# Quantum Field Theory Unitary Transformations

I am currently reading through Itzyskon and Zuber for my quantum field theory class, and I came across this regarding the unitary transformations of the Dirac bispinors in chapter 2. They show that the transformation from one frame to another: $$\psi'(x') = S(\Lambda)\psi(x)$$ has the property:

$$S(\Lambda) \gamma^{\mu} S(\Lambda^{-1}) = (\Lambda^{-1})^{\mu}_{\nu} \gamma^{\nu}$$

Which I understand. Then, if one were to make an infinitesimal transformation:

$$\Lambda^{\mu}_{\nu} = \delta^{\mu}_{\nu} + \omega^{\mu}_{\nu} \hspace{1mm} , \hspace{1mm} (\Lambda^{-1})^{\mu}_{\nu} = \delta^{\mu}_{\nu} - \omega^{\mu}_{\nu}$$

Then we can say that the unitary transformations have the form:

$$S(\Lambda) = 1 - \frac{i}{4}\sigma_{\mu \nu} \omega^{\mu \nu} \hspace{1mm} , \hspace{1mm} S(\Lambda^{-1}) = 1 + \frac{i}{4}\sigma_{\mu \nu} \omega^{\mu \nu}$$

I am somewhat perplexed by the $$\frac{i}{4}$$ factors, but I figure that they're a result of a convention in anticipation of some result. My real problem comes in when attempting to use these with the first equation to derive the commutation relation $$[\gamma^{\mu},\sigma_{\alpha \beta}]$$. In the book they give this as the following:

$$[\gamma^{\mu},\sigma_{\alpha \beta}] = 2i(\delta^{\mu}_{\alpha}\gamma_{\beta} - \delta^{\mu}_{\beta}\gamma_{\alpha})$$

In an attempt to recover these relations, I did the following:

$$S(\Lambda) \gamma^{\mu} S(\Lambda^{-1}) = (\Lambda^{-1})^{\mu}_{\nu} \gamma^{\nu} \rightarrow$$

$$(1 - \frac{i}{4}\sigma_{\alpha \beta} \omega^{\alpha \beta}) \hspace{1mm} \gamma^{\mu} \hspace{1mm} (1 + \frac{i}{4}\sigma_{\alpha \beta} \omega^{\alpha \beta} = (\delta^{\mu}_{\nu} - \omega^{\mu}_{\nu}) \gamma^{\nu}$$

Multiplying the terms, discarding higher than order 1 terms, and subtracting the $$\gamma^{\mu}$$ from each side, we get:

$$\frac{i}{4} \big( \gamma^{\mu} \sigma_{\alpha \beta} \omega^{\alpha \beta} - \sigma_{\alpha \beta} \omega^{\alpha \beta} \gamma^{\mu} \big) = - \omega^{\mu}_{\nu} \gamma^{\nu}$$

We can change the above equation to get:

$$\frac{i}{4} \big( \gamma^{\mu} \sigma_{\alpha \beta} - \sigma_{\alpha \beta} \gamma^{\mu} \big) \omega^{\alpha \beta} = \delta^{\mu}_{\beta} \gamma_{\alpha} \omega^{\alpha \beta}$$

If we switch the indices $$\alpha$$ and $$\beta$$ and then use the antisymmetric property of $$\omega^{\alpha \beta}$$, then we get:

$$\frac{i}{4} \big( \gamma^{\mu} \sigma_{\alpha \beta} - \sigma_{\alpha \beta} \gamma^{\mu} \big) \omega^{\alpha \beta} = -\delta^{\mu}_{\alpha} \gamma_{\beta} \omega^{\alpha \beta}$$

So if I add these two and do the appropriate algebra, then I arrive at the above commutation relation:

$$[\gamma^{\mu},\sigma_{\alpha \beta}] = 2i(\delta^{\mu}_{\alpha}\gamma_{\beta} - \delta^{\mu}_{\beta}\gamma_{\alpha})$$

However, this can all only be done if I assume that the $$\omega^{\alpha \beta}$$ commute with the $$\gamma^{\mu}$$. But, on the other hand, it is obvious from the result of this derivation that the $$\sigma_{\alpha \beta}$$ don't commute with $$\gamma^{\mu}$$. Why is this the case? Why should I assume that? It seems like a magical property that the $$\omega^{\alpha \beta}$$ have to get some relations that we like. Does it have to do with them being infinitesimal?

• $\omega$ is a set of 6 numbers whereas $\sigma$ is a set of 6 matrices. Commented Feb 14, 2022 at 5:00

Look again at the defining equation for the $$\sigma_{\mu\nu}$$ and $$\omega^{\mu\nu}$$: $$S = 1 -\frac{\mathrm{i}}{4}\sigma_{\mu\nu}\omega^{\mu\nu}$$ Here, $$S$$ is a matrix, and we're expanding it in term of some basis of matrices $$\sigma_{\mu\nu}$$ with coefficients $$\omega^{\mu\nu}$$. So the $$\omega^{\mu\nu}$$ are just real numbers, while the $$\sigma_{\mu\nu}$$ are matrices. Real numbers commute with everything, matrices don't.