# Enlarging the action of $C$, $P$ and $T$

I am now studying QFT using Schwartz's book, and I am going through the part discussing about how the charge conjugation, parity, and time-reversal operator acting on various object should look like. He starts with how charge conjugation operator on the spinors act: $$C : \psi \rightarrow -i \gamma_{2} \psi^{*}$$

and then enlarges the action of $$C$$ on the other objects so that the Lorentz invariance is attained.

However, I find this logic to be somewhat strange. For example, suppose we have a Lagrangian $$\mathcal{L} = \phi^{*}(\Box + m^{2}) \phi + \frac{\lambda}{3!} \phi^{3}.$$ Then after imposing a transformation such that $$\phi \rightarrow - \phi$$, we see that $$\mathcal{L}$$ is not invariant under this. Then we just simply say that this Lagrangian does not have this symmetry and go on.

But the argument on the action of $$C$$, $$P$$ and $$T$$ doesn't go this way. We first declare that the Lagrangian we are interested in must be invariant under $$C$$, $$P$$ and $$T$$, and adjust the action of them on various objects to make that happen. For example, Schwartz says that the QED interaction term $$eA_{\mu} \bar{\psi} \gamma^{\mu} \psi$$ should be invariant under Lorentz transformations, and since $$C : \bar{\psi} \gamma^{\mu} \psi \rightarrow - \bar{\psi} \gamma^{\mu} \psi,$$ he declares that $$C: A_{\mu} \rightarrow -A_{\mu}$$. (Schwartz, Quantum Field Theory and the Standard Model, p.194~195))

Why can we do this for $$C$$, $$P$$ and $$T$$? Why do we not do this for others?

For example, for $$\mathcal{L} = \phi (\Box + m^{2} ) \phi + \frac{\lambda}{3!} \phi^{3},$$ and the transformation $$\phi \rightarrow -\phi,$$ why not we extend the transformation to $$\lambda \rightarrow -\lambda$$, so that the Lagrangian $$\mathcal{L}$$ is invariant under it?

• CPT symmetry follows from relativity, doesn't it? One can surely construct Lagrangians that are not obeying this symmetry, but then they are not relativistic. I would imagine that solid state physics has a lot of those systems. These are still very important emergent systems, they are just not considered "fundamental" like theories of the vacuum are. Commented May 4, 2023 at 18:36
• @FlatterMann CPT is fundamental, but the question is about C, P, T separately (which are violated eg. in standard model) Commented May 4, 2023 at 20:37
• @QCD_IS_GOOD In that case I simply misunderstood the question... exactly for the reason you mentioned. CP violation has been experimentally observed, after all. The only fundamental symmetry that is "different" because we don't get a choice if we want to keep relativity alive is CPT. At least that is my limited understanding as an experimentalist. Not that a small violation of relativity is completely out of the question... I would keep an open mind there as well. Commented May 5, 2023 at 1:57

When folks talk about symmetry transformation, usually it's the fields (and in some cases, coordinates as well) that are transformed, e.g. $$\psi$$, $$\phi$$, and $$A_\mu$$.
And in your case of $$\mathcal{L} = \phi (\Box + m^{2} ) \phi + \frac{\lambda}{3!} \phi^{3}$$ where $$\lambda$$ is NOT a field, thus can NOT be transformed.
One workaround is to promote $$\lambda$$ to be a field that can transform properly, then you regain symmetry. One historical example: Steven Weinberg promoted the mass parameter $$m$$ of standard model fermion to the Higgs field (more precisely Higgs field multiplied by the Yukawa constant) to accommodate the electroweak symmetry.