Abstract
Full Text
MATHEMATICS
N. Ya. VILENKIN, E. L. AKIM, and A. A. LEVIN
MATRIX ELEMENTS OF IRREDUCIBLE UNITARY REPRESENTATIONS OF THE GROUP OF EUCLIDEAN MOTIONS OF THREE-DIMENSIONAL SPACE AND THEIR PROPERTIES
(Presented by Academician A. N. Kolmogorov on 25 IX 1956)
The matrix elements of irreducible unitary representations of the group \(M(3, R)\) of Euclidean motions of three-dimensional space were first computed by A. M. Rodov by means of the infinitesimal method*. In the present note these matrix elements are computed by an integral method, and some properties of these matrix elements are derived.
The irreducible unitary representations of the group \(M(3, R)\) are specified by two parameters: an integer \(m\) and a positive number \(\rho\). The representation corresponding to the values of the parameters \(m\) and \(\rho\) is constructed in the space \(H_m\) of functions defined on the group \(SO(3)\) of rotations of three-dimensional space, having an integrable square with respect to the invariant measure \(d\mu(\omega)\) in \(SO(3)\) and satisfying the functional equation
\[ f(\gamma \omega)=e^{-im\alpha}f(\omega), \tag{1} \]
where \(\gamma\) is a rotation through an angle \(\alpha\) about the axis \(Oz\). To a motion \(g\), consisting of a parallel translation by the vector \(\mathbf h\) and a subsequent rotation \(\omega_0\), there is put in correspondence the operator \(Q_g\), which takes a function \(F(\omega)\) from \(H_m\) into the function
\[ Q_g F(\omega)=e^{-i\rho r\cos \alpha_\omega}F(\omega\omega_0). \tag{2} \]
Here \(r\) denotes the length of the vector \(\mathbf h\), and \(\alpha_\omega\) denotes the angle between the axis \(Oz\) and the vector \(\mathbf h_\omega\) obtained from the vector \(\mathbf h\) under the rotation \(\omega\).
Let us choose in the space \(H_m\) an orthonormal basis consisting of functions of the form
\[ \sqrt{2l+1}\,T^l_{mn}(\omega), \qquad l \geq |m|,\quad -l \leq n \leq l \tag{3} \]
(\(l\) is an integer), where \(T^l_{mn}(\omega)\) are the matrix elements of the irreducible unitary representations of weight \(l\) of the group \(SO(3)\) (see \((^1)\), p. 78, where the formula for \(T^l_{mn}(\omega)\) is given). In this basis, the operators corresponding to the rotation \(\omega_0\) are given by block-diagonal matrices whose main diagonals contain blocks of the form \(\|T^l_{nk}(\omega_0)\|\), \(l \geq |m|\). The operators \(Q_g\) corresponding to a displacement along the axis \(Oz\) by a vector of length \(r\) are given by matrices consisting of blocks \(\|J^{l_1}_l(g)\|\), \(l,l_1 \geq |m|\), moreover
* Reported by A. M. Rodov at the Third All-Union Mathematical Congress in June 1956. Previously reported in the seminar on functional analysis at Moscow State University.
the elements \(q_{ln,l_1n_1}(g)\) of the block \(\|J^{ll_1}(g)\|\) are equal to zero for \(n\ne n_1\) and are equal to \(J_{m,l,l_1,n}(\rho r)\) for \(n=n_1\), where
\[ J_{m,l,l_1,n}(x)= \]
\[ =(-1)^{m-n}\frac{\sqrt{(2l+1)(2l_1+1)}}{2} \int_{-1}^{1} e^{-i\mu x} P_{mn}^{l}(\mu)P_{-m,-n}^{l}(\mu)\,d\mu . \tag{4} \]
For the functions \(P_{mn}^{l}(\mu)\) see \((^1)\), p. 78.
Computing the integral (4), we find
\[ J_{m,l,l_1,n}(x)= \]
\[ =(-1)^{m-n}\frac{\sqrt{(2l+1)(2l_1+1)}}{2} \sum_{k=0}^{l+l_1} \frac{i^{k-1}\left[A_{m,l,l_1,n,k}e^{ix}-B_{m,l,l_1,n,k}e^{-ix}\right]}{x^{k+1}}, \tag{5} \]
where
\[ A_{m,l,l_1,n,k} = \left[P_{mn}^{l}(\mu)P_{-m,-n}^{l}(\mu)\right]_{\mu=1}^{(k)}, \tag{6} \]
\[ B_{m,l,l_1,n,k} = \left[P_{mn}^{l}(\mu)P_{-m,-n}^{l}(\mu)\right]_{\mu=-1}^{(k)}. \tag{7} \]
The computation of the constants \(A_{m,l,l_1,n,k}\) and \(B_{m,l,l_1,n,k}\) is easy to carry out by applying Leibniz’ theorem.
Another way of computing the integral (4) consists in replacing the product \(P_{mn}^{l_1}(\mu)P_{-m,-n}^{l}(\mu)\) by a linear combination of expressions \(P_{00}^{k}(\mu)\). This method leads to the equality
\[ J_{m,l,l_1,n}(x)= \]
\[ =(-1)^{m-n}\sqrt{\frac{\pi(2l+1)(2l_1+1)}{2x}} \sum_{k=|l-l_1|}^{l+l_1} C_{-n,n}^{k,0}C_{-m,m}^{k,0}J_{k+1/2}(x)i^{k}, \tag{8} \]
where \(J_{k+1/2}(x)\) are Bessel functions of half-integer index and \(C_{ij}^{k,i+j}\) are Clebsch–Gordan coefficients \((^2)\). In particular,
\[ J_{0,l,0,0}(x)=\sqrt{\frac{\pi(2l+1)}{2x}}J_{l+1/2}(x)i^{l}. \tag{8'} \]
From formulas (5) and (8′) one easily obtains an expression of Bessel functions of half-integer index in terms of trigonometric functions. Formula (8) was obtained by another method by A. M. Rodov.
The functions \(J_{m,l,l_1,n}(x)\) satisfy various relations that generalize the corresponding relations for Bessel functions. We shall indicate here the form of the addition theorem for the functions \(J_{m,l,l_1,n}(x)\).
Addition theorem. If the parameters \(r_1,r_2,\theta\) and \(r,\theta_1\) are related by the relations
\[ r=\sqrt{r_1^2+r_2^2+2r_1r_2\cos\theta}, \tag{9} \]
\[ \cos\theta_1=\frac{r_1\cos\theta+r_2}{r}, \tag{10} \]
then
\[ \sum_{k=-N}^{N} J_{m,l_1,l_2,k}(r)\, P_{m k}^{l_1}(\cos\theta_1)\, P_{k n_2}^{l_2}(\cos(\theta-\theta_1)) = \]
\[ = \sum_{l=-\infty}^{\infty} J_{m,l_1,l,n}(r_2) J_{m,l,l_2,n_2}(r_1) P_{n_1 n_2}^{l}(\cos\theta), \tag{11} \]
\[ N=\min(l_1,l_2). \]
\[ J_{m,l_1,l_2,n}(r)= \]
\[ = \sum_{l=-\infty}^{\infty}\sum_{s=-l_1}^{l_1}\sum_{t=-l}^{l}(-1)^{s-t}J_{m,l_1,l,s}(r_2)J_{m,l,l_2,t}(r_1)P^{l_1}_{sn}(\cos\theta_1)\times \tag{12} \]
\[ \times P^l_{st}(\cos\theta)P^{l_2}_{nt}(\cos(\theta-\theta_1)). \]
From the addition theorem (11) there follow recurrence relations for the functions \(J_{m,l_1,l_2,n}(x)\). For example, putting \(\theta=0\) in it, differentiating both sides with respect to \(r_1\), and putting \(r_1=0,\ r_2=x\), we find that
\[ J'_{m,l_1,l_2,k}(x)=\sum_{l=l_2-1}^{l_2+1}J_{m,l_1,l,k}(x)J'_{m,l,l_2,k}(0). \tag{13} \]
Putting \(\theta=\pi/2\), differentiating both sides with respect to \(r_1\), and putting \(r_1=0,\ r_2=x\), we find
\[ \frac{1}{x}P^{l_2}_{n_1,n_2}(0)J_{m,l_1,l_2,n_1}(x)- \]
\[ -\frac{i}{2x}\left[\sqrt{(l_1+n_2)(l_1-n_2+1)}\,P^{l_1}_{n_1,n_2-1}(0)J_{m,l_1,l_2,n_2-1}(x)+\right. \]
\[ \left.+\sqrt{(l_1+n_2+1)(l_1-n_2)}\,P^{l_1}_{n_1,n_2+1}(0)J_{m,l_1,l_2,n_2+1}(x)\right]= \]
\[ =\sum_{l=l_2-1}^{l_2+1}P^l_{n_1n_2}(0)J'_{m,l,l_2,n_2}(0)J_{m,l_1,l,n_1}(x). \tag{14} \]
The functions \(J_{m,l_1,l_2,n}(x)\) satisfy a second-order differential equation found by A. M. Rodov.
Let us decompose the representation of the subgroup of two-dimensional motions, induced by the representation (2) of the group \(M(3,R)\), into irreducible representations. Then one obtains a formula that is a special case of Sonine’s integral ((\(^3\)), p. 406) for \(\nu=-1/2\) and integral \(\mu\). Sonine’s integral for arbitrary integral and half-integral values of \(\mu\) and \(\nu\) is obtained by considering representations of the group of \(n\)-dimensional motions. Consideration of the group of \(n\)-dimensional motions also makes it possible to give a group-theoretic interpretation of the second Sonine integral ((\(^3\)), p. 410), as well as of the Gegenbauer–Sonine integral ((\(^3\)), p. 400).
Some relations in the theory of Bessel functions are obtained by changing the basis in the representation space. In this way, for example, Sonine’s expansion ((\(^3\)), p. 153) can be obtained.
Military Engineering Academy
named after V. V. Kuibyshev
Received
1 VIII 1956
REFERENCES
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- L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Part I, 1948, p. 407.
- G. N. Watson, Theory of Bessel Functions, Part I, 1949.