Author: dabacon

Nothing is more mysterious in quantum theory than the fact that states are rays in a Hilbert space and that the probability law comes from the modulus squared of overlap between the input and output states. I like to phrase this question as “Why Hilbert space?” Of course there may be no “why”! To quote Feynman: “Do not ask yourself, if you can possibly avoid that, ‘how can it be like that?’ because you will lead yourself down a blind alley in which no one has ever escaped.” But let’s assume that there is something “beyond quantum theory.” What could such a structure look like? There are many paths we can imagine for what such a structure could look like. But all of these structures must in some limit or even exactly given an explanation for the Hilbert space structure and measurement postulate for quantum theory. So here it makes a certain sense to begin thinking about what exactly quantum theory is and what exactly quantum theory is not before we embark on exploring what is beyond quantum theory. But I think today, thanks in large part to years of foundational people yelling and screaming as well as the comfort developed with quantum theory from practicing quantum information science, we understand intimately what quantum theory is and what quantum theory is not. Perhaps it is time to move on!
After going through many phases of thinking about where quantum theory comes from, I’ve now entered a new phase. My earliest phases in thinking about quantum theory stressed the information theoretic notions of quantum theory. Thinking like a computer scientist, statistician, or information theorist leads one to a much cleaner idea of what quantum theory is and what quantum theory is not. The quantum state should never, for example, be mixed up with a realistic description of a system. Noncontextuality and the nonlocal nature of quantum correlations are best understood as telling us how we can and can’t think about the information in quantum systems. And, while these points of view are certainly enlightening, this point of view can be taken too far. For example, I have spent a considerable amount of time trying to understand if the correlations produced by measuring entangled quantum states can be seen to arise because these correlations are best for, say, winning some information theoretic game. The best success of this type of reasoning, I think, is the result of William Wootters (two ohs two tees), who showed in his Ph.D. thesis that for real quantum theory the quantum measurement postulate follows from the question of how to best send distinguishable signals through a channel with angular symmetries. But it may be, and this is where my change of heart has occured, that quantum theory does not arise because it is “best at some game” or “natural under information constraints.” This does not mean that we don’t listen to what quantum theory is and isn’t saying from an information theory perspective, but it does mean that we need to move on and look for a deeper structure behind quantum theory.
How might we do this? Well my new phase is based on a philosophical argument I have discussed here before: the nonlocal nature of quantum correlations implies that any deeper theory which explains quantum theory must take seriously that our notions of spacetime topology are wrong. If all our descriptions of quantum theory must have parts which explain nonlocality, then what is the difference in such a description between having nonlocal quantities and saying that our notion of spacetime topology is wrong. In fact I might go so far as to suggest that the failure to quantize gravity (shut up string theorists…just kidding) is evidence that this is the correct approach. Since general realtivity is our theory of spacetime structure, the reason, in this view, for why we can’t quantize general relativity is that general relativity, or some deeper theory of spacetime, is what gives rise the quantum theory. So now, in my new phase, instead of looking for the game quantum theory is best at playing, I think about the geometric constructions which might give birth to Hilbert space and the quantum probability law. I think the most inspiring connection to date of this idea are results in topological field theories, where the topology of the manifold is a dynamic quantity. And there are many who argue that gravity might be a similar such theory where we have a topological field theory with the extra structure of local degress of freedom. A beautiful paper along these lines (but not far enough along these lines) is Quantum Quandaries: a Category-Theoretic Perspective by John Baez.