# Quantum mathematics?

"Quantum" as a term/prefix used to be genuinely physical: what was supposed to be physically continuous turned out to be physically quantized.

What sense does this distinction make inside mathematics?

Especially: Is "quantum algebra" a well-chosen name? (According to Wikipedia, it's one of the top-level mathematics categories used by the arXiv, but it's not explained any further.)

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There is a very clear and important mathematical distinction between classical probability theory and quantum (non-commutative) probability. –  Gil Kalai Mar 25 '11 at 18:09
Actually, it is explained further, like all the arXiv categories: front.math.ucdavis.edu/categories/math –  Ben Webster Mar 25 '11 at 18:10
What does "well-chosen name" mean? Quantum is catchy, historically motivated, sufficiently distinct from other terms that one usually knows what it is intended to mean when one sees is, considerably much better that "basic", and so on.... –  Mariano Suárez-Alvarez Mar 25 '11 at 18:17
@Ben: "Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory" isn't very much of an explanation. –  Hans Stricker Mar 25 '11 at 18:21
This is not to disagree with Gil's comment; but I think Connes has taken issue (informally, in casual interviews) with some uses of "quantum X" as a more fundable synonym of "noncommutative X". My memory is that he politely points out that he can't see what is actually being "quantized" in some of these cases. –  Yemon Choi Mar 25 '11 at 18:57

I think that the basic intuition relating quantum algebra and quantum physics is something like:

quantum stuff = classical stuff + $\hbar$ (something complicated)

where $\hbar$ is a "small" formal variable. In other words, the point is to consider that the mathematical objects everybody knows are only approximations of more complicated objects. Hence, quantum mathematics has something to do with perturbation theory, because most of the interesting objects in quantum mathematics are perturbations of trivial solutions of some problems/equations. Here, perturbation means that these objects are formal power series in $\hbar$ whose constant term is a trivial solution (eg: 1 :) ) of some equation (eg: the Yang Baxter equation).

Hence, as John pointed out, quantum algebra involves the study of objects for which classical properties (eg: commutativity) are "almost" true (ie: true modulo $\hbar$).

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Personally, I like your answer very much: "quantum" involves a small parameter like $\hbar$ or a $q$. But I think that many people working in e.g. noncommutative geometry would disagree: they usually neglegt the deformation aspect and study "hard" quantum algebras where no classical limit is in sight. So one probably should be open also for this point of view. In NCG the analogy with ordinary (=classical) geometry is more important than the deformation aspect. –  Stefan Waldmann Mar 26 '11 at 10:31

Working in "quantum mathematics" myself, I should tend to defend this teminology a bit ;) The term is clearly motivated by the usage in physics and, nowadays, is typically used in situations where you have a "classical" mathematical object (ring, algebra, group, whatever) which traditionally is viewed in a commutative context. Then the "quantum" version means to transfer things into a noncommutative context and see what happens.

Of course, this is all very vague, but why do you call groups "groups" and fields "fields"? I guess, it is the intuition which makes this notion useful for the community. The intuition from physics is the transition from commutative to noncommutative, and I think that is really what people usually think if they hear from some "quantum blablabla" in math. So I guess, it is not a completely irritating notion :)

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On first reading I (mis?)read this as saying that 'traditionally ring, algebra, group are viewed in a commutative context'. Inserting 'a certain' in front of ring could avoid this. –  quid Mar 25 '11 at 19:11
@unknown (goolge) Of course, noncommutative rings etc have been studied before any sort of "quantum". I should have been a bit more specific here: it is the analogy (perhaps even a deformation) with a commutative object which makes the noncommutative object "quantum". Clearly not a mathematical definition :) –  Stefan Waldmann Mar 26 '11 at 10:33
Thank you for the clarification. –  quid Mar 26 '11 at 10:46

I would hold that the term non-commutative algebra is usually used to refer to the study of general noncommutative algebras. Quantum algebra involves the study of certain types of non-commutative algebras, not all non-commutative algebras. It's not black and white, but reasonably well-defined subfamily. The algebras quite often involve a parameter $q$ st when $q=1$ or $0$ the algebra is commutative - take for example Drinfeld--Jimbo algebras. The parallels with quantum theory here are obvious.

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