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What you've shown here is for perturbation theory. What about the virtual particles that are supposed to be everywhere, the ones that get ripped apart by black hole horizons or Unruh acceleration, or that are supposed to be involved in the Casimir effect, etc? What is the math for these?Well, that's easy. What's called "virtual particle" in popular science books is symbolized by internal lines of Feynman diagrams, and they stand for free-particle Green's functions (in usual perturbation theory; sometimes they can have a different meaning, e.g., in the context of resummation schemes like the ##\Phi##-derivable approximation or the functional RG methods), but that doesn't matter too much on the level of this discussion.