Strong Conservation Laws and Equations of Motion in Covariant Field Theories

Abstract

The relationship between the covariance of a field theory and the equations of motion is discussed in both the Lagrangian and the Hamiltonian formalism. All theories whose field equations are derivable from a variational principle and are covariant under arbitrary (curvilinear) coordinate transformations posses Bianchi identities and, hence, conservation laws. Because the strong conservation laws are ordinary divergences equal to zero, whether or not the field equations are satisfied, there exist certain skew-symmetric expressions whose divergences yield the components of the energy-momentum tensor. These superpotentials, as the skew-symmetric expressions are called, enable us to write the energy and momentum content of the field as two-dimensional surface integrals. Also, by using the superpotentials together with the field equations, one can find certain surface integrals which are independent of the surface of integration and which yield the equations of motion for the singularities enclosed by the surface. If the Einstein-Infeld approximation method is applied in the general theory of relativity, the above surface integrals reduce to integrals which are equivalent to those used by Einstein and Infeld to obtain the equations of motion for the field sources.In the Hamiltonian formalism the covariance of the theory is revealed in the existence of a number of constraints between the momenta and the field variables. Therefore, we examine the relationship between the constraints and the Bianchi identities, which lead to the strong conservation laws. We find that the first time derivative of the constraints leads to the existence of four linear combinations of field equations which are free of the time derivatives of the canonical field variables. These four expressions are the secondary constraints. The second time derivative of the primary constraints leads to the Bianchi identities in terms of the canonical field variables. Thus, we are able to establish the existence of the strong conservation laws in the Hamiltonian formalism and by the same arguments as in the Lagrangian formalism establish the existence of the superpotentials. The superpotentials are written out only for the gravitational theory.