PHYSICS

G. A. SOKOLIK, N. P. KONOPLEVA

CONSTRUCTIVE THEORY OF COMPENSATING FIELDS

(Presented by Academician N. N. Bogolyubov, 2 IX 1963)

1. In this paper an attempt is made to find the effective form of compensating fields as connection coefficients in the space of internal degrees of freedom \((^{1-3})\).

Consider the gauge local group \(\psi' = S(x)\psi\),

\[ S=\exp[\varepsilon_a(x)I_a]; \qquad [I_a I_b]=C^c_{ab}I_c . \]

Generalizing Noether’s theorem to the case of a local group \((^{1-3})\), we have:

\[ \mathcal{L}=\mathcal{L}(\psi;\nabla_i\psi), \qquad \nabla_i=\partial_i-A_i^a I_a, \]

\[ \delta\mathcal{L} = \frac{\partial\mathcal{L}}{\partial\psi}\delta\psi + \frac{\partial\mathcal{L}}{\partial\nabla_i\psi}\delta\nabla_i\psi =0. \tag{1} \]

To compensate the terms arising owing to \(\varepsilon_a=\varepsilon_a(x)\), it is necessary to introduce a nontensor field \(A_i^a\):

\[ \delta A_i^a=\varepsilon_b C^a_{bc}A_i^c+\partial_i\varepsilon_a; \tag{2} \]

the derivative \(\nabla_i\) can be obtained from the condition of covariance of the wave equation \(\mathcal{L}_i\partial_i\psi+m\psi=0\) with respect to \(\psi'=S\psi\). Then

\[ S\mathcal{L}_iS^{-1}=\mathcal{L}_i, \qquad \partial_i\to\nabla_i=\partial_i-A_i^a I_a, \qquad \overline{A_i^a}=e^{\varepsilon_e C^b_{ea}}\left(A_i^b+N_c^b\partial_i\varepsilon_c\right), \]

\[ N_a^b=\int_0^1 \exp[-t\varepsilon_e C^b_{ea}]\,dt \]

(passing to infinitesimal transformations, we obtain (2)).

Expanding \(\delta\mathcal{L}\) and taking into account

\[ \nabla_i\frac{\partial\mathcal{L}}{\partial\nabla_i\psi} - \frac{\partial\mathcal{L}}{\partial\psi} =0, \qquad [\delta\partial]=0, \]

we obtain

\[ \frac{\partial}{\partial x_i} \left( \frac{\partial\mathcal{L}}{\partial\nabla_i\psi}I_a\psi \right) = A_i^c C^a_{cb}J_i^b . \tag{3} \]

Thus the conservation current has the form

\[ J_i^a=\frac{\partial\mathcal{L}}{\partial\nabla_i\psi}I_a\psi, \]

where

\[ \nabla_i J_i^a=\partial_i J_i^a+A_i^c C^a_{cb}J_i^b . \tag{4} \]

From \(\mathcal{L}=\bar{\psi}(\mathcal{L}_i\partial_i\psi+m)\psi\) it follows that

\[ J_i^a=\bar{\psi}\mathcal{L}_i I_a\psi . \]

Taking into account the Lagrangian of the free field \(\mathcal{L}_0[A_i^0]\), in which \(A_i^0\) enters only in the form of the tensor

\[ F_{ik}^a=(\partial_i A_k^a-\partial_k A_i^a)-\frac{1}{2}C^a_{bc}(A_i^b A_k^c-A_k^b A_i^c), \]

transforming according to the adjoint group of the gauge group \(\psi'=S\psi\) (1),

\[ \delta F_{ik}^a=\varepsilon_b(x)C^a_{bc}F_{ik}^c, \]

as a result of which \(\mathscr L_0\) preserves gauge invariance, and the homogeneous conservation law \(\partial_i j_i^a=0\) is replaced by the inhomogeneous one
\(\partial_i J_i^a+f^a=A_i^c C_{cb}^a I_i^b\), where
\(f^a=\dfrac{\partial\mathscr L_0}{\partial F_{ik}^b}C_{ac}^b F_{ik}^c\).

It is interesting that in the case of the gravitational field, for \(\mathscr L_0=F_{ik}^aF_{ik}^a\), \(f^a\equiv0\), since \(f^a=C_{ac}^bF_{ik}^aF_{ik}^c=0\) \((^1)\), because for compact groups \(C_b^{ac}\) are completely antisymmetric: \(C_{bc}^a=-C_{cb}^a\); \(C_{bc}^a=-C_{ba}^c\). Thus only a deviation from the Euclidean metric, which is connected with allowance for gravitation, leads to \(f^a\ne0\). In this case:

\[ \mathscr L_0=g_{bc}(x)F_{ik}^bF_{ik}^c. \]

1. In order to express \(A_i^a\) effectively, we use the relation
   \(\delta A_a=\varepsilon_e C_{eb}^a X_b\), where \(X_a\) define the basis of the adjoint group \((^4)\): \(X_a=C_{ac}^b X_b\partial_c\). Then

\[ \begin{gathered} A_i^a=C_{ac}^b\Omega_b\partial_i\Omega_c,\\ C_{ac}^{b'}C_{ec}^{b}\Omega_{b'}\Omega_{c'}=\delta_{ae}. \end{gathered} \tag{5} \]

The quantities \(\Omega_a\) may be regarded as a frame specified in the space of internal degrees of freedom. The adjoint group

\[ \delta\Omega_a=\varepsilon_e(x)C_{eb}^a\Omega_b, \]

where \(S\) is regarded as its representation, induces, as is easy to see, the transformation \(A_i^a\) (2), if one takes into account the Jacobi identity.

The results obtained are close to the ideas of \((^5)\) on the connection between ordinary and isotopic space. Indeed, note that \(A_i^a\) can be represented as a mass operator by substituting (5) into

\[ \mathscr L_i(\partial_i-A_i^a I_a)\psi+m\psi=0 \]

and introducing the coordinates of isospace \(\omega\) \((^{6,7})\) and the periodicity condition
\(\omega_a=\exp\left[\dfrac{2m_a}{\Lambda}\right]\). Then

\[ \mathscr L_i\partial_i\psi(x,\omega)+M\psi(x,\omega)=0. \tag{6} \]

The expression for the mass operator is:

\[ M=m+\frac{m_i}{\Lambda}\mathscr L_i I_a C_{ab}^c \left(\omega_b\frac{\partial}{\partial\omega_c}-\omega_c\frac{\partial}{\partial\omega_b}\right), \]

where \(\Lambda\) is a constant with the dimension of length; it coincides with that given in \((^6)\) for particles of arbitrary spatial and isotopic spins, and in the present work the term corresponding to the isospin group arises because of its locality, i.e., through the connection with ordinary space.

1. In the case of an Abelian gauge group \(S=\exp[ia(x)]\), the compensating field has the form

\[ A_i=\overline{\Omega}\partial_i\Omega,\qquad \overline{\Omega}\Omega=1. \]

For \(\Omega'=S\Omega\), \(A_i'=A_i+i\partial_i\alpha\) (gradient transformations). Passing to the variables \(\Omega=e^{i\varphi}\) \((\varphi'=\varphi+\alpha)\), we have \(A_i=i\partial_i\varphi\). In other words, in the case of an Abelian group, \(A_i\) is given by a gradient \((^8)\).

1. Comparing expression (5) for \(A_i^a\) as a connection coefficient in an isospace with the Ricci coefficient

\[ \Delta_\sigma(i,k)=\Omega^\tau(j)\Omega^\lambda(k)M_{\sigma\tau\lambda}^{\gamma\alpha\beta}\Omega_\gamma(j)\partial_\alpha\Omega_\beta(j), \]

\[ M_{\sigma\tau\lambda}^{\gamma\alpha\beta} =\frac{1}{2}\left(\delta_\sigma^\gamma\delta_\tau^\alpha\delta_\lambda^\beta +\delta_\lambda^\gamma\delta_\tau^\alpha\delta_\sigma^\beta +\delta_\tau^\gamma\delta_\sigma^\alpha\delta_\lambda^\beta -\delta_\sigma^\gamma\delta_\lambda^\alpha\delta_\tau^\beta -\delta_\lambda^\gamma\delta_\sigma^\alpha\delta_\tau^\beta -\delta_\tau^\gamma\delta_\lambda^\alpha\delta_\sigma^\beta\right), \]

which is the compensating field induced by the local Lorentz group:

\[ \mathcal{L}_{ii'}=\exp\left[\varepsilon_{lk}(x)M_{ii'}^{lk}\right], \]

\[ \delta\Delta_\sigma(i,k)=\varepsilon_{ln}C_{ik;sp}^{\,ln}\Delta_\sigma(s,p)+\partial_\sigma\varepsilon_{ik} \]

or

\[ \Delta'_\sigma(i,k)=\mathcal{L}_{ii'}\mathcal{L}_{kk'}\Delta_\sigma(i',k')+M_{sp}^{ik}\partial_\sigma\varepsilon_{sp}, \]

where

\[ [I_{ik}I_{ls}]=C_{ik;ls}^{\,nm}I_{nm},\qquad M_{sp}^{ik}=\frac{1}{2}\left(\delta_s^i\delta_p^k-\delta_p^i\delta_s^k\right). \]

Let us note that gravitation is the only compensating field induced by a group specified in the same space as the field itself. In other words, gravitation has a universal character. If this circumstance is connected with the idea of a hierarchy of interactions, according to which the weakening of an interaction is associated with an enlargement of the symmetry group, then it follows from the universality of gravitation that the gravitational interaction is minimal and does not correspond to any degree of homogeneity.

In conclusion, the authors express their gratitude to Prof. K. P. Staniukovich for his interest in the work and for valuable discussions.

Scientific Research Institute of Electromechanics

Received
31 VIII 1963

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