# Homework Help: Number of degrees of polarization of a spin 1 or 2 particle

1. Jul 16, 2009

### jacobrhcp

#degrees of polarization of a spin 1 or 2 particle

On page 32 of Quantum Field Theory by A. Zee, he expects you to remember the concept of polarization, specifically how to extract the number of degrees of polarization of a spin 1 or 2 particle.

As I seem to remember from EM class, polarization is something like the dipole moment per unit volume, or the number dimensions perpendicular to a given line of propagation in a wave. Nothing like Zee suggests. My memory is fading here.

Can anyone give me a good reference for this, so I can read up on it? Or explain, if possible, why polarization is so important here, and how to determine the number of degrees of polarization for a spin m particle?

2. Jul 16, 2009

### cepheid

Staff Emeritus
Re: #degrees of polarization of a spin 1 or 2 particle

I don't know any QFT, but I can tell you that this is the wrong type of polarization you're thinking of (the word is used in more than one context). That's electric polarization (of a dielectric medium).

Polarization of light in classical EM theory is as follows:

A plane wave propagating in a vacuum must have E and B vectors pointing transversely to the direction of propagation. However, that doesn't constrain things much, because there is a whole PLANE tranverse to the direction of propagation in which E and B could be pointing. What direction, specifically, in this plane, the E vector points is the polarization of the EM wave.

I know that there is some relationship between polarization of light and photon spin, but I can't help you further.

3. Jul 16, 2009

### jacobrhcp

Re: #degrees of polarization of a spin 1 or 2 particle

that illuminates things a bit, thank you. If anyone has for me a brilliant reference that explains this, I'd be grateful. So far I've tried griffiths, wikipedia and google (three of my most used sources of information =P)... but they do not explain this relation.

4. Jul 16, 2009

### turin

Re: #degrees of polarization of a spin 1 or 2 particle

Try J. J. Sakurai, "Advanced Quantum Mechanics".

I will try to help you out with spin 1. One thing that you need to realize in QFT is that the vector potential, rather than the field strength, is the object that couples to the fermion current. And, while even in QFT the field strength is the physical object whereas the vector potential suffers from gauge ambiguity, the typical approach (the only one that I'm aware of) is to (carefully) quantize the vector potential. In fact, in modern terms, the vector potential is fundamentally considered as a consequence of gauge invariance (of the fermion current), and the field strength is then the gauge invariant object required for the kinetic energy of the gauge field.

The vector potential has, naively, four degrees of freedom: one from φ and three from A. However, gauge invariance constrains the (physically effective) number of degrees of freedom to only two. Consider, e.g., the gauge choice of ∇⋅A=0 and A0=0.

The spin is actually a quantum property. You must act the angular momentum operator on a uniform field, and observe the resulting instrinsic angular momentum. (However, see, e.g., Problem 7.27 in J. D. Jackson, "Classical Electrodynamics", 3rd ed.)

Unfortunately, that's also the wrong polarization. The polarization of the QED photon is the polarization of the vector potential (A), not the polarization of the tensor field (E and B).

Last edited: Jul 16, 2009