UDC 512.865

MATHEMATICS

A. N. GOLUBYATNIKOV

NONLINEAR SPINOR FUNCTIONS

(Presented by Academician L. I. Sedov, 3 IV 1965)

1. A spinor of weight \(j\) and rank \(r\) is a quantity that transforms according to the \(r\)-th power of an irreducible representation of weight \(j\) of the full orthogonal group \(W\) of transformations of three-dimensional Euclidean space \((^1)\). The representation of half-integral weight, which is two-valued, coincides with a representation of the universal covering group of \(W\), which is single-valued. A quantity transforming according to this representation is a spinor of the corresponding weight.

We shall say that a spinor of weight \(j\) and rank \(r\) admits a symmetry group \(G \subseteq W\), whose representation is realized by some collection of matrices of the representation of weight \(j\) of the group \(W\), if for each of these matrices the equalities

\[ A^{i_1 \ldots i_r} = a^{i_1}_{l_1}\cdots a^{i_r}_{l_r} A^{l_1 \ldots l_r}. \tag{1} \]

hold.

Consider a spinor function \(A(B_1,\ldots,B_n)\), which is a spinor of weight \(j\) and rank \(r\) and depends on \(n\) spinors of weights \(j_1,\ldots,j_n\) and ranks \(r_1,\ldots,r_n\). Let the spinor \(B_k\) of weight \(j_k\) and rank \(r_k\) \((k=1,\ldots,n)\) admit the symmetry group \(G_k\); then the spinor function \(A(B_1,\ldots,B_n)\) will admit the symmetry group \(G\), which is the intersection of the symmetry groups of the arguments \(G_1,\ldots,G_n\). This follows from the fact that the components of the function are functions of the components of the arguments, which admit the symmetry group \(G\); therefore the components of \(A(B_1,\ldots,B_n)\) also admit the same symmetry group.

This allows us to write the general form of the function, if all invariant linearly independent spinors of weight \(j\) and rank \(r\) with respect to the group \(G\), \(J_1,\ldots,J_m\), are known:

\[ A=\sum_{p=1}^{m} k_p J_p, \tag{2} \]

where the scalar coefficients \(k_p\) are functions of the joint scalar invariants of the arguments.

2. Consider the representation of weight \(1/2\). Knowing the matrices of this representation, one can find, by the method of averaging over the group, spinors of various ranks invariant with respect to subgroups of the group \(W\), but one can also, by requiring the invariance of certain spinors, obtain from equations (1) the matrix elements \(a_l^i\) of the representation of subgroups of the full orthogonal group \(W\). It is obvious that any invariant spinor with respect to some group of transformations is a function of spinors specifying this group in the indicated way, and therefore is obtained, by means of the operations of spinor algebra, from its arguments as a spinor function.

The number \(m\) in formula (2) is determined, as is known from the theory of characters \((^1)\), by averaging over the group the function \((a_i^i)^r\), where \(r\) is the exponent. For odd \(r\), \(m=0\), since under averaging, along with the terms

the term \((-a_i{}^i)^r\) enters, i.e., there are no improper invariants among spinors of rank \(2k-1\) \((k=1,2,\ldots)\).

Let the spinors \(e_1, e_2\) form a covariant canonical basis; then the fundamental spinor has the form:

\[ G=g^{ij}e_i e_j=e_1e_2-e_2e_1. \]

The requirement of its invariance selects, from the set of 4-parameter matrices,

\[ \left\|\begin{matrix} \alpha & \beta\\ -\overline{\beta} & \overline{\alpha} \end{matrix}\right\| \]

matrices with determinant equal to 1, i.e. we obtain a representation of the groups \(\infty/\infty\cdot m,\ \infty/\infty\).

Below, when considering subgroups of the group \(W\), the invariance of \(G\) will be assumed.

The requirement of invariance of the spinor \(e_1e_2^2e_1+e_2e_1^2e_2\) gives a representation of the groups \(m\cdot\infty:m,\ \infty:2,\ \infty\cdot m\), and the invariance of \(e_1e_2\)—a representation of the groups \(\infty:m,\ \infty\).

Similarly, invariance of \(e_1^2,\ e_1e_2\) gives a representation of the groups \(1,\ \overline{2};\ e_1e_2,\ e_1^4\)—of the groups \(m,\ 2,\ 2:m;\ e_1^4+e_2^4,\ e_1e_2^2e_1+e_2e_1^2e_2\)—of the groups \(2\cdot m,\ 2:2,\ m\cdot2:m;\ e_1e_2,\ e_1^6\)—of the groups \(3;\ e_1^6-e_2^6,\ e_1e_2^2e_1+e_2e_1^2e_2\)—of the groups \(3:2;\ e_1e_2,\ e_1^{12}\)—of the groups \(6,\ 6,\ 3:m,\ 6:m;\ e_1e_2^2e_1+e_2e_1^2e_2,\ e_1^6+e_2^6\)—of the groups \(3\cdot m;\ e_1e_2^6e_1+e_2e_1^6e_2,\ e_1^{12}+e_2^{12}\)—of the groups \(6\cdot m,\ 6\cdot m,\ 6:2,\ m\cdot3:m,\ m\cdot6:m;\ e_1e_2,\ e_1^8\)—of the groups \(\overline{4},\ 4,\ 4:m;\ e_1e_2^2e_1+e_2e_1^2e_2,\ e_1^8+e_2^8\)—of the groups \(4\cdot m,\ 4:2,\ \overline{4}\cdot m,\ m\cdot4:m\) (all prismatic groups are obtained in the same way).

The condition of invariance of the spinor \(\overline{e_1^5e_2}-e_1\overline{e_2^5}\), where \(\overline{\phantom{a}}\) denotes summation over all isomers, gives

\[ \alpha^4|\alpha|^2-5\alpha^4|\beta|^2-5\overline{\beta}^{\,4}|\alpha|^2+\overline{\beta}^{\,4}|\beta|^2=1, \]

\[ -6\alpha^5\overline{\beta}+6\overline{\beta}^{\,5}\alpha=0. \]

Solving this system under the condition \(|\alpha|^2+|\beta|^2=1\), we obtain either

\[ \alpha=0,\quad \beta=e^{\,i\frac{\pi}{2}k}\quad (k=1,2,\ldots), \]

or

\[ \beta=0,\quad \alpha=e^{\,i\frac{\pi}{2}k}, \]

or

\[ \alpha^4=\overline{\beta}^{\,4},\quad \alpha=e^{\,i\frac{\pi}{4}(2k-1)}, \]

i.e. a representation of the groups \(\overline{6}/2,\ 3/2\).

Since there is a relation between the “natural” basis of the vector representation (weight 1) and the spinor canonical basis (weight \(1/2\)) \((^1)\)

\[ e_x=\frac{1}{\sqrt{2}}(e_1^2-e_2^2),\qquad e_y=\frac{1}{\sqrt{2}}(e_1^2+e_2^2),\qquad e_z=\frac{1}{\sqrt{2}}(e_1e_2+e_2e_1), \]

the tensors defining the vector representation of the group \((^2)\), written in the spinor canonical basis, will be invariant with respect to the spinor representation, which facilitates the finding of invariants; the tensors \(O_h\) and \(Y_h\), defining respectively the groups \(-\overline{6}/4\) and \(3/10\), have the form:

\[ O_h=e_x^4+e_y^4+e_z^4, \]

\[ Y_h={}^{2}/_{25}\,[2(5e_x^6+\overline{e_x^4e_y^2}+\overline{e_x^2e_y^2}+5e_y^6)+(\overline{e_x^5e_z}+\overline{e_x^3e_y^2e_z}+\overline{e_xe_y^4e_z})+ \]

\[ +(\overline{3e_x^4e_z^2}+\overline{e_x^2e_y^2e_z^2}+\overline{3e_y^4e_z^2})+(\overline{e_x^2e_z^4}+\overline{e_y^2e_z^4})+13e_z^6]. \]

1. Let us write the general form of some spinor functions \((2)\).

Groups \(\infty/\infty\cdot m,\ \infty/\infty:\quad A^{\alpha\beta}=k_1g^{\alpha\beta},\quad A^{\alpha\beta\gamma\delta}=k_1g^{\alpha\beta}g^{\gamma\delta}+k_2g^{\alpha\gamma}g^{\beta\delta}.\)

Groups \(\infty:m,\ \infty:\quad A^{\alpha\beta}e_\alpha e_\beta=k_1e_1e_2+k_2e_2e_1,\quad A^{\alpha\beta\gamma\delta}e_\alpha e_\beta e_\gamma e_\delta=\)

\[ =k_1e_1^2e_2^2+k_2e_1e_2e_1e_2+k_3e_1e_2^2e_1+k_4e_2e_1^2e_2+k_5e_2e_1e_2e_1+k_6e_2^2e_1^2. \]

Groups \(m\cdot\infty:m,\ \infty:2,\ \infty\cdot m:\quad A^{\alpha\beta}=A^{\alpha\beta}\ (\infty/\infty),\quad A^{\alpha\beta\gamma\delta}e_\alpha e_\beta e_\gamma e_\delta=\)

\[ =k_1(e_1^2e_2^2+e_2^2e_1^2)+k_2(e_1e_2e_1e_2+e_2e_1e_2e_1)+k_3(e_1e_2^2e_1+e_2e_1^2e_2). \]

Groups \(1,\bar{2}\): the most general form.

Groups \(m,\,2,\,2:m\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty),\qquad A^{\alpha\beta\gamma\delta}e_{\alpha}e_{\beta}e_{\gamma}e_{\delta} =k_1e_1^4+k_2e_2^4+ \]
\[ {}+A^{\alpha\beta\gamma\delta}(\infty)e_{\alpha}e_{\beta}e_{\gamma}e_{\delta}. \]

Groups \(2\cdot m,\,2:2,\,m\cdot2:m\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta}e_{\alpha}e_{\beta}e_{\gamma}e_{\delta} = k_1(e_1^4+e_2^4)+A^{\alpha\beta\gamma\delta}(\infty\cdot m)e_{\alpha}e_{\beta}e_{\gamma}e_{\delta}. \]

Group \(3\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty). \]

Group \(\bar{3}\cdot m\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty\cdot m). \]

Groups \(\bar{6},\,6,\,3:m,\,6:m\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty). \]

Group \(3:2\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty\cdot m). \]

Groups \(6\cdot m,\,\bar{6}\cdot m,\,6:2,\,m\cdot3:m,\,m\cdot6:m\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty), \]
\[ A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty\cdot m). \]

Groups \(\bar{4},\,4,\,4:m\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty). \]

Groups \(4\cdot m,\,\bar{4}\cdot m,\,m\cdot4:m,\,4:2\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta} \]
\[ {}=A^{\alpha\beta\gamma\delta}(\infty\cdot m). \]

Groups \(\bar{6}/2,\,3/2\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty/\infty), \]
\[ A^{\alpha\beta\gamma\delta\varepsilon\xi}e_{\alpha}e_{\beta}e_{\gamma}e_{\delta}e_{\varepsilon}e_{\xi} = A^{\alpha\beta\gamma\delta\varepsilon\xi}(\infty/\infty) +k_6(e_1^5e_2-e_1e_2^5). \]

Groups \(3/\bar{4},\,\bar{6}/4,\,3/4\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\beta\gamma\delta}(\infty/\infty), \]
\[ A^{\alpha\beta\gamma\delta\varepsilon\xi}=A^{\alpha\beta\gamma\delta\varepsilon\xi}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta} = A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta}(\infty/\infty) +k_{15}O_h^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta}. \]

Groups \(3/\overline{10},\,3/5\):
\[ A^{\alpha\beta}=A^{\alpha\beta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta}=A^{\alpha\delta\gamma\beta}(\infty/\infty), \]
\[ A^{\alpha\beta\gamma\delta\varepsilon\xi} = A^{\alpha\beta\gamma\delta\varepsilon\xi}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta} = A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta ix} \]
\[ = A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta ix}(\infty/\infty),\qquad A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta ix\lambda\mu} = A^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta ix\lambda\mu}(\infty/\infty) + \]
\[ {}+k_{133}Y_h^{\alpha\beta\gamma\delta\varepsilon\xi\eta\vartheta ix\lambda\mu}. \]

Received
25 III 1965

CITED LITERATURE

1. G. Ya. Lyubarskii, Group Theory and Its Application in Physics, Moscow, 1957.
2. V. V. Lokhin, L. I. Sedov, Prikl. matem. i mekh., 27, no. 3, 393 (1963).
3. G. Weyl, Classical Groups, IL, 1947.
4. F. Murnaghan, Theory of Group Representations, IL, 1950.