xkcd: The Search
After I had spent half a year doing little more than watching babies grow and writing a review article on the minimal length, I got terribly bored with myself. So I’m apparently one of the world experts on quantum field theories with a minimal length scale. That was not exactly among my childhood aspirations.
As a child I had a (mercifully passing) obsession with science fiction. To this day contact to extraterrestrial intelligent beings is to me one of the most exciting prospects of technological progress.
I think the plausible explanation why we have so far not made alien contact is that they use a communication method we have no yet discovered, and if there is any way to communicate faster than the speed of light, clearly that’s what they would use. Thus, we should work on building a receiver for the faster-than-light signals! Except, well, that our present theories don’t seem to allow for such signals to begin with.
Every day is a winding road, and after many such days I found myself working on quantum gravity.
So when the review was finally submitted, I thought it is time to come back to superluminal information exchange, which resulted in a paper that’s now published
Quantum superpositions of the speed of light
arXiv:1207.1002
The basic idea isn’t so difficult to explain. The reason that it is generally believed nothing can travel faster than the speed of light is that Einstein’s special relativity sets the speed of light as a limit for all matter that we know. The assumptions for that argument are few, the theory is extremely well in agreement with experiment, and the conclusion is difficult to avoid.
Strictly speaking, special relativity does not forbid faster-than-light propagation. However, since in special relativity a signal moving forward in time faster than the speed of light for one observer might appear like a signal moving backwards in time for another observer, this can create causal paradoxa.
There are three common ways to allow superluminal signaling, and each has its problems:
First, there are wormholes in general relativity, but they generically also lead to causality problems. And how creation, manipulation, and sending signals through them would work is unclear. I’ve never been a fan of wormholes.
Second, one can just break Lorentz-invariance and avoid special relativity altogether. In this case one introduces a preferred frame and observer independence is violated. This avoids causal paradoxa because there’s now a distinguished direction “forward” in time. The difficulty here is that special relativity describes our observations extremely well and we have no evidence for Lorentz-invariance violation whatsoever. There is then explaining to do why we have not noticed violations of Lorentz-invariance before. Many people are working on Lorentz invariance violation, and that by itself limits my enthusiasm.
Third, there are deformations of special relativity which avoid an explicit breaking of Lorentz-invariance by changing the Lorentz-transformations. In this case, the speed of light becomes energy-dependent so that photons with high energy can, in principle, move arbitrarily fast. Since in this case everybody agrees that a photon moves forward in time, this does not create causal paradoxa, at least not just because of the superluminal propagation.
I was quite excited about this possibility for a while, but after some years of back and forth I’ve convinced myself that deformed special relativity creates more problems than it solves. It suffers from various serious difficulties that prevent a recovery of the standard model and general relativity in the suitable limits, notoriously the problem of multi-particle states and non-locality (which we discussed here).
So, none of these approaches is very promising and one is really very constrained in the possible options. The symmetry-group of Minkowski-space is the Lorentz-group plus translations. It has one free parameter and that’s the speed of massless particles. It’s a limiting speed. End of story. There really doesn’t seem to be much wiggle room in that.
Then it occurred to me that it is not actually difficult to allow several different speeds of lights to be invariant, as long as can never measure them at the same time. And that would be the case if one had particles propagating in a background that is a superposition of Minkowski-spaces with different speeds of light. Because in this case then you would use for each speed of light the Lorentz-transformation that belongs to it. In other words, you blow up the Lorentz-group to a one-parameter family of groups that acts on a set of spaces with different speeds of lights.
You have to expect the probability for a particle to travel through an eigenspace that does not belong to the measured speed of light to be small, so that we haven’t yet noticed. To good precision, the background that we live in must be in an eigenstate, but it might have a small admixture of other speeds, faster and slower. Particles then have a small probability to travel faster than the speed of light through one of these spaces.
If you measure a state that was in a superposition, you collapse the wavefunction to one eigenstate, or let us better say it decoheres. This decoherence introduces a preferred frame (the frame of the measurement) which is how causal paradoxa are avoided: there is a notion of forward that comes in through the measurement.
In contrast to the case in which Lorentz invariance is violated though, this preferred frame does not appear on the level of the Lagrangian - it is not fundamentally present. And in contrast to deformations of special relativity, there is no issue here with locality because two observers never disagree on the paths of two photons with different speeds: Instead of there being two different photons, there’s only one, but it’s in a superposition. Once measured, all observers agree on the outcome. So there’s no Box Problem.
That having been said, I found it possible to formulate this idea in the language of quantum field theory. (It wasn’t remotely as straight forward as this summary might make it appear.) In my paper, I then proposed a parameterization of the occupation probability of the different speed of light eigenspaces and the probability of particles to jump from one eigenstate to another upon interaction.
So far so good. Next one would have to look at modifications of standard model cross-sections and see if there is any hope that this theoretical possibility is actually realized in nature.
We still have a long way to go on the way to build the cell phone to talk to aliens. But at least we know now that it’s not incompatible with special relativity.
