Casimir force and the Vacuum Field

[Swamp Thing]

The Casimir force is often explained in connection with the zero point energy of the quantized EM field, and the fact(?) that metal boundaries modify this spectrum in such a way that a force is created.

However, other sources (including some discussions on this forum) say that this is not the deepest explanation. For example, models based on Lipschitz are based on the behavior of electrons within the boundaries, and are more accurate in certain cases where the zero-point EM model would fail. This discrepancy shows up as extra terms containing the fine structure constant via the measured material properties. So the apparent existence of the zero point EM field within a Casimir type structure is merely an emergent phenomenon and not an inherent property of "The Vacuum".

But this, too, seemed a bit worrying to me for a moment -- since it could mean that the entire idea the quantized EM field could then be written off as an emergent phenomenon. Of course, this path hits a roadblock as soon as we look at the behavior of single photons, entangled photon pairs etc.

So the point of the above is that I'd just like to confirm if the following statement is correct: "The quantization of the EM field is a fundamental thing in its own right, but in the context of Casimir force we are actually dealing with an emergent phenomenon that just happens to resemble and mimic the quantized field in some limited ways."

 

[Demystifier]

So the point of the above is that I'd just like to confirm if the following statement is correct: "The quantization of the EM field is a fundamental thing in its own right, but in the context of Casimir force we are actually dealing with an emergent phenomenon that just happens to resemble and mimic the quantized field in some limited ways."

Let me put it this way. The EM field (E,B) is indeed fundamental. But in the Casimir case, you do not deal with the fundamental field E. You deal with the emergent field (electric displacement) D, which is a sum of electric field E and polarization P. So in this sense, you are right. For more details see my http://de.arxiv.org/abs/1702.03291 .

 

[Swamp Thing]

Thank you.

 

[Swamp Thing]

I just started reading your paper, but I had to pause and come back to say that I just love this:

Casimir vacuum is the ground state for ... only those phenomena for which (i) the existence of Casimir plates is given and (ii) the distance y between the plates is fixed.

Stating the obvious gets a bad press, but it's often necessary to avoid the pitfalls of hidden assumptions.

 

[vanhees71]

The Casimir effect is one of the most incorrectly explained phenomena in quantum theory. The reason is that the simple calculation with the plates is NOT referring to the calculation of "vacuum flucuations" but is the interaction between charges within the plates in the strong-coupling limit $\alpha_{\text{em}} \rightarrow \infty$:

https://arxiv.org/abs/hep-th/0503158

 

[Demystifier]

The Casimir effect is one of the most incorrectly explained phenomena in quantum theory. The reason is that the simple calculation with the plates is NOT referring to the calculation of "vacuum flucuations" but is the interaction between charges within the plates in the strong-coupling limit $\alpha_{\text{em}} \rightarrow \infty$:

https://arxiv.org/abs/hep-th/0503158

Well, that's a part of the story. This interaction between the charges in the strong-coupling limit also involves quantum fluctuations in a certain quantum state. And this quantum state is also a kind of a "vacuum", provided that one is careful about what exactly one means by the word "vacuum". What turns out is that this "vacuum" is not a photon vacuum (a state with zero number of photons) but a polariton vacuum (a state with zero number of polaritons, which are quasiparticles that can be thought of as mixtures of photons and polarization quanta, with polarization quanta being quasiparticles themselves.)

 

[Swamp Thing]

From the above, I'm getting that the term "vacuum" in this context is a misnomer justified by an analogy. If we had the opportunity to popularize a better term, what would that be?

 

[Demystifier]

From the above, I'm getting that the term "vacuum" in this context is a misnomer justified by an analogy. If we had the opportunity to popularize a better term, what would that be?

I would propose quasivacuum, for it is a state with zero number of quasiparticles.

 

[Swamp Thing]

Demystifier, couple of questions re. your paper.

You introduce Equation (7) by saying,

For reasons which will become clear soon, instead of saying that those frequencies do not contribute, it is much more appropriate to say that those frequencies are zero.

I am having trouble visualizing how the E-field would look for a case where $k_y\neq n\pi/y$. I have pasted my attempt below. We're basically saying that we can't fit an integer number of waves into the gap, but we're going to try anyway. There is a definite wavenumber, hence a definite (fractional) number of waves, with an evanescent wave oozing into the metal / dielectric. But if the frequency is zero as per Equation (7), then the pattern in the figure has to be static in time.

My naive take on this is that such a static but wavy field is impossible. However, this difficulty doesn't seem to affect the final conclusion we are heading for, namely that the distance d can modulate the "effective" dielectric constant seen by the whole wave -- and this applies also to the cases where $k_y = n\pi/y$. As we reduce d, the evanescent part becomes a larger fraction of the total environment in which the wave exists, so I see how that can become the basis for your toy model via $\epsilon_k(y)$.

Another question: is it correct to say that your model is nonlinear?

..

 

[Demystifier]

My naive take on this is that such a static but wavy field is impossible. However, this difficulty doesn't seem to affect the final conclusion we are heading for, namely that the distance d can modulate the "effective" dielectric constant seen by the whole wave -- and this applies also to the cases where $k_y = n\pi/y$. As we reduce d, the evanescent part becomes a larger fraction of the total environment in which the wave exists, so I see how that can become the basis for your toy model via $\epsilon_k(y)$.

I think it's basically correct.

Another question: is it correct to say that your model is nonlinear?

Yes, absolutely. That reflects the fact that QED (in which charges interact with the EM field) is also nonlinear.

 

[Swamp Thing]

Wow, I had no idea that QED is nonlinear! I haven't studied it formally, so all I know about it is from Feynman's popularizations like his "Strange Theory..." But I somehow assumed that it is linear in the sense that QM is linear, i.e. in the sense of being applied linear algebra.

But using only the ideas from Feynman's "Strange Theory", is it possible to build a simplified picture of why QED becomes nonlinear, as opposed to plain vanilla QM?

 

[Demystifier]

But I somehow assumed that it is linear in the sense that QM is linear

In that sense QED is indeed linear, as is my toy model. Obviously, you are confused about two different notions of linearity. One refers to the evolution of the state $|\psi(t)\rangle$, which, in the Schrödinger picture, is governed by the linear Schrödinger equation. Another refers to the evolution of observables such as $x(t)$ and $p(t)$, which, in the Heisenberg picture, may be governed by either linear or nonlinear equations of motion.

 

[Swamp Thing]

Is the inverse square law in electrostatics an example of the kind of nonlinearity we are talking about? That is, QED is designed to fundamentally encompass things like the inverse square law etc.? But the time evolution of any state is still a linear computation?

 

[Demystifier]

Is the inverse square law in electrostatics an example of the kind of nonlinearity we are talking about?

Not quite. For an actual example see https://en.wikipedia.org/wiki/Nonlinear_optics

But the time evolution of any state is still a linear computation?

Yes.