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The proper time formalism for a particle propagator in an externalelectromagnetic field is combined with the path-dependent formulation of thegauge theory to simplify the quasiclassical propagator. The latter is achieveddue to a specific choice of the gauge corresponding to the use of the classicalpath in the path-dependent formulation of the gauge theory, which leads to thecancellation of the interaction part of the action in the Feynman pathintegral. A simple expression for the quasiclassical propagator is obtained inall cases of the external field when the classical equation of motion in thisfield is integrable. As an example, new simple expressions for the propagatorsare derived for a spinless charged particle interacting with the followingfields: an arbitrary constant and uniform electromagnetic field, an arbitraryplane wave and, finally, an arbitrary plane wave combined with an arbitraryconstant and uniform electromagnetic field. In all these cases thequasiclassical propagator coincides with the exact result.

The propagator of a relativistic particle via the path-dependent vector potential