Extra material for “Quantum Theory for Mathematicians”

then so is the curve x(−t), as is easily verified by direct computation. (Note that d/dt(x(−t)) = ẍ(−t), since the two derivatives pull out two minus signs.) In terms of trajectories in phase space, we note that dx(−t)/dt = −ẋ(−t). This means that the momentum of the particle along the time-reversed trajectory is the negative of the momentum along the original trajectory. We may therefore formulate time-reversal in phase space as follows.