# David Albert on symmetries of motion in quantum mechanics

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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Couldn’t you also reject the premiss that the instantaneous microscopic state of a system at t is the complete dynamical condition for that system at t? What is at stake in that premiss?

You could reject that as well. That’s basically the assumption that the equations of motion are deterministic. You could also give up the assumption of time-translation invariance. But then, at least in the case of the Schrödinger evolution, the equations of motion actually are deterministic for well-posed systems. And giving up time-translation invariance means giving up local energy conservation, which isn’t a very plausible option either…

Does rejecting the premiss that the instantaneous state gives the complete dynamics really amount to rejecting determinism or just one variety of determinism? If you have in mind Laplacean-style determinism, where every state is determined by a complete characterization of any single state, then sure. But couldn’t you give up on the premiss at stake and still think that every state is determined by, say, two or three or n other states? And if so, wouldn’t that qualify as determinism?

That’s interesting. I suppose in principle you could have dynamical laws like that. Did you have any particular examples in mind?

I don’t know … I occasionally read stuff by Peirce when I’m bored (or when I want to remind myself how dumb I am), and he has some remarks to the effect that Laplace says such and so but the only things that are real are point particles with their masses and spatial arrangements (or relative displacements or whatever) and the laws. But those things together don’t specify velocities of the particles. To get velocities, you need all of that at two different times.

I don’t have any other examples in mind, but I’m guessing that you could rig up something similar that required n different time slices.

Bryan,

>> the conclusion that “nothing ever happens” is obviously false.

H|psi> = 0 is the Wheeler-deWitt equation for quantum gravity and it is not clear that it is obviously false.

Thanks Wolfgang, that’s a really interesting point — amazingly, the Wheeler-deWitt equation doesn’t have an explicit time parameter. And there is that literature of people like Carlo Rovelli who argue that, in quantum gravity, the best interpretation of time will be as an illusion. But then, it seems that the question of time reversal for these people would be a moot point: if there’s no time, then there’s no need to figure out what it means to

reversetime.Bryan,

>> the best interpretation of time will be as an illusion

I would prefer the term ‘approximation’ instead of ‘illusion’.

Notice that we dont know how it would be if there is no time (e.g. living in the Euclidean sector of a QFT). Obviously it would not be just ‘everything standing still’, because this experience of ‘everything standing still’ already assumes some internal time of the observer imho.