ have no eigenvalues $\varepsilon^{2r}$, $p=k_0+1,k_0+2,\ldots$. The normal representations corresponding to the pairs $(m,A\varepsilon)$ and $(-m,A'_{\varepsilon^{-1}})$ are equivalent.

IV. If a normal representation belongs to the class $K_0$, then it is equivalent to one of the representations of the canonical model.

(The proof is analogous to that indicated in (1) for irreducible representations.)

In order to describe all representations of the class $K_0$, define in the ring $X$ a two-sided ideal $\widetilde X$ by means of the following set of conditions:

[
\lambda_x(m,-il)=0,\qquad l=0,\pm1,\pm2,\ldots,\qquad m=|l|+1, |l|+2,\ldots
]

To every irreducible representation $\gamma$ there corresponds a maximal ideal—the kernel of the homomorphism $\gamma$. The ideal $\widetilde X$ is the intersection of all maximal ideals corresponding to special representations.

V. In the case of representations of the class $K_0$, the linear span of vectors $\xi=T_x\eta$, where $x$ runs through $\widetilde X$ and $\eta$ through all subspaces $\mathfrak M_{j_1}^{k_1}$, is dense in $R$, and the analogous set of vectors is dense in $R'$.

The ideal $\widetilde X$ has a simpler structure than the ring $X$: it contains no relations (3b) between the functions $\lambda_{j_1j_2}^{k_1k_2}(m,\rho)$. It decomposes into a discrete—

* With a proper definition of the direct sum of Banach spaces (see (${}^2$), p. 74).

the direct sum of ideals (\tilde X_m):

[
\tilde X_m={x:\lambda_x(m',\rho)=0 \text{ for } |m'|\ne |m|}.
]

From this fact follows the unique decomposability of vectors of the form (\xi=T_x\eta), (x\in\tilde X), into components (\xi_m=T_{x_m}\eta,\ x_m\in\tilde X_m), and, taking Proposition V into account, it follows that:

VI. Every representation of class (K_0) decomposes into a discrete direct sum of normal representations (which also belong to the class (K_0)).

This completes the proof of Theorem 2. Let us note that one can construct a single model for representations of class (K_0)—without decomposing them into normal ones.

The present work was carried out under the supervision of M. A. Naimark, who also proposed the canonical model and the proof of the equivalence criterion; the author expresses deep gratitude to him for his constant attention.

Moscow State University
named after M. V. Lomonosov

Received
24 I 1958

REFERENCES

({}^{1}) M. A. Naimark, Dokl. Akad. Nauk SSSR, 97, No. 6, 969 (1954).
({}^{2}) M. A. Naimark, Linear Representations of the Lorentz Group, 1958.