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)?

This post imported from StackExchange Physics at 2014-08-30 08:02 (UCT), posted by SE-user JeffDror

Comments

Why do you say that "Amplitudes in QFT are typically real" ? Probability amplitudes are complex numbers.

This post imported from StackExchange Physics at 2014-08-30 08:02 (UCT), posted by SE-user Trimok

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@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.

This post imported from StackExchange Physics at 2014-08-30 08:02 (UCT), posted by SE-user JeffDror

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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).

This post imported from StackExchange Physics at 2014-08-30 08:02 (UCT), posted by SE-user Trimok

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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?

This post imported from StackExchange Physics at 2014-08-30 08:02 (UCT), posted by SE-user JeffDror

Answer 1

At higher loops it certainly isn't true that amplitudes are real. By the optical theorem the imaginary part of an amplitude is related to the intermediate states that can go on-shell. At higher loops lots of intermediate particles can go on shell so one gets an interesting imaginary part.

But here's one interesting way of getting a phase just at tree level. At tree level one would conclude from the optical theorem that there's no phase except at discrete momenta where the Mandelstam variables are equal to some mass-squared in the theory. But suppose we have a theory with an infinite tower of particles with increasing masses, with some small spacing between the masses. Then if we average over momentum scales that are large compared to this spacing, we could get a large imaginary part.

The classic example is high energy, small angle scattering in flat space string theory (or QCD), where the four-point amplitude (averaged over a range of energies that's much larger than the string scale) is

\(\Gamma(-1-\alpha' t)e^{-i\pi t}s^{\alpha' t},\)

where \(\alpha' \) is the Regge slope. The imaginary part is related to the fact that there is an infinite number of resonances that can be produced in the s-channel.

Another way you can get a phase in field theory is if there is a time delay. For example consider quantum field theory in the curved background of a highly boosted particle (a shockwave). In this background there is the Shapiro time-delay, see http://arxiv.org/pdf/1407.5597.pdf for a recent discussion. At large impact parameter and high energy (the eikonal approximation) one can resum the loops to get the answer

\(\exp(iG_{\text{N}}s\log b)\)

and differentiating with respect to the energy gives the standard answer for the Shapiro time-delay.