# Plane wave expansion of massive vector boson

1. Oct 4, 2014

### guest1234

I'm trying to derive Feynman rules for massive vector boson and its antiparticle. It all boils down to plane wave expansion of the bosons which atm is a little bit confusing.

Should I account for two different set of ladder operators (as in the case of complex KG or spinors, cf Peskin&Schröder p58) and use both frequency modes, or should I use single set of ladder operators and positive (negative) frequency mode for the (anti-)particle (as in this thesis).

Which one is correct?

2. Oct 4, 2014

### vanhees71

It depends on what kind of particles you like to describe. If your massive vector bosons are charged, you can use a complex field which in the quantized version is written as a mode decomposition in terms of an annihilation operator $\hat{a}(\vec{p},\sigma)$ of the "particles" and a creation operator $\hat{b}^{\dagger}(\vec{p},\sigma)$ of anti-particles. Both have the same mass and opposite charges. An example in Nature are the W bosons: The $W^+$ and the $W^-$ are the corresponding particle-anti-particle "partners". Note, however, that they are special in the sense that they are part of the non-Abelian gauge fields gaining their mass from the Higgs mechanism in the electroweak part of the standard model.

You can as well start with a real massive vector field. Then, in the quantized version, you must set $\hat{b}=\hat{a}$ for all field modes. This implies that you have (strictly) neutral particles. An example is the Z boson, which is its own antiparticle. It's also part of the gauge bosons of the electroweak standard model and the same caveat applies as to the W bosons.

As you see, that's analogous to the case of scalar bosons. The only difference is that in addition you have three polarization states, according to the vector nature of the particles.

3. Oct 4, 2014

### dextercioby

You can use the path integral approach where you need no a, astar, b, bstar going to a, adagger, b, bdagger. All you need is the Lagrangian action for 2 types of fields, phi and phistar and the machinery of Green functions,

4. Oct 4, 2014

### guest1234

So... I think I'll go with two sets of ladder operators as usual in complex scalar field/fermions. It seems that the thesis uses them implicitly -- in Wick contraction the other operator is neglected because of the normal ordering requirement.

About path integral.. thanks but I'll skip it for the moment. Too much overhead involved.