system
Given the following post, respond as if you're commenting on Reddit:
human
What are virtual particles? How are they theoretically real yet undetectable?
gpt
Virtual particles are undetectable by definition. They are mathematical artifacts of *certain* methods of calculating *certain* observables. Specifically, they show up in perturbation theory.

In quantum mechanics, when you want to calculate the probability amplitude for a system to evolve from some initial state to some final state, you apply the time evolution operator to the initial state, and project it onto the final state. You can then break down the time evolution operator into a product of infinitesimal time evolutions, express this as a sum over all possible intermediate states.

This is how you derive the Feynman path integral formulation of QM, which is unrelated to the question, but it helps to understand what’s going on in a calculation in perturbation theory. In perturbation theory, you expand the matrix elements of the S-matrix (time evolution operator from t = - infinity to t = infinity) in a similar kind of series, where the terms in the series can be represented by Feynman diagrams. Each Feynman diagram starts with the same asymptotic initial and final states, but they contain some number of intermediate states, where some particles may have been created or destroyed. The “internal lines” in the diagrams, or the particles which don’t exist initially and will never interact with your detector in the final state, are virtual particles. They’re just part of an infinite sum over all possible intermediate states. You can’t say that any one of those intermediate processes is the one that “really happened”, you have to include contributions from all of them.

Because your quantum field theory probably conserves energy and momentum, four-momentum conservation is respected at each vertex in every diagram in your perturbation expansion. So the virtual particles in each diagram have whatever energy and momentum is necessary to respect the conservation laws. So to make things even weirder, if you try to evaluate the “mass” of a virtual particle by calculating m^(2) = E^(2) - **p**^(2), you don’t get the mass of the real version of that kind of particle. If you interpret the virtual particle as something that literally exists, you find nonsensical results, like photons with nonzero mass, or even particles with **imaginary** mass (negative mass-squared).

You may have heard layperson explanations about virtual particles “popping into existence”, or “borrowing energy from the vacuum”, but these are oversimplified, and not meant to be taken literally. You may have also heard of phenomena like the Casimir effect and Hawking radiation, which are described to lay audiences in terms of virtual particles, but the truth is that any phenomenon which can be explained in terms of virtual particles can be explained without ever referencing virtual particles. They only show up in certain calculation methods. You could in principle do the exact same calculation another way, and never have to reference virtual particles. And physics is invariant under the way we choose to calculate things. Therefore, virtual particles should not be interpreted to literally exist.