Scalar time parametrization of relativistic quantum mechanics: The covariant Schrödinger formalism

Abstract

We investigate the implications of a manifestly covariant formulation of relativistic quantum mechanics by using a relativistic version of Schrödinger's equation with an independent scalar evolution parameter canonically conjugated to the variable mass. This approach, which yields ordinary relativistic Klein-Gordon quantum mechanics as an evolution-time stationary solution, can be shown to lead to an alternative quantum field theoretical formulation which could be devoid of the divergency problems of the standard quantum field theory. This covariant Schrödinger formalism as derived from a stochastic variational principle is shown to yield a natural Hilbert space construction in direct analogy with non-relativistic quantum mechanics. It is then demonstrated that this theory can be interpreted as a consequence of a space-time metric fluctuation and that it can be related to mass ensemble theories. Furthermore a Lagrangian formulation of the theory is presented and the relation between the field Hamiltonian and the components of the energy-momentum tensor is established. Some interesting conclusions can be drawn by requiring the local U(1) invariance of the theory, as e.g. the emergence of an additional scalar potential V related to the variable mass and the corresponding field equations. A possible generalization of the theory is finally presented by introducing Clebsch parameters which generalize the quantum motions without introducing new forces.