Monday, June 16, 2014

Convergent spaghetti worlds

3:AM Magazine had an interview with Alastair Wilson, a philosopher who thinks about the Everettian Quantum Mechanics (also known as the Many Worlds Interpretation, henceforth referred to as EQM).  It's nice to know that some philosophers are thinking seriously about it, since physicists generally aren't trained or paid to do so.  Wilson plays to the strengths of his academic discipline by identifying the most interesting questions of Many Worlds.  I'd like to discuss a few of them.

Branching or parallel worlds?

Wilson says:
In informal or popular discussions, people use two metaphors pretty much interchangeably to describe the Everettian multiverse: branching worlds and parallel worlds. Of course, these two metaphors are in tension: the former suggests mereological overlap of worlds, like a branching tree, whereas the latter suggests mereological distinctness of worlds, like a packet of spaghetti.
Wilson goes on to note (correctly) that the many worlds of EQM are an emergent macroscopic structure, from the much more complex fundamental structure of quantum physics.  Therefore, the question of a branching tree vs a packet of spaghetti is really a question of which is more useful.

Wilson prefers the spaghetti metaphor because it partially resolves the "probability problem" of EQM:  If all of the many worlds exist, then how does it make sense to assign probabilities to each of the many worlds?  In the spaghetti metaphor, when we speak of probabilities, the probabilities represents our belief that we are in any particular noodle.

I am agreement with Wilson, but it's worth poking at that answer.  Let's consider the simplest kind of world-splitting.  No, not the double slit experiment, even simpler!  Consider a beam splitter.

A beam splitter takes a beam of light, and reflects half of it while transmitting the other half.  It's a rather standard optic, and we have several of them on the laser table in our lab.

But suppose that instead of a beam of light, we had a single photon of light.  Even though there's only one photon, this single photon will still split paths, becoming a superposition of path 1 and path 2.  We've generated two distinct worlds, one where the photon is on path 1, and one where it's on path 2.

But these worlds are not very different from one another.  Put another way, they are very close to each other in parameter space--all their parameters are exactly the same, except for the one parameter describing the location of the photon.  Because the worlds are very close to each other, it is possible for them to interact, and experimentally feasible to make them interact in a controlled manner.  In the EQM picture, we would say that the worlds have not yet diverged. Note that the distinction between divergent worlds and not-yet-divergent worlds is an emergent distinction rather than a fundamental one, since it partly has to do with what is experimentally feasible.

The beamsplitter in reverse

Yes, it's entirely possible to join these two worlds together again, to perform the beamsplitter experiment in reverse.  Simply place mirrors on the two beam paths.  This is a very common setup, called an interferometer (used in many experiments, but best known for the one that led to Relativity theory).

It's called an interferometer is because the light can actually end up on two paths: path A or path B.  Whether it ends up on path A or B depends on whether the waves of light are in sync or out of sync, which is to say, whether they interfere constructively or destructively.  Let's say that they interfere in such away that they only go down path B.

This presents a problem for the metaphor of the packet of spaghetti.  Let's treat path 1 as a packet of spaghetti, and path 2 as another packet of spaghetti.  If we have just path 1 on its own, half of its spaghetti goes down path A, and half of it goes down path B.  And if we have just path 2 on its own, the same thing happens.  And yet, if we have the photon go down path 1 and path 2 simultaneously, all of the spaghetti ends up on path B.

There is spaghetti from 1 that goes down A, and spaghetti from 2 that goes down A, but rather than adding the quantities of noodles in A, we subtract them.  Here the metaphor of spaghetti just breaks down, since there's no reason you would ever subtract quantities of spaghetti if we took the metaphor seriously.

But we had already admitted that the spaghetti was just a metaphor for an emergent macroscopic structure of EQM, so it's unsurprising to see the metaphor break down on the microscopic level.  It's just good to keep in mind how exactly it happens, and maintain skepticism about our metaphors.

Why divergence?

Now that I've illustrated how in EQM worlds can converge as well as diverge, we can ask why divergence is so much more common than convergence.  Ultimately, it has to do with the Second Law of thermodynamics, and the increase of entropy over time.

If you have two strands of spaghetti, there are many ways for them to be apart from each other, and only a few ways for them to be stuck to each other.  That is to say, pulling strands of spaghetti apart is associated with an increase in entropy.  Thus, strands are more likely to diverge than converge.

But there's also an issue of dimensionality.  Imagine that the spaghetti strands are close together in parameter space, in that they only differ by one parameter: the location of a single photon.  As long as this is the only parameter which distinguishes the strands of spaghetti, it's as if they're trapped in a low-dimensional space.  They're like cars, trapped on a 2d surface, prone to crashing into one another if we're not careful.  But once the worlds differ by more parameters, we add more dimensions.  It's much harder for two airplanes to collide into each other than cars, because they live in a 3-dimensional space.

However, not impossible

Now imagine that the two spaghetti strands differ by billions of billions of parameters.  This can easily happen, if for instance the photon on path 1 hits a screen, a screen which is composed of about 10^23 electrons and nuclei.  So now we're talking about airplanes not in 3-dimensional space, but airplanes in 10^23-dimensional space.  It's not surprising if they hardly ever collide.

And that's why the many worlds metaphor works.  If macroscopic worlds converged as often as they diverged, then we'd constantly face this problem of interfering noodles.  But convergence only tends to occur on the microscopic levels.

Update: I have a followup post talking about the Bohmian interpretation