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The S-Matrix, Time Reversal, And All That

The S-Matrix the description of physical scattering.  The way it works is this:  there is an "incoming" Hilbert space of states, and an "outgoing" Hilbert space of the same size of (possibly different) states.  These states are asymptotic, in the sense that they are thought to be infinitely far away from the region where any interaction occurs.

Suppose a generic incoming state is described by

so that the vector  ⃗α contains complete information about an incoming state, and that a generic outgoing state is described by

so that the vector ⃗a contains complete information about an outgoing state.

The S matrix is the matrix which takes incoming states and maps them to outgoing states:

Conservation of probability implies that S is unitary, ie S(S†)=1.

If we express the incoming and outgoing states in a momentum eigenbasis, it is easy to consider the action of time reversal.  First, the two Hilbert spaces change roles, so that the outgoing states become incoming states, and vice versa.  The coefficients ⃗a and ⃗α get complex conjugated (and change roles), and the momentum of each state gets reversed.  There is some other unitary matrix   ̃S satisfying

Time-reversal invariance is the statement processes that happen can also legitimately happen in reverse, meaning   ̃S = S.  If we multiply both sides of the original S-matrix relation by S† and then complex conjugate that relation, we conclude that S = ST.

That is, time reversal invariance implies that S is symmetric under transposition.