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June 8, 2007

The n-Café Quantum Conjecture

Posted by Urs Schreiber

Conjecture (roughly): Quantum mechanics, which has historically been considered as a structure internal to 0Cat0\mathrm{Cat}, has a more natural formulation as a structure internal to 1Cat1\mathrm{Cat}. This involves refining functions to “bundles of numbers” (namely fibered categories, or, dually, refining (action) functors to pseudofunctors) and it involves refining linear maps by spans of groupoids.

In this natural formulation, quantization will no longer be a mystery – but a pushforward: the quantum propagator (tU(t))(t \mapsto U(t)) is the pushforward to a point of the classical action (pseudo)-functor.

nn-Dimensional quantum field theory works the same way, but internal to nCatn\mathrm{Cat}.

Evidence.

1) The Tale of groupoidification (or “Tale of spanification”): The shift in degree is real.

Linear algebra and linear representation theory is in many respects better thought of as an unrefined version of what should really be the study of spans in groupoids. Taking this seriously and applying it to the linear algebra appearing in quantum mechanics might be responsible for the categorical refinement in the above conjecture.

John Baez has been reporting on joint work on this with Jim Dolan and Todd Trimble in the last couple of installments of This Week’s Finds: 247, 248, 249, 250, 251, 252.


2) It does make sense when applied to quantum mechanics.

Before the Narn i Groupoidification appeared, Jeffrey Morton had already applied the central ideas appearing there to the quantum harmonic oscillator. While there remain lots of open questions, the main insight is that it does make good sense. Indeed, many concepts in quantum mechanics become simpler (conceptually) this way.

Jeffrey Morton
Categorified Algebra and Quantum Mechanics
arXiv:math/0601458v1

(see also Jamie Vicary’s work, discussed in our recent thread Categorifying Quantum Mechanics)

The example John emphasizes a lot is: the canonical commutation relations for Fock operators which play such a crucial role throughout quantum theory [a,a ]=Id [a,a^\dagger] = \mathrm{Id} becomes a mere combinatorial statement: it says that there is one more way to first insert one item into a box of similar items and then remove one, than there is to first remove one and then put it back in.

This is the kind of “simplification of concepts” that is to be expected. But it should only be a start.


3) The process of quantization becomes simpler, conceptually.

As John Baez has explained in great detail in his lectures on Quantization and Cohomology ( week 18 21, and others) a classical action functional is really a functor.

In particular, the non-kinetic part of the action is a parallel transport functor. The kinetic part is, I think, better thought of as being part of the measure which appears once we realize the shift in categorical degree and then push-forward this parallel transport functor to a point.

This should be what quantization is: classical (non-kinetic) action quantization quantum theory parallel transport functor push-forward to a point quantum propagation functor. \array{ \text{classical (non-kinetic) action} &\stackrel{\text{quantization}}{\rightarrow}& \text{quantum theory} \\ \text{parallel transport functor} &\stackrel{\text{push-forward to a point}}{\rightarrow}& \text{quantum propagation functor} } \,.

One can check that with everything conceived in its usual un-refined formulation, this push-forward does produce the right answers on (k<n)(k \lt n)-morphisms.

For instance, for n=1n=1 it is easy to see that the push-forward computes the space of states of the quantum particle: The First Part of the Story of Quantizing by Pushing to a Point….

Then, for n=2n=2 it does, apparently, produce the right kinematics for the string: the D-branes over the ends of the open string appear as 2-spaces of states of the 2-particle : Quantum 2-States: Sections of 2-vector bundles, QFT of Charged n-particle: Chan-Paton Bundles.

(For n=3n=3 one can see states of the 3-particle in the context of the Chern-Simons membrane reproduce aspects of 2-dimensional conformal field theory. I will summarize some of the aspects of this in a future entry.)

But in the unrefined version of quantum mechanics, the canonical push-forward to a point does not even exist for top level nn-morphisms. One has to fake it by introducing measures and intervening by hand.

However, once we do refine the parallel transport from functors to pseudofunctors, it seems that also the usual kinetical part of the quantum theory is produced by a canonical god-given push-forward to a point: The Canonical 1-Particle.

And it turns out that in fact even without any intent to quantize it, a parallel transport functor is very naturally thought of in terms of its curvature pseudo-functor. As described in nn-Curvature In fact, this does clarify a couple of things in gauge and higher gauge theory.

I expect that this classical passage from parallel transport functors to pseudofunctors roots in the same general abstract nonsense which the shift appearing in the “Tale of Groupoidification” does: judging from what I understood, it seems that in both cases the deal is the Grothedieck construction, or a variation of that theme, which relates fibered (1-)categories p:PX p : P \to X to pseudofunctors XCat. X \to \mathrm{Cat} \,.

Posted at June 8, 2007 9:41 AM UTC

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28 Comments & 7 Trackbacks

Re: The n-Café Quantum Conjecture

Urs, what’s your position on the point behind the question we finally got articulated here:

If different forms of mechanics are all just matrix mechanics over different rigs, where’s the superposition principle in classical mechanics?

The idea might be expressed as that classical mechanics should be cast at the same category theoretic level as quantum mechanics. It’s just a rig change away.

Posted by: David Corfield on June 8, 2007 12:06 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Ah, you mean that maybe this puzzle about how to think of classical mechanics properly as a deformed-rig version of quantum mechanics becomes maybe clearer once we do take care of the “categorical refinement” also in the classical case?

Could be. Should be, in fact. If the conjecture is true. But I don’t know.

Posted by: urs on June 8, 2007 12:15 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

“classical mechanics should be cast at the same category theoretic level as quantum mechanics” –

this seems as plausible, without detailed construction, as –

“Newtonian mechanics should be cast at the same category theoretic level as (Special/General) Relativity.”

That is, the normative “should be” as a goal for n-categorification rather than a claim about the Physics of the cosmos we happen to inhabit?

Posted by: Jonathan Vos Post on June 8, 2007 11:45 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Narn i Groupoidification

HAH!

Actually, I was just noting the other day that I must be a geek, since as I pack up my apartment I have two boxes labeled “Tolkien”.

Posted by: John Armstrong on June 8, 2007 3:33 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Narn i Groupoidification

HAH!

Do you think I have the grammar about right?

I bought the Narn recently when I missed a train, at the railway station. I made the interesting observation that it had the greatest impact on me when I read it after thinking about technical things all day. Then I found the language alone really moving.

This effect was less pronounced after a leisurely spent day of a weekend.

Maybe that’s a reason why it’s a geeky thing. It acts as a counterbalance of something, somehow. :-)

Posted by: urs on June 8, 2007 6:02 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Narn

Is this where Lewis got the name Narnia?

Posted by: Mike Stay on June 8, 2007 7:45 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Is this where Lewis got the name Narnia?

Unfortunately, my book containing Lewis’ correspondence with Tolkien is packed away, so I can’t check. It wouldn’t surprise me, though. The root of Narn was glossed as “story” back to the oldest of the Quenya lexicons (see HoME vol.5), and it’s just the sort of thing it strikes me Lewis would lift as an homage.

Urs: That grammar should be as correct as is possible. Obviously there’s no root in the lexicons even remotely describing “groupoid”, so it would have to come in as an undeclined loanword.

Posted by: John Armstrong on June 8, 2007 10:08 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

OR there was that night where the Inklings had a few pints with a Mathematical Physicist and tried a complex parameterization:

Narn^(ia)

Posted by: Jonathan Vos Post on June 8, 2007 11:48 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

n-Dimensional quantum field theory works the same way, but internal to nCat.

Why do objects internal to nCat require mathematically ill defined renormalization for their proper definition?

Posted by: Thomas Larsson on June 9, 2007 3:44 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Why do objects internal to nCat require mathematically ill defined renormalization for their proper definition?

While renormalization need not be ill defined (see Connes-Kreimer) it is true that quantization tends to be a subtle and difficult step.

Hence one might wonder:

How is that non-straightforwardness reflected in the category-theoretic formulation we are discussing here?

This has a nice answer:

the action functor which we are talking about can naturally be pulled back. But what we need for quantization is its push forward.

This is an operation defined only indirectly, by means of a property.

For one, the push-forward here need not exist. If it exists, it may be hard to find.

There will, in general, not be a strightforward prescription to compute such a push-forward. The best one can in general hope for is a direct prescription to check whether a given guess satisfies all the properties.

So when I say that “quantization is a push-forward” (slightly simplifying, by the way, since it is really a combination of pullbacks and push forwards) this doesn’t mean that, if true, all problems in quantum field theory become trivial instantly.

Posted by: urs on June 10, 2007 4:19 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Perhaps it is a bit like the group String GString_G - if we try to construct it as an object in a category, it is defined only up to homotopy. But if we define it as an object in a 2-category, it is (Frechet-) smooth.

Posted by: David Roberts on June 12, 2007 5:57 AM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Question from the peanut gallery…

Would this change in viewpoint be more or less natural formulated in a “discrete” spacetime, e.g. an n-diamond?

I’m still grappling with the basic ideas.

Posted by: Eric on June 9, 2007 5:58 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Would this change in viewpoint be more or less natural formulated in a “discrete” spacetime, e.g. an n-diamond?

Yes. In fact, that’s currently part of the best evidence I have that the above conjecture is actually true:

In

QFT of the Qharged nn-Particle: The canonical 1-particle

I try to work out the push forward which we are talking about for the case of an ordinary charged particle 1-particle which propagates on a 1-dimensional space, which is modeled by a discrete lattice of points \simeq \mathbb{Z}.

The main result there is that, indeed, by going throught the pull-push quantization prescription, we do indeed obtain the (discretized version of) the (euclidean) quantum propagator texp(tΔ), t \mapsto \exp(- t \Delta) \,, where Δ\Delta is the covariant Laplace operator in one dimension.

This may have sensible continuum analogs, but in the discretized setup it is an elementary computation.

Posted by: urs on June 10, 2007 4:28 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

I see. There you phrased the exercise as:

Exercise. Find (1) such that the entirely canonical quantization procedure applied to it, in particular using the Leinster measure as described above, reproduces (a discrete approximation to) the ordinary textbook quantum theory of the charged 1-particle.

Your use of the word “approximation” demonstrates a bias. Who is to say that the ordinary textbook theory is not the continuum approximation to this? ;)

Eric

Posted by: Eric on June 10, 2007 8:14 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

At least I’m consistent (if memory challenged!). As I continue reading down through the comments there, I see I made almost the exact same comment there :D

Posted by: Eric on June 10, 2007 8:29 PM | Permalink | Reply to this

Re: The n-Café Quantum Conjecture

Your use of the word “approximation” demonstrates a bias. Who is to say that the ordinary textbook theory is not the continuum approximation to this?

Well, indeed. That’s one of the things here which might be exciting: by tuning our configuration space category such that the induced Leinster measure matches the measure we expecxt to see, we might happen to find out what the “true” category structure on our spacetime really is.

It may turn out that the measure (say the Wiener measure) which we keep using ordinarily, and which assumes an ordinary continuum spacetime, will have to be regarded as a mere approximation to something slightly different.

Whether that “slightly different” thing is really nothing but (the category generated from) a finite graph, or something more subtle I can’t tell yet. But at least the measure induced from a finite graph is approximately that of the continuum. If that means that the former approximates the latter or the latter the former, I don’t dare to tell, at this point.

Posted by: urs on June 11, 2007 2:58 PM | Permalink | Reply to this

Second Quantization

While reading Jamie Vicary’s formulation of the Fock space construction I felt the urge to finally figure out what second quantization means in terms of the above setup.

I think it nicely fits in. In fact, I want to add to the above conjecture the statement:

With quantization regarded as the push-forward of a transport functor, second quantization becomes exactly that: an iteration of this process, which pushes the quantum propagator itself to a point.

Recall, the idea is that we start with a parallel transport functor tra:𝒫 1(X)Vect \mathrm{tra} : \mathcal{P}_1(X) \to \mathrm{Vect} which sends paths in configuration space XX to their classical “phases”. (Let me be cavalier with the precise definition of the codomain, since part of the conjecture is that we actually have a nice 2-category here and are talking about pseudofunctors, really. I don’t want to look at the deatails of that at the moment, but consider the general structure. )

Next, we consider “abstract paths”, which are morphism in 1Cob S1\mathrm{Cob}_S, where SS denotes some unspecified extra structure, like Riemannian structure, for instance.

By looking at the space of “histories” hist[1Cob S,𝒫 1(X)] \mathrm{hist} \subset [1\mathrm{Cob}_S,\mathcal{P}_1(X)] we expect to get a functorial quantization prescription which sends functors on paths in configuration space (classical actions) to functors on abstract paths (quantum propagators) tra(Q(tra):1Cob SHilb). \mathrm{tra} \mapsto (Q(\mathrm{tra}) : 1\mathrm{Cob}_S \to \mathrm{Hilb} ) \,. I have described in detail how this is supposed to work in QFT of charged nn-Particle: Dynamics and went through an example in The canonical 1-Particle.

For what I want to say here about second quantization, the details of this are not that important. What is important is the main statement:

we have a prescription which sends transport functors to transport functors (here “transport” means: has the properties required in this business, in particular smoothness).

So the point is: we can repeat this process systematically!

My claim here is that it looks as if repeating this process (i.e. going along the “quantization” edge of the cube twice) is exactly what is known as “second quantization”.

Namely, consider what will happen: after we have quantized once, the new configuration space is that of objects of 1Cob S1\mathrm{Cob}_S: but this are just disjoint unions of the one abstract point: these are parameterized by the natural numbers.

So, when quantizing again, we get “wave functions” on the natural numbers.

These will have to take values in the fibers of the original functor – but that means in symmetric powers of the Hilbert space assigned to the single point by the original quantum functor Q(tra)Q(\mathrm{tra}).

So we do obtain the Fock space construction of objects, this way. I think.

Notice that “second quantization” is really usually two things: first passing to the Fock space of the single system, and then, seciondly, adding interactions between several copies of the original thing (otherwise it would be just a free theory).

Here we don’t have this freedom of adding interaction by hand. Instead, the value of the second quantized functor on morphisms will be entirely determined by the push-forward operation from the behaviour of the original functor.

If there were no interactions to start with, then none will appear. Otherwise, they will fix the interaction in the second quantized theory.

For instance, start with the standard 1-particle on 1Cob Riem1\mathrm{Cob}_{\mathrm{Riem}}. Since 1-dimensional such cobordisms cannot split or join, there is no interaction at all going on.

But then, start with 2Cob S2\mathrm{Cob}_S. Now we do have interactions in the game, given by, essentially, the pair-of-pants (three-holed sphere) and its opposite. Accordingly, second quantizing a functor on that by push-forward will produce a theory with unique interactions now.

It looks to me like this should be exactly the category-theoretic formulation of the claim in string theory that interactions of the string are fixed by the fact that it is a string (while for the particle we are faking it, passing to graphs which fail to be manifolds at the vertices and adding interactions by hand there).

I belive that second quantizing a transport functor on 2Cob2\mathrm{Cob} as described here will lead to “string field theory”. It’s a matter of working it out in concrete examples. Right now it is part of my conjecture.

Posted by: urs on June 10, 2007 5:00 PM | Permalink | Reply to this

Re: Second Quantization

Hi Urs,

I know my gushing is probably getting old to some people, but I think you are doing some fantastic stuff. My gut reaction after reading this is that you have essentially found a nice graphical way to teach quantum mechanics purely with pictures and no need at all to resort to formulas. In this regard, you could almost get away with teaching this to high school students (or earlier) who can draw directed graphs.

That was one of the very cool things that comes out of elementary electromagnetic theory via differential forms on a “discretization” of spacetime, i.e. it becomes trivial to explain Maxwell’s equations without ever writing down a formula simply be looking at stuff flowing out of the boundary of a 4-dimensional domain (maybe projected down to 2+1-dimensions for drawing purposes :)).

I am so insanely busy with work (and can’t forget family) that I haven’t been able to sit down and learn the arrow theoretic stuff yet, but I am looking forward to calmer days when I finally can.

One unfortunate thing about all this is that you are just too smart. Your brain operates on a level so far beyond mere mortals that you’ve, to a certain extent, lost touch with mere mortal’s (like mine) ability to understand stuff.

What you need is a Rosetta stone for arrow theory. I’ve grown to trust my gut to a certain extent and my gut tells me that what you are doing is not so insanely unreachable for mere mortals and, at its heart, it is probably quite simple once you break through the linguistic barrier.

Simple yet powerful. If I understand, by turning a fairly simple crank on some, almost laughably trivial, structures you end up with an arrow theoretic (or picture theoretic) description of simple quantum mechanics (0-quantization?). Turning the crank once again gives an elegant/simple desctiption of quantum field theory (1-quantization?). Iterating the process once more gives second quantization.

I’m sure I’ve got the details muddled, but it is probably not completely wrong.

Beautiful!

Then, moving along a different axis of the cube you end up with quantum strings, quantum field of strings, and second quantized string field theory.

Sounds like what you were after from the beginning to me :)

So when is the book coming out?

By the way, when the history books are written can you give my grandkids (who haven’t been born yet) a gift and mention my name ;)

Eric

Posted by: Eric on June 10, 2007 7:49 PM | Permalink | Reply to this

Rosetta Stone: arrow theory of quantum mechanics

What you need is a Rosetta stone for arrow theory.

At least I am aware of the need of such a stone (in fact this metaphor was suggested to me by Bruce Bartlett), hence there is a section 1.2 in my talk notes On 2d QFT: From Arrows to Disks which is titled

A rosetta stone: arrow theory of quantum mechanics

If you could look at that and tell me what you find understandable and which parts still look weird, I could try to improve on that exposition.

On the other hand, I am certainly already aware myself of a couple of things that should be improved on eventually, since a couple of further developments occurred to me since the writing of that document. One of these is the realization that the translation operators which I discuss in the “rosetta stone” section actually nicely fit snugly into one unifying structure which also contains the gauge transformations and the position operators. That is described in QFT of Charged n-Particle: Algebra of Observables.

Posted by: urs on June 10, 2007 8:10 PM | Permalink | Reply to this

Phases

The Tale of Groupoidification teaches us that, for GG a group,

instead of the category

Rep Vect(G):=Hom(ΣG,Vect) \mathrm{Rep}_{\mathrm{Vect}}(G) := \mathrm{Hom}(\Sigma G, \mathrm{Vect})

we can just as well consider the bicategory

Rep Set(G):=Spans ΣG(Grpd) \mathrm{Rep}_{\mathrm{Set}}(G) := \mathrm{Spans}_{\Sigma G}(\mathrm{Grpd})

whose objects are groupoids fibered over ΣG\Sigma G and whose morphisms are spans of these.

In quantum theory, we are dealing with complex vector spaces, hence with U(1)U(1)-representations. So the above tells us that instead of complex vector spaces we may consider groupoids fibered over ΣU(1)\Sigma U(1).

We want to obtain the non-finite U(1)U(1) here from something “more fundamental”. Probably from \mathbb{Z}.

Let Σ 2\Sigma^2 \mathbb{Z} be the 1-object 1-morphism 2-groupoid which has \mathbb{Z} as its 2-morphisms.

Notice that by sending the single Hom-groupoid of Σ 2\Sigma^2 \mathbb{Z} to simplicial spaces by taking its nerve, and then sending it further to topological spaces by taking its realization, this Hom-groupoid becomes U(1)U(1).

So Σ 2\Sigma^2 \mathbb{Z} is something like a categorical refinement (I am not sure I should say “categorification” here) of ΣU(1)\Sigma U(1).

Therefore it seems that one way to think about groupoids fibered over ΣU(1)\Sigma U(1) is to think about 2-groupoids fibered over Σ 2\Sigma^2 \mathbb{Z}.

Therefore

Speculation (very roughly): The complex vector spaces in quantum physics are really to be thought of as 2-groupoids fibered over Σ 2\Sigma^2 \mathbb{Z}.

Evidence:

The nn-particle whose wave function naturally takes values in 2-groupoids is the 2-particle. If the above conjecture is really right, it’s the 2-particle for which “uncategorified” quantum mechanics applies.

We know a couple of things of the quantum theory of the 2-particle. So we can check. As a first rough check, notice that 2-groupoids over Σ 2\Sigma^2 \mathbb{Z} should be equivalent to pseudo-2-functors Σ 22Cat\Sigma^2 \mathbb{Z} \to 2\mathrm{Cat}.

But notice: one such 2-functor is an object in a braided monoidal category.

That sounds familiar…

Posted by: urs on June 15, 2007 6:03 PM | Permalink | Reply to this

Re: Phases

Another way to see what I just said (or at least another aspect of what should be the same phenomenon) is to remember that U(1)U(1)-bundles are the same as Σ\Sigma \mathbb{Z}-2-bundles.

(We once talked about that here).

U(1)U(1)-1-torsors play the same roles as Σ\Sigma \mathbb{Z}-2-torsors.

Here I am saying: U(1)U(1)-representations should play the same roles as Σ\Sigma \mathbb{Z}-2-representations.

Posted by: urs on June 15, 2007 10:11 PM | Permalink | Reply to this

Re: Phases

Is this a tautology?
Notice that bundles with fibre U(1) are NOT the same as principal U(1)-bundles.
Also Groupmaps H –> G
are NOT the same as maps BH –> BG
even up to the usual equivalences

Posted by: jim stasheff on June 16, 2007 1:57 PM | Permalink | Reply to this

Re: Phases

Is this a tautology?

I wouldn’t call it a tautology. But some people do seem to regard it as one.

For mortals, it’s supposed to be a theorem that, for G (2)G_{(2)} a topological 2-group and |G (2)||G_{(2)}| its nerve, as a category – which then is automatically a topological group – the following to things have the same classification

- principal G (2)G_{(2)}-2-bundles

- principal |G (2)||G_{(2)}|-1-bundles .

Notice that bundles with fibre U(1) are NOT the same as principal U(1)-bundles.

Yes, right. I am really talking about principal bundles here.

Posted by: urs on June 17, 2007 6:33 PM | Permalink | Reply to this

Re: Phases

Jim wrote:

Also Groupmaps HGH \to G are NOT the same as maps BHBGBH \to BG even up to the usual equivalences

If HGH \to G is a crossed module I think the situation is nice - Loday talks about precisely such a thing in his “Spaces with finitely many homotopy groups” JPAA paper. Unfortunately I can’t recall the details, and it’s quite cold here today - and I don’t want to go to the library ;-)

From first principles, I would imagine the `missing’ maps of groups would be the generalised morphisms (spans) - at the level of simplicial objects:

BH X BG ,BH_\bullet \stackrel{\sim}{\leftarrow} X_\bullet \to BG_\bullet,

the left map being a weak homotopy equivalence.

Posted by: David Roberts on June 19, 2007 3:05 AM | Permalink | Reply to this

Re: Phases

Let me get this straight. Are you reconstructing QM without the voodoo? :)

Posted by: Eric on June 15, 2007 11:18 PM | Permalink | Reply to this

Re: Phases

Let me get this straight. Are you reconstructing QM without the voodoo?

Who? Me? The Wizard is! I am just being inspired and can’t keep my mouth shut.

Posted by: urs on June 17, 2007 6:50 PM | Permalink | Reply to this

Re: Phases

Here is yet another aspect of what I was saying about getting U(1)U(1)-phases from \mathbb{Z}.

The following is essentially the triviality U(1)/ U(1) \simeq \mathbb{R}/\mathbb{Z} but interpreted in “the right way”, if you wish.

Namely, a fancy way to restate this triviality is to say that there is an equivalence of strict 2-groups

U(1)(), U(1) \simeq (\mathbb{Z} \to \mathbb{R}) \,, where on the left the group U(1)U(1) is regraded as a 2-group in the obvious way and where on the right we have the 2-group coming from the obvious crossed module of groups, coming from the inclusion \mathbb{Z} \hookrightarrow \mathbb{R}.

So, this is a triviality. But luckily we are not afraid of the trivial here.

A slightly less trivial sounding statement I can make now is the following:

First, as a roughly correct slogan:

Principal ()(\mathbb{Z} \to \mathbb{R})-2-bundles with connection are the same as principal U(1)U(1)-bundles with connection.

Here is the more precise statement:

let P 2 t(X)P^t_2(X) be a version of the 2-path 2-groupoid of XX defined as follows:

- objects are the points of XX

- morphisms are parameterized paths in XX cobounding two points

- 2-morphisms are thin homotopopy classes of thin maps [0,1] 2X[0,1]^2 \to X cobounding two paths.

Then:

locally trivializable smooth 2-functors P 2 t(X)()Tor \P_2^t(X) \to (\mathbb{Z} \to \mathbb{R})\mathrm{Tor} are equivalent to locally trivializable smooth 1-functors P 1(X)U(1)Tor, \P_1(X) \to U(1)\mathrm{Tor} \,, which in turn we know are equivalent to smooth principal U(1)U(1)-bundles with connection.

Posted by: urs on June 19, 2007 11:34 AM | Permalink | Reply to this

Re: Phases

I think I can further improve on what I just said:

For XX some manfold, let now P 2(X)P_2(X) be the ordinary (weak) 2-path 2-groupoid:

- objects are points of XX

- morphisms are parameterized paths

- 2-morphisms are thin homotopy classes of smooth maps [0,1] 2X[0,1]^2 \to X cobounding such paths.

Denote, as usual, by INN(G (2))=INN() \mathrm{INN}(G_{(2)}) = \mathrm{INN}(\mathbb{Z} \to \mathbb{R}) the Lie 3-group of inner automorphisms of the 2-group coming from the crossed module ()(\mathbb{Z} \hookrightarrow \mathbb{R}).

Notice that in the present case, this happens to be strict Lie 3-group.

Proposition. Smooth locally trivializable pseudo 2-functors curv:P 2(X)INN()Tor \mathrm{curv} : P_2(X) \to \mathrm{INN}(\mathbb{Z} \to \mathbb{R}) \mathrm{Tor} are the same as smooth principal U(1)U(1)-bundles with connection.

Proof. Exactly same reasoning as before, only that by passing to INN()\mathrm{INN}(\cdots) a certain constraint vanishes such that we may extend the 2-functor even to non-thin surfaces. \Box

As the notation suggests, this 2-functor is actually the curvature 2-transport of the corresponding 1-transport, following the general logic of nn-curvature.

If you like this can be regarded as

“modelling a phase bundle (with connection) without mentioning phases”.

Originally the hope was that we can also do away with the 1-dimensional continuum \mathbb{R} and use just \mathbb{Z}, nothing else.

This certainly works for principal U(1)U(1)-bundles without connection. These are “the same” as (1)(\mathbb{Z} \to 1)-2-bundles.

For this to generalize to principal U(1)U(1)-bundles with connection I had to throw a copy of \mathbb{R} into the game, as above, such as to make room for the continuous smooth nature of the parallel transport.

But I could imagine that by working suitably in a world of homotopy functors or the like, we might even get U(1)U(1)-bundles with connection from just homotopy 2-functors locally with values in the 2-group (1)(\mathbb{Z} \to 1). But that’s just a guess.

Posted by: urs on June 19, 2007 5:27 PM | Permalink | Reply to this
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