# Casimir plates prevent photon existence only perpendicular to them?

1. Feb 25, 2014

### Edi

Does Casimir plates prevent photon existence only perpendicular to them?
I mean, Casimir attraction arises from the fact that the plates prevent some wavelenghts of photons to exist in between them, so an imbalance arises and pushes the plates together, right?
But what about photons in other than 90 degrees angle to the plates? say, at 45 degrees - the length of the photon in such a way could be longer then a perpendicular photon, but still limited, no?
.. if so, even further - what about photons perpendicular to the plates? Photons that would otherwise go straight trough the gap between the plates?
Some say that photons are supposed to width-less and height-less, but, then again - photon polarization works kinda with the principle that a photons have width (at least in 2D or maybe even in 1D perpendicular to its path line) for its electric and magnetic field.

2. Feb 25, 2014

### Staff: Mentor

The Casimir force is not limited to parallel plates. Does that answer the question?

For parallel plates, only the component perpendicular to the plates is limited, while the other directions of a photon wave vector are free. This also limits "45°-photons".

3. Feb 26, 2014

### gburkhard

The polarization interpretation isn't quite correct: The fields change in amplitude in the direction of polarization, but that is the field along the axis of the photon -- the field is not defined away from that axis. The picture of a wave going up and down along the propagation direction is not one of a field as a function of transverse distance, but rather field amplitude as a function of distance along that line.

If you ask what is the lateral spatial extent of a photon, then that is a really complicated question, because of what boils down to fourier analysis and what you can do spatially to define the narrowest beam possible. I don't know what the fundamental limit for this would be, but you can certainly transmit photons into cavities smaller than their wavelength by coupling to, for instance, traveling plasmon modes inside of a nanostructured gap.

4. Mar 2, 2014

### Edi

Maybe, somehow.. the main question was about photons passing parallel between the plates, but this begs me to ask more questions along the line : Two particles (say, electrons or simply atoms(?)) in space would be attracted to each other not only by gravity, but also do to the [very small] Casimir force.. yes?
Between asteroids.., planets,.. stars.. - Casimir force would work there too?

Different question:
If the attraction arises from the difference of pressure inside the gap of the plates and outside of the plates, then it really matters where the experiment takes place, doesn't it? I men - if it happens in a evacuated metal container, in a human made lab, then the outside pressure/ photon wavelengths are limited to the size of the container containing the experiment. If done in, say, outer space, with no container and other things blocking the wavelengths, the difference would be larger, because .. well.. there is much more space in outer space, wouldn't it?

Last edited: Mar 2, 2014
5. Mar 2, 2014

### Staff: Mentor

The Casimir force is an effective force between conducting objects. Elementary particles are not "conducting objects" in that sense, so that view is a bit problematic.

Asteroids and planets are bad conductors, but in principle: yes. They are completely negligible.

If you change other forces, you change the net force between objects, but that is not related to the Casimir force between the two objects.

6. Mar 2, 2014

### Born2bwire

The Casimir force can occur between non-conducting dielectrics, Lifshitz did the early seminal work on this. Also to expand on what you said, the Casimir force occurs due to the correlation between molecules due to the fluctuating electromagnetic fields from the orbiting electrons. In other words, it's a van der Waals force. So in this sense it does not exist between elementary particles and like any other van der Waals force, it is only important on very small scales.

So atoms do experience the Casimir force, it's specifically called the Casimir-Polder potential, but not elementary particles. The effective force between a macroscopic body of molecules due to the Casimir-Polder potential is the Casimir force. This makes it interesting because we can have different interpretations of the Casimir force that are all valid. Photon radiation pressure is one way to look at it, but if you are having trouble with it you can think of it in the classical electromagnetic field picture (this is valid due to the fact that the Casimir force is between macroscopic systems, at least systems too big to be true quantum systems, and because Maxwell's Equations are still valid in QED). As I stated above, the force arises because the orbiting electrons around the nucleus create fluctuating electromagnetic fields, primarily as a fluctuating dipole moment. You can think of this as a byproduct of the electron cloud. This fluctuating dipole moment couples with the fluctuating dipole moments of the neighboring atoms. This coupling creates a potential that is spatially dependent and thus a force results (a simpler model of this is the London force). I'll skip a lot of explanation to say that we can also calculate the Casimir force using classical electromagnetics. We find all of the electromagnetic modes that a system can naturally support with its boundary conditions. Then, the Casimir energy is found by adding up all the energy of these modes.

With two infinite parallel plates, it is easy to see how the plates restrict the modes that can exist in the system. The dependence on the density of state of these modes with the separation of the plates is what gives rise to the Casimir force. Now, as mfb was discussing earlier, we can describe the modes of this system by the wavevector. This wavevector is continuous in the transverse direction but is quantized in the normal direction due to the plates. So actually, we can have electromagnetic waves in any direction contributing to the energy in the system. However, it is the magnitude of these waves (the frequency associated with a given direction of propagation) that is constricted and the energy of the system is dependent upon these frequencies.

But you are correct in saying that the Casimir force could be affected by the confines of our system. For example, placing our plates (which are now finite) inside a sealed metallic box would further restrict the modes that can exist. Now we not only have the boundary conditions imposed by the plates, but also by the walls of the cavity to contend with. This is actually a system that I have used to calculate the Casimir force because it has the nice property of being lossless if we assume a finite cavity and only perfect conductors (bringing the size of the cavity to infinity requires an infinitesimal loss due to causality). However, though I describe the Casimir force as being macroscopic, the lengths scales over which it is significant are on the order of micrometers and nanometers. So if we assume a reasonably sized cavity in the lab in which we do our experiment, it will be so large that the cavity modes will be, for all intents and purposes, more or less the same as an infinite vacuum. That is, unless you confine your objects to a cavity or semi-enclosed container that is on the order of the length scales in which the Casimir force is important, then you are not going to see much influence.

7. Mar 3, 2014