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I've recently been trying to wrap my head around the notion of virtual particles, which as far as I understand live in quantum histories which can never be observed directly and which are not bound by certain laws of physics, but which may evolve into physically possible histories and thus produce a noticeable effect on the expectation value of quantum observables through interference. The equation I've seen which separates "real" states which we can potentially observe from "virtual" ones which we can't is the "mass shell equation"

$$E^2=m_0^2c^4+{\bf p}^2c^2.$$

This reminds me of Richard Feynman's informal derivation of the action principle towards the end of this lecture https://www.feynmanlectures.caltech.edu/II_19.html , where by treating both physically possible and impossible histories on equal footing, he gives a reason why histories with non-stationary action should be vanishingly unlikely. Can a relativistic version of this approach give us the mass shell equation on top of the Euler-Lagrange equations? Can the mass shell equation be derived from the action principle somehow? Am I seeing a connection where there isn't one and the virtual particles of QFT are different from Feynman's non-stationary paths?

Qmechanic
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Without a Compass
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  • The imprecise, hand-waving answer is "yes, it can be". 1960's era textbooks talked about 2nd quantization, but did not mention path integral formulations, the 1980's era textbooks do cover path integral formulations, and some try to bridge this divide. In the modern era, mathematicians have gotten into the game, and have generated lots of detailed and arcane results. Sadly, I know of no surveys that would provide an overview. Super-short answer: think of virtual particles as perturbative corrections. That's all. They're a formal trick cause we don't know how to solve the eqns exactly. – Linas Jun 06 '21 at 17:44

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Since Feynman considers point particles let us discuss this case. Yes, OP's speculations can indeed be realized. The Hamiltonian action $$\begin{align}S_H[x,p,e]~=~&\int d\tau ~L_H, \cr L_H~=~&p_{\mu}\dot{x}^{\mu}-\frac{e}{2}(p_{\mu}p^{\mu}+m^2), \end{align} $$ for a relativistic point particle has the mass-shell constraint $$p_{\mu}p^{\mu}+m^2~\approx~ 0$$ as one of its Euler-Lagrange (EL) equations, cf. e.g. this Phys.SE post.

Qmechanic
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