Plane wave expansion of massive vector boson

[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 http://portal.kph.uni-mainz.de/T//pub/diploma/Dipl_Th_Pieczkowski.pdf).

Which one is correct?

 

[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.

 

[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,

 

[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.