The LHC is coming, and it’s time to place your bets. What do you do? (Fun though it may be, shooting the hostage doesn’t really help here.) We’re committed Bayesians (for the sake of this post, at least), and we want to assign a probability that the LHC will see supersymmetry. More generally, we have a set of possibilities for our observable physics, and we would like to assign probabilities to each. This is called the problem of finding a measure. Since the theory of eternal inflation with its “bubbling universes” is the context where the multiverse often comes up, this is often referred to as “measures for eternal inflation”.
To approach this problem, let’s return to the drug side effects situation of the previous post on this subject. To come up with the probability of 5%, we tested the drugs on a number of people and saw how many experienced side effects. Can we do this with our multiverse? Well, the obvious thing to do is to look at each possible universe (or “valley”) in the multiverse and assume that are all equally likely. This is called a “counting measure”, and one can use it to derive probability distribution for physical parameters. There is, however, one major problem with it. If there are an infinite number of possible universes, a counting measure does not exist. Since the total probability must be one, the probability for any given universe is zero. (There’s no continuum here, so probability densities can’t save you.) Oops.
Now, you might argue that assigning an equal probability to each universe isn’t the right thing to do. After all, we’re not universes; we’re observers. Thus, we should weight each universe by the number of observers. This is hard to do, but one can guess various proxies for the number of observers. Popular examples are the amount of free energy or the amount of cold matter. People have tried to implement this sort of weighting, but I don’t see how it escapes the above problem. Instead of using a counting measure on universes, you’re using a counting measure on observers. But it’s still a counting measure, and if there are an infinite number of observers, you’re still SOL. It seems to me that to proceed along these lines, one either must choose something other than a counting measure (and justify your choice somehow) or postulate a finite number of possible universes or observers.
There’s another aspect of the story that we have neglected, however. As discussed previously, in many theories, the multiverse is populated by a sort of reproductive process. It seems reasonable that this would affect your choice of measure. In Lee Smolin’s hypothesis of cosmological natural selection, for example, black holes produce new universes and, consequently, one might imagine that universes with large number of black holes are more plentiful. But, as with so many things in this story, this is hard to make precise. If the multiverse keeps reproducing forever, one has an infinite number of universes. In order to obtain a measure from this (and solve the above problem with counting measures), we need a limiting process that we can compute with. An intuitive idea would be to look at the (possibly finite) distribution of universes at a given time and then take the large time limit. The problem with this is that, in general relativity, there is no global notion of time. Depending on how we decide to choose the sequence of time slices in our limit, we can obtain completely different answers.
I make no claims to originality with any of these arguments. They are well-known to most of the experts on this subject, and they have a variety of responses. I think some people believe that there is a right answer — that if we understand the theory and the philosophy well enough, we will able to say that this is the correct measure and begin to take our bets. Most people I talk to, however, take a much more pragmatic approach. Given some abstract principle that leads to a measure, some will say that we should consider the choice of principle as part of the theory. Thus, if the experiments turn out other than how we made our bets, the combination of physical theory and measure principle is falsified. (Lee Smolin’s principle wherein one postulates that we live at a local maximum of the measure/fitness function is a variation on this, avoiding questions of probability by sheer force of axiomatics.) Others say in a similar vein that we should plug away, and if we find something that works, we get to be happy. It’s possible, I suppose, that one of these measure papers will lead to a series of predictions that get vindicated at each opportunity. Who would want to argue with success? Finally, some have tried to look for quantities that end up being independent of the choice of measure, thus sidestepping much (but perhaps not all) of this philosophical morass.
I want to end with an odd consequence of such reasoning called the Doomsday argument. As I argued above, there’s nothing particularly special about a given point in time, so if we are to consider ourselves as generic human, we should consider all humans throughout time. Human population has been undergoing exponential growth recently, and if this were to continue, there would be far more people in our future than in our past. Since the average person should have roughly half of humanity born before and half of humanity born after them, it would seem that either we’re not very average, or the exponential growth of humanity will come to an end. Soon. In other words, the end is nigh. Doomsday approacheth. Repent.
Or not. I just can’t make myself take this very seriously, although many people do. Take Ken Olum’s version of this. Consider all observers in the multiverse. Surely a fair fraction of them would live in an interstellar civilization that will have many more people than our measley six billion. Well, then, where are my spaceships? I find the lack of spaceships very disappointing. Or take the “simulation argument” mentioned in John Tierney’s latest column and discussed subsequently around the internet. If our hyperadvanced descendents are performing lots of simulations, who’d be simulating boring stuff like us? Wouldn’t there be lots of simulations a whole lot cooler? You know, magic, spaceships, lots of kick-ass martial arts maybe. Fun stuff.
Why stop with simulations, anyways? The philosopher David Lewis has put forth the idea of modal realism, that all possible worlds exist. The mother of all multiverses. Greg Egan’s Permutation City on some seriously strong steroids. Counterfactuals become a piece of cake, but oy vey. It all seems like it could be a lot of fun, I suppose, but it’s not for me. As for where you stop, well that’s up to you. If not in this universe, then in the next.