I need to show that
u^{+}_{r}(p)u_{s}(p)=\frac{\omega_{p}}{m}\delta_{rs}
where
\omega_{p}=\sqrt{\vec{p}^2+m^{2}}
[itex]u_{r}(p)=\frac{\gamma^{\mu}p_{\mu}+m}{\sqrt{2m(m+\omega_{p})}}u_{r}(m{,}\vec{0})[\itex] is the plane-wave spinor for the positive-energy solution of the Dirac equation...
Weinberg shows how to derive the commutators directly, not how to derive one from the other. I know I should still be able to figure this problem out from what he shows in the book, but I can't seem to make it work.
I don't have access to Ramond.
Given [M^{\mu \nu},M^{\rho\sigma}] = -i(\eta^{\mu\rho}M^{\nu\sigma}+\eta^{\nu\sigma}M^{\mu\rho}-\eta^{\mu\sigma}M^{\nu\rho}-\eta^{\nu\rho}M^{\mu\sigma})
and [ P^{\mu},P^{\nu}]=0
I need to show that
[M^{\mu\nu},P^{\mu}] = i\eta^{\mu\rho}P^{\nu} - i\eta^{\nu\rho}P^{\mu}We've been given the...
Not sure how to do that. Everything I try seems to just cancel back out, e.g.
\gamma^{0}\gamma^{\mu}\gamma^{0} = \gamma^{0}(2\eta^{\mu 0}-\gamma^{0}\gamma^{\mu})
=\gamma^{0}2\eta^{\mu 0}-(\gamma^{0})^{2}\gamma^{\mu}
=...
As they are unitary matrices, it means the eigenvalues are +/- i, +/- 1. Hooray!
I am still stuck on how to show 1.
Using the defining property I can generalize, (\gamma^{\mu})^{2}=\eta^{\mu\mu}. I am given the conjugate transposes for 0 and i, putting those together I get...
I've just started a masters in physics after a 4-year break and am having some real trouble getting back into the swing of things! We have been asked to prove some properties of gamma matrices, namely:
1. \gamma^{\mu+}=\gamma^{0}\gamma^{\mu}\gamma^{0}
2. that the matrices have eigenvalues...