As a quick check, we look at the non-relativistic limit.
In free case, this is trivial: the time and space parts of the kernel separate.
In the interacting case we:
Replace with expectation of , which is :
Since non-relativistic: is approximately one:
Therefore can replace all surviving bits of with . Potentials dependent on become dependent on , and derivatives with respect to become derivatives with respect to :
Applying these rules to the Lagrangian, we replace:
With:
Using:
Factor the quantum time part in the measure:
Using:
Factor out the kernel:
Using:
The four dimensional integrals are now products of the free kernel in quantum time and the usual path integral kernel in three dimensional.
The potentials are the familiar ones.
The Lagrangian is now the usual Lagrangian in the non-relativistic case, as in [Feynman-1965d].
This reduction fails if relativistic velocities are involved, if there is significant time dependence in the potentials, or if the wave functions have large initial uncertainties in time.