Recently I was reading Fulvio Melia's book "Cracking the Einstein Code", and I was struck by his discussion of an old (very old) "paradox" known as "Zeno's Arrow". The paradox goes like this: imagine the passage of time as a succession of frozen snapshots of each moment. In any given snapshot there is no movement, since it shows only a single instant. Any moving object just appears frozen at its current position.

Then the question is, if there's no motion at a given instant, where does motion come from? Or, what "connects" these motionless states? There appears to be no information in the individual snapshots that would allow them to be stitched together into a physical time flow. Zeno argued that motion was, therefore, impossible - one of several arguments he used to support the idea that universe was, contrary to appearance, timeless and changeless.

Of course, I should hasten to say that I'm not sure there's any real paradox. A pictorial "snapshot" of a moment in time simply doesn't capture all of the relevant information; each object also has an instantaneous velocity, which isn't captured in an image. There appears to be no problem modeling this situation in a mathematically consistent way, as embodied in the classical physics of Newton.

All the same, something doesn't seem quite right. It is troubling that motion, in the classical view, seems to only be definable in terms of the change that occurs between two successive times, yet it also needs to exist as a property of a single point in time. It is, then, rather interesting to note that quantum mechanics precisely removes this dichotomy, making it so that full information about an object's position (the "snapshot") actually contains full information about its movement as well.

In quantum mechanics, all the information about an object's position is contained in its "wave function", which is just a function which shows how likely the object is to be found at each position.* One might think that each object would also have a second wave function showing how likely it is to have any given velocity, but this isn't the case.

Quantum mechanics represents a much deeper change than just adding some probabilities to classical physics, and the real quantum trick is that all the information about an objects's motion is also encoded in the same wave function which describes its position. The "motion" of an object is, in effect, just another way to look at its "position"; in mathematical terms, it is the fourier transform. The type of "position-only" snapshot imagined by Zeno simply doesn't exist.

Clearly what this is saying is that motion and position are both fundamentally quite different from how we imagine them, and perhaps unified in a way which isn't yet fully reflected in our theories of spacetime. It also shows the true content of the uncertainty principle, as I blogged about earlier: http://www.letstalkphysics.com/2009_11_01_archive.html.

And I do find it quite compelling that quantum mechanics so exactly resolves this ancient dilemma of Zeno; perhaps it is telling us that the "paradox" really was paradoxical after all?

* This is an oversimplification since the wave function is a complex number and hence contains more information than is strictly necessary to give the probability of being at a particular position. In fact it contains exactly twice as much information, which is not surprising since it's encoding velocity too. Nevertheless, the fact remains that both position and velocity are inextricably intertwined in a single function, and can't be separately specified.