Index Concordia

Relativity – Week 04

Posted in Relativity by Index Guy on September 29, 2007

[For all the details look at sections IIA7, IIIA1, IIIA4 and IIIB1 of part 1 of Fields.]

This week saw the conclusion of spinor notation and the beginning of action principles. On Monday we wrote Maxwell’s equation in spinor notation. Then me mentioned briefly chirality and duality.

After some words on actions and notation we wrote down the action for a scalar particle (obeying the p^2 + m^2 = 0 relation), and for a massless Dirac spinor, obeying \gamma \cdot \Psi = 0.

If we want to describe a charge particle then we need to consider complex fields; this doubles the fields (one complex field is equal to two real fields). For complex fields the action is invariant under phase transformation of the fields. When one considers the phase to be a function of the coordinates, one needs to add a gauge field in order to retain phase invariance. This leads to weak coupling. For the complex field we calculated the conserved current, which changes sign under charge conjugation.

For the Maxwell equations in spinor notation we also derived the energy-momentum tensor, which has a simpler form. It is then more clear that this tensor is invariant under a duality transformation.

Then we considered fields with arbitrary integer spin and looked at the sign of the potential. For spin 1 the “charges” appear with unit powers times (-1)^{1+1}, so we see that objects with same sign in charge will repel while attraction occurs for opposite signs. This is what happens in electromagnetism, with the photon being the spin 1 particle. For spin 2 the “charges” appear with powers of 2 and we get a factor of (-1)^{2+1}. Hence we only have attraction. This explain while gravity only attracts all particles (even antiparticles), with the graviton expected to be a spin 2 particle.

Finally on Friday we considered the action for a free, spinless particle and described it relativistically as the invariant length of the worldline. We considered parameterization of this worldline and different gauge transformations.

Next week: external fields, conservation and pair creation. Then Yang-Mills theory!

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