Origin of phases in amplitudes in QFT

Question

Amplitudes in QFT are typically real. I'd like to understand the physical meaning of an amplitude having a phase. I know of three ways that amplitudes can get a phase:

• If the couplings have an imaginary component
• If there is a trace over the spin matrices, $ \gamma _\mu $ producing a $ i \epsilon _{ \alpha \beta \gamma \delta } $.
• If a particle has a significant decay width we allow its propagator to have an imaginary contribution, $$ \frac{i}{p^2-m^2} \rightarrow \frac{i}{p^2-m^2+i m \Gamma } $$

I have heard many times that phases have to do with CP violation, but I'm not able to make the connection. In particular I'd like to know

1. What is the meaning behind the different sources of phases in the amplitude mentioned above?
2. Are there any other sources of phases that I'm missing?
3. Is it true that if the above sources were gone then all amplitudes to all orders would be real (or is it possible for extra $i$'s to sneak in due to things like Wick rotation)?

Answer 1

My apologies if this answer is more simplistic that you were looking for, but you did ask why phases are related to CP-violations.

Most QFTs are CPT invariant, so a CP-violation is also a T-violation.

If you consider an S-matrix element to be the amplitude for a transition from $a$ to $b$, i.e. $\langle a|b\rangle$, then the T-version should be $\langle b|a\rangle$, so if no T-violation, the two amplitudes should be the same, so the amplitude is real.

All the sources of phase you mention seem to be about the introduction of phases into a perturbative approximation to the true amplitude, so the same argument should apply if a perturbative approximation up to a particular order is designed in a way where it (the perturbative approximation) is also CPT-invariant.