# Vacuum fluctuations of a free field and closed diagrams

1. Sep 7, 2005

### hellfire

I am confused with the Scharnhorst and Casimir effects. The confusion started here. As it seams that there is no response to my questions in post #9, I will try here.

I stated there my understanding about perturbation theory which leads to Feynman diagrams: The amplitudes for the fluctuations in the vacuum state of a free field can be computed as a time ordered product of fields in <0|…|0>. According to Wick's theorem, this can be decomposed as a product of Feynman propagators, leading to Feynman diagrams with no loops. Only if an interaction exists, a term $$e^{-i\int dt H_I}$$ appears within the product <0|...|0> due to the fact that one does not consider |0> anymore but the vacuum state of an interacting field expressed in terms of |0>. Only the term with the integral leads to loops in the Feynman diagrams after expanding the exponential as a power series (this is basically what I understood from P&S).

I think this summarizes to the statement that virtual particles appear only in case of interactions. However, what I wrote above seams to be incorrect. Otherwise, claims like this:

Make no sense to me. As far as I know the EM field inside the plates of a Casimir effect is not interacting. Phenomena like the Scharnhorst effect (FTL of photons in a Casimir vacuum) make also no sense as it seams to imply that there is some contribution of virtual particles to the propagation of light in a normal free vacuum (a contribution which is smaller in the Casimir vacuum).

2. Sep 7, 2005

### marlon

Virtual particles arise because of two distinct reasons. You only named one (force carriers). I refer to the https://www.physicsforums.com/journal.php?s=&action=view&journalid=13790&perpage=10&page=4 [Broken] entry of my journal.

one loop diagrams express the second posibility : ie vacuum fluctuations: particles that pop out of the vacuum and die shortly after at the same position of their origin.

regards

marlon

Last edited by a moderator: Apr 21, 2017 at 7:32 PM
3. Sep 7, 2005

### vanesch

Staff Emeritus
I would tend to agree completely with what you write. The strict free field is completely solvable and all its Feynman diagrams are trivial (lines connecting ingoing and outgoing particles) because there are no vertices.

What I know is that the expression for the Casimir pressure does not contain the EM interaction constant, and that the contribution is the same for all fields (not necessarily the photon field). As such it seems to be a property of the free field (that's btw how it is calculated, no ? You calculate a (cut-off) energy density of the vacuum (the 1/2 hbar omega term of each mode) of the allowed modes in the box, slightly move one of the walls, and calculate it again, and from the dE over dx you calculate the force. This is purely based upon the free field, EXCEPT OF COURSE that you need to assume an interaction with the walls which will give you the allowed modes. Hence, that's why only EM modes (up to a certain rather low energy) are taken into account (gamma rays are not reflected by the metal walls, nor are gluons or Z-bosons).
In all this, I don't see where a 1-loop diagram comes in...
But I'm out of my depth anyways.