Geometrical formulation of quantum fields

Abstract

Path integral formulation of quantum mechanics defines the wavefunction associated with a particle as a sum of phase-factors, which are periodic functions of classical action. In the present article, this periodicity is shown to impart the corresponding periodicity to a one parameter family of wavefunctions generated by the translations of arclength used to parameterize the trajectories. Translation parameter is adjoined to the base to obtain an extended manifold. Periodicity of the family of wavefunctions with respect to the translation parameter together with solutions of the generalized Klein-Gordon equation, which is deducible from the path integral formulation, is used to define a quantized field with zero vacuum energy in the extended manifold with the particle being its quantum. Classical description of essentially the same particle is obtained in the extended higher dimensional space using its properties. This manifold can replace the base in this treatment to continue the program for higher dimensional manifolds generated in the process. Results are illustrated by taking the three-dimensional Euclidean space the base, which yields the classical and quantum particle descriptions of photon in the base and the field description in the resulting extended manifold, which is identified with the Minkowski spacetime. The field formulation yields the quantized Maxwell's equations. A novel interpretation of time as the corresponding translation parameter results in the process. Classical description in the extended manifold, i.e., the Minkowski spacetime, results in the relativistic description of a massive particle related to the photon. The results are further illustrated for this massive particle in the Minkowski spacetime obtaining parallel results.