About the Bargmann-Wigner Method

[HeavyWater]

TL;DR Summary

By example, show how one obtains a Bargmann-Wigner wave function.

See Reference: ArXiv: 1208.0644

The Bargmann Wigner Method seems to be a means to obtain wave equations for higher spins, I.e. j=1, 3/2, 2, etc. In this method, one uses a Dirac-like equation to operate on a wave function. For example, equations, 2.2 and 2.3 show Dirac-like equations that operate on wave functions. Usually in QM, we SOLVE for the wave function, but to my surprise equation 2.4 GIVES the wave function (here for j=1). In addition, I am surprised to see Dirac Gamma matrices (which pertain to Fermions) in an equation that will be used to derive a bosonic (j=1) wave equation. Can you talk about how someone came up with equation 2.4 and show how one derives one (or both) of the terms in 2.4.

 

[samalkhaiat]

https://www.physicsforums.com/threa...motion-for-a-scalar-field.957092/post-6071636https://www.physicsforums.com/threa...theory-from-the-ground-up.941219/post-5954706

 

[HeavyWater]

I know the responders are busy...but no one has responded to the last sentence in my query--it's about how to arrive at both terms in equation 2.4 in the (above) specified reference. Please read it again--if you don't know the answer it's ok to say so.

 

[samalkhaiat]

I have already answered your question. Read the argument which led to Eq(2.1) in
https://www.physicsforums.com/threa...theory-from-the-ground-up.941219/post-5954706Any totally symmetric (BW) multi-spinor can be expanded in terms of the symmetric subset $\{ \gamma^{\mu} C , \sigma^{\mu\nu}C \}$ of the Clifford algebra
$$\Gamma^{a}C = \big\{ \{ C , i\gamma_{5}C , \gamma^{\mu}\gamma_{5}C \} , \{ \gamma^{\mu}C , \sigma^{\mu\nu}C \}\big\} .$$
The spin-1 case that you are talking about is even simpler because the multi-spinor is of rank-2. So, you can expand $\Psi_{mn}(x) = \Psi_{nm}(x)$ as follow
$$\Psi_{mn}(x) = \left( \gamma^{\mu}C \right)_{mn} \ B_{\mu}(x) + \frac{1}{2} \left( \sigma^{\mu\nu}C \right)_{mn} \ G_{\mu\nu}(x) .$$

 

[HeavyWater]

Seeing these great responses just reminds me of my frustrations being a small time physicist. Thank you Samalkhaiat and others for taking the time and patience to answer my questions.