Extended Translation Invariance and Associated Gauge Fields

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 Schwarzschild'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.