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Hi,

I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra).

Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##, ##\text{Tr}## being the trace and ##g^{\mu\nu}## the Minkowski metric.

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

##\text{Tr}(\alpha A) = \alpha\text{Tr}(A)##

##g_\mu g^\mu = 4I_4##

I get : ##\text{Tr}(\gamma^\mu \gamma^\nu) = \text{Tr}(g^{\mu\nu} \gamma_\nu \gamma^\nu) = g^{\mu\nu} \text{Tr}(4I_4) = 16 g^{\mu\nu}##

What am I missing there?

Edit : I think this is related to the space in which you evaluate ##\gamma## (which I still have a hard time to understand - Minkowski space vs 4-vect space I think? where ##\gamma## is a set of matrices in the first and a vector in the second?). In fact, Griffith says that ##\gamma_\mu\gamma^\mu=4## while on Wikipedia it is rather ##\gamma_\mu\gamma^\mu=4I_4##. My proof would work if I considered Giffith's identity, but then how do I distinguish in which context ##\gamma_\mu\gamma^\mu## is equal to 4 rather than ##4I_4## ?

Thanks!

Alex

I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra).

1. Homework Statement1. Homework Statement

Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##, ##\text{Tr}## being the trace and ##g^{\mu\nu}## the Minkowski metric.

## Homework Equations

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

##\text{Tr}(\alpha A) = \alpha\text{Tr}(A)##

##g_\mu g^\mu = 4I_4##

## The Attempt at a Solution

I get : ##\text{Tr}(\gamma^\mu \gamma^\nu) = \text{Tr}(g^{\mu\nu} \gamma_\nu \gamma^\nu) = g^{\mu\nu} \text{Tr}(4I_4) = 16 g^{\mu\nu}##

What am I missing there?

Edit : I think this is related to the space in which you evaluate ##\gamma## (which I still have a hard time to understand - Minkowski space vs 4-vect space I think? where ##\gamma## is a set of matrices in the first and a vector in the second?). In fact, Griffith says that ##\gamma_\mu\gamma^\mu=4## while on Wikipedia it is rather ##\gamma_\mu\gamma^\mu=4I_4##. My proof would work if I considered Giffith's identity, but then how do I distinguish in which context ##\gamma_\mu\gamma^\mu## is equal to 4 rather than ##4I_4## ?

Thanks!

Alex

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