Lattice QED, most likely path of fields --> no interactions?

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Main Question or Discussion Point

Say I have a large spacetime lattice set up on a supercomputer where I calculate the scattering cross section of two spinless electrons of equal and opposite momentum via lattice QED. To get the right results we must add the amplitudes for every possible "path" the field can evolve from initial conditions?

Will the most likely "path" of the electron field be one where the spinless electrons do not interact at all (or "hardly at all", small delta p) because of the smallness of electric charge?

The interesting physics comes from the relatively small fraction of "paths" where electrons scatter?

Thanks for any help!

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Now I think I'm confused above as we evolve the fields between initial and final conditions in the path integral?

Forget field theory for a moment and consider the path integral formulation for a single particle in regular QM. We specify a starting and ending position for the electron and ask for the amplitude for the electron to propagate from the given starting position to the given ending position. In most situations, the greatest contribution to this amplitude ends up coming from the set of paths that are fairly close to the classical path that connects the given starting and ending positions.

The same principle applies in field theory, but it's less useful because we don't usually specify two field configurations and ask for the amplitude to go from one to the other. We are usually interested in scattering amplitudes of particles, but particles are complicated quantum excitations of the fields. A one-particle state, for example, is actually a superposition of an infinite number of field configurations.

In practice to compute a scattering amplitude on the lattice we use the LSZ formula to relate a scattering amplitude to a vacuum expectation value of some product of field variables. A vacuum expectation value can be written as a path integral where we sum over *all possible* initial and final field configurations. So this is what we do in lattice calculations.