In Time and Chance, David Albert writes that since the Schrödinger equation involves a first (instead of a second) derivative, “the dynamical laws that govern the evolutions of quantum states in time cannot possibly be invariant under time-reversal” (p.132). I’ve always struggled with his argument for this claim, which he gives in a footnote on the same page. Here’s what he writes.
The idea is this: suppose that the instantaneous microscopic state of a certain physical system at time t is also that system’s complete dynamical condition at t, and suppose that the dynamical equations of motion of that system are invariant under time-reversal. Then whatever it is that those equations entail about times other than t is patently going to have no alternative whatsoever but to be symmetrical about t. Suppose (moreover) that the equations of the motion of this system are invariant under time-translations … . Then (if you think it over) the state of this system is going to have no alternative but to be entirely unchanging in time. And so any theory for which instantaneous states are also invariably complete dynamical conditions, and for which the equations of motion are invariant under time-reversal, and for which the equations of motion are invariant under time-translation, is necessarily a theory according to which nothing ever happens. (Albert 2000, 132.)
I’ve thought it over, and I think I’ve finally got it.
The key to the passage is this: Albert takes time-reversal to transform the state ψ(t) to the state ψ(-t). This is a highly non-standard view; in particular, time reversal in quantum mechanics is normally taken to involve conjugation as well. But consider the consequences of Albert’s view for the Schrödinger equation — it means that if ψ(t) is a solution, then so is ψ(-t):
But substituting t → -t into the original Schrödinger equation, we find that:
Adding these two equations, we now have that:
and hence that ψ(-t) = 0. In other words, the state of the world is constant, and “nothing ever happens.” And indeed, this result is more general than the Schrödinger equation — as Albert suggests, it seems to hold for any deterministic first order equation of motion, with only a single first derivative, which is both time-translation and time-reversal invariant (according to Albert’s definition).
Now, the conclusion that “nothing ever happens” is obviously false. So, one of these premises must be false as well. Albert rejects the premise that quantum mechanics is time reversal invariant. But of course, there is a plausible alternative. We can reject Albert’s picture of time reversal instead.
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