# Casimir effect and vacuum energy and a bit of relativity....

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A. Neumaier
2019 Award
this is a hijack of the thread
... because what you say is not connected to the Casimir effect.
As @A. Neumaier has pointed out on numerous occasions
and in particular here, with the connection to the Casimir effect pointed out here.

... because what you say is not connected to the Casimir effect.

and in particular here, with the connection to the Casimir effect pointed out here.
Thank you!

haushofer
I've printed the papers by Jaffe and Demystifier, since I'm very interested in this Casimir-description. I think I get the reasoning, but I'm still wondering why the QFT 'vacuum fluctuation' calculation yields the same answer as the Van Der Waals calculation for the Casimir force. Is there any simple/intuitive way of seeing why these two seemingly completely different approaches lead to the same answer?

A. Neumaier
2019 Award
why the QFT 'vacuum fluctuation' calculation yields the same answer as the Van Der Waals calculation for the Casimir force.
Perturbation theory in quantum field theory always produces results expressed in terms of vacuum expectation values, no matter which problem is treated. Thus the coincidence of the results is not surprising. What is surprising is only that many people think something significant is expressed by talking about these vacuum expectation values in terms of vacuum fluctuations.

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Staff Emeritus
2019 Award
why these two seemingly completely different approaches
Why do you think they are completely different? In both cases, the force arises through the boundary conditions. The issue comes about when one tries to paint a mental picture of the results of this calculation.

stevendaryl
Staff Emeritus
Why do you think they are completely different? In both cases, the force arises through the boundary conditions. The issue comes about when one tries to paint a mental picture of the results of this calculation.
If I understand what the two approaches to deriving the Casimir force are, they sure seem very different:
1. Treat the electromagnetic field classically, but assume that the charges inside the metal plates are undergoing random internal motion. Then you get a Van der Waals type force between the plates.
2. Treat the plates classically (as just boundary conditions for the electromagnetic field), and treat the field modes of the E&M field quantum mechanically.
The two approaches seem to be focusing on completely different subsystems, and are making completely different approximations about what to treat classically and what to treat quantum mechanically.

Haelfix, haushofer and vanhees71
haushofer
Why do you think they are completely different? In both cases, the force arises through the boundary conditions. The issue comes about when one tries to paint a mental picture of the results of this calculation.
Well, I have to think about that, but the VdW-force I associate with dipole-interactions, giving a 1/r^6 law. For me that's really different from 'vacuum interactions' as described by Feynman diagrams.

-edit Steven's post above me states my issue a lot clearer, I guess.

Haelfix
Just so people are clear, this business is deeply controversial in the literature and not settled. There are quite a few subtleties involved, and really requires going through the entire analysis with a scalpel.

haushofer
Just to make sure i get it: in Jaffe's paper fig.3, those external legs are the conducting charges of the plates, right?

What I still can't reconcile is the dependancy of the Casimir force on the distance between the plates, i.e. eqn.(3) v.s. the VdW expression between eqn.5 and 6 (top of page 5). Shouldn't one expect F~1/d^6? What am I missing?

Staff Emeritus
2019 Award
Shouldn't one expect F~1/d^6? What am I missing?
The field from a charged plane is not the same as a field from a charged point.

Demystifier and haushofer
haushofer
Ok, that's something I definitely missed, thanks! :P

Demystifier
Gold Member
Just to make sure i get it: in Jaffe's paper fig.3, those external legs are the conducting charges of the plates, right?
No. Those are dielectric functions ##\epsilon## treated as a classical non-dynamic background field.

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Demystifier
Gold Member
If I understand what the two approaches to deriving the Casimir force are, they sure seem very different:
1. Treat the electromagnetic field classically, but assume that the charges inside the metal plates are undergoing random internal motion. Then you get a Van der Waals type force between the plates.
2. Treat the plates classically (as just boundary conditions for the electromagnetic field), and treat the field modes of the E&M field quantum mechanically.
The two approaches seem to be focusing on completely different subsystems, and are making completely different approximations about what to treat classically and what to treat quantum mechanically.
Or
3. Treat both charges and EM field as quantum dynamical mutually correlated fields.
Eq. (14) in my https://arxiv.org/abs/1702.03291 explains the equivalence of the 3 pictures in a very simple way.

Demystifier
Gold Member
Just so people are clear, this business is deeply controversial in the literature and not settled. There are quite a few subtleties involved, and really requires going through the entire analysis with a scalpel.
That's exactly why I have written
https://arxiv.org/abs/1702.03291

Demystifier
Gold Member
Is there any simple/intuitive way of seeing why these two seemingly completely different approaches lead to the same answer?
Yes. See Secs. IV.B and IV.C, as well as the discussions around Eqs. (14) and (78) in my https://arxiv.org/abs/1702.03291

Demystifier
Gold Member
The issue comes about when one tries to paint a mental picture of the results of this calculation.
I hope my https://arxiv.org/abs/1702.03291 helps to get a better mental picture.

Demystifier
Gold Member
Sorry all for over-advertising my most recent paper, it's probably irritating.
But I have written it precisely with intention to answer the kind of questions which are asked here, so I couldn't sleep well if I didn't point that out to you.

vanhees71
Gold Member
2019 Award

Demystifier
Demystifier
Gold Member
eloheim, bhobba and stevendaryl
vanhees71
Gold Member
2019 Award
Congratulations!

bhobba and Demystifier
Demystifier
Gold Member
At a recent conference I have presented an invited talk entitled

The origin of Casimir effect: Vacuum energy or van der Waals force?

Abstract:
In the literature on Casimir effect there are two approaches that make the same measurable predictions but offer very different explanations on the conceptual level. According to one approach the effect has origin in vacuum energy, while according to another it has origin in van der Waals forces. To resolve the resulting conceptual confusion, I discuss the conceptual aspects of Casimir effect from several different points of view. This includes fundamental particle physics (general principles of quantum electrodynamics), condensed matter physics (electrodynamics in continuous media) and non-relativistic quantum mechanics (a toy model with only a few degrees of freedom). All points of view lead to the conclusion that, at the fundamental microscopic level, Casimir effect originates from van der Waals forces, while the vacuum energy approach is an effective theory valid only at the macroscopic level.

The pdf of the presentation is attached.

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vanhees71 and bhobba