Schrodinger’s Cat Lives again!

This article concerns a new paper I just submitted and now published . It concerns a peculiar feature of quantum mechanics (and also of classical mechanics).

The feature is this. The Laws of Physics appear indifferent to the direction of time. If you play a video of two balls colliding elastically with each other, you could play the video backwards and the event would look perfectly reasonable. Of course, if you scale the event up and show the reverse of a video of two balls made of glass colliding and shattering into a million little pieces, such a video would look mighty weird. No one has actually seen a million little pieces of glass crashing into each other and coalescing into an object shaped like a ball.

On the other hand, if the ball broke up into, say, two pieces, it just might look possible, though rare, for an event to occur where two pieces of glass collided and got glued into one ball.

The distinction is merely of size. If there are a very small number of particles involved in a physical event, you could well have both the event and its time-reversed version occur without problem, However, if there were a macroscopic number of particles involved, there would be an infinitesimal chance of recurrence, though it would be possible in principle.

Suppose there were $N$ particles involved that could (each) be either in one unique ordered state or $M$ other disordered (broken up in various ways) states. Then there is one way for them to be together and $N^M$ ways of them being broken up. From pure arithmetic, if $N \sim 100, M \sim 1000$ , this is a number with $10000$ digits to the left of the decimal point – it is absolutely humongous! So the chance that a system that starts in one of those large number of states happens to, by chance, end up in the one, unique ordered state, is exceedingly unlikely?

How unlikely? If we sampled one final state every second, it would take us $100^{1000}$ seconds, which is $10^{2000}$ years. The universe is only $10^{10}$ years old or so, so this is $10^{1990}$ universe lifetimes. If you stick around long enough, it will come back to order, but you might need a lot of pizza while you are waiting!

Similarly, in quantum mechanics, the equations are time-reversal invariant. Just, for clarity, the universe (and the laws that govern our universe) doesn’t happen to be time-reversal invariant. However, the basic laws of quantum mechanics, which is a framework that allows one to write down the laws that govern physics in our universe, are time-reversal invariant. We just have, in our universe, some $\: \: {\bf {particular \: \: interactions}}\: \:$ that operate differently and explicitly break this symmetry.

It is a mathematical tour de force to obtain, starting with basic equations that are time-reversal invariant, phenomena that are clearly irreversible (in any practical sense). In a very precise sense, the Schrodinger’s cat experiment is one such. If you don’t know the experiment, read this. People have tried to explain the peculiar consequences of this experiment in many ways, including crazy ideas such as one needs to be a conscious being to know that a cat is dead. In the paper mentioned above, I analyzed the classic double slit experiment to check whether an electron has gone through one slit or another and figured out how, as the measuring apparatus gets bigger and bigger, one sees that the measuring apparatus itself gets driven to one (electron’s gone through slit $1$ ) or the other (electron’s not gone through slit $1$ ) state. And the key in this is that measuring apparata add energy to a system – without adding energy and amplifying weak signals, one doesn’t know if one is measuring an event or measuring noise.

And it just means that if the cat in Schrodinger’s eponymous experiment were just a few atoms big, one could look inside, see it “dead” (properly defined for a 5-10 atom-sized cat) and then look again a few minutes later and see it “not-dead”. Of course, for a real cat, that will take much too long and for all practical purposes, the cat is indeed “dead” or “alive”.

Is there something like a fundamental limit of how little energy you need to input to “collapse” a wave-function? That is something the uncertainty relationship points to, but it is worth thinking about. Clearly, it means one should define what “collapse” means? For our lifetime? For the universe’s lifetime? For a large multiple of the universe’s lifetime?

Read on!