On Gravity and the Path Integral

Question

The path integral, in the simplest case, usually attributes a classical action to every conceivable trajectory a particle can take between to points in spacetime. This assumes a flat, Minkowski background. Is it possible to use the path integral to attribute a classical action to the formation of one or more virtual particles?

If so, then the trajectory of a particle would be altered by the effect of the mass-energy of such particles on the metric of background space. That is, a very low-probability event of a very energetic particle-antiparticle pair forming and annihilating at some point in space and time might alter the trajectory of a particle through the bending of spacetime.

Answer 1

If you create a virtual electron with 4 momentum:

$$ p^{\mu}_- = (E_-, \vec p_-) $$

then the positron has:

$$ p^{\mu}_+ = (-E_-, -\vec p_-) $$

so the total is:

$$ p^{\mu}=p^{\mu}_-+p^{\mu}_+ = (0, 0, 0, 0) $$

which has a stress energy tensor proportional to:

$$ T^{\mu\nu} \propto p^{\mu}p^{\nu} = 0 $$

Path integrals are relativistic QFT, and energy and momentum are conserved at all vertices. Moreover, you don't just rando get a high $|E|$ pair, you have to integrate over all $d^4p$..it's a loop diagram, so it is what it is..always.

When you get into high energy contributions (aka ultraviolet), you then get into cut-offs and renormalization and all that.