The field equations in a three-dimensional commutative space

Abstract

The field equations in a three-dimensional commutative space based on a set of commutation relations are derived. In this space, the commutation relation of the position and kinematic momentum of a particle is generalized to include a metric tensor field in addition to a vector field. The introduction of a metric tensor is a generalization of the commutation relation for Feynman’s proof of the Maxwell equations. In this paper, as the equations of motion and the field equations are classical, the Poisson bracket and not the commutation relation is used in the calculations. As the commutative space is defined by the Poisson bracket, the equations of motion for the particle and the field equations for the metric tensor and vector are derived from the Poisson bracket in Hamiltonian mechanics. The Helmholtz conditions, which express the existence of a Lagrangian for a particle in the space, are also derived from the Poisson bracket. Then the field equations are calculated explicitly by two approaches. One is to calculate the Helmholtz conditions using the equations of motion. The other is to calculate the Jacobi identity for the kinematic momentum or velocity of the particle. In addition to the homogeneous Maxwell equations, the generalized field equations are obtained to define the generalized electric and magnetic fields of the tensor field. Just like the usual electric and magnetic fields, the generalized fields are invariant under a local gauge transformation and should play significant roles in physics. Finally, the homogeneous Maxwell equations of the vector field are seen to exhibit similarities with the generalized field equations for the tensor field. This similarity provides a useful theoretical framework for constructing gravitoelectromagnetism, which is based on analogies between the equations for electromagnetism and relativistic gravitation. It remains to establish the usefulness of the theoretical framework with applications of the field equations.