  The proper time formalism for a particle propagator in an external
electromagnetic field is combined with the path-dependent formulation of the
gauge theory to simplify the quasiclassical propagator. The latter is achieved
due to a specific choice of the gauge corresponding to the use of the classical
path in the path-dependent formulation of the gauge theory, which leads to the
cancellation of the interaction part of the action in the Feynman path
integral. A simple expression for the quasiclassical propagator is obtained in
all cases of the external field when the classical equation of motion in this
field is integrable. As an example, new simple expressions for the propagators
are derived for a spinless charged particle interacting with the following
fields: an arbitrary constant and uniform electromagnetic field, an arbitrary
plane wave and, finally, an arbitrary plane wave combined with an arbitrary
constant and uniform electromagnetic field. In all these cases the
quasiclassical propagator coincides with the exact result.
