Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

April 23, 2007

Who’s on the Right Track?

Posted by David Corfield

Our 300 th300^{th} post at the Café.

In this interview, Alain Connes mentions work he has carried out with Matilde Marcolli on a book which treats physics for three hundred pages, and number theory for the second three hundred. Regarding an analogy they have pursued, concerning spontaneous symmetry breaking in the two fields, he writes

We know that the universe has cooled down, well, it suggests that when the universe was hotter than, say, at the Planck’s temperature, there was no geometry at all, and that only after the phase transition was there a spontaneous symmetry breaking which selected a particular geometry and therefore the particular universe in which we are. (p. 8)

This also suggests that

…the people who are trying to develop quantum gravity in a fixed space are on the wrong track.

If this latter claim were true, which quantum gravity theorists would not be ruled out?

Posted at April 23, 2007 1:08 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1250

39 Comments & 0 Trackbacks

Re: Who’s on the Right Track?

I think there are really two issues here:

1) can we handle “nongeometric” spacetimes (i.e. spacetimes which are not manifolds, but something more general, like a noncommutative space)?

2) can we handle the quantum theory of all these nonperturbatively, i.e. without trying to expand the theory around any given point in the space of all these spacetimes?

It seems that when Connes and Marcolli write

[…] there was no geometry at all, and that only after the phase transition was there a spontaneous symmetry breaking […]

they are referring to point 1), while when they say

[…] the people who are trying to develop quantum gravity in a fixed space are on the wrong track.

they are referring to point 2).

There is a proposal in which for every 2-dimensional SCFT of central charge 15 we get a point in a space of possibly noncommutative/generalized geometries, and a prescription for how to make an expansion from there into the rest of the space of geometries.

But it is true that this proposal has been mainly studied at points where the given geometry is commutative.

Posted by: urs on April 23, 2007 2:52 PM | Permalink | Reply to this

Re: Who’s on the Right Track?

the people who are trying to develop quantum gravity in a fixed space are on the wrong track.

I should look up the context in which this appears.

The way it appears here I find it kind of odd, given that Connes spend so much energy on trying to identify the “fixed” (noncommutative) geometry which is associated to the standard model.

Of course I know the buzzword governing everybody’s emotions lurking behind this sentence, namely “background independence”. As I have probably said before, my impression is that in most of the cases that “background independence” is demanded, more (or less, as in the above quote) explicitly, what is really meant is instead “nonperturbative formulation” (my point 2)).

Posted by: urs on April 23, 2007 3:24 PM | Permalink | Reply to this

Re: Who’s on the Right Track?

I would very much expect that Connes is doing something beyond merely raising the ‘background independence’ question. He says before the quotations I give:

In fact there is an analogy, a conversion table, between the formalism of spontaneous symmetry breaking which is used for arithmetic systems, zeta functions, dual systems etc. and a formalism which seems extremely tempting to people who are trying to quantize gravity. While establishing this dictionary, we found out in the literature that the notion of KMS state, which plays a fundamental role in our work on symmetry breaking for arithmetic systems, also plays a role in the electroweak symmetry breaking which gives masses to particles in the standard model.

Posted by: David Corfield on April 23, 2007 4:11 PM | Permalink | Reply to this

Re: Who’s on the Right Track?

I would very much expect that Connes is doing something beyond merely raising the ‘background independence’ question. He says […]

Thanks! Sounds very interesting.

Of course I cannot know what Alain Connes thinks, but personally I find the following remarkable:

A spectral triple is to a good degree nothing but an axiomatization of the idea of the effective background geometry as seen by a quantum superparticle: the entire geometry is reconstructed from, essentially, just the spectrum of the supercharge of that particle.

If the particle has N=1N=1-supersymmetry and is in fact nothing but an ordinary spinning point particle on some Riemannian manifold, then this supercharge is nothing but the Dirac operator associated to a spinor bundle on that manifold, and we recover ordinary spin geometry.

(So I am talking about worldline susy, not target space susy! Those who can should think NSR here as opposed to GS.)

This perspective of spectral triples as susyQM systems is very manifest, but it has been particularly amplified in a series of old papers by Jürg Fröhlich.

But if this is so, then it would suggest (stronly so, I think, but I realize that this is a matter of taste as long as it has not been made more precise) that the spectral action – a functional on the space of all these supercharges – is nothing but the effective target space action of a field theory whose perturbative expansion comes from summing such superparticle amplitudes over graphs.

In fact, in the old days people like Chamseddine, Fröhlich and others have thought about lifting spectral triples from points to strings, which would make this analogy even stronger.

As soon as I am done finishing onethousandthreehundredfiftyfour other things, I’ll work on that… :-)

Posted by: urs on April 24, 2007 2:13 PM | Permalink | Reply to this

Hot times; Re: Who’s on the Right Track?

The Planck temperature, from the eponymous Max Planck, is the unit of temperature, denoted by T_P, in the system of natural units known as Planck units, which any intelligent beings would use, free of arbitrary units (i.e. the kilogram defined by a metal cylinder in Paris).

It is one of the Planck units that represent a fundamental limit of quantum mechanics. The Planck temperature is the fundamental upper limit of temperature; modern science considers it nonsensical to conjecture about anything hotter. It is also, almost inevitably in most theories, the temperature of the Universe during the first unit of Planck time of the Big Bang according to current cosmology.

T_P = {m_P c^2}/k =
sqrt{hbar c^5}/{G k^2}}
= 1.41679(+-.0000011) x 10^32 K

where:

m_P is the Planck mass

c is the speed of light in a vacuum

hbar is the reduced Planck constant (or Dirac’s constant)

k is the Boltzmann constant

G is the gravitational constant

Now, even fairly conservative scientists suggest that the geometry of space-time could have been quite different at this first clock-tick after Big Bang.

LISA will search for the energy released when, hypothetically, additional dimensions curled up into neat Calabi-Yau spaces, breaking whatever symmetry was present beforehand.

Planck temperature is also a kind of threshold regarding quantum foam, hot enough for wormholes to emerge everywhere, and ruining the pseudoreimannian manifold, i.e. by topologically eliminating a well-behaved sense of neighborhood of a point, or even a proper metric for distance at small (plank-length) scales.

The next step, as urs points out, is to generalize to some noncommutative space. I’m also puzzled that he seems (previous thread) to suspect that the Standard Model plus plausible neutrino generations and mass (i.e. oscillation) pops out when one adds a pinch of quaternions and a dash of 3x3 matrices to the good old complex numbers.

First, not having read the 300 pages of cosmology and 300 pages of pure math, I have no idea how this goes beyond well-known use of quaternions to get rotaions of spheres in 3-space, and Pauli matreices for spin in QM.

If they’d needed to go to, say, 5x5 or 11x11 matrices I’d be more surprised.

No question about the leadership of Alain Conne’s noncommutative geometry program. Simply that nothing has trickled down to folks like me, who’ve published on Mathematical Physics in minor venues, and attended the occasional lecture on noncommutative geometry (in my case, at Caltech seminars by visitors in that area).

I’d be interested in Conne’s said that the Baez-Egan work is on the right track.

Congratulations on 300. That means that these threads can now fight off an army of a million Persian theorists, right?

And Happy Shakespeare’s Birthday. And death day. As he put it, in Lear, about the Big Bang: “nothing shall come from nothing.”

Posted by: Jonathan Vos Post on April 23, 2007 3:54 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

Jonathan wrote:

I’d be interested in Conne’s said that the Baez-Egan work is on the right track.

Four comments on this sentence. First, it’s not grammatical. Second, the dude’s name is “Connes”, not “Conne”. Third, everything I did with Greg Egan was also done with Dan Christensen. Fourth, I’d be interested too if Connes thought our work was on the right track, but he’s never hinted that he thinks this — even though spin foam models are not “trying to develop quantum gravity in a fixed space”.

Connes has his own approach to physics, and I’m pretty sure he thinks that is on the right track.

Posted by: John Baez on April 24, 2007 6:05 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

Connes has his own approach to physics […]

which need not be incompatible with other approaches.

In particular, since his work on the “spectral action” is nothing but the observation: “hey, look, the entire Lagrangian of the standard model coupled to gravity may be rewritten as a very natural functional on the space of generalized Dirac operators”.

Rewriting a classical action may suggest a new approach to physics (and this one certainly does, I think, even though it is not entirely clear which one ;-), but by itself is not new physics yet.

In particular, whatever Connes’ reformulation of our world’s configuration space as a space of spectral triples hints at, it will ultimately need a nonperturbative description, and since it involves gravity, it will need a nonperturbative description of quantum gravity – which may well be a spin foam description. Or it may not. But right now it could just as well be.

In particular, I find the following noteworthy: a Dirac operator is, as Quillen once remarked, nothing but “a quantized connection”. (Namely a connection postcomposed with the symbol map which sends differential form (graded commutative things) to Clifford algebra elements (non-graded commutative)). So if we think that a good way to think of a space of connections is as a space of holonomy maps, or parallel transport maps – as we do! – then it is quite natural to speculate that it makes good sense to conceive the “space of all Dirac operators” (I am being deliberately vague here, hope that’s clear) in terms of a space of maps from loops to something.

(It’s pretty clear how this would work for Dirac operators on ordinary spin bundles. It will require a little more thinking generalizing this away from Riemannian geometry to spectral geometry.)

So, I think, Connes’ approach in fact nicely vindicates the premises of those loopy approaches to quantum gravity: namely that it should be helpful to conceive the space of Riemannian structures as a space of (the associated Levi-Civita-)connections.

Connes’ “spectral action principle” in effect shows that the graviton, the gauge bosons and the Higgs boson (!) nicely unify as different components of one single connection.

At the level of the action functional that is. No more. So this makes us want to apply our best tools for quantizing gauge theories at this action functional!

Wilson networks, aka spin networks, are a great tool for doing just that. So there we go.

But, to my mind, it gets even better. To me, the fact that Connes’ rewriting of the action functional of the standard model as a functional on the space of Dirac operators evaluated at a special point where the spectral geometry of that operator defines a noncommutative space with spectral dimension 4+0 and K-dimension 4+6 is one more hint for the idea that those Dirac operators on spacetime really have to be conceived as point-particle limits of a 2-Dirac operator.

Just pretend for a moment that you accept that idea. What’s the configuration space of 2-Dirac operators like? Well, it should be like a space of 2-connections.

If we use Wilson line networks to quantize connections, we’ll use Wilson surface networks, aka spin foams, to quantize configuration spaces of 2-connections.

In fact, I think the theory of 2-Dirac operators and the parallel surface transport associated with the corresponding 2-connections is no longer out of reach.

For a long time I was puzzled by the right categorification of Dirac operators should be. Part of the question is:

“What is a Clifford algebra, really?”

Only if one knows that, it seems, does one have a chance to categorify and obtain 2-spinor bundles and the like.

Maybe last weekend I had an idea for how to proceed here, maybe even a good idea. I am waiting for that idea to crystallize a little more, then I might entertain you all (read: bore to death :-) with a blog entry on that.

Posted by: urs on April 24, 2007 7:09 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

urs wrote:

Connes’ “spectral action principle” in effect shows that the graviton, the gauge bosons and the Higgs boson (!) nicely unify as different components of one single connection.

Does this play a significant role in the prediction of the Higgs mass that these models are said to make? It’s well known that if one assumes the Standard Model is valid up to very high scales, the Higgs mass must lie in a fairly small window. It has never been clear to me if the claimed Higgs mass predictions of the noncommutative geometry approach are just a version of this well-known fact, or if they contain deeper information. (I haven’t spent much time trying to figure this out, so maybe it’s clearly explained somewhere.)

Posted by: Matt Reece on April 24, 2007 8:11 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

Does this play a significant role in the prediction of the Higgs mass that these models are said to make?

I don’t know! Hopefully somebody reading this does and gives us a hint..

Posted by: urs on April 24, 2007 8:33 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

Looking into the papers a bit more, I see that they do predict a relationship between the Higgs quartic coupling and the gauge couplings at the GUT scale. So they aren’t just using the known bounds to locate the Higgs mass. I take it that their claim is that only a subset of the SM parameter space is noncommutative-geometry-friendly. My earlier impression had been that they were mostly just reformulating the SM in a mathematically nicer language, but apparently there’s a bit more to it than this.

Of course, the electroweak/GUT hierarchy is put in by hand in these models, so one must simply assume the correct W mass to learn about the top and Higgs masses. (As far as I can tell, the place where the EW/GUT tuning is located is in the moments of the arbitrary function f(D/Λ)f(D/\Lambda).)

Posted by: Matt Reece on April 24, 2007 11:43 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

Looking into the papers a bit more, I see that they do predict a relationship between the Higgs quartic coupling and the gauge couplings at the GUT scale.

The Higgs self-coupling at what scale (since it runs)?

Also, what do we mean by “the GUT scale”? With the current value of α 3\alpha_3, the couplings don’t unify, unless there are extra degrees of freedom, charged under SU(3)×SU(2)×U(1)SU(3)\times SU(2)\times U(1) (beyond those of the Standard Model).

Posted by: Jacques Distler on April 25, 2007 2:48 AM | Permalink | PGP Sig | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

Their logic appears to be that they find a relationship among the gauge couplings, α 2=α 3=(5/3)α 1\alpha_2 = \alpha_3 = (5/3) \alpha_1, as one would expect in a GUT. So they interpret their Lagrangian as being a GUT-scale Lagrangian. So the Higgs quartic is predicted at the GUT scale. Of course, we know the SM does not unify, and it appears Connes knows this as well, so there are some comments in the paper to the effect that there is probably other physics at a lower scale that modifies the running. Given that, it doesn’t appear that their model is meant to be a serious phenomenological proposal, though whether they treat it as one or not seems to vary from page to page. Anyhow, I’m just trying to get some handle on what sort of relations among couplings they claim to be able to obtain in their approach, and why….

Posted by: Matt Reece on April 25, 2007 4:19 AM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

… so there are some comments in the paper to the effect that there is probably other physics at a lower scale that modifies the running.

Physics that’s not present in the GUT-scale Lagrangian?

Given that, it doesn’t appear that their model is meant to be a serious phenomenological proposal,

No?

The question that arises, then, is: what would it take to make it one?

Anyhow, I’m just trying to get some handle on what sort of relations among couplings they claim to be able to obtain in their approach, and why….

A worthwhile effort, whether or not what we’re discussing constitutes a ‘serious phenomenological proposal.’

Posted by: Jacques Distler on April 25, 2007 4:44 AM | Permalink | PGP Sig | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

I take it that their claim is that only a subset of the SM parameter space is noncommutative-geometry-friendly. #

For what it’s worth, that’s my impression, too.

…so there are some comments in the paper to the effect that there is probably other physics at a lower scale that modifies the running.

Physics that’s not present in the GUT-scale Lagrangian? #

One thing they explicitly say, related to how their model would have to be completed by something else, is that the assumption that the noncompact part of their NCG space is just an 4\mathbb{R}^4 seems unjustified, given the NCG context of everything. My impression was that the idea was that a noncommutative geometry of spacetime would kick in beyond some energy scale.

Well, that’s a trivial comment on an NCG model, I guess. :-) But anyway one should keep in mind that the current model assumes the non-compact space to be commutative at all scales.

In general, I think the following is important: you may regard all the spectral triples and noncommutative geometry here as a way to concisely package a huge Lagrangian. But once you have your Lagrangian, and are just interested in particle physics, you may just as well forget that it may be encoded in some spectral triple.

In particular, since this Lagrangian contains Einstein-Hilbert gravity, you will then have to deal with quantizing your gravitational theory. The mere fact that you have written it down in a more concise form does not change that.

So then, if we think that by itself this quantization is not possible, then it follows that the spectral action needs to be “completed” by something…

Posted by: urs on April 25, 2007 12:42 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

I should probably clarify that when I say “it doesn’t appear that their model is meant to be a serious phenomenological proposal”, I don’t mean to sound disparaging. It’s just that given that the model from the outset is the SM with gauge coupling unification, which doesn’t seem to describe the real world, it’s not clear to me why they exert so much effort working out precise numbers for the top and Higgs masses.

Physics that’s not present in the GUT-scale Lagrangian?

The question that arises, then, is: what would it take to make it one?

Maybe I should just quote what Chamseddine, Connes, and Marcolli say in hep-th/0610241:

The fact that the experimental values show that the coupling constants do not exactly meet at unification scale is an indication of the presence of new physics. A good test for the validity of the above approach will be whether a fine tuning of the finite geometry can incorporate additional experimental data at higher energies.

I can’t speculate about what they have in mind with “fine tuning of the finite geometry,” though whatever modification one makes to the algebra would surely modify the GUT-scale Lagrangian. Of course the easiest way to fix the running would be to add some new matter with electroweak charges but no SU(3) charges; it’s not clear to me whether their framework can accommodate this.

Posted by: Matt Reece on April 25, 2007 2:49 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

I can’t speculate about what they have in mind with “fine tuning of the finite geometry,” though whatever modification one makes to the algebra […]

Just for the record, let me emphasize that there is much more to specify than just the algebra (you are probably well aware of this, but I’ll mention it nevertheless).

To define the spectral triple, we need the algebra itself, and then a representation of that algebra on a Hilbert space, as well as a choice of grading and of real structure of that Hilbert space.

There is a considerabe amount of data encoded in the choice of representation. (I once tried to summarize some of these aspects here).

In fact, only in his latest papers has Connes suggested a way to see how the intricate definition of the representation necessary for the spectral action to spit out the Standard Model can be understood more concisely as coming from a direct sum over all inequivalent irreducible odd bimodules of the internal algebra.

(Which thus looks like, by the way, again as yet another step in condensing the information contained in the standard model Lagrangian, which we would hardly be willing to recognize as a coincidence, I guess. But I don’t know.)

Similar comments apply to the details of the choice of the grading and the real structure. Lots of standard model data enters there, and things looked really messy for a long while, until Connes explained now recently that all this can be understood simply as defining an isomorphism of the above-mentioned bimodule with its contragradient one.

And then of course one has to specify the Dirac operator itself.

Posted by: urs on April 25, 2007 8:28 PM | Permalink | Reply to this

Re: Hot times; Re: Who’s on the Right Track?

There are lots of ways to make various features of the SM pop out of some simpler formulation. For example, embedding SU(3)×SU(2)×U(1)SU(3)\times SU(2)\times U(1) in a GUT gauge group.

By itself, that doesn’t particularly impress me.

What happens however, whenever you come up with such an explanation, is that it implies relations among previously a priori unrelated couplings.

Connes et al find relations between the Yukawa couplings (and the Higgs self-coupling) and the GUT gauge coupling, which seem to be new.

That’s interesting.

The thing that seems puzzling, at first blush, is that they find constraints on the Yukawa couplings, while still having the CKM matrix being completely arbitrary.

Do you have any intuition as to why there is no constraint on the form of the CKM matrix?

Posted by: Jacques Distler on April 25, 2007 10:00 PM | Permalink | PGP Sig | Reply to this

A Grand Unified Theory of Theores of Everything

Urs wrote:

I am waiting for that idea to crystallize a little more, then I might entertain you all (read: bore to death :-) with a blog entry on that.

Go ahead! Bore me to death!

So now you’re planning to unify string theory, loop quantum gravity, Connes’ noncommutative geometry approach to physics and higher gauge theory in one nice package! Excellent! If I didn’t know you well, I’d suspect it was a trick to get as many disciples as possible.

One small comment: before I forced Andre Henriques to speak about his paper on integrating L L_\infty-algebras at the Fields Institute conference on nn-categories, and before he got a high fever and escaped this task, he had wanted to speak on categorified Clifford algebras and their relation to elliptic cohomology.

So, maybe this fits into your grand scheme!

Here’s what Henriques told me… I don’t think this is top secret, so I’ll quote him:

So my title could have been “A 3-category of conformal nets”, or maybe also “Higher Clifford algebras”. Here’s what it’s about:

Let vN2 be the symmetric monoidal 2-category of von Neuman algebras and bimodules, composition being given by Connes fusion, and the monoidal structure given by tensor product. Stolz and Teichner conjectured the existence of an interesting symmetric monoidal 3-category C with the property that Hom C(1,1)Hom_C(1,1)=vN2.

The invertible objects in that 3-category should then play the same role for Elliptic cohomology that the finite dimensional Clifford algebras play in K-theory. Namely, the group of invertible objects should be a small abelian group that is somehow related to the periodicity of Elliptic cohomology.

Conformal nets are the objets of our candidate symmetric monoidal 3-category. And there’s also a candidate for the generator of the group of invertible objects: the net of local fermions. We even have an argument (provided mostly by Peter Teichner) that shows that the net of local fermions is of order at least 24!

This is joint work with Chris Douglas and Arthur Bartels. Chris is not that far from you (he’s at Stanford), so I guess you can always ask him to tell you about it.

Posted by: John Baez on April 24, 2007 9:00 PM | Permalink | Reply to this

Re: A Grand Unified Theory of Theores of Everything

This is joint work with Chris Douglas and Arthur Bartels.

Hah! In Oberwolfach Peter Teichner mentioned this to me, and the remainder of the day I got on Arthur Bartels’ nerves making him explain it to me.

I did think it was top secret, so I didn’t mention it.

What is about to drive me nuts, though, is that this is so closely related to something which is top secret that it is hard to bear.

The invertible objects in that 3-category should then play the same role for Elliptic cohomology that the finite dimensional Clifford algebras play in K-theory. Namely, the group of invertible objects should be a small abelian group that is somehow related to the periodicity of Elliptic cohomology.

Exciting times.

If you are looking for entertainment, try this:

have a brief look at what I just wrote about supersymmetric quantum mechanics. Notice the 2-group INNU(H)\mathrm{INN}U(H) appearing.

Ponder the “canonical 2-representation” of that. Then imagine categorifying from superpoints to superstrings. Then compare with the above paragraph.

Then become enlightened…

;-)

(Not that I have become enlightned about this entirely yet. But I recognize a Koan when I see one…)

Posted by: urs on April 24, 2007 9:16 PM | Permalink | Reply to this

Re: A Grand Unified Theory of Theores of Everything

Hm, actually maybe I am making a mistake here. This may sound unrelated, but: what can be said about the relations between Clifford algebra and the group algebra of Spin\mathrm{Spin}?

Posted by: urs on April 24, 2007 9:20 PM | Permalink | Reply to this

Head for Beer and Spin-Foam; Re: Hot times; Re: Who’s on the Right Track?

Dear John Baez:

First, I’m sorry for the non-grammatical posting, embarassing for me as a professional writer. Second, my error with location of apostrophe, which thus mangled the possessive of “Connes.” Third, I apologize for inadvertently omitting your coauthor Dan Christiansen in the citation. Fourth, is there any spin-foam connection to the von Nemann approach and recent improved formula for foams in Nature and Sci.Am.:

April 25, 2007
Mathematicians Point the Way to a Perfect Head of Beer
Simple formula may lead to a host of improved materials and the perfect pour
By JR Minkel
[URL omitted as causes errors in this form entrry]

A HEAD WITH NUMBERS: A long-sought mathematical formula may allow researchers to predict the evolution of beer foam and other materials made from tiny domains….

According to the new equation, the change in volume of such a tension-driven domain is essentially the sum of the lengths of the domain’s edges (imagine a honeycomb) minus six times the average width of the domain, all multiplied by a constant that is particular to the material in question.

Robert MacPherson of the Institute for Advanced Studies in Princeton, N.J., and David Srolovitz of Yeshiva University in New York, N.Y., published their findings online today in Nature….

Posted by: Jonathan Vos Post on April 25, 2007 8:40 PM | Permalink | Reply to this

Time and clocks

Before trying to combine GR and QM, one should first understand the concepts of time in both theories:

1. In GR, time is what clocks show.

2. In QM, time is a parameter rather than an observable. It was apparently shown by Pauli in the 1920s that a self-adjoint time operator leads to an unbounded Hamiltonian. (Thanks to Wolfgang for pointing this out.)

There is obviously a conflict here, since what is observed in a physical clock-reading experiment must be an observable. It seems to me that any approach to QG which does not explain this apparent paradox is still-born.

Posted by: Thomas Larsson on April 23, 2007 4:50 PM | Permalink | Reply to this

Re: Time and clocks

I have been troubled since learning from Feynman, at Caltech circa 1969, that, in Thomas Larsson’s words: “In QM, time is a parameter rather than an observable.” I passed the courses, and published in the field, but never “got it” at the deepest level, and stayed deeply puzzled by the different and apparently contradictory models of time in GR and QM

Time is not “in” the content of QM, but, rather, the evolution of Unitary matrix-represented systems is parameterized by a time “outside” the content of the QM system. This applies even in the simplest examples where, for instance, we have an e^-t decay as formally associated with an imaginary component of mass.

Differing interpretations of QM often involved different models of time, i.e. Coperhagen versis Everett many worlds versus Tegmark multiverse as Mathematical structure versus John Cramer’s interactionist model, etcetera. These models have different topologies – i.e. a tree-structure in many-worlds is profoundly different from a 3+1 space-time.

If one grapples with a “no gemoetry” Planck-time Planck-temeperature cosmos, and hope to have signature 3+1 geometry as some sort of emergent property of evolution in a noncommutative superspace of possible geometries, then how on Earth can one build in a time parameter as demanded by QM? I mean, if there’s no space-time geometry one Planck-clock-tick after Big Bang, what is the clock that ticks, and what parameterizes the evolution in the superspace?

I’m way out of my depth here, but trying to be in a mixed state of keeping and open mind AND being skeptical of metaphysical paradoxes that brush time under the rug.

Feynman path integration over all possible histories presupposes a space-time geometry in which there ARE histories, and how ncan that be done without time in the gemoetry? I do find the Baez use of Joyal in that regard quite exciting.

What are the “labeled structures” over a superspace of geeometries out of which manifolds emerge?

Posted by: Jonathan Vos Post on April 23, 2007 5:12 PM | Permalink | Reply to this

Re: Time and clocks

one should first understand the concepts of time in both theories

Would you say that the “arrow of time” (and its many incarnations) is a derived concept (namely, from the two pillars), or would you set it as a #3 issue?

One thing that I find often disregarded is the fact that the statistical mechanics of gravitational systems is not a completely understood discipline, although being a classical one. If a quantum theory of gravity is reached, should it not address the peculiarities seen in N-body gravitational physics? Specially considering that quantum gravity systems would be N-body quantum gravity systems.

Christine

Posted by: Christine on April 23, 2007 5:39 PM | Permalink | Reply to this

Re: Time and clocks

> any approach to QG which does not explain this

I think QG could very well be ‘time-less’, as proposed by Julian Barbour.
The Wheeler-DeWitt equation is
H |psi> = 0 and ‘time’ does not appear.
But we know that the Hartle-Hawking prescription formally provides for a |psi> and it is defined in the ‘time-less’ Euclidean sector.

Posted by: wolfgang on April 24, 2007 2:08 AM | Permalink | Reply to this

Barbour, Egan; Re: Time and clocks

“The Edge” [8.16.99] introduces Julian Barbour thus:

In a profile in The Sunday Times (October, 1998), Steve Farrar wrote: “Barbour argues that we live in a universe which has neither past nor future. A strange new world in which we are alive and dead in the same instant. In this eternal present, our sense of the passage of time is nothing more than a giant cosmic illusion. ‘There is nothing modest about my aspirations,’ he said. ‘This could herald a revolution in the way we perceive the world.’” Cosmologist Lee Smolin notes that Barbour has presented “the most interesting and provocative new idea about time to be proposed in many years. If true, it will change the way we see reality. Barbour is one of the few people who is truly both a scientist and a philosopher.”

In “Permutation City” and related fictions, Greg Egan gives us this theory, or one like it, in dazzlingly plotted subjectivity.

Einstein also said “Reality is merely an illusion, albeit a very persistent one.”

Time may indeed be illusory, and thus not necessary to a correct physical theory. Is that possible?

One poem notes:
“Only time will tell /
if my work is representational. /
Only time will tell if time will tell.”

[The Lichtenberg Figures
Ben Lerner
Copper Canyon Press, 2004
Softcover, ISBN: 1556592116,
winner of the 2003 Hayden Carruth Award from Copper Canyon Press]

Posted by: Jonathan Vos Post on April 24, 2007 6:02 AM | Permalink | Reply to this

What is super, really?

I wrote:

Maybe last weekend I had an idea for how to proceed here, maybe even a good idea. I am waiting for that idea to crystallize a little more,

Oops, looks like I am disregarding my own rule of conduct here. After all, the best way to help an idea crystallize is to make a fool of yourself and start talking about it to others. ;-)

So here is the question: “What is a Dirac operator, really? What is supergeometry, really?”

(Here “really” is defined like this: you know you really know what it is if and only if it can be blindly categorified.)

I had vaguely thought about this question every now and then in the past. But recently I was alerted again to the urgency of this question by the following observation:

The superpoint is essentially nothing but the infinitesimal interval TT of synthetic differential geometry.

This is in the sense that, if \nearrow denotes either the superpoint or the infinitesimal interval, then, for any space XX, the morphism space TX:=Hom(,X) T X := \mathrm{Hom}(\nearrow,X) behaves like the tangent bundle to XX.

For =T\nearrow = T the synthetic infinitesimal interval, this is true essentially by construction. For = 0|1\nearrow = \mathbb{R}^{0|1} the N=1N=1 superpoint, this is a well known fact, appearing for instance as Lemma 3.2 on p. 29 of the thesis

Florin Dumitrescu, Superconnections and parallel transport.

The only slight lie about this statement is that Hom( 0|1,X)\mathrm{Hom}(\mathbb{R}^{0|1},X) is really the odd tangent bundle. That is an important issue, which however I will no further get into here.

What I find exciting about this is that it suggests a way to naturally approach supergeometry in categorical terms. That’s because I think I know a nice way to approach synthetic differential geometry in categorical terms.

Here is how I conceive the “infinitesimal interval” this way: I think of it as the terminal 1-category, that with a single morphism. Accordingly, I shall argue that this is the N=1N=1-superpoint for me.

More generally, I would then regard the (N=n)(N=n)-superpoint as the terminal nn-category: that with a single nn-morphism.

I’ll try to convince you in a moment that this actually might make good sense. But before continuing, just notice that for (N=0)(N=0) this just says that the non-super point is just the 1-element set.

Which it is.

(If Toby is reading this, we could indulge in using this to favour the world with a discussion of (N=1)(N=-1) and (N=2)(N=-2) supersymmetry .)

The idea was this:

think of any space XX in terms of its strict \infty-groupoid of paths P(X). P(X) \,. nn-morphism are thin-homotopy classes of maps [0,1] nX[0,1]^n \to X cobounding two (n1)(n-1)-morphisms.

Then P n(X):=Hom nCat( n,P(X)), P_n(X) := \mathrm{Hom}_{n\mathrm{Cat}} ( \nearrow_n , P(X) ) \,, with slight abuse of notation, where I now write n\nearrow_n for my (N=n)(N=n)-superpoint, as before.

What looks like notational overkill here is turned into a cool fact by using this cool fact: strict smooth pp-functors f:P p(X)Σ p(U(1)) f : P_p(X) \to \Sigma^p(U(1)) are in bijection with differential pp-forms.

But this tells us that in the world of categories, functors, transformations, modifications, etc (i.e. in the best of all worlds ;-) we are entitled to identitfy T nX:=P n(X) T^{\vee n} X := P_n(X) the nnth exterior power of the tangent bundle with the nn-path nn-groupoid of XX.

In fact, we can just as well use the nn-groupoid (U(1)) n(U(1))^n coming from the nn-crossed module U(1)U(1)U(1)U(1) \to U(1) \to \cdots \to U(1) and find the entire exterior bundle up to degree nn i=0 nΩ n(X)Hom nCat(P n(X),(U(1)) n). \oplus_{i=0}^n \Omega^n(X) \simeq \mathrm{Hom}_{n\mathrm{Cat}}( P_n(X),(U(1))^n) \,.

(Only thing I am still puzzled about is how to naturally see the wedge product operation on these functors. Must be something obvious, but I don’t see it yet).

Time is running out. I need to be more brief.

So what’s the superline then? Do functors on superlines reproduce superconnections? In particular, do we get supersymmetric quantum mechanics from functors on super-1-cobordisms?

Here is a brief sketch:

Like our superpoint is really a point with lots of paths emanating from it, with the magic of smooth functors on these paths ensuring that only the tangent vector of these paths at that point matters, so a supercurve is a superpoint in curve space, i.e. a supercurve with a tangent vector at every point.

As you have noticed, superification here is actually categorification. The super 0-point is actually a 1-category.

Same for the supercurve now.

It’s a curve as usual \array{ \downarrow \\ \downarrow } but now we remember tangency classes of surfaces emanating from that (or rather: this is what our smooth 2-functors will see).

So, schematically, our supercuve looks like an infinitesimally thin strip of surface \array{ &\to& \\ \downarrow &\Downarrow& \downarrow \\ &\to& \\ \downarrow &\Downarrow& \downarrow \\ &\to& \\ \downarrow &\Downarrow& \downarrow \\ &\to& \\ \downarrow &\Downarrow& \downarrow } (rather, like a finite such surface, but our smooth 2-functors will only care about the tangents).

Suppose we think of this as the worldline of a superparticle. If it were an ordinary particle we’d want to look at smooth functors from the curve to U(H)U(H), for HH some Hilbert space of states.

Now we need a 2-functor mimicking that. The obvious choice is to use instead the 2-group coming from the crossed module id:U(H)U(H), \mathrm{id} : U(H) \to U(H) \,, i.e. the 2-group INN(U(H))\mathrm{INN}(U(H)). Let’s assume that’s right.

Then what’s a smooth 2-functor applied to the above supercuve?

Now some magic happens, which is making me think that this is on the right track.

We know that such a smooth 2-transport comes from

- a 1-form D˜\tilde D on the strip (which represents our supercurve) with values in u(H)=LieU(H)u(H) = \mathrm{Lie}U(H)

- a 2-form H˜\tilde H on the strip, with values in u(H)u(H).

First thing to notice: it looks like these two differential forms take values in the same Lie algebra. But if we are sufficiently pedantic, then in fact they don’t! Rather, D˜\tilde D takes value in u(H)u(H) regarded as being odd graded, while H˜\tilde H takes values in u(H)u(H) regarded as being even graded.

The categorification of Lie algebras, and its equivalence to L L_\infty-algebras naturally brings a grading into the game here. Without us having put this in by hand, the formalism knows that

D˜\tilde D is odd.

H˜\tilde H is even.

It’s god-given.

Second thing to notice:

the formalism tells us that these two differential forms are not independent.

Since we are thinking quantum mechanics here, and time-independent quantum mechanics for simplity, we assume this 1-form D˜\tilde D and 2-form H˜\tilde H are in fact constant along our curve. Then the compatibility condition (known sometimes as “fake flatness”) demands that H˜=D˜ 2. \tilde H = \tilde D^2 \,.

And that’s god-given. It’s not put in by hand.

Third thing to notice is this:

our 2-functor which we apply to the strip can be regarded as follows: it assigns a 1-form to each curve on the strip, and then assigns to the entire strip the path-ordered (in curve space!) exponential of that 1-form over the strip.

That 1-form turns out to be, in turn, the path-ordered exponential, along the curve, of H:=H˜(t,), H := \tilde H(t,\cdot) \,, where tt is the horizontal tangent on our strip.

But since we are just looking at that one curve, and care about the strip emanating from it only in as far as this tangent vector is concerned, we finally find that our 2-functor with values in INN(U(H))\mathrm{INN}(U(H)) (here HH denoted a Hilbert space, recall) sends the supercurve γ\gamma to e |γ|H e^{ |\gamma| H} where the Hamiltonian HH is an even operator that is required (that’s precisely what distinguishes our 2-functor on the strip with values in INN(U(H))\mathrm{INN}(U(H)) from a mere 1-functor with values in U(H)U(H)!) to be that square of the odd operator DD H=D 2. H = D^2 \,.

And there we go: U(H)U(H)-transport along supercurves is in bijection with supersymmetric quantum mechanics.

Okay, so much for now.

Posted by: urs on April 24, 2007 8:32 PM | Permalink | Reply to this

Re: What is super, really?

What is supergeometry, really?

This question reminds me of the following paragraph from math-ph/0202025

“1.0.1. What a Lie superalgebra is.
Dealing with superalgebras it sometimes becomes useful to know their definition. “

Somewhat more elaborated:

“Lie superalgebras were distinguished in topology in 1930’s or earlier. So when somebody offers a “better than usual” definition of a notion which seemed to have been established about 70 year ago this might look strange, to say the least. Nevertheless, the answer to the question “what is a Lie superalgebra?” is still not a common knowledge. Indeed, the naive definition (“apply the Sign Rule to the definition of the Lie algebra”) is manifestly inadequate for considering the (singular) supervarieties of deformations and applying representation theory to mathematical physics, for example, in the study of the coadjoint representation of the Lie supergroup which can act on a supermanifold but never on a superspace (an object from another category). So, to deform Lie superalgebras and apply group-theoretical methods in “super” setting, we must be able to recover a supermanifold from a superspace, and vice versa.”

Posted by: Thomas Larsson on April 25, 2007 7:23 AM | Permalink | Reply to this

Re: What is super, really?

I wrote:

[…] and find the entire exterior bundle up to degree nn i=0 nΩ n(X)Hom nCat(P n(X),(U(1)) n). \oplus_{i=0}^n \Omega^n(X) \simeq \mathrm{Hom}_{n\mathrm{Cat}}( P_n(X),(U(1))^n) \,.

(Only thing I am still puzzled about is how to naturally see the wedge product operation on these functors. Must be something obvious, but I don’t see it yet).

This is even more puzzling since I can see also the exterior derivative canonically acting on these nn-functors (namely coming from canonically forming their nn-curvature).

So what’s the exterior product, really? Given a smooth functor F:P n(X)T F : P_n(X) \to T and another one G:P m(X)T G : P_m(X) \to T' how do I canonically, arrow theoretically, define FG:P n+m(X)something? F \wedge G : P_{n+m}(X) \to \mathrm{something}?

I mean, I know that, when all TTs here are Σ kU(1)\Sigma^k U(1)s, then FF is equivalent to an nn-form ω F\omega_F, GG is equivalent to an mm form ω G\omega_G and I can simply wedge these and find the (n+m)(n+m)-functor coming from integrating and exponentiating ω Fω G\omega_F \wedge \omega_G. But that’s pedestrian and doesn’t generalize. What’s the general mechanism at work here?

Or is the wedge product, unlike the exterior derivative, not a natural thing, but just a caprice of Nature?)

Posted by: urs on April 26, 2007 5:55 PM | Permalink | Reply to this

Re: What is super, really?

Hi Urs,

Remember me? :)

In my opinion, the wedge product is an unnatural operation that only makes sense in a continuum limit of something more fundamental. That something is probably the cup product or something like it, e.g. the product we looked at :)

Should 0-forms really commute with 1-forms or is that an approximation you recovery after making some gross assumptions?

Just a random thought…

Eric

Posted by: Eric on April 26, 2007 9:31 PM | Permalink | Reply to this

Re: What is super, really?

Plus, I could be totally confused, but it would seem that “concatenation” is a natural product for an arrow-theoretic something. We know that concatenation, as viewed as a product, preserves the graded Leibniz rule and becomes the wedge product in the continuum limit.

Posted by: Eric on April 26, 2007 10:55 PM | Permalink | Reply to this

Re: What is super, really?

it would seem that “concatenation” is a natural product for an arrow-theoretic something

Well, depends. On the cubical graph, it was possible to identify edge-paths of length nn with nn-cubes. This gives the close relation between mere concatenation (which naively wouldn’t seem to change the degree as far as dimension is concerned) and the wedge product.

I think I am looking for a more general way to understand this. But maybe I shouldn’t…

Posted by: urs on April 27, 2007 9:36 AM | Permalink | Reply to this

Re: What is super, really?

Oh no!

Perhaps time and the ages is reducing your ability to deduce what I mean rather than depend on what I say. Too bad :)

“Concatenation” was meant in a broader sense to try to stimulate some better ideas. Clearly concatenation in its typical form is not going to help you.

How about this? In the arrow-theoretic approach (that I still don’t understand), can you express things like this:

(1)αβ,γ=α,βγ \langle\alpha\wedge\beta,\gamma\rangle = \langle\alpha,\beta\vee\gamma\rangle

??

I think it may be natural to think about how one might define \wedge in terms of \vee or vice versa.

Still just thinking out loud…

Eric

Posted by: Eric on April 27, 2007 7:46 PM | Permalink | Reply to this

Re: What is super, really?

Remember me? :)

Sorry! Please don’t give up on me yet! :-)

There seem to be several ways to look at all this. And I am still not sure how they all hang together.

I know what you are referring to: if we pick a certain discretized model of P n(X)P_n(X), where everything is generated from some graph, then the formula for the wedge product of transport nn-functors has a rather natural algebraic formulation.

What I am looking for, here, is a way to see this in a way that does not assume so many details about the nature of P n(X)P_n(X).

Like for the exterior derivative: I think I now know what it means to take the curvature of an nn-functor of about any sort. If that happens to have domain the discrete paths we talked about, then it reproduces the notion of (covariant) exterior derivative that we were discussing.

I am wondering if something similar holds for the wedge product. Is there a general nonsense which would give me a very general operation on nn-functor that I would address as “wedge product”, and such that in special situations it reduces to known definitions of wedge product (like the standard one, or the non-graded commutative one which you love so much), or is there not?

Currently it seems to me that there is not. But possibly I am just being dense.

Posted by: urs on April 27, 2007 9:33 AM | Permalink | Reply to this

Re: What is super, really?

if \nearrow denotes either the superpoint or the infinitesimal interval, then, for any space XX, the morphism space TX:=Hom(,X) T X := \mathrm{Hom}(\nearrow,X) behaves like the tangent bundle to XX.

Then what kind of thing should we expect to measure the symmetry of the tangent bundle? A 2-group?

Posted by: David Corfield on April 30, 2007 8:16 AM | Permalink | Reply to this

Re: What is super, really?

Then what kind of thing should we expect to measure the symmetry of the tangent bundle? A 2-group?

Ah, good question.

First let me emphasize that I am saying that the tangent bundle of XX can be thought of as “being” the path groupoid 𝒫 1(X) \mathcal{P}_1(X) of XX. That this may, by some notational gymnastics, also be regarded as maps from here to there is actually secondary to your question. So I will not focus on that.

Okay, so what are automorphisms Aut Cat C (𝒫 1(X)) \mathrm{Aut}_{\mathrm{Cat}_{C^\infty}}(\mathcal{P}_1(X)) of the path groupoid?

If we require these to be smooth, which we should, then they need to come from diffeomorphisms on the space of objects, i.e. from diffeomorphisms of XX.

But this then already fixes the behaviour on paths: they are simply being pushed forward along the diffeomorphism.

So in terms of the tangent bundle these automorphisms are simply bundle maps induced from diffeomorphisms of the base and pullback of the bundle along that.

Still, we can have nontrivial 2-morphisms between two such diffeomorphisms. If you look at what these are, you find that they correspond to “fields of paths” (“arrow fields” in Chris Isham’s terminology) on XX – one path originating at every point of XX connecting that point with its image under the diffeomorphisms.

It is good to think of these as flow-lines of the flow of a vector field on XX. This way we find that two diffeomorphisms here are connected by a 2-morphisms if they differ by a diffeomorphisms that is connected to the identity.

At the beginning of this entry I recall how I argued that this can be used to extract an explicit arrow-theoretic notion of vector field from this notion of tangent bundle.

Later it occurred to me that this notion of vector field (think: “translation operator”) is just one out of three components of an automorphism of a section of a vector bundle – when the latter is conceived in a similar arrow-theoretic manner, as for instance described here.

The other two components are “position operators” and “gauge transformations”. Together they form the algebra of observables of the object charged under that vector bundle – as described here.

Posted by: urs on April 30, 2007 1:51 PM | Permalink | Reply to this

Re: What is super, really?

So what do your automorphisms (diffeomorphisms) and nontrivial 2-morphisms between them form? Do they make a 2-group?

I’m asking for rather oblique reasons. Reading Greg Egan’s nice concrete example of a group acting on a set, e.g., rotations of a cube, it made me regret that we didn’t have nice concrete things for 2-groups to act on during the Klein 2-geometry days. I was thinking about a ‘bundle’ over the cube’s vertices, perhaps a 2 element set at each vertex.

This got me thinking about other bundles, such as the one you had just mentioned here. Bundles cropped up periodically on the Klein 2-geometry thread.

Posted by: David Corfield on April 30, 2007 2:56 PM | Permalink | Reply to this

Re: What is super, really?

So what do your automorphisms (diffeomorphisms) and nontrivial 2-morphisms between them form? Do they make a 2-group?

Sure. For any category CC, the category Aut Cat(C) \mathrm{Aut}_{\mathrm{Cat}}(C) is a 2-group. Same if CC is smooth and we consider only smooth automorphisms Aut Cat C (C). \mathrm{Aut}_{\mathrm{Cat}_{C^\infty}}(C) \,. So Aut(𝒫 1(X))\mathrm{Aut}(\mathcal{P}_1(X)) is guaranteed to be a 2-group. All we have to do is work out how it looks like.

I argued that its 1-morphisms are diffeomeorphisms of XX, and 2-morphisms go between two diffeomorphisms that differ by one that is connected to the identity and contain the extra data of a “path field” being the flow of a vector field realizing that diffeo.

Horizontal composition of 1-morphisms and vertical composition of 2-morphisms is the obvious composition of the respective diffeomorphisms.

One checks that horizontal composition of 2-morphisms comes from first pushing one path field forward along one diffeomorphism and then composing.

I made crucial use of the 2-group property of Aut(𝒫 1(X))\mathrm{Aut}(\mathcal{P}_1(X)) when I gave that definition of vector fields:

consider the sub-1-group of Aut(𝒫 1(X))\mathrm{Aut}(\mathcal{P}_1(X)) which consists of those 2-morphisms that start at the identity diffeomorphism and with composition being horizontal composition of 2-morphisms.

Then, I think, vector fields on XX are in bijection with Lie group homomorphisms from the reals into that sub-1-group.

Of course if you are just looking for toy examples of 2-groups acting on something, it might be more useful to just cook up some small category CC and look at the 2-group Aut(C)\mathrm{Aut}(C).

Posted by: urs on April 30, 2007 3:10 PM | Permalink | Reply to this

Re: Who’s on the Right Track?

On the occasion of his 150th birthday, here Max Planck on film.

Posted by: Thomas Riepe on April 23, 2008 11:01 AM | Permalink | Reply to this

Post a New Comment