1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Gamma matrices

  1. Sep 30, 2011 #1
    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. [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.
  2. jcsd
  3. Sep 30, 2011 #2


    User Avatar
    Science Advisor

    You will need to use the defining property of the gamma matrices, namely:
    [tex]\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu\nu}[/tex]
    Most of these should follow from this. For example, this tells you that
    [tex] (\gamma^0)^2 = 1, (\gamma^i)^2 = -1[/tex]
    What can you conclude about the eigenvalues from this?
  4. Oct 3, 2011 #3
    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, ([itex]\gamma^{\mu}[/itex])[itex]^{2}[/itex]=[itex]\eta^{\mu\mu}[/itex]. I am given the conjugate transposes for 0 and i, putting those together I get


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


    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 [itex]\gamma^{0}\gamma^{\mu}[/itex] but it didn't seem helpful.
  5. Oct 3, 2011 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Use the anticommutation relation to switch the order of [itex]\gamma^0[/itex] and [itex]\gamma^\mu[/itex] in [itex]\gamma^0\gamma^\mu\gamma^0[/itex]. It doesn't matter which [itex]\gamma^0[/itex] you use.
  6. Oct 5, 2011 #5
    Not sure how to do that. Everything I try seems to just cancel back out, e.g.

    [itex]\gamma^{0}\gamma^{\mu}\gamma^{0}[/itex] = [itex]\gamma^{0}[/itex](2[itex]\eta^{\mu 0}[/itex]-[itex]\gamma^{0}\gamma^{\mu}[/itex])

    =[itex]\gamma^{0}[/itex]2[itex]\eta^{\mu 0}[/itex]-([itex]\gamma^{0}[/itex])[itex]^{2}\gamma^{\mu}[/itex]

    = [itex]\gamma^{0}[/itex]([itex]\gamma^{\mu}\gamma^{0}+\gamma^{0}\gamma^{\mu}[/itex])+[itex]\gamma^{\mu}[/itex]



    How does this help me?
  7. Oct 5, 2011 #6


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    What does the second line evaluate to when [itex]\mu=0[/itex] and when [itex]\mu=i[/itex]?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook