UDC 539.12.01

PHYSICS

Yu. V. NOVOZHILOV, I. A. TERENT’EV

ON THE THEORY OF \(SU_6\)-SYMMETRY OF ELEMENTARY PARTICLES

(Presented by Academician V. A. Fock on 28 VI 1965)

Recently the question of relativistic \(SU_6\)-symmetry has been widely discussed. In order to obtain an \(S\)-matrix possessing such a symmetry, it is necessary to construct unitary representations of an inhomogeneous group \(P(SU_6)\), for which the small group is \(SU_6\). The construction of unitary representations of the group \(P(SU_6)\) is the subject of the present paper.

The algebra of generators of the group \(P(SU_6)\) was studied by the authors \((^1,^2)\). It turned out that the derivation of the commutation relations has a general character and is readily generalized to the case of groups \(P(SU_r)\) with \(SU_r\) as the small group. Bearing in mind that the comparison of relativistic \(SU_6\)-symmetry with the symmetries \(SU_r\) is also of interest, we shall carry out the calculations for arbitrary \(r\). The Poincaré group is the special case \(P(SU_r)\) for \(r=2\). The commutation relations in \(P(SU_r)\) have the form

\[ \begin{aligned} [M_a,M_b]&=iF_{abc}M_c, & [P_\alpha,M_b]&=iF_{\alpha b\sigma}P_\sigma,\\ [N_a,M_b]&=iF_{abc}N_c, & [P_\alpha,N_b]&=D_{\alpha b\sigma}P_\sigma,\\ [N_a,N_b]&=iF_{abc}M_c, & [P_\alpha,P_\beta]&=0. \end{aligned} \tag{1} \]

Here Latin indices run through the values \(1,\ldots,r^2-1\), and Greek indices \(0,1,\ldots,r^2-1\). As before \((^1,^2)\), the generators \(M_a\) describe “spatial” rotations, \(N_a\) generate Lorentz-like transformations, and \(P_\alpha\) are displacement operators (\(r^2\) components).

Introduce a complete system of \(r^2\) complex Hermitian matrices \(\Lambda_\alpha\), having \(r\) rows and \(r\) columns, with the normalization condition \(\operatorname{Sp}\Lambda_\alpha\Lambda_\beta=\nu\delta_{\alpha\beta}\), where \(\Lambda_0=(\nu/r)^{1/2}E\). Then the structure constants \(F\) and \(D\) can be found with the aid of the matrices \(\Lambda_\alpha\)

\[ [\Lambda_\alpha,\Lambda_\beta]=2iF_{\alpha\beta\sigma}\Lambda_\sigma;\qquad \{\Lambda_\alpha,\Lambda_\beta\}=2D_{\alpha\beta\sigma}\Lambda_\sigma; \]

\[ F_{\alpha\beta\gamma}=-\frac{1}{2\nu}\operatorname{Sp}(\Lambda_\alpha[\Lambda_\beta,\Lambda_\sigma]);\qquad D_{\alpha\beta\gamma}=\frac{1}{2\nu}\operatorname{Sp}(\Lambda_\alpha\{\Lambda_\beta,\Lambda_\sigma\}). \tag{2} \]

The subgroup \(LS(r)\), constructed on the generators \(M_a\) and \(N_a\), is an analogue of the homogeneous Lorentz group. The representations of this group can be obtained in exactly the same way as the representations for the case \(r=6\) \((^2)\).

The key point of the theory of symmetry based on the group \(P(SU_r)\) should be regarded as the construction of the explicit form of the generalized Wigner operator \(\alpha\) \((^3)\). The operator \(\alpha\) is defined as a transformation of the group \(LS(r)\) which takes the standard momentum \(p^0\) into \(p\)

\[ \alpha_{\mu\nu}(p)p_\nu^0=p_\mu. \tag{3} \]

For representations with a positive-definite invariant form \(\det \Lambda_\mu p_\mu=x^r\), the standard momentum may be chosen in the form \(p_0^0=x\), \(p_a^0=0\). Equation (3) determines \(\alpha(p)\) up to a transformation \(R\) belonging to the small group, since \(Rp^0=p^0\). Making use of

thereby, we choose as \(\alpha\) a pure Lorentz transformation generated by the generators \(N_a\). If the momentum \(p\) is written by means of the matrix \(\hat p=\Lambda_\alpha p_\alpha\), then

\[ \hat\alpha(p)\hat p^{\,0}\hat\alpha(\hat p)=\hat p, \tag{3′} \]

where \(\hat\alpha(p)\) is a square \(r\times r\) matrix.

Consider the basis \(SU_r\)-vector \(n_a\), for which the relation

\[ (\Lambda_a n_a)_{ij}=\xi_i\xi_j^*-\frac1r\delta_{ij}\equiv\theta_{ij},\qquad i,j=1,2,\ldots,r, \tag{4} \]

holds, where \(\xi_j\) belongs to a set of \(r\) complex numbers \(\xi_1,\ldots,\xi_r\) with the normalization \(\sum \xi_j^*\xi_j=1\). The matrix \(S_{ij}=\xi_i\xi_j^*\) has the property \(S^m=S\). Therefore

\[ \theta^m=f(m,r)\left(\theta+\frac1r\right)+\left(-\frac1r\right)^m, \qquad f(m,r)=\left(\frac{r-1}{r}\right)^m-\left(-\frac1r\right)^m . \tag{5} \]

These formulas are sufficient for calculating \(\hat\alpha(p)\).

The simplest form of the Hermitian transformation \(\hat\alpha(p)\) is

\[ \hat\alpha(p)=\exp[(\Lambda_a n_a)\beta]=\exp(\theta\beta), \tag{6} \]

where the parameter \(\beta\) depends on \(p\). From (4) and (5) it follows that

\[ \hat\alpha(p)=\Phi(r,\beta)\left(\theta+\frac1r\right)-\exp\left(-\frac1r\beta\right), \]

\[ \Phi(r,\beta)=\exp\frac{r-1}{r}\beta-\exp\left(-\frac1r\beta\right). \tag{7} \]

Substituting (7) into (3′), we obtain

\[ \frac{\chi}{r}\Phi(2\beta)+\chi\exp\left(-\frac2r\beta\right) = \sqrt{\frac{\nu}{r}}\,p_0\equiv\rho_0, \]

\[ p_k=\rho n_k,\qquad \rho=\chi\Phi(2\beta). \tag{8} \]

Consequently,

\[ \beta=-\frac r2\ln\frac q\chi,\qquad q=\rho_0-\frac{\rho}{r}. \tag{9} \]

Formula (9) together with (7) gives the solution of the problem of the Wigner operator for the case of the group \(P(S\hat U_r)\).

For the construction of irreducible unitary representations it is necessary to know the invariant \(Z=\det\Lambda_\mu p_\mu\), characterizing momentum space. The expression for \(Z\) in the case \(r=6\) in terms of the \(\hat M\)-invariants composed of the momenta was obtained earlier \({}^{(1)}\) without using the properties of \(n_a\). From the form of \(\hat\alpha\) it is clear that in the theory an essential role belongs to the quantities

\[ t=\rho_0+\frac{r-1}{r}\rho,\qquad q=\rho_0-\frac{\rho}{r}. \tag{10} \]

We shall seek \(Z\) as a function of \(q\) and \(t\) from the condition of invariance of \(Z\) with respect to Lorentz-like transformations generated by \(N_a\). This means that \(Z\) commutes with \(N_a\). When acting on a scalar function of the momentum, the generator \(N_a\) is equivalent to \(N_a^0\)

\[ N_a^0=-D_{\alpha\beta a}p_\beta\frac{\partial}{\partial p_\alpha}. \tag{11} \]

It is convenient to introduce \(N=n_aN_a\). Then from (10) and (11) we have

\[ [N,q]=\frac1r q,\qquad [N,t]=-\frac{r-1}{r}\,t. \tag{12} \]

The determinant \(Z\) is expressed simply as

\[ Z = q^{r-1}t. \tag{13} \]

Let us proceed to the study of the generators \(M_a\) and \(N_a\). With the aid of (6) and (9) it can be shown that, in the unitary representation, the generators \(M_a\) and \(N_a\) contain orbital parts \(N_a^0\) and \(M_a^0\)

\[ M_a = M_a^0 + \mathfrak{M}_a;\qquad N_a = N_a^0 + K_{ab}\mathfrak{M}_b, \tag{14} \]

where \(\mathfrak{M}_a\) is the generator of “spin” transformations; moreover

\[ M_a^0 = - i F_{abc} p_b \frac{\partial}{\partial p_c}. \tag{15} \]

For \(K_{ab}\) one may write

\[ K_{ab} = \frac{1}{\nu}\operatorname{Im}\operatorname{Sp} \left(\Lambda_b \hat{\alpha}^{-1}(p)[2N_a^0+\Lambda_a,\hat{\alpha}(p)]\right) = \]

\[ = 2\left[ \frac{q}{\rho}\left(2-b-\frac{1}{b}\right)+1-b \right]F_{acb}n_c,\qquad b=\left(\frac{q}{\chi}\right)^{1/2}. \tag{16} \]

Equation (14) is a generalization, to the case of the group \(P(SU_r)\), of the relations for the Poincaré group \((r=2)\) that were obtained by Yu. M. Shirokov \((^4)\).

In the course of computing (16) it is convenient to use the following relations:

\[ [N_a^0,\varphi(q)] = \frac{n_a\nu}{r-1}\,q\,\frac{\partial\varphi}{\partial q}, \]

\[ [N_a^0,\theta] = \frac{n_a\nu r}{r-1} \left[ \frac{r-2}{r}+\frac{\rho^0}{\rho} \right]\theta - \left(\frac{q}{\rho}+\frac{1}{2}\right)\Lambda_a. \]

Having the Wigner operator, one can solve a number of the problems posed earlier \((^{1,2,5})\). With the aid of this operator one can obtain equations of motion in various forms and additional conditions, and write out their solutions explicitly. This operator is also important for the investigation of Bargmann–Wigner-type equations of motion and for determining the covariant vector \(B_a\), entering the invariant \(B_aP_a=\chi^2\) \((^5)\).

Leningrad State University
named after A. A. Zhdanov

Received
22 I 1965

REFERENCES

\(^1\) Yu. V. Novozhilov, I. A. Terent’ev, DAN, 165, No. 3 (1965).
\(^2\) Yu. V. Novozhilov, I. A. Terent’ev, Vestn. LGU, No. 10, 5 (1965).
\(^3\) E. P. Wigner, Ann. Math., 40, 149 (1939).
\(^4\) Yu. M. Shirokov, DAN, 94, 857 (1954); C. Fronsdal, Phys. Rev., 113, 1367 (1959).
\(^5\) Yu. V. Novozhilov, Phys. Lett., 16, No. 3 (1965).