Masters in Physics: Proving Properties of Gamma Matrices

[bubblehead]

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 +/- 1, +/- i

3. The trace of $\gamma^{\mu}$ is zero

4. if $\gamma_{5} = -i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ then $\gamma_{5},\gamma^{\mu}$= 0, $\gamma^{2}_{5}$=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 $\gamma^{\mu}$? By $\gamma^{\mu}$ 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.

 

[phyzguy]

You will need to use the defining property of the gamma matrices, namely:
$$\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu\nu}$$
Most of these should follow from this. For example, this tells you that
$$(\gamma^0)^2 = 1, (\gamma^i)^2 = -1$$
What can you conclude about the eigenvalues from this?

 

[bubblehead]

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

$\gamma^{\mu+}$=-$\eta^{\mu\mu}\gamma^{\mu}$

I can substitute ($\gamma^{\mu}$)$^{2}$ for $\eta^{\mu\mu}$. For -1 I can substitute the square of gamma zero, giving

$\gamma^{\mu+}$=$\gamma^{\mu}\gamma^{\mu}\gamma^{0}\gamma^{0}\gamma^{\mu}$

How do I continue from here? I am stuck on how to rearrange this because the matrices are not commutative. I tried substituting a rearranged defining property for $\gamma^{0}\gamma^{\mu}$ but it didn't seem helpful.

 

[vela]

Use the anticommutation relation to switch the order of $\gamma^0$ and $\gamma^\mu$ in $\gamma^0\gamma^\mu\gamma^0$. It doesn't matter which $\gamma^0$ you use.

 

[bubblehead]

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}$

= $\gamma^{0}$($\gamma^{\mu}\gamma^{0}+\gamma^{0}\gamma^{\mu}$)+$\gamma^{\mu}$

=$\gamma^{0}\gamma^{\mu}\gamma^{0}$+$\gamma^{0}$)$^{2}\gamma^{\mu}$+$\gamma^{\mu}$

=$\gamma^{0}\gamma^{\mu}\gamma^{0}$

How does this help me?

 

[vela]

What does the second line evaluate to when $\mu=0$ and when $\mu=i$?