I just got back from a conference in Reykjavik, Iceland (!), on “Foundational Questions in Physics and Cosmology.” Photos and trip report coming soon. For now, please content yourself with the following remarks, which I delivered to the assembled pontificators after a day of small-group conversation in a geothermally-heated lagoon.
I’ve been entrusted to speak for our group, consisting of myself, Greg Chaitin, Max Tegmark, Paul Benioff, Caslav Brukner, and Graham Collins.
Our group reached no firm conclusions about anything whatsoever.
Part of the problem was that one of our members — Max Tegmark — was absent most of the time. He was preoccupied with more important matters, like posing for the TV cameras.
So, we tried to do the best we could in Max’s absence. One question we talked about a lot was whether the laws of physics are continuous or discrete at a fundamental level. Or to put it another way: since, as we learned from Max, we’re literally living in a mathematical object, does that object contain a copy of the reals?
One of us — me — argued that this is actually an ill-posed question. For it’s entirely consistent with current knowledge that our universe is discrete at the level of observables — including energy, length, volume, and so on — but continuous at the level of quantum amplitudes. As an analogy, consider a classical coin that’s heads with probability p and tails with probability 1-p. To describe p, you need a continuous parameter — and yet when you observe the coin, you get just a single bit of information. Is this mysterious? I have trouble seeing why it should be.
We also talked a lot about the related question of how much information is “really” in a quantum state. If we consider a single qubit — α|0〉 + β|1〉 — does it contain one bit of classical information, since that’s how many you get from measuring the qubit; two bits, because of the phenomenon of superdense coding; or infinitely many bits, since that’s how many it takes to specify the qubit?
You can probably guess my answer to this question. You may have heard of the “Shut Up and Calculate Interpretation of Quantum Mechanics,” which was popularized by Feynman. I don’t actually adhere to that interpretation: I like to discuss things that neither I nor anyone else has any idea about, which is precisely why I came to this wonderful conference in Iceland. I do, however, adhere to the closely-related “What Kind of Answer Were You Looking For?” Interpretation.
So for example: if you ask me how much information is in a quantum state, I can show you that if you meant A then the answer is B, whereas if you meant C the answer is D, etc. But suppose you then say “yes, but how much information is really there?” Well, imagine a plumber who fixes your toilet, and explains to you that if the toilet gets clogged you do this; if you want to decrease its water usage you do that, etc. And suppose you then ask: “Yes, but what is the true nature of toilet-ness?” Wouldn’t the plumber be justified in responding: “Look, buddy, you’re paying me by the hour. What is it you want me to do?”
A more subtle question is the following: if we consider an entangled quantum state |ψ〉 of n qubits, does the amount of information in |ψ〉 grow exponentially with n, or does it grow linearly or quadratically with n? We know that to specify the state even approximately you need an exponential amount of information — that was the point Paul Davies made earlier, when he argued (fallaciously, in my opinion) that an entangled state of 400 qubits already violates the holographic bound on the maximum number of bits in the observable universe. But what if we only want to predict the outcomes of those measurements that could be performed within the lifetime of the universe? Or what if we only want to predict the outcomes of most measurements drawn from some probability distribution? In these cases, recent results due to myself and others show that the amount of information is much less than one would naïvely expect. In particular, the number of bits grows linearly rather than exponentially with the number of qubits n.
We also talked about hidden-variable theories like Bohmian mechanics. The problem is, given that these theories are specifically constructed to be empirically indistinguishable from standard quantum mechanics, how could we ever tell if they’re true or false? I pointed out that this question is not quite as hopeless as it seems — and in particular, that the issue we discussed earlier of discreteness versus continuity actually has a direct bearing on it.
What is Bohmian mechanics? It’s a theory of the positions of particles in three-dimensional space. Furthermore, the key selling point of the theory is that the positions evolve deterministically: once you’ve fixed the positions at any instant of time, in a way consistent with Born’s probability rule, the particles will then move deterministically in such a way that they continue to obey Born’s rule at all later times. But if — as we’re told by quantum theories of gravity — the right Hilbert space to describe our universe is finite-dimensional, one can prove that no theory of this sort can possibly work. The reason is that, if you have a system in the state |A〉 and it’s mapped to (where |A〉, |B〉, and |C〉 are all elements of the hidden-variable basis), then the hidden variable (which starts in state |A〉) is forced to make a random jump to either |B〉 or |C〉: you’ve created entropy where there wasn’t any before. The way Bohm gets around this problem is by assuming the wavefunctions are continuous. But in a finite-dimensional Hilbert space, every wavefunction is discontinuous!
We also talked a good deal about the many-worlds interpretation of quantum mechanics — in particular, what exactly it means for the parallel worlds to “exist” — but since there’s some other branch of the wavefunction where I told you all about that, there’s no need to do so in this one.
Oh, yeah: we also talked about eternal inflation, and more specifically the following question: should the “many worlds” of inflationary cosmology be seen as just a special case of the “many worlds” of the Everett interpretation? More concretely, should the quantum state you ascribe to your surroundings be a probabilistic mixture of all the inflationary “bubbles” that you could possibly find yourself in?
Other topics included Bell inequalities, the definition of randomness, and probably others I’ve forgotten about.
Finally, I wanted to take the liberty of mentioning a truly radical idea, which arose in a dinner conversation with Avi Loeb and Fotini Markopoulou. This idea is so far-out and heretical that I hesitate to bring it up even at this conference. Should I go ahead?
Well, OK then. The idea was that, when we’re theorizing about the nature of the universe, we might hypothetically want some way of, you know, “testing” whether our theories are right or not. Indeed, maybe we could even go so far as to “reject” the theories that don’t succeed in explaining stuff. As I said, though, this is really just a speculative idea; much further work would be needed to flesh it out.