Extended translation invariance and associated gauge fields

Abstract

Gauge fields together with non-linear field equations to govern them are introduced by requiring that the Lagrangian should be invariant under an extended translation in space-time, i. e. a translation in which four parameters are replaced by four arbitrary coordinate-dependent functions. A prescription is given to convert a non-invariant canonical (“pseudo”) energy-momentum tensor into an invariant one. The symmetric part of these field equations is examined for the two cases: (1) under linear and non-relativistic approximation, it reduces to the classical gravitational-field equation, (2) for static and spherically symmetric field, its solution is shown to correspond to Schwarz-schild’s solution. The antisymmetric part has no classical analogues, for there are no sources of skew-symmetric energy-momentum tensors in the classical experiments. A reasonable method is proposed to eliminate this redundant field.