Compare and contrast:
1) Categorified algebra and quantum mechanics, Jeffrey Morton.
and
2) A categorical framework for the quantum harmonic oscillator, Jamie Vicary.
Posted at June 7, 2007 9:30 AM UTCTrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1307
Surely SGpd is a bicategory, rather than a strict 2-category, and you can’t just forget about the 2-cells since horizontal composition of 1-cells is not associative on the nose?
OK, fair enough. But couldn’t we just skeletalise Gpd before we begin, and avoid those issues?
There’s a lot to say about this subject! Assuming we agree that quantum mechanics might be better understood by categorifying it, there are still lots of decisions to make about how to go about it! It’s great to see a couple proposals on the table. Now people can start to examine their merits.
Alas I don’t have much time now — I’m just about to teach my last class for the spring quarter. So, I’ll just make a couple technical remarks.
Jamie Vicary wrote:
The only thing that’s not obvious about the category SGpd — for now, just considered as a 1-category — is what to do about composition of morphisms. We do this using pullbacks.
It’s really important that we think of as a 2-category. More importantly, we compose spans of groupoids using weak pullbacks — also known as pseudo-pullbacks. Ordinary pullbacks are no good!
A groupoid decategorifies to give a vector space, and a span of groupoids decategorifies to give an operator between vector spaces. Composing spans using weak pullback then decategorifies to give the usual composition of linear operators. This would not work if we composed spans using ordinary pullbacks!
To understand the significance of composing spans of groupoids via weak pullback, it helps to think about these spans as invariant witnessed relations. Here’s how composing spans works in that language:
Suppose we have a composable pair of spans of groupoids:
S
/ \
/ \
F/ \G
/ \
v v
X Y
and
T
/ \
/ \
H/ \I
/ \
v v
Y Z
Suppose is a witness to the fact that and are -related:
Suppose is a witness to the fact that and are -related
And, suppoose we have an isomorphism
Then, we get a witness to the fact that and are -related! So, the objects of the weak pullback are triples .
If we used ordinary pullbacks instead of weak ones here, we’d be demanding that , and thus demanding that and be equal. Equations between objects are evil — and our sin would be punished: composition would no longer decategorify to give composition of linear operators!
I’m afraid this barely begins to explain how cool the subject really is: for example, how this formalism leads swiftly to the categorified Hecke algebras, quantum groups, and so on. I’ll say much more about all this in future episodes of the Tale of Groupoidification.
For now, folks can learn how composition of spans works starting on page 31 in Jeff’s paper. Jeff uses weak pullbacks to compute the inner product of stuff types, the action of a stuff operator on a stuff type, and the composition of stuff operators. The last of these three is a special case of composing spans of groupoids. All three are special cases of ‘contracting indices’, where you think of an -legged span of groupoids as the categorified version of a rank- tensor.
Weak pullbacks are also explained in my my course notes, starting on the last 2 pages of the notes from week 4.
If you look carefully, you’ll see that weak pullbacks are crucial in this approach to categorifying the harmonic oscillator… for example, in getting the canonical commutation relations!
… the fact that we started with the category of groupoids doesn’t seem important. If groupoids are more important than any other sort of algebraic structure, then there must be some special reason to use them.
Groupoids are important because they’re a fundamental way to describe symmetries. A groupoid is like a set with symmetries. A span of groupoids is like an invariant witnessed relation between such sets with symmetries.
So, it’s wonderful that we can get Hilbert spaces and linear operators by decategorifying such primordial structures — so primordial that they don’t even mention the complex numbers! It’s even better that we can derive specific aspects of quantum mechanics like the canonical commutation relations, their -deformed versions, and quantum groups, starting from very natural examples of groupoids and spans between groupoids.
Nonetheless, I’m glad you’re trying something else. The more people start trying to categorify quantum mechanics, the sooner the revolution will occur.
John Baez said:
If we used ordinary pullbacks instead of weak ones here, we’d be demanding that , and thus demanding that and be equal. Equations between objects are evil — and our sin would be punished: composition would no longer decategorify to give composition of linear operators!
That’s amazing! I never guessed the weakness would be so important. I’ll have to look at that much more closely.
So what I said above certainly needs to be modified; we need to use weak pullbacks to define composition in the category. But once the category’s defined, it still seems to be an instance of the structure that I describe in my paper, except maybe now some things need to be done ‘weakly’ rather than strictly. I’ll need to think about that.
John Baez also said:
Groupoids are important because they’re a fundamental way to describe symmetries. A groupoid is like a set with symmetries. A span of groupoids is like an invariant witnessed relation between such sets with symmetries.
Okay — but mathematically, why must we use groupoids? Is there anything particular that doesn’t work nicely if we use some other suitable 2-category, instead of Gpd? Like the category of small categories?
In terms of model categories, if we use the usual `folk’ model structure on Cat, the fibrant-cofibrant objects (if I recall correctly) are precisely the groupoids. Thus hom-objects for groupoids behave as they `should do’ from the point of homotopy theory - on decategorifying i.e. passing to connected components/equivalence classes of things one gets the right things.
John Baez wrote:
Groupoids are important because they’re a fundamental way to describe symmetries.
Jamie Vicary wrote:
Okay — but mathematically, why must we use groupoids? Is there anything particular that doesn’t work nicely if we use some other suitable 2-category, instead of Gpd? Like the category of small categories?
Mathematically, symmetry is a very nice thing. As a result, the whole theory of Fock spaces, annihilation and creation operators, the -deformed version of all these, Hecke algebras, quantum groups, and so on can be categorified way using some very natural spans of groupoids. We get the original uncategorified theory back when we convert these spans of groupoids back into linear operators using ‘groupoid cardinality’.
This isn’t a reason why we “must” use spans of groupoids. Other spans are interesting too. I’m just saying that spans of groupoids give a profoundly fascinating theory, with connections to lots of important mathematics. We’re systematically categorifying lots of the ‘quantum algebra’ people have developed so far, and making it simpler in the process.
For example, the Fock space for the 1-dimensional harmonic oscillator corresponds to the groupoid of finite sets! The canonical commutation relations receive a simple combinatorial interpretation: there’s one more way to put a ball in a box and then take one out, than there is to take one out and then put one in. And so on… this is just the beginning of a long tale.
But suppose this isn’t enough fun for you. Suppose you wanted to generalize all this. There are many different directions you could go; it’s important to pick one with a good chance of big payoffs. We don’t want to generalize just for the sake of generalization: we want to dig in directions where we’re likely to find gold.
So, I would suggest replacing groupoids by -groupoids or -categories.
Why? First, these are fundamental entities, and obvious generalizations of groupoids. Second, the concept of groupoid cardinality played a key role in getting back linear operators from spans of groupoids. A good generalization should involve something with the same nice properties. But, groupoid cardinality generalizes beautifully to -groupoids: the resulting concept is called homotopy cardinality. It also generalizes nicely to categories: that’s Tom Leinster’s concept of Euler characteristic for categories. And if you read his paper carefully, you’ll see he also explains how to define an Euler characteristic for -categories!
I suspect all the abstract nonsense about getting linear operators from spans of groupoids will generalize to spans of -categories. It would be good to prove this.
But, I think the really fun thing will be to find nice examples of concepts and results in linear algebra that can be categorified using spans of -categories… examples where plain old groupoids aren’t enough.
Just to make this explicit:
the expectation would be that spans of -groupoids decategorify in some sense (I guess we decategorify only “one step”) to -vector spaces.
Right?
weak pullbacks – also known as pseudo-pullbacks
Of course terminology is a contentious and awkward issue, but that statement makes me nervous all the same!
The thing that you’re calling a weak pullback is what has traditionally been called an iso-comma-object. I think this terminology originates in Max Kelly’s Elementary Observations on 2-Categorical Limits (Bull. Austral. Math. Soc. 39 (1989)).
Now, “weak pullback” is not an unreasonable name for such a thing, though I don’t think I agree with the implicit suggestion that the traditional terminology is so bad that it needs to be changed. But pseudo-pullback seems dangerous: there is a general notion of pseudo-limit of a diagram in a 2-category, and it is surely reasonable to expect a pseudo-pullback to be the pseudo-limit of a cospan. But that’s not the same thing! A pseudo-pullback is a universal diagram like this:
with invertible 2-cells in the triangles.
In the context of the 2-category of groupoids (where all 2-cells are invertible) why not just say “comma category” or “comma groupoid”? Surely comma categories are more widely understood than two-dimensional limits?
Reading the above, it sounds very grumpy. Sorry! I really don’t feel grumpy. But aren’t two-dimensional limits confusing enough without confusing the terminology too?
Also, the term weak pullback is often used to refer to something that is like a pullback except that the fill-in arrow is not necessarily unique. That’s another potential source of confusion.
Robin wrote:
But pseudo-pullback seems dangerous: there is a general notion of pseudo-limit of a diagram in a 2-category, and it is surely reasonable to expect a pseudo-pullback to be the pseudo-limit of a cospan. But that’s not the same thing!
I bet they’re essentially the same.
Consider the case of plain old pullbacks. Following the official definition of ‘colimit’, Mr. Colimit says the colimit of a diagram shaped like this:
is a commutative diagram like this:
with some universal property.
On the other hand, Mr. Pullback says the pullback of a diagram like this:
is merely a commutative diagram shaped like this:
with some universal property. In other words, Mr. Pullback leaves out the southeasterly arrow!
Technically these guys are getting different results. But, the difference is ‘inessential’, because there’s a unique way to fill in that southeasterly arrow while keeping the diagram commutative.
Now consider the categorified situation. This time Mr. Weak Colimit will produce a diagram like this:
where the two triangles commute up to specified 2-isomorphisms, and some weak universal property holds. Mr. Weak Pullback will merely produce a square like this:
commuting up to a specified 2-isomorphism, and satisfying a weak universal property.
This time we can’t uniquely reconstruct Mr. Colimit’s diagram from Mr. Pullback’s diagram… but I bet we can reconstruct it uniquely up to a canonical isomorphism of the appropriate sort. And that’s all we should ever want, at this level of the categorical ladder!
Clearly there’s stuff to be checked here… I don’t claim I’ve checked it, but I feel pretty darn sure it’s true.
As you note, it’s difficult to discuss terminological issues like these without sounding grumpy. I have my own convention, quite hardened by now, which is to use ‘weak’ to indicate the style of categorification where equations are replaced by specified isomorphisms satisfying coherence laws. So, I call those things I was talking about ‘weak pullbacks’.
But of course the phrase ‘weak pullback’ is inherently a bit vague until an actual definition is specified! After all, there are various equivalent approaches to defining pullbacks — witness Mr. Colimit and Mr. Pullback above. These naturally lead to different definitions when we categorify and weaken them — so we should check to see if the resulting different definitions are ‘essentially the same’ in some precise sense, or not.
Regarding other possible terms:
I only mentioned ‘pseudo’ because I consider that the Australian word for ‘weak’. Now I’m regretting it.
I can imagine using the term ‘comma category’, as you suggest — except for a couple of small problems.
First, I dislike the term ‘comma category’, as does its accidental inventor Lawvere, since the term fails to convey any clear idea of what it means.
Second, and far more importantly, I’m always studying the analogy between spans of sets, spans of spaces, and spans of groupoids. The first we compose using pullbacks, the second we compose using homotopy pullbacks, and the third we compose using… here I want to say weak pullbacks, not ‘comma categories’. The analogy between strict constructions, homotopy constructions and weak constructions is so powerful that I want a terminology that makes it clear!
Indeed, both the strict and weak constructions are special cases of homotopy constructions: specialized to homotopy 0-types (sets) and homotopy 1-types (groupoids) respectively.
It’s true (I too am convinced) that the pseudo-pullback and the iso-comma-object are equivalent. In a sense that’s not good enough, since both those things are unique only up to isomorphism; in another sense it is good enough since who, after all, really minds about the difference between equivalent things? Actually this raises an interesting point. You wrote:
I have my own convention, quite hardened by now, which is to use ‘weak’ to indicate the style of categorification where equations are replaced by specified isomorphisms satisfying coherence laws. So, I call those things I was talking about ‘weak pullbacks’.
If one takes that literally, then you ought to be interested neither in pseudo-pullbacks nor in iso-comma-objects, but in bi-pullbacks, which are unique only up to equivalence. And yet, to quote John Power (from Why Tricategories?, Example 5.1):
The details are ghastly. […] To express the one-dimensional part of the limiting property requires at least nine isomorphisms. […] Moreover, to account for the two-dimensional property requires at least eighteen isomorphisms and four other 2-cells, all subject to numerous equations.
So, when we can get away with it, in practice we prefer not to deal with bi-pullbacks directly. If we can take a pseudo-pullback – or even just an iso-comma-object – then we can relax in the knowledge that we do have a (special kind of) bi-pullback, and that we’ve found it in a comparatively painless way.
From that point of view, the distinction between pseudo-pullbacks and iso-comma-objects really doesn’t matter, since both are just dodges for finding a bi-pullback without too much hassle.
If nothing else, this exchange has certainly helped me to get some of these subtleties straighter in my head!
This is interesting, and I’ll have to look at what you’re doing in more detail later - tomorrow I have to defend my dissertation, so I’ll be brief. However, there are a couple of points that touch on things I’m actively thinking about, so I’ll say something about them.
One is this business about composition of spans of groupoids by weak pullback. In the categorifying-TQFT’s project I’m working on, this turns out to be important, as John commented, in getting composition to work “nicely” (i.e. weakly - in other words, for a 2-linear map got from a composite of two spans to be isomorphic to the composite of the 2-linear maps obtained from each). If you don’t use the weak pullback, you undercount the objects in the composed span. Then the 2-linear map you get from the composite is strictly “included” by a 2-morphism into the composite of the pair of 2-linear maps from each span. Not a desirable feature - I was very angry with it for several weeks until I finally got it right.
You could also think of this as “overcounting” objects in the groupoid you’re pulling back over (the bottom of the weak pullback square). This is related to the point about groupoid cardinality, which is how I first saw this in the harmonic-oscillator setting. Objects with symmetry groups of size N should contribute less (to the tune of 1/N) to the cardinality of the groupoid than they do to the cardinality of the set of equivalence classes. (As opposed to a set of equivalence classes equipped with symmetry groups). Those terms are then the ones which throw off the composition in just the same way as the undercounting I mentioned above.
Last thing: my interest was piqued that you wanted to use some category with biproducts to construct your model of the oscillator, since there’s a lot of important properties involving the biproduct in Vect in the quantization stage of this TQFT business I’m looking at now. But on second look, it doesn’t seem to be used in the same way, so I don’t know what to think about it.
Last thing: my interest was piqued that you wanted to use some category with biproducts to construct your model of the oscillator, since there’s a lot of important properties involving the biproduct in Vect in the quantization stage of this TQFT business I’m looking at now. But on second look, it doesn’t seem to be used in the same way, so I don’t know what to think about it.
That sounds interesting — what are you using the biproduct for? For my purposes, the biproduct structure mostly just sits there and conveniently enriches the category in commutative monoids. I think it works the same way in SGpd, as well.
The other important thing that it allows you to write down, and study, the isomorphism , which is pretty important; I require that this isomorphism has to be unitary. This works quite nicely in the case of SGpd, I think, where we have , 1 being the 1-object 1-morphism category, 2 being the 2-object 2-morphism category, and being the Fock space functor. We take to be the free symmetric monoidal category functor, and the free symmetric monoidal category over 2 is unitarily isomorphic (in SGpd) to the product (in Gpd) of the free symmetric monoidal category over 1 with itself.
Is categorification of QM physically important? I.e., is there some experiment which could distinguish between categorified and ordinary QM, or is it just a mathematical reformulation of ordinary QM?
I think the philosophical point is (as expressed in other comment threads on this weblog) that if we can categorify QM we have refomulated it to bring out “what’s really going on”. That is, we don’t properly understand QM as it stands, and part of categorifying the theory will be teasing out that understanding.
If so, the sense I get from the kinds of attempts at categorifying quantum theories I’ve thought about is that one should pay plenty of fairly literal-minded attention to sum over histories and also symmetries of histories.
Jeffrey wrote:
If so, the sense I get from the kinds of attempts at categorifying quantum theories I’ve thought about is that one should pay plenty of fairly literal-minded attention to sum over histories and also symmetries of histories.
If this turns out to be the case, then it’ll be tremendously exciting. But there are lots of problems with path integrals in the first place. It might be hoped that these problems would be cured by a categorical description — but, since the conventional path integrals can’t actually be made well-defined, we would probably have to create something that’s qualitatively different than conventional quantum mechanics.
There are also some philosophical peculiarities with path integrals: why are the classical solutions so important at a fundamental level?
Also, it mustn’t be forgotten that we still can’t actually do quantum mechanics in a 2-category yet. There’s no prescription available that tells us how to take a physical system, encode it 2-categorically, and extract physical predictions from the formalism. Of course, we have this 1-categorically already, because conventional quantum mechanics is 1-categorical: you find the Hilbert space for your system, find the state in the category of Hilbert spaces that gives your starting configuration, find a self-adjoint Hamiltonian , then apply to to get your final state, finishing up by taking inner products, using the -functor, of this final state with what you’re going to measure. But this fits the 1-categorical framework too perfectly: there’s surely no point just blindly carrying it through in a 2-category, unless there’s the 2-ness is making a difference somehow.
we would probably have to create something that’s qualitatively different than conventional quantum mechanics.
If so, this wouldn’t be interesting. The interesting conjecture is that ordinary quantum particles, as perceived in nature, have a categorical description which captures more of their true nature.
We are talking here about the usual kind of refinement in physical theory: the new theory supersedes the old one, but includes it as a special (possibly limiting) case.
we still can’t actually do quantum mechanics in a 2-category yet.
It’s certainly not well developed yet. But people are already doing it.
It’s called extended quantum field theory. See for instance Hopkins Lecture on TFT: Infinity-Category Definition and references given there (also the comment section).
We just had an entire workshop essentially entirely about how to grasp -dimensional quantum field theories in terms of -functors on extended -cobordisms.
Of course it’s best understood for comparaitvely easily tractable quantum field theories (Dijkgraaf-Witten, Chern-Simons, and things like that).
Bruce Bartlett will chime in and tell you more about that…
There’s no prescription available that tells us how to take a physical system, encode it 2-categorically, and extract physical predictions from the formalism.
Not that there is nothing left to be understood, but quite some things are known. It’s a matter of active research right this very moment we speak.
As I may have mentioned, I have a proposal for a systematic definition, whose working title is The charged quantum -particle: kinematics, dynamics.
This is supposed to capture all those -dimensional quantum field theories which can be thought of as generalized Sigma-models describing -dimensional objects (points, strings, membranes, etc.) coupling to an -gauge connection (gauge field, Kalb-Ramond-Gerbe, Chern-Simons 2-gerbe, etc). (In principle, this means it can also be applied to (super)-gravity, in as far as this is indeed a theory of morphisms into a 3-groupoid
Peter Stolz and Stephan Teichner meanwhile have a concept of extended quantum field theory which has large overlap with that description. More on that in an upcoming entry.
Of course, we have this 1-categorically already, because conventional quantum mechanics is 1-categorical: you find the Hilbert space for your system, find the state in the category of Hilbert spaces that gives your starting configuration, […] finishing up by taking inner products, using the functor, of this final state with what you’re going to measure. But this fits the 1-categorical framework too perfectly: there’s surely no point just blindly carrying it through in a 2-category, unless there’s the 2-ness is making a difference somehow.
It turns out there is. I describe how to conceive precisely what you describe here, namely the expression in -categorical terms, using the -fold Hom, in D-Branes from Tin Cans: Arrow Theory of Disks and show that this 2-categorical correlator does reproduce the structure of the disk amplitude for the string coupled to a Kalb-Ramond field and sitting on a D-brane with Chan-Paton bundle. This involves doing the categorical path integral: Disk Path Integral for String in trivial KR Field
Alternatively, one can directly describe the CFT quantum amplitude this way by using FRS formalism. This is in fact reproduced this way, as explained in detail in FRS formalism from Transport and summarized comprehensively in On 2-dimensional QFT: from Arrows to Disks.
we still can’t actually do quantum mechanics in a 2-category yet.
It’s certainly not well developed yet. But people are already doing it. It’s called extended quantum field theory.
There’s clearly a whole body of work here that I’m not familiar with. It seems that you’re saying quantum mechanics can indeed be categorified, and early indications are that this categorification naturally resembles quantum field theory, in an appropriate sense.
So where does this leave quantum mechanics as used to study non-field-theoretic problems, such as simple spin systems? Are you suggesting that categorified quantum mechanics is not useful for studying these? Such a conclusion would perhaps not be philosophically terrible — after all, we believe quantum field theory is fundamental, not bog-standard quantum mechanics — but it would at least deserve to be stated explicitly.
If these non-field-theoretic systems can be categorified, then shouldn’t there be some much simpler 2-categorical framework for them, simpler than the extended QFT stuff that you describe in the same way that conventional quantum mechanics is simpler than conventional quantum field theory?
It seems that you’re saying quantum mechanics can indeed be categorified, and early indications are that this categorification naturally resembles quantum field theory, in an appropriate sense.
Right. Actually, I can separate this into two statements:
1) it is possible to take ordinary -dimensional quantum field theories, defined for instance in terms of path integrals or similar procedures, and extract from that (at least parts of, depending on how far people pushed this) the data of an -functor which
i) sends points of parameter space to -vector spaces
ii) sends arcs to morphisms of -vector spaces
etc.
i.e. an -functor of the kind
Apparently the first attempts in this direction are those by Dan Freed, see the references given in Freed on Higher Structures in QFT, and also his Slide’s for Freed’s Andrejewski Lecture.
Since then, this has been refined by various people. See for instance section 5.3 of What is an elliptic object?.
Few of these developments are really completed, as far as I am aware, to a fully precise, fully detailed -functor. There is a lot of progress for finite group models (Dijkgraaf-Witten theory). I know of at least two major works on this topic which should appear in the near future.
2) the data thus obtained (the “extended quantum field theory”) can indeed be regarded as a categorification of quantum mechanics in the obvious sense that QM is a 1-functor while these -dimensional quantum field theories yield -functors
Are you suggesting that categorified quantum mechanics is not useful for studying these? Such a conclusion would perhaps not be philosophically terrible after all, we believe quantum field theory is fundamental, not bog-standard quantum mechanics — but it would at least deserve to be stated explicitly.
Over the weekend, I will have to have a closer look at your paper, so see exactly what it is you are doing.
I am actually making a distinction here between two different things one could do to quantum mechanics, which might both be called “categorification”, but which actually achieve different things.
1) As I said above, the kind of categorification which takes one from point-particle quantum mechanics to quantum field theory (or the quantum machanics of extended objects) is obtained by passing from evolution 1-functors to evolution -functors.
2) On top of that, there are indications that the vector spaces which appear in ordinary quantum mechanics are themselves already best thought of as placeholders for structures which are really better realized categorical. That’s what John Baez is talking about in the “tale of groupoidification” and that’s how I am currently thinking Jeffrey Morton’s work would be best interpreted.
This would mean that all -functors here really ought to be pseudo -functors, i.e. taking actually values in -categories.
(This, in turn, should have a reformulation in terms of fibred -categories, which would get rid of the appearance of the , in case we don’t like that.)
Urs wrote:
… the obvious sense that QM is a 1-functor 1Cob1Hilb.
Hmm… I’m not sure I know exactly what you mean here. Presumably, 1Cob is the category of ‘Feynman diagrams’, with objects n-fold collections of points and morphisms equivalance classes of graphs up to topology. I can’t see how a functor 1Cob1Hilb is equivalent to defining a quantum mechanical system in 1Hilb; i.e., choosing a Hilbert space and a Hamiltonian.
Wouldn’t cobordification be enough?
Jamie Vicary wrote:
Presumably, 1Cob is the category of ‘Feynman diagrams’, with objects -fold collections of points and morphisms equivalance classes of graphs up to topology.
No, 1Cob is the category of 1-dimensional cobordisms. The objects are compact 0-dimensional manifolds (finite collections of points). Given two of these, say and , a morphism is an equivalence class of 1-dimensional manifolds with boundary equal to .
A symmetric monoidal -functor
is called a unitary -dimensional TQFT. In the case , this turns out to amount to nothing more than a finite-dimensional Hilbert space!
But, Urs is in the habit of equipping his cobordisms with Riemannian metrics. Then we get something closer to ordinary quantum mechanics. I’ll let him explain the details, if he wants.
Urs is in the habit of equipping his cobordisms with Riemannian metrics. Then we get something closer to ordinary quantum mechanics.
I see! That’s a really nice approach, I’d never thought of that before. So I guess all of the morphisms in 1Cob have to be Hermitian, if we don’t allow any ‘pair of pants’ cobordisms, and we only allow Riemannian metrics. I can see then how a 1D unitary TQFT would be equivalent to a choice of a Hilbert space with Hermitian operator. Clever!
Yes. When set up right, smooth structure-preserving functors from Riemannian 1-cobordismsm to Hilbert spaces are in bijection with tuples consisting of
- a Hilbert space
- a Hermitean operator
The functor sends the single objkect (the point) to that Hilbert space, and it sends the cobordism of lenght to the operator .
Nothing of what I said before makes much sense if you don’t keep this formulation of quantum mechanics in mind. Sorry, I should have emphasized this before.
You can find more details in the work by Stolz and Teichner, which I linked to before.
By the way, I have now read your paper in detail.
You give what I would call an arrow-theoretic description of the Fock space construction, and various constructions associated with it: the Fock space functor you realize as a certain adjunction on the category whose objects play the role of our Hilbert spaces of states.
This becomes something like a categorification of quantum mechnaics if and when this arrow-theory is internalized into the world of 2-categories.
Say we consider some notion of 2-Hilbert spaces. Then we might try to find the 2-analogue of the kind of adjunction you consider, now on this 2-category.
But another question might be if this is something we actually want to do. Your construction is maybe best be thought not as about the quantization of the 1-particle in the harmonic oscillator, but as the functor which sends any quantum system to its second quantization (as you note).
We haven’t talked much about second quantization here with respect to the functorial definition of quantum mechanics and its categorification. But I think it does fit in nicely. I will make a comment on that in our other thread “The -Café Quantum Conjecture”.
Jamie Vicary wrote:
There are also some philosophical peculiarities with path integrals: why are the classical solutions so important at a fundamental level?
The classical solutions don’t play a fundamental role in path integration. One integrates over all classical trajectories, solutions or not.
Nonetheless, there are philosophical peculiarities with path integration, which one should ponder in conjunction with the immense practical problem of making path integrals rigorous.
The classical solutions don’t play a fundamental role in path integration. One integrates over all classical trajectories, solutions or not.
Of course, sorry — I meant it’s peculiar that the classical configuration space, and such a peculiar class of classical trajectories in it, should play such a central role.
Yes, it’s peculiar, and people who work on making quantum field theory rigorous find path integrals to be little more than a preliminary heuristic. When the going gets tough, the quantum field theory has to stand on its own two feet.
Nonetheless I feel there’s something fundamental about path integrals.
An amusing side-note: one of the main ‘virtues’ of supersymmetry is that it eliminates some divergences, and makes reasoning from path-integral considerations more powerful. But, whether this is really a virtue is not yet known. Since nobody has seen evidence for supersymmetry in our world, it could be just a delusive dream-world made to order for physicists who like path integrals.
Of course, the virtues of supersymmetry as a mathematical tool are indisputable.
I feel there’s something fundamental about path integrals.
For what it’s worth, so do I. I’ve been enamoured with them since I saw Richard Feynman’s amazing 3-part QED lecture series as an undergraduate, available online in colour video in some hidden corner of the web! Much better than just reading the transcripts in book form.
Although, many people don’t like them so much. You have to be brave to give a seminar about path integrals anywhere near my supervisor!
Most physical arguments for why we should try to categorify quantum mechanics seem to hinge on path integrals, or something in that vein. Are there any that don’t? (Of course, there are plenty of mathematical and aesthetic arguments that don’t.)
Is categorification of QM physically important? I.e., is there some experiment which could distinguish between categorified and ordinary QM, or is it just a mathematical reformulation of ordinary QM?
As we discussed recently, there are various indications that quantum mechanics itself, as currently conceived, is actually best thought of as a decategorified shadow of something with richer internal structure.
As Jeffrey indicates, aspects of which we have discussed here, it seems that the path integral finds a natural conceptual home as a certain categorical colimit once we realize that the Hilbert spaces we see are really to be thought of as something one notch higher categorical.
I don’t quite like calling this a “categorification” of quantum mechanics. I am pretty sure that categorifying ordinary quantum mechanics times leads to -dimensional quantum field theory. (We just spent an entire workshop on that idea, essentially.)
Rather, what seems to be going on here is that quantum mechanics is itself actually one categorical level deeper than we had thought, only that nobody had realized this before.
If true, this should mean first of all that a couple of assumptions that enter into quantum mechanics would be reduced to fewer assumptions. We talked about how the action, the measure and the path integral, for instance, would come out of a single colimit once we realize that configuration space is really to be thought of as a category.
As Jeffrey said, the gain would be a better understanding of “what’s really going on”. This should in particular help understanding what’s going on as we pass from quantum mechanics to quantum field theory. In fact, I believe it already does…
Urs wrote:
As Jeffrey said, the gain would be a better understanding of ”what’s really going on”. This should in particular help understanding what’s going on as we pass from quantum mechanics to quantum field theory. In fact, I believe it already does…
Could you explain how?
Could you explain how?
I had reported on a couple of insights which one may arrive at this way in a series of posts here. Mostly concerning QFT.
QFT of Charged n-particle: Chan-Paton Bundles
QFT of Charged n-Particle: T-Duality
QFT of Charged n-Particle: Disk Path Integral for String in trivial KR Field
QFT of Charged n-Particle: Sheaves of Observables
This all follows one single principle: we realize the action functional of a “charged -particle” (a particle, a string, a membrane), whose evolution is hence described by an -dimensional quantum field theoy, by
a) first writing down the parallel transport -functor which describes the coupling of this -particle to its “background fields”.
b) then quantizing this -particle by pushing the parallel transport forward to a point.
The pushforward always apparently gives the right answer on -morphisms.
But the procedure makes most natural sense once we assume that ordinary quantum mechanics really has one “categorical depth” more than usually thought. Then it also, apparently, gives the right answer on -morphisms: The Canonical 1-Particle.
I think of this push-forward procedure as being the third edge of a cube.
A more comprehensive overview of this can be found here.
A non-sequitur: Lee Smolin recently reviewed multiple books about Einstein in the NY Rev of books. Toward the end, he returns to Einstein’s point that QM is an approximation to a deeper theory we haven’t seen yet.
[…] Einstein’s point that QM is an approximation to a deeper theory we haven’t seen yet.
In the context that we are talking about here, I find that the following aspect is remarkable and hasn’t quite found the attention yet which maybe it ought to get:
The famous statement “quantization is a mystery”, which alludes to the fact that there is usually not one unique quantum theory corresponding to a given classical system, rests on the implicit premise that all classical systems one can imagine (namely: every symplectic space equipped with a Hamiltonian function) ought to be regarded as being on equal footing as far as quantization is concerned.
I have developed the suspicion that this assumption is wrong: I tend to think that there are physical systems which ought to be regarded as fundamental and others which are merely effective descriptions of conglomerates of fundamental systems.
For “effective theories” quantization has every right to be a mystery. But “fundamental systems” might admit a natural quantization functor.
The main example is the “charged particle”: the classical system which models a point-particle that propagates on a (Riemannian) space and is charged under a vector bundle with connection on .
While, in principle, this, too has many different quantizations, there is really in an obvious way only one which is natural and “sensible”:
the Hamiltonian operator should be the covariant Laplace operator and nothing else.
That’s why I keep going on about the charged -particle:
I think the particle charged under a vector bundle with connection, the string charged under a 2-bundle with connection, the membrane charged under a 3-bundle with connection, etc., all qualify as “fundamental systems”.
Their quantization should not be a mystery. But a pushforward.
When we have that quantization, we may consider second quantization.
So, I am saying, there might be a qualitative difference bwteen, say, the 2-dimensional quantum field theory which describes the worldsheet dynamics of a string, and the 10-dimensional QFT arising from its second quantization.
While the former naturally is a functor on cobordisms, the latter maybe is not. It is not a “fundamental” system itself, but rather the second quantization of a fundamental system.
So, when trying to understand what quantization really is, I believe we should not so much worry about the quantization of, say, the kicked top, as people do. Its quantization is not “natural” and hence mysterious. On the other hand, quantizations of charged -particles is natural and ought to be non-mysterious.
Re: Categorifying Quantum Mechanics
I think I can make a guess as to how these approaches are connected. I’m going to describe how I think it should all work out, but I’ve by no means worked through all the details, so let me know where I get it wrong.
An important category in JM’s paper is the 2-category SGpd of spans of groupoids, which is defined with
I’m going to concentrate on the 1-categorical structure of SGpd, ignoring the 2-morphisms. They’re by no means unimportant, but JV’s paper is entirely 1-categorical, and we’ll be able to do without them. I would love to know how they enter the scene!
The only thing that’s not obvious about the category SGpd — for now, just considered as a 1-category — is what to do about composition of morphisms. We do this using pullbacks. Given and , then the pullback of and gives a pair of functors and ; we then define .
So, what properties does SGpd have? First of all, it’s a -category: given any , we can easily obtain . It also has products and coproducts, which coincide to give -biproducts; for groupoids and , their biproduct is given by , the coproduct in Gpd. Weirdly, the product structure in Gpd has not survived to SGpd, so we explicitly retain it by putting a symmetric monoidal structure on SGpd: the tensor product of two morphisms is given by the product of the defining functors in Gpd, i.e., .
So, it seems to me that SGpd is a symmetric monoidal -category with -biproducts. But this is exactly what JV’s paper requires to construct a model of the harmonic oscillator!
For every groupoid , we have to construct a Fock space groupoid over it. (F will stand for ‘Fock space’ from now on.) Among other things, we need all of these Fock space groupoids to carry a canonical commutative monoid structure. So, defining 1 to be the trivial groupoid with only one morphism, what could be? It surely has to be FinSet! A naive answer would be the groupoid , but my guess is that would be associated to the free noncommutative monoid over 1. So, does this work? Does deserve to be called the ‘free commutative monoid groupoid over 1’?
So, what’s a state of ? It’s a morphism , where 1 is the monoidal unit object. Since is the terminal object in Gpd, is unique, and so specifying this state amounts to choosing a functor — in other words, a stuff type! Raising and lowering operators should emerge using JV’s prescription, and should turn out to be just the same as the raising and lowering stuff operators defined in JM’s paper.
As far as this goes, though, the fact that we started with the category of groupoids doesn’t seem important. If groupoids are more important than any other sort of algebraic structure, then there must be some special reason to use them. Something to do with groupoid cardinality, perhaps? What would happen if we used the category of spans of categories, rather than spans of groupoids?