Action at a distance in Quantum Field Theory

Question

Definitely, I don't mean entanglement here: Suppose we have an electron and proton situating some distance apart, there is an electrostatic force between them, and this force is mediated by virtual particles, so action at a distance is executed between them. So my question is Does Quantum field theory support action at a distance? If yes how?

Answer 1

No. And the example you provide is not action at a distance. As you stated, there is an exchange of virtual photons. Quantum field theory has laws that come from symmetries (like all of physics). Sometimes these symmetries are global or local.

In modern QFT it is believed that global symmetries are unnatural. They have an “action at a distance” feel. We now suspect that all fundamental symmetries are local gauge symmetries. Global symmetries are either all broken (such as parity, time reversal invariance, and charge symmetry) or approximate (such as isotopic spin invariance) or they are the remains of spontaneously broken local symmetries.

In the case of a certain physical phenomena that arises from a global symmetry, it is necessary that this same physical symmetry has to hold locally. That is, the symmetry must be preserved locally. In fact it is demanded that certain physical phenomena and the mathematics defining these physical phenomena therein have to be refined so that locally the same physics is preserved. This is necessary so as to discount “action at a distance” which is not physical. For example in the Higgs mechanism a mass term arises which breaks gauge symmetry. For this we add a scalar field term that couples to the gauge field via the addition of a covariant derivative.

Answer 2

The go-to formulation of an action-at-a-distance replacement for field theory, in the case of electromagnetism, is the Wheeler-Feynman Theory, which was published in the late 1940's. In contrast to non-relativistic physics, where simultaneity is absolute, in Relativity, the light cones are absolute. So action-reaction takes place along light cones. That means the action goes forward in time, the reaction backwards in time. They devised thought experiments, in one of their papers, to show how paradox can be avoided, by invoking the idea of the "glancing blow" solutions. This was later picked up by Friedman, Morris, Thorne et al. in their landmark early 1990's Time Travel Paper - without credit, I might add. They got the idea from a grad student who, in turn, very likely picked it up by osmosis from Wheeler and Feynman. It's also possible Friedman knew of it, since action-at-a-distance electrodynamics has been one of his areas of interest.

Though the Wikipedia link does not say so directly, one of the main motivations of Wheeler and Feynman's attempt at an action at a distance formulation was to get rid of the uncountable infinity of field degrees of freedom and replace them by particle degrees of freedom, as a precursor to quantizing electrodynamics as a quantum mechanics, rather than as a quantum field theory. This attempt hit road blocks and the whole endeavor went through several rounds of morphing, evolving - almost beyond recognition - eventually into Feynman's path integral formulation of quantum field theory.

Action at a distance electrodynamics is an active field of research rooted all the way back to the 19th century. Here's another formulation by Kennedy in 1967 Instantaneous Action At A Distance Formulation Of Classical Electrodynamics.

Many people tried jumping on the action-at-a-distance bandwagon early on. Kerner did a compilation of the early approaches in the 1970's in Theory of action-at-a-distance in relativistic particle dynamics. A reprint collection. Wigner took a stab at it in the 1960's, with an attempt to set up a framework for action-at-a-distance over space-like intervals.

Dirac laid forth an all-encompassing framework for relativistic many-body dynamics in the 1940's that addressed the different ways one might combine the (energy, momentum, mass, angular momentum, moment) set of one-body invariants for different bodies into composite bodies. Leutwyler put up a hard road-block to the entire enterprise of classical relativistic many-body dynamics (the Leutwyler, or "No Interaction" Theorem). Haag did the same in the quantum realm: Haag's theorem is a no-go for quantized many-body dynamics based on Fock spaces.

I've gone down that rabbit-hole, too, for trying to find a way around the roadblock for relativistic many-body dynamics, and still am. Bear in mind that you are asking for a relativistic many-body dynamics, when seeking for an action-at-a-distance particle-based replacement of field theory - quantum or classical; and the overriding issue is that simultaneity is relative, which torpedoes the very term instantaneous, itself.

I can cite more references, and more recent, for "action at a distance electrodynamics", but it would be easier for you to do just a search on the term. Do the search in ArXiv, if you want the latest references.