# Origin of phases in amplitudes in QFT

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)?
• Why do you say that "Amplitudes in QFT are typically real" ? Probability amplitudes are complex numbers. Aug 27, 2014 at 13:59
• @Trimok: True, that's why they don't need to be real. But if you calculate the amplitudes contributing to S matrix elements, ${\cal M}$, in QED for example, you will find that they are going to be real (even though they didn't apriori "have to be"). This is also mentioned in pg 232 on Peskin and Schroeder. Aug 27, 2014 at 14:05
• I found a paper, where modifications of QED are analysed in the process $e^+e^- \to \gamma\gamma$. Some modifications, like implementing a scalar boson (vertex $Se^+e^-$) seems to lead ((more precisely the pseudo-scalar part) to complex amplitudes (see end page $509$, page $510$) . See also the general structure of the cross section pages $503 \to 506$ (the array page $506$ is interesting). Aug 29, 2014 at 10:50
• Interesting, I have yet to go through it in detail, but my most naive guess is then that having a pseudo scalar boson may break CP symmetry? Aug 29, 2014 at 18:46

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.