# Source of Casimir effect energy

1. Jul 16, 2012

### gildomar

I know that the Casimir effect is a purely quantum mechanical phenomenon arising from the suppression of photons in the space between the plates and thus pushed together from the photons outside the plates. But just looking at a single cycle of the plates of the plates coming together, where exactly does the energy come to do so in a thermodynamic sense? As in does the energy effectively come from the space itself outside the plates? Or am I not looking at the situation correctly?

Last edited: Jul 16, 2012
2. Jul 18, 2012

### nonequilibrium

I've heard that the Casimir effect is actually not a quantum mechanical effect, but that the visualisation that you're talking about is merely the quantum mechanical version/calculation of the classical Van der Waals forces (aka London interaction).

In that case you also gain kinetic energy, and it's easy to answer your question in that case: it comes from the built up (electromagnetic) potential energy in the fields.

3. Jul 18, 2012

Forgive me if I am wrong, but as far as I know, the gained energy is accounted for by a negative energy density left between the plates. In thermodynamics terms....this only happens when the plates are superconducting [and thus very cold]. This means that energy is approaching equipartition - so the entropy is still increasing overall.
Hope this is useful

4. Jul 18, 2012

### JPBenowitz

When you do the calculations there is an infinite amount of energy outside the two plates and less infinite energy in between the plates so the difference leaves you with a negative energy density.

5. Jul 19, 2012

### Born2bwire

Yeah, more or less. The Casimir effect comes about from the Casimir-Polder force which is the QED correction to the London force. The implementation of the special relativity and the quantization of the electromagnetic field resulted in that a molecule can be attracted towards even PEC objects. The Casimir force is mainly just a macroscopic implementation of the Casimir-Polder intermolecular force.

As for the energy, thermodynamics does not really have to come into it due to the fact that the Casimir force exists even at absolute zero. Thermodynamic corrections are applied at non-zero temperatures. The Casimir effect can be described using statistical physics via the fluctuation-dissipation theory though (and in this way the thermodynamic effects are more apparent, relate to other examples like the Johnson-Nyquist noise). As for where the energy come from, it's kind of like where does the energy come from in many electromagnetic system. Energy is injected into the system when you assembled the system together. Like bringing an electron in from infinity to within proximity of another electron requires work, the same work that you get when you release the electron and allow the two to separate.

When calculating the Casimir energy, you have to normalize it to a finite value and this is done by normalizing against the energy of the system when the objects are at infinite separation. So you can think of the Casimir energy as the energy change in the system by bringing the objects together. The force arises due to the dependence of the energy on the relative position of the objects. So the actual value of the energy is something that can always be redefined since that represents a constant offset. It's the change in the energy that matters and so the physical meaning of the energy can be changed offhand just like how the meaning of electric potential can be changed based on how the zero potential is defined.

6. Jul 19, 2012

### gildomar

Thanks for the replies.

So if the Casimir effect can just be described by London forces, how much relevancy does describing it in terms of zero-point interactions have? What I mean is, is describing the effect in terms of zero-point interactions just a mathematical trick in order to make the calculations easier and easier to visualize? Or is it "real"? Or does the question not even have any meaning?

7. Jul 19, 2012

### alemsalem

the London force is between two objects that are supposed to be neutral but the force is there because of the "polarization" of the objects by slight separation of the charges.
(maybe) in the Casimir effect you still need the zero-point energy to do the calculation and in understanding "vacuum polarization". I think its a quantum effect with a classical analogue.

8. Jul 19, 2012

### Born2bwire

The zero point energies I believe are not strictly needed. R. L. Jaffe has a paper that shows that you can formulate the Casimir force without any reference to the quantum vacuum and its zero point energies (Phys. Rev. D. 72, 021301 (2005)). So the existence of the Casimir force is not to be taken as proof of the quantum vacuum.

One way in which the vacuum and its zero point energy can come into play is in explaining why molecules exhibit a flucutating moment. The London force assumes that the molecules have a fluctuating dipole moment modeled as an electron acting as a 3D isotropic harmonic oscillator. The magnitude of the oscillator is immaterial (we can say that it has to do with the polarizability of the molecule and use experimental results to work it out) and it gives us a potential profile on how the molecules interact. The Casimir-Polder force is a reworking of this problem using QED as opposed to London's non-relativistic quantum mechanics. Peter Milonni shows in his "Quantum Vacuum" text that the source of the fluctuations can be found to be the coupling of the molecule with the fluctuating vacuum fields. So the vacuum fields behave like a background electromagnetic field that is being applied to the objects. Even the QED electromagnetic fields must satisfy the macroscopic boundary conditions so we can then find the modes of the system where these vacuum fields can exist. In other words, the presence of scatterers forces the vacuum modes to shift to frequencies where they can naturally conform to the boundary conditions. So by finding the density of states of the vacuum field, we can find the energy of the vacuum and use this to find the Casimir energy. This actually can be done computationally although it is more popular to use the fluctuation-dissipation theory or a path integral as the means of deriving a computational method.

How does one explain the Casimir effect without the vacuum? I can't recall exactly but I seem to remember that it was done by Schwinger using his source theory approach. The disadvantage is that it is a much more complicated derivation than using the vacuum approach. Physically, I think one could probably describe the fluctuations as resulting from natural quantum fluctuations in the electron clouds that one expects over short time periods. That is, the eigenstates of the orbitals only describe the statistical behavior of the electrons. Over short time periods, one could probably expect that a symmetric orbital like the s can have distortions but over long periods the overall dipole moment will die out. I haven't actually read of a derivation that takes this approach myself but I wouldn't be surprised if it can be proven to be physically equivalent. My own research is on the macroscopic (if we can be so generous in describing the length scales) Casimir force and not the Casimir-Polder force.

9. Jul 19, 2012

### dslowik

From reading this thread it seems there are two seperate causes of an attractive force between metal plates in a vacuum:
1) The exclusion of (longer wavelenth) modes of the (vacuum)electromagnetic field between the plates which leads to fewer zero-point quantum oscillator energy contributions as the plates get closer, and so an attraction.
2) Van-der Walls type polarization fluctuations among the molecules in each of the plates attracting each other.
So it seems that by explaination 1, the answer to the OP is that as the energy density outside the plates remains constant, it decreases between the plates, effectively less pressure betweeen the plates than the ambient vacuum as they get closer. And by explanation 2, that the attraction is that of random but correlated fluctuating electric dipoles in the plates.
I thought the original Casimer effect was explanation1; that new and interesting phenomenon uncovered by the Casimer effect had to do with the vacuum energy. Is 2 really equivalent? Can't the plate material be made so as to vary the relative contribution of effects 1 and 2?

10. Jul 19, 2012

### cgk

It's a mathematical trick... those things are reformulations of each other. For obvious reasons the vacuum fluctuation explanation formulation is much more popular, however.

After all, if everyone thought this was a simple dispersive interaction effect, would you think you had ever heard of it?

The original formulation is actually #2: http://prola.aps.org/abstract/PR/v73/i4/p360_1

So if you are not terribly interested in the QED corrections (those arise due to the retardation of the Coulomb interaction), you can obtain this force by standard non-relativistic QM and nothing more than the Schroedinger equation. The London dispersion force is an electronic correlation effect, and no reference to any vacuum is needed. Note that between molecules such forces are calculated routinely using the methods of quantum chemistry. They do not receive any special handling: correlated electron methods like CCSD(T) produce them automatically (together with all the other dynamic correlation effects) to an accuracy of <0.5%. The simplest methods which can account for dispersive interactions are MP2 (Moeller-Plesset perturbation theory, 2nd order) and RPA (random phase approximation). If you google MP2, you will see that its formula can be written in a single line and contains nothing more than a sum over electron repulsion integrals between orbitals and orbital energies.

11. Jul 20, 2012

### dslowik

Thanks cgk, that 1947 abstract of Casimir and Polder discusses the calculaltion of the effects of retardation on the interacton between two neutral atoms. They first do a simpler calculation of the interaction between a neutral atom and a perfectly conducting plane. In either case, it seems that fluctuations of dipoles are central.

My question now is, in the absence of any atomic polarization fluctuations, would there be a force of attraction between two perfectly conducting planes that is due solely to the exclusion of (longer wavelength) modes of the (vacuum)electromagnetic field?

12. Jul 20, 2012

### Born2bwire

No, they are one and the same. Without the fluctuating moments, the vacuum field would not interact with the plates.

13. Jul 20, 2012

### dslowik

oh, I thought maybe they(Casimir Polder) were calculating the effects of the quantum fluctuations of the vacuum field (in particular the excluded long modes) on the interaction of the fluctuating moments. I mean I see how fluctuating mooments can attract each other in classical electromagnetism, and the exclusion of long modes is not required for that. Now, with each mode having a zero-point energy from QM, excluding some of them will change the energy density of the vacuum between the platesleading to an attraction that has nothing to do with the fluctuating moments in the atoms of the plates. To me there seems to be two quite different phenomoman going on which each lead to an attraction between the plates, And yes, one may effect the other too...

14. Jul 20, 2012

### Born2bwire

Ok, so Casimir-Polder works because atoms and molecules are generally polarizable. So the fluctuating vacuum fields will couple with the molecules and induce fluctuating dipole moments. The dipole moment of one molecule induced by the vacuum creates a secondary electromagnetic field that induces a dipole moment in a neighboring molecule. If you want to know more about how a force arises from this, take a look at the van der Waals induction effect (this assumes permanent dipole moments and polarizable molecules but similar physics are involved). But conceptually, the polarization of the molecules by the vacuum fields is no different than if we applied an external electromagnetic field.

So that's a microscopic picture of the what is happening, but what about macroscopically? Well if we applied an electric field over a bulk material, we can incorporate the net effect of all the induced dipole moments with the Polarization Field. This polarization field, when added to the electric field, gives us the electric displacement field (electric flux density). That is, we already have a simple macroscopic model for what happens when we apply a field that induces dipole moments in the constituent molecules of a bulk, the permittivity and permeability.

That means that the fluctuating vacuum field is going to behave in the same way in the macroscopic picture, to whit, it will satisfy the same boundary conditions that would apply to an explicitly applied electromagnetic field. The Casimir energy is going to be a measure of the energy contained in the interactions between the molecules due to the vacuum induced fluctuating dipole moments. The baseline energy is thus going to be the energy of the system when there is no interaction between the induced moments. This is the system when the objects are infinitely separated from each other (which includes the energy of the field interactions between the molecules within the objects but these do not factor in the Casimir force). The effects of the interaction of the induced dipole moments are going to be encapsulated into the change in the energy contained in the fields when we bring these objects together.

That is why we can simply find all the possible modes of electromagnetic fields that naturally satisfy the boundary conditions of the system and add the energies of the modes up to find the Casimir energy (after normalizing it as described above).

One of the main points here is that the Casimir force comes about due to the reliance of the energy of the system on the relative separation of the scatterers. So the Casimir energy is defined with that in mind. The normalization removes the divergence in the energy density and it removes some of the energies that do not contribute to the Casimir effect (the energy contained in the induced dipole moments and their self-interactions within the objects themselves).

Basically what I was getting at in my previous response is that the interaction of the vacuum fields with the scatterers is mediated, microscopically, by their induction of dipole moments in the molecules. The induction of these dipole moments forces the fields to conform to the macroscopic boundary conditions. Thus, in this interpretation you cannot have one without the other.

Another way to find the macroscopic forces is to use the Maxwell stress-tensor. The stress-tensor gives the pressure distribution across a body due to the electromagnetic fields along the surface. This involves knowing the time-averaged correlation functions of the fields. The fluctuating vacuum fields have are zero mean but have non-zero correlations. The value of these correlations can be found by using the fluctuation-dissipation theorem. So in this case, the force can be found by finding all the possible field distributions of the vacuum fields that satisfy the boundary conditions and summing up the pressures. In terms of calculation, this becomes essentially the same problem though (in a manner of speaking, instead of solving for the eigenfrequencies of the system you solve for the dyadic Green's function of the system). Path integral formulations can be used by starting with a Lagrangian and action that describes the energies of the vacuum fields in the presence of the scatterers. The path integrals can be used to find the energy of the vacuum field states.

15. Jul 21, 2012

### Darwin123

If you want to analyze it thermodynamically, then you have to add a term in the expression for potential energy that includes the energy stored in the Casmir force. The Casmir force is a conservative force that would vary with distance between the two plates. The potential energy is the negative of the work it takes to move the plates relative to each other.
The potential energy can be calculated by an integral of force with distance. There potential energy term that can be derived from the force.
Note that one does not have to know what causes the Casmir force to determine the potential energy. All one needs is the expression for force. Once one knows the force, one can determine the work needed to move something. The negative of the work is potential energy.
The van der Waals force is a force between two atoms without a dipole. The van der Waals force can be derived from the zero point energy. A lot of computer simulations use a potential energy function based on the van der Waals force. The potential energy decreases as the sixth power of distance between the atoms.
One doesn't need to know what causes a force to know what the potential energy is.

16. Jul 22, 2012

### dslowik

born2bwire thanks for all that information. Seems you have been working in this area for some time..

However I am still puzzled because it does seem to me that there should in fact be a force of attraction between two idealized perfectly conducting planes due to the excluded EM vacuum modes. If you look at
http://en.wikipedia.org/wiki/Casimir_effect
this is described under the section titled "Effects"
and there is a calculation of this purely vacuum effect given under the section "Derivation of Casimir effect assuming zeta-regularization".
This wikipedia page also discusses some of the corrections to this due to the atomic structure of the plates, but even in the absence of those corrections, there is a force that is purely vacuum related..

Surely the atomic fluctuations intertwine with the vacuum, and the meat of the realistic calculations must be there as you described. But I'm just wondering philosophically if the vacuum by itself would have an effect and it seems that it does.

Last edited: Jul 22, 2012
17. Jul 22, 2012

### Born2bwire

The point I am making is that the change in the energy density of the vacuum comes about because the vacuum fluctuations couple with the atoms and molecules. The change in the energy density of the vacuum incorporates the effects of the interactions between the induced dipoles on one object with the induced dipoles on another object. Their interactions are incorporated into the boundary conditions that the vacuum fields must conform to.

18. Jul 23, 2012

### gildomar

Part of the reason that I was wondering about this was from reading about the corrections to the force that Lifgarbagez made to it concerning the effects of dielectrics with the plates (namely, that the force will be greater or less than "normal" depending on the dielectric constant), and the viewpoint of wavelength exclusion. I was wondering then what would would happen if each of the 2 plates was a different dielectric? Using the exclusion viewpoint, it seemed that there would be a different force "pushing" the 2 plates together. Or would the force would just be an average of what you would get from using the 2 dielectrics in the equation (using just the London force picture)?

19. Jul 23, 2012

### Born2bwire

The force would differ but it would not be an averaging. One of the differences between the Casimir force versus the London or Casimir-Polder force is that it has multiscattering effects. London force is a pairwise intermolecular force. But what really happens is that the induced dipole on a secondary molecule will in turn cause induced dipoles in other neighboring molecules and so on. So for this reason I do not think you can extrapolate the result based on the forces from other materials. But again, on the macroscopic scale, all of this gets encompassed into the dielectric constant of the material. As it stands however, for the simulation of NEMS/MEMS devices, the researchers that I know just use pairwise London forces. So hopefully we can get Casimir calculations to be easy, robust and fast enough to supercede this.

If you want to see how the case between dielectric slabs are handled, a derivation that is similar to Casimir's PEC plate derivation was done by van Kampen. Once again, van Kampen finds a dispersion relation to describe the allowed modes of the electromagnetic fields and uses the argument principle to add up the zero point energies of these modes. This is still an approximation because he does not take into account the propagation of light, but it can still be shown to agree with the Lifgarbagez result. Lifgarbagez has another way calculating the Casimir force but it is much more complicated.