The total energy in the Casimir effect

Question

In the usual derivation of the Casimir effect between two large conducting plates, one finds that the energy in the empty space (relative to the ground state energy one has when the plates are infinitely far apart) is negative and proportional to $1/d^3$, where $d$ is the distance between the plates. Thus, naively it would appear that the total energy of the system consisting of the vacuum and the plates (consisting, for instance, of the energy in the vacuum, the mass energy of the plates etc) can get arbitrarily small in comparison to the energy of the system when the plates are not there. However, in reality, the plates are made of particles obeying the rules of QFT. Thus, we really have an interaction between the quantum fields of the particles in the plates and the field in the space between the plates (say, the EM field). In QFT, we expect the spectrum of the Hamiltonian to be bounded from below and that there is a stable vacuum state. Presumably, this vacuum state is given by the state where the quantum fields of the particles in the plates are not excited, ie when the plates are not there. Thus, from QFT it seems we would expect that the total energy is bigger whenever the plates are there, regardless of the distance between the plates. Does this mean that the total energy in a realistic system (as opposed to the usual simplification of infinitely large perfectly conducting non-quantum plates) consisting of the plates and the space around them can not be negative (relative to the vacuum state) after all? If so, what other sources of energy in the (realistic) system "cancels" out the negative contribution from the empty space between the plates?

I would imagine that the question of the total energy (not just the energy in the empty space between the plates) of the system in a Casimir apparatus has been extensively studied in the literature. If anyone is aware of key results concerning this issue, then that would be greatly appreciated.

(I am aware that there are many questions on this site concerned with the Casimir effect, but after searching I have not found questions concerning this particular issue)

Answer 1

The usual Casimir calculation of the Casimir force in terms of EM modes and boundary condition on the perfectly conducting plates is just a formal trick, and gives only approximate value of the actual Casimir force between real plates.

The actual plate in such an experiment is not infinitely conducting and perfectly flat; it is either a conductor with non-zero resistivity, or a dielectric which can polarize inside, but in any case, there are charged particles, non-zero current density and non-zero EM field in the plate. The Casimir force is physically due to attractive van der Waals forces between atoms or molecules in the plates.

Energy in a region of volume 1m$^3$ where there is only vacuum is zero, if plates are there, then energy is positive, as there is positive mass of the plates. In the macroscopic description adopting Poynting formulae, EM energy can't be negative. It can be negative only if we abandon the Poynting definitions and redefine EM energy to give only the interaction part, but this is not usually done in these calculations. Instead, the Poynting definitions are used, and then EM energy is positive.

The plates can't get as close as to make the Casimir force potential energy negative enough to make the whole energy of the system negative; the Casimir formula ceases to be accurate at short enough distances (certainly at distances comparable to atom spacing in the plates), because then repulsive effects become important (atoms don't collapse on each other on the plate boundaries).

Flat enough plates that get close enough (atomic distances) can weld together into a thicker plate, and this will still have net positive energy; the decrease of potential energy is of the order of surface binding energy of two solid lattice flat boundaries.

Answer 2

I think your question makes a very basic assumption which is incorrect. The "total energy of the system" is not an observable quantity; only differences in energy are.

Energy measurements usually depend on measuring the flow of energy from one system to another (eg heat flow in a calorimeter) or measuring the change in property of a system which is highly correlated with change in some type of energy (eg height of a weight above ground). In no case is there a measurement procedure which can measure absolute energy, only changes in energy - conversion between one type and another, or flow from one part of an (approximately) closed system to another.

In the case of Casimir energy, it is natural to think of something like "two infinite conducting planes infinitely far apart" as having zero absolute Casimir energy; but that is an arbitrary choice, and you can equally set your absolute zero to be when the plates are 1m apart, or any other point. It would make no difference in terms of anything measurable. The thing that can you measure is that when the distance changes from $d_1$ to $d_2$, the Casimir energy changes by $1/d_1^3 - 1/d_2^3$.

That said, certainly if bringing two real conducting plates very close, at some point the approximations of the plates being a contiunuous conducting solid, and being perfectly flat plates, stop being good approximations - either the roughness on the surface, or the spacing between atoms, starts being a factor. At that point the Casimir energy behavior would deviate from ~ $-1/d^3$. How it would deviate is a really complex question which I'm not even sure how to begin answering. However at such short ranges other interactions (dispersion, Pauli repulsion etc) become many orders of magnitude stronger than Casimir forces, so the deviation from ideal behavior is extremely small compared to the other forces involved.