Plane wave expansion of massive vector boson

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Discussion Overview

The discussion revolves around the derivation of Feynman rules for massive vector bosons and their antiparticles, specifically focusing on the plane wave expansion of these bosons. Participants explore different approaches to quantization and the implications of using various sets of ladder operators.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether to use two different sets of ladder operators for massive vector bosons, similar to complex Klein-Gordon fields, or a single set for both particles and antiparticles.
  • Another participant suggests that the choice depends on the type of particles being described, noting that charged massive vector bosons can be represented with complex fields and separate annihilation and creation operators for particles and antiparticles.
  • A different perspective proposes using a real massive vector field, leading to the conclusion that the annihilation and creation operators for all field modes must be the same, resulting in neutral particles.
  • One participant mentions the path integral approach as an alternative that does not require the use of ladder operators, focusing instead on the Lagrangian action and Green functions.
  • A later reply indicates a preference for using two sets of ladder operators, referencing implicit usage in Wick contractions and expressing reluctance to adopt the path integral method due to its complexity.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate approach to quantizing massive vector bosons, with no consensus reached on whether to use one or two sets of ladder operators. The discussion remains unresolved regarding the best method for deriving the Feynman rules.

Contextual Notes

Participants highlight the dependence of their approaches on the nature of the particles (charged vs. neutral) and the implications of using different quantization methods. There are references to specific examples from the standard model, but no definitive conclusions are drawn.

guest1234
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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?
 
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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.
 
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,
 
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.
 

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