# Solving a Casimir configuration, the U - bracket

A problem related to an interesting Casimir configuration.

I have the following configuration, related to a modified Casimir experiment.  We have two parallel metal plates and an orthogonal metal plate connecting the two parallel plates. Note that this is very close to the experimental setup of the original Casimir experiment, except for the orthogonal plate (connecting the two parallel plates at one end). I would like to know all the Casimir  forces that exist in this configuration.

For clarification, you can see the following picture: My naive intuition could be incorrect, but my impression is that the asymmetric radiation  pressure due to quantum fluctuations would lead to the emergence of a certain force (parallel to the parallel plates), but I don't know if the QFT / QED calculations justify my intuition (where in fact  these virtual particles are just an intuitive representation of links in Feynman diagrams, linked to terms in infinite perturbation series).

Heuristic derivation of the Casimir effect from  the Generalized  Uncertainty Principle

Closely following the arguments in these two papers, that force mentioned above does indeed exist (for the three plate system mentioned above).

I hope somebody here can quickly settle this problem, so I don't spend a long time digging for an answer. Thanks.

## 1 Answer

It would be a nice experiment to make. I could say that an uniform Lorentz-invariant vacuum energy density must look like $$\rho(\omega) \propto \omega^3$$, so it should locally appear to be mostly condensed in modes that are shorter than your cavity. If you want to make the force observable, you need the cavities to be as tightly-packed as your engineering permits. I don't think this particular setup has been attempted because it is extremely hard to measure without zero gravity.

• Connected question , to your answer physics.stackexchange.com/q/32485 – Cristian Dumitrescu Sep 28 '19 at 17:46
• physics.stackexchange.com/q/109115/31339 The stress-energy tensor is Lorentz invariant, and it is connected to energy density . – Cristian Dumitrescu Sep 28 '19 at 20:03
• physics.stackexchange.com/q/154842/31339 Some answers to the three questions above clarify some of the problems related to the Lorentz invariance of energy density. – Cristian Dumitrescu Sep 28 '19 at 20:21
• I agree that this design must be seriously considered, and experiments must be performed in this direction. I followed the arguments in the papers linked in my question (heuristic derivation of the Casimir effect based on Heisenberg uncertainty principle ), but I only have superficial knowledge of the mathematical framework of QFT. Therefore, I am pretty sure that the Casimir U - bracket shows that quantum fluctuations can do work, but I cannot reliably give a quantitative analysis of this system (I mean a good estimate of the magnitude of that force). I hope somebody will help. – Cristian Dumitrescu Sep 28 '19 at 20:39