Positive and negative plane wave solutions of Dirac equation

[Gene Naden]

I continue to be occupied with the first chapter of Lessons on Particle Physics by Luis Anchordoqui and Francis Halzen. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf.

I am on page 24, where they derive equations 1.5.67, which are:
$(\gamma^\mu p_\mu-m)u(p)=0$ and
$(\gamma^\mu p_\mu+m)v(p)=0$
Here the plane wave solution is given by equation 1.5.65, which is
$\psi(x)=u(p)e^{-ipx}+v(p)e^{ipx}$

They make the remark "The two negative energy solutions $u^{(3,4)}$ are to be associated with an antiparticle..."

I wondered what the solutions $u^{(3,4)}$ looked like, why there were two of them and what the superscripts (3,4) referred to.

I found a discussion I could understand at https://quantummechanics.ucsd.edu/ph130a/130_notes/node488.html

The (3,4) refer to the third and fourth components of the spinor, which are associated with negative energy by the zero-momentum relation $(i\gamma^0\partial_t -m)\psi=0$

I think I am beginning to like the Dirac equation (a little bit)

 

[AndyW]

!Dear fellow scientist,

Thank you for sharing your progress with the first chapter of Lessons on Particle Physics. I am glad to hear that you are finding the material interesting and engaging.

Regarding your question about the solutions $u^{(3,4)}$ and the superscripts, I would like to offer some clarification. The solutions $u^{(3,4)}$ refer to the negative energy solutions of the Dirac equation, which are related to the antiparticle solutions as you mentioned. The superscripts (3,4) correspond to the third and fourth components of the spinor, as you correctly found in your research. These components are associated with negative energy by the zero-momentum relation, as you mentioned.

As for what these solutions look like, they are complex-valued spinors that satisfy the Dirac equation. The exact form of these solutions depends on the specific system and conditions being studied. However, in general, these solutions can be thought of as describing the spin and momentum of a particle or antiparticle.

I am glad to hear that you are beginning to appreciate the Dirac equation. It is a fundamental equation in particle physics and has been instrumental in our understanding of the subatomic world. Keep up the good work and don't hesitate to reach out if you have any further questions.