UDC 513.82

MATHEMATICS

V. A. ZHELNOROVICH

REPRESENTATION OF SPINORS IN \(n\)-DIMENSIONAL SPACE BY SYSTEMS OF TENSORS

(Presented by Academician L. I. Sedov on 1 III 1966)

§ 1. Basic definitions. Let us first consider the even-dimensional complex Euclidean space \(R_n^+\), \(n=2\nu\), referred to an orthonormal basis \(e_i\). Let \(\gamma_1,\gamma_2,\ldots,\gamma_{2\nu}\) be matrices of dimension \(2^\nu\), which by definition satisfy the equation

\[ \gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij}I, \tag{1} \]

where \(\delta_{ij}\) is the Kronecker symbol, and \(I\) is the identity matrix of dimension \(2^\nu\). Introduce the notation \(\gamma_{i_1,i_2\ldots i_k}=i^{k(k-1)/2}\gamma_{i_1}\gamma_{i_2}\cdots\gamma_{i_k}\), \(i_1<i_2<\cdots<i_k\). The matrices \(I,\gamma_i,\gamma_{i_1\gamma i_2},\ldots,\gamma_{i_1\gamma i_2\ldots i_n}\) are linearly independent and form a group of \(2^{2\nu}\) elements. As is known, any two solutions \(\gamma_i,\bar{\gamma}_i\) of equation (1) are connected by the equality \(\bar{\gamma}_i=T\gamma_iT^{-1}\), \(\det T\ne0\), and the matrices \(\gamma_i\) may be chosen Hermitian and in such a way that \(\gamma_1,\gamma_2,\ldots,\gamma_\nu\) are symmetric, while \(\gamma_{\nu+1},\gamma_{\nu+2},\ldots,\gamma_{2\nu}\) are antisymmetric \((^1)\).

Let \(L=\|l_q^p\|\) be an orthogonal transformation of the space \(R_{2\nu}^+\). The set of unimodular matrices \(S\), defined by the equation

\[ \gamma_p=l_p^qS\gamma_qS^{-1}, \tag{2} \]

forms a group realizing a representation of the group \(L\), called the spin representation.

An object \(\psi=\{\psi^i\}\) with components \(\psi^i\), defined up to sign and transforming according to the representation \(S\), is called a spinor of first rank in the space \(R_{2\nu}^+\).

If \(\gamma_i\) is a solution of (1), then, obviously, \(\gamma_i^\tau\) (\(\gamma_i^\tau\) are the transposed \(\gamma_i\)) is also a solution of (1); therefore there exists a matrix \(C\) such that \(\gamma_i^\tau=C\gamma_iC^{-1}\), \(\det C=1\).

If \(\nu\) is odd, then it is easy to see that \(C=\gamma_\nu\gamma_{\nu-1}\cdots\gamma_1\); if \(\nu\) is even, then \(C=\gamma_{2\nu}\cdots\gamma_{\nu+1}\) in the case when \(\gamma_1,\gamma_2,\ldots,\gamma_\nu\) are symmetric and \(\gamma_{\nu+1},\gamma_{\nu+2},\ldots,\gamma_{2\nu}\) are antisymmetric. Hence it follows that \(C^\tau=(-1)^{\nu(\nu-1)/2}C\). If \(\psi_i\) are the covariant components of the spinor \(\psi\), then, by definition, \(\psi_i=e_{ij}\psi^j\), where \(E=\|e_{ij}\|=i^{n(n-1)/2}\gamma_n\gamma_{n-1}\cdots\gamma_1C\).

In the space \(R_n^+\), select a pseudo-Euclidean space \(R_n^{(s)}\) of index \(s\), fixing the basis \(ie_1\ldots ie_s e_{s+1}\ldots e_n\). Introduce the Hermitian matrix \(\Pi\), defined by the equations

\[ \Pi\gamma_i\Pi^{-1}=\pm\gamma_i,\qquad \Pi\Pi^*=(-1)^{(\nu-s)(\nu-s+1)/2}I; \tag{3} \]

here the minus sign is for \(i=1,\ldots,s\), and the plus sign for \(i=s+1,\ldots,2\nu\). A dot over a letter denotes complex conjugation. The spinor \(\dot{\psi}=\Pi\psi\) is called conjugate with respect to \(\psi\).

Let us now consider odd-dimensional spaces \(R_n^+\), \(n=2\nu+1\). We denote \(\gamma_{2\nu+1}=i^\nu\gamma_{1,2,\ldots,2\nu}\). The matrices with an even number of indices \(I,\gamma_{i_1i_2},\ldots,\gamma_{i_1i_2\ldots i_{2\omega}}\) \((i_p=1,2,\ldots,2\nu+1)\) are linearly independent. The spin representation of the proper orthogonal group \(L\) of transformations of the space \(R_{2\nu+1}\) is given by the group of matrices \(S\), defined from equation (2), in which the indices \(p,q\) take values from 1 to \(2\nu+1\).

The covariant components of the spinor \(\psi_i\) are determined by the matrix \(E\):

\[ (-1)^\nu \gamma_i^{\mathrm T}=E\gamma_iE^{-1}. \]

The conjugate spinor \(\bar\psi\) in the space \(R_{2\nu+1}^{(s)}\) is determined by the matrix \(\Pi\)

\[ \pm \gamma_i=(-1)^{\nu-s}\Pi\gamma_i\Pi^{-1}, \]

where the minus sign is for \(i=1,2,\ldots,s\).

§ 2. Representation of spinors by a system of complex tensors

Consider the complex matrix \(\Psi=\{\psi^{ij}\}\) of dimension \(r\). If \(\psi^{ij}=\psi^i\psi^j\), then the \(\psi^{ij}\) satisfy the equalities

\[ \psi^{ij}\psi^{kl}=\psi^{ik}\psi^{jl}=\psi^{il}\psi^{kj},\qquad \psi^{ij}=\psi^{ji}, \tag{4} \]

among which the independent ones are \(r(r+1)/2-r\) equations
\(\psi^{\nu\nu}\psi^{ij}=\psi^{\nu i}\psi^{\nu j}\) \((i,j\ne \nu,\ \psi^{\nu\nu}\ne0)\) and
\(\frac12 r(r-1)\) equations \(\psi^{ij}=\psi^{ji}\). In all there are \(r^2-r\) independent equations. Conversely, it follows from (4) that there exists a system of \(r\) components \(\psi^k\), determined up to sign, such that \(\psi^{ij}=\psi^i\psi^j\). Indeed, if \(\psi^{\nu\nu}=0\) for all \(\nu\), then from (4) it follows that \(\psi^{ij}=0\) for all values of the indices \(i,j\). If \(\psi^{\nu\nu}\ne0\), then set

\[ \psi^k=\psi^{\nu k}/\pm\sqrt{\psi^{\nu\nu}}. \tag{5} \]

In view of (4), this definition of \(\psi^k\) does not depend on the value of the index \(\nu\).

Suppose now that the \(\psi^{ij}\) are the components of an object \(\Psi\) transforming according to the representation \(S\times S\), where \(S\) is any representation of some group, and let the equations (4) be invariant with respect to the group \(S\times S\). Then the components \(\psi^k\), defined according to (5), transform according to the representation \(S\). Indeed, it follows from (5) that the transformation of \(\psi^k\) is determined by the transformation of \(\psi^{ij}\) in a unique way. Obviously, the invariance of the identities (4) is preserved if the \(\psi^k\) transform according to the representation \(S\); consequently, by uniqueness, the \(\psi^k\) can transform only according to the representation \(S\). Thus, the object \(\psi^{ij}\) satisfying the identities (4) is equivalent to the object \(\psi^k\).

Let \(S\) be the spinor representation of the \(2\nu\)-dimensional orthogonal group. It is known that

\[ S\times S\sim \sum_{k=0}^{2\nu}D^k, \]

where \(D^k\) is the representation according to which the tensor of rank \(k\), antisymmetric in all indices, transforms. Hence, in this case the object \(\psi^{ij}\) is equivalent to the tensor aggregate
\[ \Lambda=\{c_0,\ c_i,\ c_{i_1i_2},\ldots,\ c_{i_1i_2\ldots i_{2\nu}}\}. \]
If \(S\) is the spinor representation of the \((2\nu+1)\)-dimensional group, then
\[ S\times S\sim \sum_{k=0}^{\nu}D^{2k}; \]
therefore, the object \(\psi^{ij}\) is equivalent to a tensor aggregate consisting of tensors of even ranks.

The components of antisymmetric tensors \(C_{i_1i_2\ldots i_k}\) may be defined in the following way:

\[ c_{i_1i_2\ldots i_k}=(A_{i_1i_2\ldots i_k})_{\alpha\beta}\psi^{\alpha\beta},\qquad A_{i_1i_2\ldots i_k}=E\gamma_{i_1}\gamma_{i_2}\cdots\gamma_{i_k}. \tag{6} \]

Since \(\det E\ne0\) and the \(\gamma_{i_1i_2\ldots i_k}\) are linearly independent, the \(A_{i_1i_2\ldots i_k}\) are also linearly independent, and, consequently, the aggregates \(\Lambda\) are indeed equivalent to the objects \(\psi^{ij}\).

Using the symmetry properties of \(C\), one can show that the matrices \(A_{i_1i_2\ldots i_k}\) have the following symmetry properties:

\[ (A_{i_1i_2\ldots i_k})^{\mathrm T} = (-1)^{(\nu(\nu+1)+k(k+1))/2} A_{i_1i_2\ldots i_k}. \]

Therefore, if \(\psi^{ij}\) satisfies the identities (4), then the part of the tensors
\(c_{i_1i_2\ldots i_k}\) for which \([\nu(\nu+1)+k(k+1)]/2\) is odd vanishes. If \(\psi^{ij}\) satisfy the identities (4), then the tensors \(c_{i_1i_2\ldots i_k}\) satisfy
\(\frac12 2^\nu(2^\nu-1)\) independent bilinear identities, all of which contain—

be obtained in generalizing the Pauli identity for the case of an \(n\)-dimensional space [2]

\[ \begin{aligned} 2^\nu(\psi^+\theta\psi)(\psi^+\theta'\psi) &= \sum_{k=1}^{2\nu}\ \sum_{i_1<i_2<\cdots<i_k}^{2\nu} (\psi^+\gamma_{i_1 i_2\ldots i_k}\psi) (\psi^+\theta'\gamma_{i_1 i_2\ldots i_k}\theta\psi) \\ &\quad+(\psi^+\psi)(\psi^+\theta'\theta\psi), \qquad n=2\nu. \end{aligned} \tag{7} \]

\[ \begin{aligned} 2^\nu(\psi^+\varepsilon\psi)(\psi^+\theta'\psi) &= \sum_{k=1}^{2\nu}\ \sum_{i_1<i_2<\cdots<i_k}^{2\nu+1} (\psi^+\gamma_{i_1 i_2\ldots i_{2k}}\psi) (\psi^+\theta'\gamma_{i_1 i_2\ldots i_{2k}}\theta\psi) \\ &\quad+(\psi^+\psi)(\psi^+\theta'\theta\psi), \qquad n=2\nu+1, \end{aligned} \]

where \(\theta'\), \(\theta\) are arbitrary matrices of dimension \(2^\nu\); \(\psi^+\), \(\psi\) are the covariant and contravariant components of a spinor.

Thus, a spinor \(\psi^k\) in the space \(R_n^+\), \(n=2\nu,\,2\nu+1\), is equivalent to a tensor aggregate \(\Lambda\) consisting of complex antisymmetric tensors satisfying the \(\tfrac12 2^\nu(2^\nu-1)\) bilinear identities (7). In view of this, any spinor equation can be written in an equivalent manner as an equation in the components of the tensors \(c_{i_1 i_2\ldots i_k}\).

We note that formula (5) determines the components \(\psi^k\) in any coordinate systems, but the transformation of the components \(\psi^k\) to curvilinear coordinates turns out to be nonlinear with respect to \(\psi^k\).

§ 3. Representation of spinors by a system of real tensors. Consider an \(r\)-dimensional complex matrix \(\psi^{\dot p q}\). Let \(\psi^{\dot p q}=\psi^{\dot p}\psi^q\). Then \(\psi^{\dot p q}\) satisfy the identities

\[ \psi^{\dot p q}=(\psi^{\dot q p})^{\cdot}, \tag{8} \]

among which the independent ones are \((r-1)^2\) real equations
\(\psi^{\dot\nu\nu}\psi^{\dot p q}=\psi^{\dot\nu q}\psi^{\dot p\nu}\)
\((p,q\ne\nu,\ \psi^{\dot\nu\nu}\ne0)\) and \(r^2\) real equations
\(\psi^{\dot p q}=(\psi^{\dot q p})^{\cdot}\). Obviously, if the components \(\psi^k\) determine the matrix \(\psi^{\dot p q}\), then the components \(\psi^k e^{i\varphi}\), and only they, determine the same matrix \(\psi^{\dot p q}\).

Conversely, it follows from (8) that there exists a system of components \(\psi^k\), determined up to a phase \(e^{i\varphi}\), such that \(\psi^{\dot p q}=\psi^{\dot p}\psi^q\). Indeed, if \(\psi^{\dot\nu\nu}=0\) for all \(\nu\), then from (8) it follows that \(\psi^{\dot p q}=0\) for all \(p,q\). In this case we put \(\psi^k=0\). If \(\psi^{\dot\nu\nu}\ne0\), then set

\[ \psi^k=\frac{\psi^{\dot\nu k}}{\pm\sqrt{\psi^{\dot\nu\nu}}}\,e^{i\varphi}, \qquad \varphi \text{ is an arbitrary real number.} \tag{9} \]

In virtue of (8), such a definition of the set \(\psi^k\) does not depend on the value of \(\nu\). It can be shown that only the components \(\psi^k\) determined by (9) satisfy the equation \(\psi^{\dot p q}=\psi^{\dot p}\psi^q\).

Let \(\psi^{\dot p q}\) be the components of an object transforming according to the representation \(S^{\cdot}\times S\), where \(S\) is any representation of some group, and let the equalities (8) be invariant with respect to the group \(S^{\cdot}\times S\). Then, obviously, one can specify such a transformation law \(\varphi\) that the components \(\psi^k\) transform according to the representation \(S\).

Thus, specifying an object \(\psi^{\dot p q}\) satisfying the identities (8), and the argument \(\varphi\) of one of the components \(\psi^k\), completely determines the object \(\psi^k\). Let \(S\) be the spinor representation of the \(2\nu\)-dimensional orthogonal group of transformations of the space \(R_{2\nu}^{(s)}\). It is known that

\[ S^{\cdot}\times S \sim \sum_{k=0}^{2\nu} D^k. \]

Hence the object \(\psi^{\dot p q}\) is equivalent to the tensor aggregate \(\Omega=\{\Omega_0\Omega_i\ldots \Omega_{i_1 i_2\ldots i_2}\}\), consisting of antisymmetric tensors.

If \(S\) is a spinor representation in the space \(R_{2\nu+1}^{(s)}\), then \(S^{*}\times S\sim \sim \sum_{k=0}^{\nu}D^{2k}\), and in this case the object \(\psi^{pq}\) is equivalent to the aggregate
\(\Omega=\{\Omega_0\Omega_{i_1i_2}\ldots \Omega_{i_1i_2\ldots i_{2\nu}}\}\), consisting of antisymmetric tensors of even rank.

The components of these tensors may be defined as follows:

\[ \Omega_{i_1i_2\ldots i_k}=(D_{i_1i_2\ldots i_k})_{\alpha}^{\cdot}\psi^{\alpha\beta}, \qquad D_{i_1i_2\ldots i_k}=E\Pi\gamma_{i_1}\gamma_{i_2}\ldots\gamma_{i_k}i^{k(k+1)/2}. \]

Using (3), one can show that the matrices \(D_{i_1i_2\ldots i_k}\) are Hermitian.

Therefore, if \(\psi^{pq}\) satisfy the identity \(\psi^{pq}=(\psi^{qp})^{*}\), then the components \(\Omega_{i_1i_2\ldots i_k}\) are real. If \(\psi^{pq}\) satisfy the identities (8), then the components of the tensors \(\Omega_{i_1i_2\ldots i_k}\) satisfy \((2^\nu-1)^2\) bilinear identities, each of which is contained in identity (7).

Thus, specifying the aggregate \(\Omega\) and the argument \(\varphi\) of one of the components \(\psi^k\) completely determines the spinor. Hence, the spinor equations can be written in an equivalent form in the components of the aggregate \(\Omega\) and \(\varphi\). Eliminating the argument \(\varphi\) from such equations, one can then obtain a closed system of equations in the components of the aggregate \(\Omega\).

I express my sincere gratitude to L. I. Sedov for valuable advice in the course of this work.

Moscow State University
named after M. V. Lomonosov

Received
24 II 1966

REFERENCES

\({}^{1}\) J. A. Shouten, Indag. Math., 11, 3, 4, 5 (1949).
\({}^{2}\) K. M. Case, Phys. Rev., 97, 3, 810 (1955).