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Fibre Bundle Formulation of Relativistic Quantum Mechanics: I. Time-dependent Approach

A new fibre bundle formulation of the mathematical base of relativistic quantum mechanics is presented. At the present stage, the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, first, part of our investigation we consider the time-dependent or Hamiltonian approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) liftings of paths or sections along paths of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the evolution transport along paths in the space-time) of the state liftings/sections in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations.