Tuesday, June 17, 2014

More literal spaghetti

Yesterday, I wrote about the spaghetti metaphor for Everettian Quantum Mechanics (aka Many Worlds).  Unfortunately, the metaphor breaks down because different worlds in quantum mechanics can constructively and destructively interfere with each other on a microscopic level, although we generally don't need to worry about this on a macroscopic level.

But there is another interpretation of quantum mechanics for which the spaghetti metaphor is more exact.  I'm speaking of the Bohmian interpretation of quantum mechanics.

The Bohmian interpretation is usually the go-to example for how we can have a deterministic theory of quantum mechanics.  In Bohmian theory, every particle has a well-defined trajectory, and only occupies one position at any given time.  The fact that we can't predict exactly where the particle will be just has to do with the fact that we do not (and cannot) know with certainty the particle's initial position.  Bohmian theory makes all the same predictions as the other major interpretations of quantum mechanics.  But this comes at a cost: faster-than-light information transfer.

I'm largely a Many Worlds partisan, but I give the Bohmian interpretation credit, because it basically starts with the Many Worlds interpretation.  In Bohmian theory there is no wavefunction collapse.  Instead, the wavefunction simply evolves according a single equation, and splits into many worlds just like in the Many Worlds interpretation.

The difference is that the wavefunction is not interpreted as a description of reality (and therefore there aren't really many worlds).  While the wavefunction is a real object, it is seen as distinct from all the particles we see around us.  The wavefunction is interpreted as a pilot wave which merely guides the motion of particles.*  All particles have a definite position, and we just need is this complicated pilot wave object to determine their motion.

*For those who have studied quantum mechanics, it's actually quite simple to understand.  Even in standard quantum mechanics, we speak of the probability current.  We can obtain the probability "velocity" by dividing the probability current by the probability density.  Bohmian theory interprets this literally, by having the particle velocity equal the probability velocity.

To relate this to the spaghetti metaphor, let me consider the classic double slit experiment.  We send light through two slits, and the waves coming from the two slits interfere with each other.

 Waves of light go through two slits, located at the bottom, and travel upwards.  (Technical details: I'm just showing the real part of the wavefunction, with blue positive, red negative, and green zero.)

At some points, the waves interfere constructively, and at other points they interfere destructively.   This creates alternating dark and light fringes.  In more ordinary quantum interpretations, this is because the wavefunction of the light determines the probability that the light is in any particular location.

 The colors here show the probability that light is in any particular location.

But in the Bohmian interpretation, we do not interpret the wavefunction as probability.  Instead, we interpret it as a pilot wave.  The light goes on a well-defined trajectory, it's just hard to predict which particular trajectory it's on.

Here I show many possible trajectories for the light, as calculated using Bohmian theory.  As you can see, particles don't respect conservation of momentum, and even though light only goes through one slit, it is obviously affected by the presence of the other slit.  (You can find lots of similar images on the net, but this whole post is really an excuse for me to write a Bohmian calculator, so I'm giving you my image.)

Perhaps now you can see how this follows the spaghetti metaphor.  Each possible trajectory for the light is a single strand of spaghetti.  When we speak of probabilities in quantum mechanics, we're really talking about our degree of belief that we are in any particular strand of spaghetti.

The difference is that in the Bohmian interpretation, there is only one strand of spaghetti, the one that includes us.  And in my Literal Spaghetti interpretation, there are many strands of spaghetti, of which we are just one.