# Should the trace of a product of gamma matrices depend on the convention I use?

I am trying to work out $$\text{Tr}[\gamma_5\gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta]$$ using the same convention as J.J. Sakurai (Advanced Quantum Mechanics), what I get is $$\text{Tr}[\gamma_5\gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta]=4\epsilon_{\mu\nu\alpha\beta},$$ what one gets using the same convention as Peskin & Schroeder instead is $$\text{Tr}[\gamma^5\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta]=-4i\epsilon^{\mu\nu\alpha\beta}.$$ I understand the difference in sign but shouldn't the imaginary unit be present also in the first case? What I mean is: shouldn't I get the same result using either of these conventions?

## 1 Answer

You are halfway correct. The trace of a combination of $\gamma$-matrices does not depend on the representation in which they are expressed. Sakurai primarily uses the Dirac-Pauli representation, while Peskin and Schroeder use the Weyl chiral representation. This difference in representation should no affect the traces of matrix combinations; the traces can be determined entirely from the dimension of the Dirac matrices and their anticommutators.

The reason for the difference between the expressions is that Sakurai actually takes the Dirac matrices to be a representation of a different Clifford algebra than the algebra used in Peskin and Schoeder. Sakurai uses an old-fashioned convention, in which $\{\gamma_{\mu},\gamma_{\nu}\}=2\delta_{\mu\nu}$. The convention used in Peskin and Schoeder (and, as I believe they point out in their preface, all modern field theory texts; Sakurai, on the other hand, describes this convention as "deplorable") has $\{\gamma_{\mu},\gamma_{\nu}\}=2g_{\mu\nu}$. The convention in Sakurai has its advantages; it makes all the $\gamma$-matrices Hermitian. However, since it involves the non-covariant $\delta_{\mu\nu}$ instead of $g_{\mu\nu}$, it is much harder to work with in fully relativistic problems (which are not the primary focus of Sakurai's book, so it's not too much of a problem). The more modern formalism is fully covariant, which is why it is preferred, but at the cost of making the spacelike $\gamma$-matrices skew-Hermitian.

As I recall, both books use a Hermitian $\gamma_{5}$, which corresponds to the helicity for massless spinors. Then the difference between the two traces is that in Peskin and Schroeder, the three skew-Hermitian spacelike $\gamma$-matrices bring in an extra factor of $i$ each. That is the origin of the factor of $i$ in your last expressions.

• But since the only informations regarding the convention i need to evaluate the trace are the anti-commutation relations and the definition of the $\gamma_5$ matrix, when i plug (1234) into $(\mu\nu\alpha\beta)$ i don't get the imaginary unit because Sakurai defines $\gamma_5$ as $\gamma_1\gamma_2\gamma_3\gamma_4$, as the i in this case is found in the $\gamma_k$ k=1,2,3 matrices definition, this way i get 4 instead of 4i, so am i making some kind of mistake or since the algebra is different it's ok for me to get a different result? @Buzz – Egosphere May 14 '16 at 9:07
• Your formulas seem to be correct. (I can't check the overall signs right now, since both of the textbooks are at my office at work, and I'm at home.) The key is that Sakurai's convention gives each of the spacelike matrices an extra factor of i, which explains the difference between the two results. – Buzz May 14 '16 at 13:36