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

[stevendaryl]

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.

 

[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?

 

[Vanadium 50]

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.

 

[haushofer]

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

 

[Demystifier]

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.

 

[Demystifier]

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]

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]

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]

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]

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]

Well, it's self-advertisement for something the readers can download for free. That's ok, I'd say

 

[Demystifier]

If someone is still interested, now a version accepted for publication in Ann. Phys. is available:
https://arxiv.org/abs/1702.03291

 

[vanhees71]

Congratulations!

 

[Demystifier]

The published version:
http://www.sciencedirect.com/science/article/pii/S0003491617301410

 

[Demystifier]

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.