# Square of the Feynman amplitude for $a +b\to c+d$ and its reverse

In quantum field theory, if a process $$a +b\to c+d$$ is allowed by a certain interaction Lagrangian (hermitian), the reverse process, $$c+d\to a+b$$, must also be allowed (as far as I understand) by the Lagrangian. Here, I use the word "allowed" rather loosely to mean that the amplitude is nonzero (without taking into account the phase space factor).

However, I have a question unanswered. Is it true that the reverse process, $$c+d\to a+b$$, also has the same squared Feynman amplitude as that of the forward process, $$a+b\to c+d$$? If yes, can we understand this from the behaviour of the $$S$$-matrix element? If not necessarily true, give me a counter-example and point out how this might fail to be true.

This would only be true if the theory describing the scattering of $${a,b,c,d}$$ is time reversal invariant.
It can be shown that if $$S$$ is the S-matrix operator, and $$T$$ is the time reversal operator, then $$S_{T} = TST^{-1} = S^{\dagger}$$.
Then for any states $$|\eta \rangle$$ and $$|\zeta\rangle$$ we have $$\langle \zeta_{T}|S|\eta_{T}\rangle = \langle \eta|S|\zeta\rangle$$, where $$|\eta_{T}\rangle =T|\eta\rangle$$ and the same for $$|\zeta_{T}\rangle$$.
Thus if the theory which tells you how to compute $$S$$ is time reversal invariant, then the amplitudes for the forward and backwards process will be equal.
And example is the Standard Model, CP symmetry is violated due to the complex phase in the CKM matrix, which relates different mass eigenstates of quarks to each other in processes such as $$t\rightarrow b_{i}W^{+}$$ where $$b_{I}$$ is a bottom like quark $$(d,s,b)$$. Since $$CP$$ is violated, but $$CPT$$ is not, this must mean that $$T$$ is violated, and thus time reversed processes don't have the same amplitudes.
• Thanks! I've two follow-up questions. First, can you explain how you proved the time-reversal property of the S-matrix? Second, if $|\zeta\rangle$ is the initial state, does $|\zeta\rangle_T$ represent the final state? Commented Jul 12 at 18:37