# Difficulty understanding Srednicki's derivation of the Lie Algebra Commutators of the Lorentz Group

I have just started reading through Srednicki's QFT in preparation for a few courses I am about to take. I have taken courses that have covered the basics of special relativity including contravariance and covariance, basic tensors, etc but not in enormous depth. For those who have a copy of the text, my question is about how he obtained equation 2.13 on page 32. The purpose of this discussion in the text seems to be constructing the lie algebra commutators of the Lorentz group. However I am stuck with some working in the middle.

Srednicki defines a Lorentz transformation as a linear change of coordinates that satisfies $$\ g_{\alpha\beta} \Lambda_\gamma^\alpha \Lambda_\sigma^\beta = g_{\gamma \sigma}\tag{2.6}$$

and proceeds to write an infinitesimal Lorentz transformation in the form: $$\ \Lambda_\beta^\alpha = \delta_\beta^\alpha + \delta w_\beta^\alpha \tag{2.7}$$

Then, if $\ U$ is a unitary operator representing a quantum symmetry where: $\ U(x x')= U(x)U(x')$ he writes:

$$\ U(1+\delta w) = I+({i/2h})\delta w_{\alpha\beta}M^{\alpha\beta}\tag{2.12}$$

Where $\ M^{\alpha\beta}$ is a generator of the Lorentz group.

Then he takes $$\ U(\Lambda)^{-1} U(\Lambda')U(\Lambda)=U(\Lambda^{-1}\Lambda'\Lambda) ,$$ sets $$\Lambda'=1+\delta w'$$ and expands both sides in linear order in $\ \delta w$ to obtain:

$$\ \delta w_{\alpha\beta}U(\Lambda)^{-1}M^{\alpha\beta}U(\Lambda) = \delta w_{\alpha\beta} \Lambda_\gamma^\alpha \Lambda_\sigma^\beta M^{\gamma\sigma} \tag{2.13}$$

My question is how this last expression was obtained? How did he expand to linear terms in $\ \delta w$?

Apologies if that explanation is difficult to follow. To be honest, I was able to follow it only by blindly accepting a few of his statements. I am hoping that for people familiar with the content what I have written is enough to follow Srednicki's discussion, as he didn't give much more information.

He moved faster than I expected in this chapter and I was struggling to keep up. I think I am lacking background knowledge he was whizzing over.

• Hopefully this www2.ph.ed.ac.uk/~s0948358/mysite/… should explain what Srednicki is saying better. Jan 11, 2018 at 14:21
• Who recommended Srednicki to you? There are far better books out there on this particular topic. I like the treatment by Pierre Ramond in his "Field Theory. A Primer" very much. Jan 11, 2018 at 15:13
• @blobteppa Fantastic, thank you, those notes look like cover exactly what was confusing me. Hopefully once I've gone through them I'll be able to understand Srednicki's derivation more clearly.
– leob
Jan 12, 2018 at 6:03
• @DanielC I'd heard online that Srednicki is used as the text for many current QFT courses. I skimmed the first chapter and quite liked it, but have found those that follow a little fast moving. I've also had Zee, Peskin and Ramond suggested online so might give them a look.
– leob
Jan 12, 2018 at 6:09
• @DanielC For better or worse my college moves particularly fast through topics like QM. As a result I am after a book that pushes me and isn't too thick. However neither do I want to be incapable of keeping up. I have been thinking Srednicki might just be a little challenging at the start while I fill in some gaps in my knowledge. However I will also skim a few other texts and perhaps swap.
– leob
Jan 12, 2018 at 6:12