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.
O brave new quantum world!
Let me philosophize:
Isn’t it too early to talk about categories in [quantum] physics?
I get confused when I ask myself similar questions, because on the one hand quantum physics is understood and taught as a structure/approximation on how we understand nature; It might as well be “inapplicable” [scientists’ euphemism for wrong] here or there. On the other hand quantum mechanics comes fully equipped with a mathematical structure, Functional Analysis, itself based on “artifacts” like
real number and the Axiom of Choice [Zorn’s lemma], which is a beauty to behold and already ripe for applying category theory as a coup de grace!
Now suppose one gets a theory such as Quantum Gravity or “Glass Bead Game” theory that contains mathematical structure of quantum theory. Will it not also be based on some artifacts, and thus still an abstraction? And still craving a “why” question?
I guess this has more to do with walking the path than with what is supposed to be at the end. It is the journey itself that counts.
I’m not certain I follow.
I certainly agree that the discovery of a theory which contained quantum theory as some limit or from some point of view would still leave us with a “why” question. But we can always ask “why” about any theory. So I don’t motivate looking for a deeper structure as a “why” question, but as a question of a possible way to reconcile gravity and quantum theory.
just for my own edification, what’s the status of the category-theoretic perspective in string theory and loop quantum gravity? i read some pretty mind-bending things on stringification as categorification on the string table…
Well, I was trying to motivate my first sentence.
Categories are not necessarily deeper, in the same sense that, for example, General Relativity is deeper than Newtonian law of the inverse squared attraction. In fact most of the times categories turn out to be a common language connecting otherwise different mathematical hierarchies.
A common language is nice but is not deep. [I am being stubborn here]
Well I don’t think the point of Baez’s article is that category theory is anything deeper having to do with quantum and general relativity. In fact I think Baez mistitle his article. The point of the article is the deep connection between topological field theories and the machinery of quantum theory.
Joe wrote:
I am in the process of writing a PhD thesis which will try to discuss aspects of this in part. There is only a rough sketch so far but if you are interested you can find the current version with already plenty of information concerning your question here.
As you will see, I am claiming this is also related to categorfying spectral geometry, see also here.
Dave wrote:
To my mind the, at least philosophically, interesting point made in John Baez’s ‘quantum quandaries is that he reviews how the dynamics of an extended object is naturally described in a category of cobordisms (which indeed is how, following Segal etc., mathematicians set up topological but also more general field theories) and that once you are working in that category it is a much smaller step to the category of Hilbert spaces than when you start with any description of point particles.
I find this is a cute general-abstract-nonsense argument for strings (but of course that’s not the aspect of the conclusion emphasized in John Baez’s paper).
And category theoretic reasoning does seem to play a role here. You can hardly argue that quantum field theory serves to motivate quantum mechanics!
I recently attained my associates degree and barely understand basic wave packets. I like geometry. I didn’t believe in GOD until one day I was doing basic multiplication and my numbers started talking to me. Yes, they talk to me, and they are going to start talking to you too!! I can assure much life and many equations beyond quantum theory.
ok this is all nice…but what does it all mean to the layman who doesn’t understand this? Can you explain it in words that the every day Joe such as I would understand, without the million dollar words?
Please break it down for us non-scientist people!
Well, my english is not so good, im from peru, and here there is not that much aproach to quantum science, niether lot of books and teachers to ask.
ive read all the answers and questions, and i have my own theory.
Beyond understanding all in a way a measure, meters, cm, km, gr, homs, etc etc etc, we must FIRST understand really the basic of our comunication system, undestand that all we talk and say ITS LIMITED, and that we must also create new words to describe new things. REMEMBER THAT words are kind of a sistem, made from diferent units that finaly make a resultant.
understanding abstact things can only be made if diferent concret things colide each other in one kind of way and generate a new sistem than we call abstaction but in real is one pak ok ideas, words and conceptualisations we have put together to get that thing we can describe with a word.
that simple thing of attaching packs and conceptualize them, is waht we really should fully undestand, becouse making more systems and creating more complex scenarios for our mind will never get the anserws we want. OF COURSE THER IS ANOTHER DIMENSION AND TIMES AND PARALEL NATURES, of course WE CANOT SEE X RAYS AND HAVE THE HEART RATE OF A FLY, and see at the frame rates they see, even if the monitor in front of you does so.
our limitations are the only things that make us stay at all aour sences.
reality exist for every unit “we can measure” but that unit and so on have its own reality with its own time and its own energy and its own rules.
BUT WE ALWAYS MEASURE that units IN OUR WAY, AND IN OUR TIME.
tha unit with lots of units make systems that for a reason end up in a unit. WITH ITS OWN TIME AND REALITY.
the day we fully understand the COMUNICATION PROCES of comuinicating, and the time, the sence the measure THAT, THAT UNIT that we analise uses for “living”, then there ONLY THERE we will have that EMULATION way to understand it in our own HUMAN way.
all things in the universe and sorroundings are unique and uses their own lenguage and comunication way.
just see animal, humans, trees, etc, we all are diferente adn we all must undestarnd each other cos we all use diferent “lenguajes”
we only use THE HUMAN WAY, thats our basic LIMITATION.
ALL THINGS ARE POSSIBLE WE JUST HAVENT FIND THE WAY YET.
BEYOND HUMAN LIMITS we have find the way always to break them with machines and stuff.