How Do Dirac Gamma Matrices Satisfy Their Anticommutation Relations?

[McLaren Rulez]

Homework Statement

Given that [itex]\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}*1[/itex] where [itex]1[/itex] is the identity matrix and the [itex]\gamma[/itex] are the gamma matrices from the Dirac equation, prove that:

[itex]\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}*1[/itex]

Homework Equations

[itex]g^{\mu\nu}\gamma_{\nu}=\gamma^{\mu}[/itex] and [itex]g_{\mu\nu}\gamma^{\nu}=\gamma_{\mu}[/itex]

The Attempt at a Solution

I'm not sure what to start with. I tried expressing the terms of the relation to be proved as follows

[itex]\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=g_{\mu\alpha}\gamma^{\alpha}g_{\nu \beta}\gamma^{\beta}+ g_{\nu\beta}\gamma^{\beta}g_{\mu\alpha}\gamma^{ \alpha }[/itex]

but that isn't going anywhere. So how do I approach this?

 

[dextercioby]

Hmm, just replace mu and nu with their possible values and see what you get. Don't forget that the metric tensor is diagonal (probably diag(+,-,-,-)).

 

[George Jones]

Given that [itex]\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}*1[/itex]

Another way to do it is to multiply both sides of this equation by [itex]g_{\alpha \mu} g _{\beta \nu}[/itex].

 

[McLaren Rulez]

Thank you George Jones! That did the trick nicely. Using [itex]g_{\mu\alpha}g^{\alpha\nu}=\delta^{\mu}_{\nu}[/itex] the result follows easily.

dextercioby, thank you for replying. I think your method also works but I must assume the metric is diag(1, -1 , -1, -1) which is not always the case right?

 

[paranormal]

I would approach this problem by first understanding the properties of the gamma matrices and the identity matrix. The gamma matrices are a set of four 4x4 matrices that satisfy the anti-commutation relation given in the homework statement, and the identity matrix is a diagonal matrix with all diagonal elements equal to 1.

Next, I would use the given equations to manipulate the terms in the relation to be proved. For example, we can substitute g^{\mu\nu}\gamma_{\nu}=\gamma^{\mu} into the first term to get:

\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=g_{\mu\nu}\gamma^{\nu}\gamma_{\nu}+ \gamma_{\nu}\gamma_{\mu}

Then, we can use the anti-commutation relation to simplify the expression:

g_{\mu\nu}\gamma^{\nu}\gamma_{\nu}+ \gamma_{\nu}\gamma_{\mu}= 2g_{\mu\nu}\delta^{\nu}_{\nu}= 2g_{\mu\nu}*1

Finally, we can use the properties of the identity matrix to get the desired result:

2g_{\mu\nu}*1= 2g_{\mu\nu}\delta^{\mu}_{\mu}= 2g_{\mu\nu}\gamma_{\mu}\gamma_{\nu}= \gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}

Therefore, we have proved the given relation:

\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}*1

In conclusion, as a scientist, I would approach this problem by first understanding the properties of the given matrices and then using algebraic manipulations and properties to prove the desired relation.