- #1

- 6

- 0

1. [itex]\gamma^{\mu+}=\gamma^{0}\gamma^{\mu}\gamma^{0}[/itex]

2. that the matrices have eigenvalues +/- 1, +/- i

3. The trace of [itex]\gamma^{\mu}[/itex] is zero

4. if [itex]\gamma_{5} = -i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}[/itex] then [itex]\gamma_{5},\gamma^{\mu}[/itex]= 0, [itex]\gamma^{2}_{5}[/itex]=I, eigenvalues = +/- 1

For #1 we are to use the Clifford algebra. We have not been given the definitions of the the gamma matrices -- I don't know if we are expected to know these or if they are irrelevant for the proof. We are also given that gamma 0 is equal to its conjugate transpose and gamma i (i = 1,2,3) is equal to its conjugate transpose times -1. I don't even know where to start on this one -- not quite clear on how the multiplication of the matrices works.

#2 I can do if I take each matrix individually, but how do it do it for the 'general' case of [itex]\gamma^{\mu}[/itex]?

**By [itex]\gamma^{\mu}[/itex] does it mean I need to take all gammas at once as a set/group/4-vector (how can it be a vector if its components are matricies?), or does it mean for a general gamma mu, where mu = 0,1,2,3?**

#3 Same as above, fine if I take each matrix individually, but how to do it generally?

# 4 I can do the eigenvalues and the gamma-five-squared = identity, but I'm not sure about the commutator? Again, do I use some 'general' gamma mu?

I have never had to use tensors before so that whole area is still not quite clear to me -- I understand the concept but not really how tensor operations work.