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PHYSICS
G. A. SOKOLIK
SPINOR NOTATION FOR THE GRAVITATIONAL FIELD
(Presented by Academician N. N. Bogolyubov on 2 IX 1963)
- In the present paper the method of compensating fields is extended to spinor quantities.
In work \((^1)\) attempts were made to construct a spinor theory of gravitation, but the results of \((^1)\) are incorrect, since, according to \((^2)\), tensor and spinor quantities can be related by the formulas
\[ g^{p\dot\alpha\beta}g_{\dot\alpha\gamma}^{q} + g^{q\dot\alpha\beta}g_{\dot\alpha\gamma}^{p} = 2g^{pq}\delta_\gamma^\beta, \qquad \Phi_{\dot\alpha\beta}=g_{p\dot\alpha\beta}x^p \]
only in an orthogonal basis. In what follows we shall use the method of compensating fields \((^3)\); moreover, the relation between the invariant derivative induced by the local Lorentz group \(\mathcal L\) and spinor invariant derivatives will be indicated.
- The expression for the invariant derivative that specifies the gravitational interaction \((^3)\) follows from the condition of covariance of the wave relativistic equation in an orthogonal basis:
\[ (\Omega_\sigma(i)\mathcal L_i\partial_\sigma+m)\psi=0, \tag{1} \]
where \(\Omega_\sigma(i)\) are Lamé coefficients relating orthogonal and world coordinates, with respect to the local Lorentz group,
\[ \mathcal L_{ii'}=\exp[\varepsilon_{lk}(x)M_{ii'}^{lk}], \]
\[ M_{ii'}^{lk}=\frac12(\delta_i^k g_{i'}^{\,l}-g_i^{\,l}\delta_{i'}^k) \]
(\(u_\sigma\) are world coordinates; \(x_i=\Omega_\sigma(i)u_\sigma\)),
\[ \partial_\sigma\to\nabla_\sigma=\partial_\sigma-\frac12\Delta_\sigma(i,k)I_{ik}, \]
where \(\Delta_\sigma(i,k)\) are the Ricci coefficients:
\[ \Delta_\sigma(i,k)=\Omega^\tau(i)\Omega^\lambda(k)M_{\sigma\tau\lambda}^{\gamma\alpha\beta} \Omega_\gamma(i)\partial_\alpha\Omega_\beta(j), \]
\[ M_{\sigma\tau\lambda}^{\gamma\alpha\beta} = \frac12( \delta_\sigma^\gamma\delta_\tau^\alpha\delta_\lambda^\beta + \delta_\lambda^\gamma\delta_\tau^\alpha\delta_\sigma^\beta + \delta_\tau^\gamma\delta_\sigma^\alpha\delta_\lambda^\beta - \delta_\sigma^\gamma\delta_\lambda^\alpha\delta_\tau^\beta - \delta_\lambda^\gamma\delta_\sigma^\alpha\delta_\tau^\beta - \delta_\tau^\gamma\delta_\lambda^\alpha\delta_\sigma^\beta ), \]
which transform according to the formula
\[ \delta\Delta_\sigma(i,k) = \varepsilon_{ln}C_{ln;\,sp}^{ik}\Delta_\sigma(s,p) + \partial_\sigma\varepsilon_{ik} \]
\((^3)\); \(I_{ik}\) are the generators of the representation \(\mathcal L\) according to which \(\psi\) transforms:
\[ \varepsilon_{ik}=\varepsilon_{ik}(u), \qquad [I_{ik}I_{ls}]=C_{ik;\,ls}^{nm}I_{nm}. \]
- We shall use the well-known decomposition that reduces the Lorentz group to the contracted direct product of second-order unimodular groups, according to which dotted and undotted semispinors \((1/20)\) and \((01/2)\) transform \((^2)\). Then the generators of the representation are reduced to
\[ \tau_i=iI_{4i}+I_{jk}; \qquad \nu_i=iI_{4i}-I_{jk} \quad (ijk=\text{cycl}); \qquad [\tau_i\nu_j]=0. \]
\[ [\tau_i\tau_j]=\tau_k \quad\text{and}\quad [\nu_i\nu_j]=\nu_k \]
correspond to the sets of weights \((j_1j_2)\) and \((j_2j_1)\).
Equation (1) in this case takes the form of a second-order equation:
\[ \mathscr{L}_{\alpha\dot{\alpha}}\Omega_{\sigma}(\alpha)\Omega_{\dot{\sigma}}(\dot{\alpha})\, \partial_{\sigma\dot{\sigma}}\psi+m\psi=0, \tag{2} \]
where \(\Omega_{\sigma}(\alpha)\) and \(\Omega_{\dot{\sigma}}(\dot{\alpha})\) are the spinor components of \(\Omega_{\alpha}(i)\); \(u_{\sigma}=\Omega_{\sigma}(\alpha)\xi_{\alpha}\), \(\partial_{\sigma}=\partial/\partial u_{\sigma}\), while \(u_{\dot{\sigma}}=\Omega_{\dot{\sigma}}(\dot{\alpha})\xi_{\dot{\alpha}}\) (a 4-component vector has the meaning of the bispinor \(x_{\alpha}=\xi_{\gamma}\xi_{\dot{\delta}}\)).
Assuming the locality of \(\varepsilon_a(u_{\sigma})\) and \(\dot{\varepsilon}_a(u_{\dot{\sigma}})\), \(S=1+\varepsilon_a\tau_a\), \(\dot S=1+\dot{\varepsilon}_a\nu_a\), and requiring covariance of (2) with respect to \(S,\dot S\), we obtain:
\[ \partial_{\sigma}\to \partial_{\sigma}-A_{\sigma}^{a}\tau_a,\qquad \partial_{\dot{\sigma}}\to \partial_{\dot{\sigma}}-A_{\dot{\sigma}}^{a}\nu_a, \]
where \(A_{\sigma}^{a}\) and \(A_{\dot{\sigma}}^{a}\) transform in such a way as to compensate the terms:
\[ (\partial_{\sigma}\varepsilon_a)\tau_a\partial_{\dot{\sigma}},\qquad (\partial_{\dot{\sigma}}\dot{\varepsilon}_a)\nu_a\partial_{\sigma}, \]
\[ \tau_a\tau_c\varepsilon_b C^{c}_{pb}A_{\sigma}^{a}A_{\dot{\sigma}}^{a},\qquad \tau_c\nu_p\varepsilon_b A_{\sigma}^{a}A_{\dot{\sigma}}^{p}C^{c}_{ab}, \tag{3} \]
where \(C^{a}_{bc}\) define the algebra of the 3-dimensional orthogonal group: \([A_iA_j]=A_k\) \((ijk=\mathrm{cycl})\). The covariant form of (2) is
\[ \mathscr{L}_{\alpha\dot{\alpha}}\Omega_{\sigma}(\alpha)\Omega_{\dot{\sigma}}(\dot{\alpha}) (\partial_{\sigma}-\tau_a A_{\sigma}^{a}) (\partial_{\dot{\sigma}}-\nu_a A_{\dot{\sigma}}^{a})\psi +m\psi=0. \tag{4} \]
The formula \(\nabla_{\sigma\dot{\sigma}}=\nabla_{\sigma}\nabla_{\dot{\sigma}}\) expresses \(\Delta_{\sigma}(i,k)\) through \(A_{\sigma}^{a}\) and \(A_{\dot{\sigma}}^{a}\). The latter can be written as connection coefficients in spinor spaces:
\[ A_{\sigma}^{a}=C^{b}_{ac}\Omega_{\alpha}(b)\,\partial_{\sigma}\Omega_{\alpha}(c), \]
\[ A_{\dot{\sigma}}^{b}=C^{b}_{ac}\Omega_{\dot{\alpha}}(b)\,\partial_{\dot{\sigma}}\Omega_{\dot{\alpha}}(c). \tag{5} \]
Then \(A_{\sigma}^{a}\) and \(A_{\dot{\sigma}}^{a}\) transform according to formula (3), if \(\Omega_{\alpha}(b)\) and \(\Omega_{\dot{\alpha}}(b)\) transform according to the adjoint groups of the unimodular groups \(S\) and \(\dot S\),
\[ \delta\Omega_{\sigma}(b)=\varepsilon_e C^{b}_{ec}\Omega_{\sigma}(b), \]
\[ \delta\Omega_{\dot{\sigma}}(b)=\dot{\varepsilon}_e C^{b}_{ec}\Omega_{\dot{\sigma}}(c), \]
isomorphic to \(S\) and \(\dot S\) themselves.
Research Institute of Electromechanics
Received
31 VIII 1963
CITED LITERATURE
- L. Witten, Phys. Rev., 113, 357 (1959).
- É. Cartan, Theory of Spinors, IL, 1947.
- G. A. Sokolik, DAN, 148, No. 3 (1963).