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UDC 530.12:531.18
MATHEMATICS
A. N. GOLUBYATNIKOV, V. V. LOKHIN
TENSOR INVARIANTS OF SUBGROUPS OF THE LORENTZ GROUP
(Presented by Academician L. I. Sedov, December 2, 1968)
For every subgroup $\mathcal{L}_k$ of the Lorentz group $\mathcal{L}$ that can be specified by the requirement of invariance of some finite set of tensors $H_{(1)}, \ldots, H_{(n)}$, any other tensor invariant $T$ of it, of arbitrary rank, can be represented in the form of a linear combination of tensors $T_{(1)}, \ldots, T_{(p)}$ constructed by means of the operations of tensor multiplication and contraction from the basic tensors $H_{(1)}, \ldots, H_{(n)}$:
\[ T = k_1 T_{(1)} + \ldots + k_p T_{(p)}, \]
where the tensors $T_{(1)}, \ldots, T_{(p)}$ form a basis of the linear space of tensors of rank $r$ invariant with respect to the group $\mathcal{L}_k$, the dimension of which is denoted by $p$.*
Thus, the construction of tensor invariants for subgroups of the Lorentz group $\mathcal{L}$ reduces to finding the basic tensors $\{H_m\}$ for all subgroups $\mathcal{L}_k$ of the Lorentz group $\mathcal{L}$ that can be specified with the aid of tensors. All these subgroups $\mathcal{L}_k$ are Lie subgroups of the Lorentz group with a finite component group, whose classification is given in (⁶). The method for choosing all subgroups $\mathcal{L}_k$ and obtaining their basic tensors $\{H_m\}$ is indicated in (⁵).
Table 1 lists all subgroups $\mathcal{L}_k$ of the Lorentz group $\mathcal{L}$ that are specified with the aid of tensors, and for all these subgroups the corresponding sets of tensors are written out. In some cases the group can be specified by a smaller number of invariant tensors; however, the tensors indicated are sufficient to obtain the entire algebra of tensor invariants.
At the beginning of each series of formulas the infinitesimal operators of the maximal connected subgroup $\mathcal{G}_l$ of the group $\mathcal{L}_k$ under consideration are indicated, while the symbols of A. V. Shubnikov** are used to denote its finite component group (⁶).
The infinitesimal operators $X_q$ of the proper Lorentz group of transformations of the variables $x^i$ $(i = 1, 2, 3, 4)$, which leave invariant the quadratic form $x^{12} + x^{22} + x^{32} - x^{42}$, have the form
\[ X_1 = x^2 \partial_3 - x^3 \partial_2,\quad X_2 = x^3 \partial_1 - x^1 \partial_3,\quad X_3 = x^1 \partial_2 - x^2 \partial_1, \]
\[ X_4 = x^1 \partial_4 + x^4 \partial_1,\quad X_5 = x^2 \partial_4 + x^4 \partial_2,\quad X_6 = x^3 \partial_4 + x^4 \partial_3, \]
where $\partial_i = \partial/\partial x^i$.
The vectors $e_1, e_2, e_3, e_4$ form an orthonormal basis. Products and powers of the vectors $e_i$ are understood as tensor products, $e_{ij\ldots k} = e_i e_j \ldots e_k$. The symbols $[\,]$, $(\,)$ attached to indices denote alternation and symmetrization.
* Basic tensors $\{H_{(m)}\}$ and tensor bases $\{T_{(s)}\}$ of the first, second, third, and fourth ranks for subgroups of the group $O(3)$ were established in (¹–⁴).
** $n$ denotes a rotation about the $x^3$ axis through the angle $2\pi/n$; $1 : 2$ denotes a rotation about the $x^2$ axis through the angle $\pi$; reflection of $x^3$ is denoted by the symbol $:m$, or by a bar over the reflected $x^3$; reflection of $x^2$ by the symbol $\cdot m$, reflection of $x^4$ by a subscript bar, $\cdot m'$ denotes reflection of $x^1$, $1 : 2'$ denotes rotation about the $x^1$ axis through the angle $\pi$. $3/2$, $3/4$, and $3/5$ denote the symmetry groups of the tetrahedron, octahedron, and icosahedron.
Table 1
| Subgroup | Parameter | Tensor invariants | Subgroup | Parameter | Tensor invariants |
|---|---|---|---|---|---|
| $\mathcal G_1(X_1,X_2,X_3,X_4,X_5,X_6)$ | $\mathfrak g,\ E$ | $1\cdot\underline m$ | $\mathfrak g,\ E,\ \mathbf e_4^2,\ \mathbf e_3$ | ||
| $\dfrac{\underline 2}{2}\cdot m$ | $\mathfrak g$ | $1\cdot m$ | $\mathfrak g,\ \mathbf e_4^2,\ \mathbf e_3$ | ||
| $\mathcal G_2(X_1,X_2,X_3)$ | $\dfrac{\underline 2}{2}\cdot m$ | $\mathfrak g,\ \mathbf e_3^2,\ \mathbf e_4$ | |||
| $1$ | $\mathfrak g,\ E,\ \mathbf e_4$ | $\dfrac{2}{\underline 2}\cdot m$ | $\mathfrak g,\ \mathbf e_{[34]},\ \mathbf e_4^2$ | ||
| $\dfrac{\underline 2}{2}$ | $\mathfrak g,\ \mathbf e_4$ | $\dfrac{\underline 2}{2}\cdot m$ | $\mathfrak g,\ E,\ \mathbf e_3^2,\ \mathbf e_4^2$ | ||
| $\dfrac{1}{2}$ | $\mathfrak g,\ \mathbf e_{[123]}$ | $\dfrac{2}{\underline 2}\cdot m$ | $\mathfrak g,\ \mathbf e_{[124]},\ \mathbf e_4^2$ | ||
| $\dfrac{\underline 2}{2}$ | $\mathfrak g,\ E,\ \mathbf e_4^2$ | $\dfrac{2}{\underline 2}\cdot m\times\underline 1$ | $\mathfrak g,\ \mathbf e_3^2,\ \mathbf e_4^2$ | ||
| $\dfrac{\underline 2}{2}\times\underline 1$ | $\mathfrak g,\ \mathbf e_4^2$ | $\mathcal G_8(X_6)$ | |||
| $\mathcal G_3(X_3,X_4,X_5)$ | $n:\underline m$ | $\mathfrak g,\ E,\ D_{nh},\ \mathbf e_{[12]}$ | |||
| $1\cdot\underline m$ | $\mathfrak g,\ E,\ \mathbf e_3$ | $\underline m\cdot n:\underline m$ | $\mathfrak g,\ E,\ D_{nh},\ \mathbf e_2^2\ (n=1)$ | ||
| $1\cdot m$ | $\mathfrak g,\ \mathbf e_3$ | $m\cdot n:\underline m$ | $\mathfrak g,\ D_{nh},\ \mathbf e_{[34]}$ | ||
| $\dfrac{\underline 2}{2}\cdot m$ | $\mathfrak g,\ E,\ \mathbf e_3^2$ | $n:\underline m\times\underline 1$ | $\mathfrak g,\ D_{nh},\ \mathbf e_{[12]}$ | ||
| $\dfrac{2}{\underline 2}\cdot m$ | $\mathfrak g,\ \mathbf e_{[124]}$ | $m\cdot n:\underline m\times\underline 1$ | $\mathfrak g,\ D_{nh},\ \mathbf e_2^2\ (n=1)$ | ||
| $\dfrac{2}{\underline 2}\cdot m\times\underline 1$ | $\mathfrak g,\ \mathbf e_3^2$ | $2n:\underline m$ | $\mathfrak g,\ \mathbf e_{[34]}D_{nh},\ \mathbf e_{[12]}$ | ||
| $\mathcal G_4(X_3,\ X_4-X_2,\ X_1+X_5)$ | $m\cdot 2n:\underline m$ | $\mathfrak g,\ \mathbf e_{[34]}D_{nh}$ | |||
| $1$ | $\mathfrak g,\ E,\ \mathbf e_3+\mathbf e_4$ | $\mathcal G_9(X_4-X_2)$ | |||
| $1:\underline 2$ | $\mathfrak g,\ \mathbf e_{[123]}+\mathbf e_{[124]}$ | $1$ | $\mathfrak g,\ E,\ \mathbf e_2,\ \mathbf e_3+\mathbf e_4$ | ||
| $\underline 2$ | $\mathfrak g,\ E,\ (\mathbf e_3+\mathbf e_4)^2$ | $2$ | $\mathfrak g,\ E,\ \mathbf e_2^2,\ \mathbf e_3+\mathbf e_4$ | ||
| $1\cdot m$ | $\mathfrak g,\ \mathbf e_3+\mathbf e_4$ | $1:\underline 2$ | $\mathfrak g,\ \mathbf e_{[123]}+\mathbf e_{[124]},\ \mathbf e_2$ | ||
| $\dfrac{1}{2}\cdot m$ | $\mathfrak g,\ (\mathbf e_3+\mathbf e_4)^2$ | $\underline 2$ | $\mathfrak g,\ E,\ \mathbf e_2^2,\ \mathbf e_2(\mathbf e_3+\mathbf e_4)$ | ||
| $\mathcal G_5(X_3,X_6)$ | $\dfrac{2}{\underline 2}:\underline 2$ | $\mathfrak g,\ \mathbf e_{[123]}+\mathbf e_{[124]},\ \mathbf e_2^2$ | |||
| $\underline 2$ | $\mathfrak g,\ E,\ \mathbf e_{[12]}$ | $1:\underline m$ | $\mathfrak g,\ E,\ (\mathbf e_3+\mathbf e_4)^2\mathbf e_2$ | ||
| $\dfrac{\underline 2}{2}\cdot m$ | $\mathfrak g,\ E,\ \mathbf e_3^2-\mathbf e_4^2$ | $2:\underline m$ | $\mathfrak g,\ E,\ \mathbf e_2^2,\ (\mathbf e_3+\mathbf e_4)^2$ | ||
| $\dfrac{\underline 2}{2}\times\underline 1$ | $\mathfrak g,\ \mathbf e_{[12]}$ | $1\cdot m$ | $\mathfrak g,\ \mathbf e_{[134]},\ \mathbf e_3+\mathbf e_4$ | ||
| $\underline 2\cdot m$ | $\mathfrak g,\ \mathbf e_{[34]}$ | $2\cdot m$ | $\mathfrak g,\ \mathbf e_2^2,\ \mathbf e_3+\mathbf e_4$ | ||
| $\underline 2\cdot m\times\underline 1$ | $\mathfrak g,\ \mathbf e_3^2-\mathbf e_4^2$ | $m'\cdot 1:\underline m$ | $\mathfrak g,\ (\mathbf e_3+\mathbf e_4)^2,\ \mathbf e_2$ | ||
| $\mathcal G_6(X_4-X_2,\ X_1+X_5)$ | $m\cdot 1:\underline m$ | $\mathfrak g,\ \mathbf e_{[134]},\ (\mathbf e_3+\mathbf e_4)^2$ | |||
| $n$ | $\mathfrak g,\ E,\ \mathbf e_3+\mathbf e_4,\ \omega_n$ | $m\cdot 2:\underline m$ | $\mathfrak g,\ \mathbf e_2^2,\ (\mathbf e_3+\mathbf e_4)^2$ | ||
| $\underline{2n}$ | $\mathfrak g,\ E,\ (\mathbf e_3+\mathbf e_4)^2,\ \omega_n\ (n\ \text{odd}),\ \omega_{2n},\ (\mathbf e_3+\mathbf e_4)\omega_n\ (n\ \text{even})$ | $\underline 2\cdot m$ | $\mathfrak g,\ \mathbf e_2^2,\ \mathbf e_{[14]}+\mathbf e_{[13]}$ | ||
| $n:\underline 2$ | $\mathfrak g,\ \mathbf e_{[123]}+\mathbf e_{[124]},\ \omega_n$ | $1:\underline 2'$ | $\mathfrak g,\ \mathbf e_{[134]},\ \mathbf e_2(\mathbf e_3+\mathbf e_4)$ | ||
| $n:\underline m$ | $\mathfrak g,\ E,\ (\mathbf e_3+\mathbf e_4)^2,\ \omega_{2n},\ (\mathbf e_3+\mathbf e_4)\omega_n\ (n\ \text{odd}),\ \omega_n\ (n\ \text{even})$ | $1\cdot m'$ | $\mathfrak g,\ \mathbf e_2,\ \mathbf e_3+\mathbf e_4$ | ||
| $n\cdot m$ | $\mathfrak g,\ \mathbf e_3+\mathbf e_4,\ \omega_n$ | $\underline 2\cdot m'$ | $\mathfrak g,\ \mathbf e_2^2,\ \mathbf e_2(\mathbf e_3+\mathbf e_4)$ | ||
| $m\cdot n:\underline m$ | $\mathfrak g,\ (\mathbf e_3+\mathbf e_4)^2,\ \omega_{2n},\ (\mathbf e_3+\mathbf e_4)\omega_n\ (n\ \text{odd}),\ \omega_n\ (n\ \text{even})$ | Finite groups | |||
| $\underline{2n}\cdot m$ | $\mathfrak g,\ (\mathbf e_3+\mathbf e_4)^2,\ \omega_n\ (n\ \text{odd}),\ \omega_{2n},\ (\mathbf e_3+\mathbf e_4)\omega_n\ (n\ \text{even})$ | $n$ | $\mathfrak g,\ E,\ D_{nh},\ \mathbf e_3,\ \mathbf e_4$ | ||
| $\mathcal G_7(X_3)$ | $\underline{2n}$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_4D_{nh},\ \mathbf e_3$ | |||
| $1$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_3\mathbf e_4$ | $n\times\underline 1$ | $\mathfrak g,\ \mathbf e_{[12]},\ D_{nh},\ \mathbf e_3$ | ||
| $\underline 1$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_3$ | $n:\underline 2$ | $\mathfrak g,\ E,\ \mathbf e_4,\ \mathbf e_3D_{nh}\ (n\ \text{odd}),\ D_{nh}\ (n\ \text{even})$ | ||
| $1:\underline 2$ | $\mathfrak g,\ E,\ \mathbf e_3^2,\ \mathbf e_4$ | $n:\underline 2$ | $\mathfrak g,\ \mathbf e_{[123]},\ \mathbf e_3\mathbf e_4,\ \mathbf e_3D_{nh}\ (n\ \text{even}),\ D_{nh}\ (n\ \text{even})$ | ||
| $1:2$ | $\mathfrak g,\ \mathbf e_{[123]},\ \mathbf e_{[34]}$ | $\underline{2n}:2$ | $\mathfrak g,\ \mathbf e_{[123]},\ \mathbf e_3\mathbf e_4D_{nh}\ (n\ \text{odd}),\ \mathbf e_4D_{nh}\ (n\ \text{even})$ | ||
| $1:\underline 2$ | $\mathfrak g,\ \mathbf e_3\mathbf e_{[12]}$ | $n:\underline 2\times\underline 1$ | $\mathfrak g,\ \mathbf e_{[123]},\ \mathbf e_3D_{nh}\ (n\ \text{odd}),\ D_{nh}\ (n\ \text{even})$ | ||
| $\underline 2$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_4$ | $\underline{2n}$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_3D_{nh},\ \mathbf e_4$ | ||
| $2$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_{[34]},\ \mathbf e_4^2$ | $\dfrac{\underline{2n}}{2n}$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_3D_{nh},\ \mathbf e_3\mathbf e_4$ | ||
| $\dfrac{\underline 2}{2}\times\underline 1$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_4^2$ | $\underline{2n}\times$ | $\mathfrak g,\ \mathbf e_{[12]},\ \mathbf e_3D_{nh}$ | ||
| $1\cdot m$ | $\mathfrak g,\ \mathbf e_{[34]},\ \mathbf e_4$ |
(continued)
| \(n:m\) | \(g,\ \mathbf e_{[12]},\ D_{nh},\ \mathbf e_4\) | \(3/2\) | \(g,\ \mathbf e_{[123]},\ T_d,\ \mathbf e_4\) |
| \(n:\underline m\) | \(g,\ \mathbf e_{[12]},\ D_{nh},\ \mathbf e_3\mathbf e_4\) | \(3/2\times\underline 1\) | \(g,\ \mathbf e_{[123]},\ T_d\) |
| \(\underline{2n}:m\) | \(g,\ \mathbf e_{[12]},\ \mathbf e_4D_{nh}\) | \(\overline 6/2\) | \(g,\ T_h,\ \mathbf e_4\) |
| \(\underline{2n}:\underline m\) | \(g,\ \mathbf e_{[12]},\ \mathbf e_3\mathbf e_4D_{nh}\) | \(\overline 6/2\) | \(g,\ E,\ T_h\) |
| \(n:m\times\underline 1\) | \(g,\ \mathbf e_{[12]},\ D_{nh},\ \mathbf e_3^2\) | \(\overline 6/2\times\underline 1\) | \(g,\ T_h\) |
| \(n\cdot m\) | \(g,\ P_{nh},\ \mathbf e_3,\ \mathbf e_4\) | \(3/4\) | \(g,\ \mathbf e_{[123]},\ O_h,\ \mathbf e_4\) |
| \(n\cdot\underline m\) | \(g,\ D_{nh},\ \mathbf e_4\mathbf e_{[12]},\ \mathbf e_3\) | \(3/\overline 4\) | \(g,\ \mathbf e_{[123]},\ \mathbf e_4T_d\) |
| \(\underline{2n}\cdot m\) | \(g,\ \mathbf e_4D_{nh},\ \mathbf e_3\) | \(3/\overline 4\times\underline 1\) | \(g,\ \mathbf e_{[123]},\ O_h\) |
| \(n\cdot m\times\underline 1\) | \(g,\ D_{nh},\ \mathbf e_3,\ \mathbf e^2\ (n=1)\) | \(3/\overline 4\) | \(g,\ T_d,\ \mathbf e_4\) |
| \(m\cdot n:m\) | \(g,\ D_{nh},\ \mathbf e_4,\ \mathbf e_2^2\ (n=1)\) | \(3/\overline 4\) | \(g,\ E,\ T_d\) |
| \(\underline m\cdot n:m\) | \(g,\ D_{nh},\ \mathbf e_4\mathbf e_{[12]}\) | \(3/\overline 4\times\underline 1\) | \(g,\ T_d\) |
| \(\underline m\cdot n:\underline m\) | \(g,\ E,\ D_{nh},\ \mathbf e_4^2,\ \mathbf e_2^2\ (n=1)\) | \(\overline 6/4\) | \(g,\ O_h,\ \mathbf e_4\) |
| \(m\cdot n:\underline m\) | \(g,\ D_{nh},\ \mathbf e_3\mathbf e_4\) | \(\overline 6/4\) | \(g,\ \mathbf e_4T_h\) |
| \(m\cdot\underline{2n}:m\) | \(g,\ \mathbf e_4D_{nh},\ \mathbf e_2^2\ (n=1)\) | \(\overline 6/\overline 4\) | \(g,\ E,\ O_h\) |
| \(m\cdot\underline{2n}:\underline m\) | \(g,\ \mathbf e_3\mathbf e_4D_{nh}\) | \(\overline 6/\overline 4\) | \(g,\ \mathbf e_4T_d\) |
| \(m\cdot n:m\times\underline 1\) | \(g,\ D_{nh},\ \mathbf e_3^2,\ \mathbf e_2^2\ (n=1)\) | \(\overline 6/\overline 4\times\underline 1\) | \(g,\ O_h\) |
| \(\overline{2n}\cdot m\) | \(g,\ \mathbf e_3D_{nh},\ \mathbf e_4\) | \(3/5\) | \(g,\ \mathbf e_{[123]},\ Y_h,\ \mathbf e_4\) |
| \(\overline{2n}\cdot m\) | \(g,\ \mathbf e_3D_{nh},\ \mathbf e_3\mathbf e_4\) | \(3/5\times\underline 1\) | \(g,\ \mathbf e_{[123]},\ Y_h\) |
| \(\overline{2n}\cdot\underline m\) | \(g,\ E,\ \mathbf e_3D_{nh},\ \mathbf e_2^2\ (n=1)\) | \(3/\overline{10}\) | \(g,\ Y_h,\ \mathbf e_4\) |
| \(\overline{2n}\cdot\underline m\) | \(g,\ \mathbf e_3D_{nh},\ \mathbf e_4\mathbf e_{[12]}\) | \(3/\overline{10}\) | \(g,\ \mathbf e_4Y_h\) |
| \(\overline{2n}\cdot m\times\underline 1\) | \(g,\ \mathbf e_3D_{nh},\ \mathbf e_2^2\ (n=1)\) | \(3/\overline{10}\times\underline 1\) | \(g,\ Y_h\) |
For some tensors special notations have been introduced: \(g=e_1^2+e_2^2+e_3^2-e_4^2\) is the metric tensor specifying the Lorentz group \(\mathcal L\); \(E=e_{[1234]}\); \(O_h=e_1^4+e_2^4+e_3^4\); \(T_d=e_{(123)}^4\); \(T_h=e_1^2e_2^2-e_2^2e_1^2+e_2^2e_3^2-e_3^2e_2^2+e_3^2e_1^2-e_1^2e_3^2\); \(Y_h=5e_1^6+e_1^4e_2^2+e_1^2e_2^4+5e_2^6-e_1^5e_3+e_1^3e_2^2e_3+e_1e_2^4e_3+2e_1^2e_3^4+2e_2^2e_3^4+2e_3^6\), where the square bracket above denotes summation over all isomers (differing only in the order of the factors); \(D_{nh}=\operatorname{Re}(e_1+ie_2)^n=e_1^n-e_1^{\,n-2}e_2^2+e_1^{\,n-4}e_2^4-\ldots\); \(\omega_n=\operatorname{Re}(e_{[31]}+e_{[41]}+ie_{[32]}+ie_{[42]})^n\).
Using the known theorems and formulas of the theory of finite-dimensional representations of classical groups (see, for example, \((^7)\)), one can, by means of the methods of analytic continuation, induction, and the theory of characters of compact groups, derive simple formulas for the dimension \(p_i=p_i(r,\mathscr G_i)\) of the space of invariant tensors of rank \(r\) for all connected subgroups \(\mathscr G_i\) of the Lorentz group: \(p_1=\bigl(C_r^{r/2}(r/2+1)\bigr)^2\) for even \(r\), and \(p_1=0\) for odd \(r\); \(p_2=p_3=p_4=C_{2r}^r/(r+1)\), \(p_5=(2C_{r-1}^{r/2})^2\) for even \(r\), and \(p_5=0\) for odd \(r\); \(p_6=(C_r^{[r/2]})^2\), \(p_7=p_8=p_9=C_{2r}^r\).
The authors express their deep gratitude to L. I. Sedov for his attention to and discussion of the work.
Scientific Research Institute of Mechanics
Moscow State University
named after M. V. Lomonosov
Received
11 XI 1968
CITED LITERATURE
\(^1\) V. V. Lokhin, DAN, 149, No. 2, 295 (1963).
\(^2\) V. V. Lokhin, DAN, 149, No. 6, 1282 (1963).
\(^3\) L. I. Sedov, V. V. Lokhin, DAN, 149, No. 4, 796 (1963).
\(^4\) V. V. Lokhin, L. I. Sedov, PMM, 27, issue 3, 393 (1963).
\(^5\) V. V. Lokhin, DAN, 186, No. 3 (1969).
\(^6\) A. N. Golubyatnikov, DAN, 186, No. 3 (1969).
\(^7\) D. P. Zhelobenko, UMN, 17, issue 1 (109), 27 (1962).