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Last time when I commented all the articles, we were impressed how interesting and serious they were. Tonight it's slightly easier to describe all the hep-th papers on the web, but some of them still look interesting:
This paper by Matthew Headrick and Joris Raeymaekers is obviously interesting. Consider the nonsupersymmetric orbifold of type II string theory - Adams-Polchinski-Silverstein (APS) type of orbifold - on C/Z_n for large n. You know that there are many tachyons in the twisted sectors. For large n, it makes sense to T-dualize around the angular direction of C/Z_n. You get some SUGRA solution. Well, many people have definitely looked at the orbifold in this way, using T-duality. But Matt and Joris finally consider the obviously interesting limit in which n is sent to infinity, but you keep n times alpha' fixed. It's some kind of zero slope limit, but the "lightest" tachyons (=closest to being massless) whose squared masses are comparable to (-1/alpha' n) survive this limit because these squared masses are exactly inverse to the quantity that is kept fixed.
Note that the tachyons come from twisted sectors. By interpreting the angle as a circle, they're winding strings. It means that in the T-dual picture, they are momentum modes of a field in the dual string theory which is effectively supergravity. Finally, the present authors calculate some interactions of the momentum modes in supergravity - which exist off-shell - and they show that they agree with the couplings of the different tachyons calculated from the CFT - which only exist on-shell. That's very interesting. The main thing I worry about is that the result is perhaps not too unexpected because instead of the orbifold, one might work directly with the "limiting CFT" on the thin cone and its T-dual.
This author constructs some new solutions of the modified Ricci-flatness equations, something that is necessary for a CFT to be well-defined. You know that Ricci flatness is the right equation of motion only if you have no fluxes and if the dilaton is constant. If the dilaton is not constant, the Ricci tensor is nonzero - Einstein's equations get a source. He or she does not quite want to talk about a non-constant dilaton. Instead, he or she focuses on another generalization of the CFT and Ricci flatness - namely a CFT with an extra complex field that has an "anomalous dimension". My understanding is that it's just a matter of notation whether you say that a field has an "anomalous dimension", or whether you redefine it by a function of the dilaton, and you allow the dilaton to vary. If my understanding is correct, Nitta has effectively found new solutions of the combined Einstein's equations with some dilaton-gradient source. These solutions have either a U(N) isometry, or an O(N) isometry. It's because the coordinates of his or her manifolds are explicitly written using U(N) or O(N) covariant coordinates, and the metric only depends on the quadratic invariants. Some of these solutions can be interpreted as generalizations or deformations of the Ricci-flat metric of the conifold with an extra linear dilaton - at least that's my impression.
We often say that the maximally supersymmetric Yang-Mills theory in four dimensions is "finite" because of powerful supersymmetric cancellations. Well, what is exactly is finite? Clearly, there are operators with anomalous dimensions that must be regulated and that are cutoff-dependent, and so forth. So it's not everything that is finite. However, the effective action is something that should be finite - it sort of computes the correlators of the elementary fields in which the divergences are supposed to cancel between the bosons and fermions. In this paper, these two guys try to prove the finiteness of the N=4 effective action in the N=1 superspace language. The effective action of N=4 can be written in this form, due to the Slavnov-Taylor identity, as long as we allow the superfields to be "dressed". They must go through some un-controversial steps to show that the R-symmetry anomaly cancels due to the N=2 supersymmetry, and they re-check that the one-loop beta-function vanishes. Nevertheless, the final result is that once you express the effective action of the N=4 gauge theory in terms of the dressed fields, all the terms become independent of the UV cutoff, and the effective theory is therefore finite.
OK, there was a Lorentz violating paper last time, too, and many comments may be repeated. I still don't understand the motivation behind these models. The space of possible non-relativistic non-stringy quantum field theories is huge. I don't really feel what constrains it. For relativistic theories, we may label all fields by their dimension - which is their dimension with respect to space as well as time. However, for non-relativistic theories, we must introduce separate spatial and temporal dimensional analyses, and I don't think that we can really distinguish "renormalizable" theories from "non-renormalizable" theories. Of course, this can be done if we write down a non-relativistic theory as a deformation of a relativistic theory, and this is what this groups does, too. Nevertheless the number of new terms, once you allow the Lorentz symmetry to be broken, is again huge. Moreover, I think that the lessons of 1905 are serious lessons, and breaking the Lorentz invariance explicitly should also be accompanied by breaking of the rotational invariance, and I see no reasons to do so. Let's stop criticism for a while.
What theory do they consider? Take the usual Maxwell action in four dimensions, and imagine that you decide to deform it in some way, by adding another action. What is the other action you know for a U(1) gauge field? Well, the Chern-Simons action. There is a problem: the usual Chern-Simons term only exists in three dimensions. However, that's not a problem for those who don't care about the Lorentz invariance. Just multiply the Chern-Simons 3-form with a general vector, i.e. a 1-form, to get a 4-form, and you can integrate the product over the spacetime. The vector picks a priviliged direction which is OK with you. Because the Chern-Simons action has an epsilon in it, you will break not only the Lorentz symmetry but also the CPT symmetry - which can happen once the Lorentz symmetry is gone. To make the things even more confusing, add an external current J that couples to the gauge field via the J.A term.
That was too natural so far. Let's make something more fancy. Reduce this four-dimensional Lorentz invariant theory to three spacetime dimensions (dimensional reduction). Moreover, don't reduce it along the priviliged vector discussed previously, but along a more general vector. In this case, the three-dimensional Lorentz symmetry will still be violated. If you write down some terms, you will discover mass terms of various types. Just do it and derive the equations of motion and solve them and draw several graphs. And don't forget to be excited that the results pick a priviliged reference frame (even though you know that it was your starting point). It's probably a good feature to violate the Lorentz symmetry.
Finally, let me admit that I am totally lost. I have no idea why they're doing what they're doing, whether it should be a physically realistic model or a mathematically interesting one: I just don't see the meaning of it all. This type of activity is what most of us would be doing today if we had no string theory. Combining random terms that apparently follow no deeper or organizational principles - terms extracted from an infinite chaotic ocean of arbitrary terms and their combinations - terms that are much more ugly and unjustified than the theories that are known to work. Sorry for being so skeptical; I might simply be dumb.
These colleagues first repeat a lot of the commercials about "Causal Dynamical Triangulations" that they've already written in many previous papers. The starting points are very obvious and sort of naive: try to define the path integral of quantum gravity in a discretized form. (It's like spin foams in loop quantum gravity, but you don't necessarily require that the details will agree.) OK, so how can you discretize a geometry? You triangulate it into simplices, and you imagine that every simplex has a region of flat Minkowski spacetime in it.
(That's not like loop quantum gravity - the latter assumes that there is no geometry "inside" the spin foam simplices - the geometry is concentrated at the singular points and edges of the spin foam.)
Then you write down the Einstein-Hilbert action many times and you emphasize that it is discretized. There are many other differences from loop quantum gravity: while the minimal positive distance in loop quantum gravity is sort of Planckian, in the present case they want to send the size of the simplices to zero and the regulator should be unphysical. Of course that if you do it, you formally get quantized general relativity with all of its problems: as soon as the resolution becomes strongly subPlanckian, the fluctuation of the metric tensor becomes large. The path integral will be dominated by heavily fluctuating configurations where the topology changes a lot and where the causal relations are totally obscured - and the results of these path integrals will be non-renormalizably divergent - at least if you expand them perturbatively. But this is simply what a correct, authentic quantization of pure gravity gives you.
> These colleagues first repeat a lot of the commercials about "Causal > Dynamical Triangulations" that they've already written in many previous > papers.
Once again, Lubos is much faster than me and I make my comments without having ready anything of the papers than the abstract. And I agree, when I saw this paper on the arxive, my reaction was "another one of those. how for into the abstract do I have to read to find the new stuff?" as was yours.
However, again once again, I am a bit less critical than you are. OK, it seems they beat the publicity drum a lot but I think this is fair if you are a small group that wants to be noticed in the stringy atmosphere of hep-th. And I should mention that Jan Ambjorn has worked on many different things including matrix models (the old ones), lattice theories and string theory.
So let me try to say a couple of words in defence of their approach: This stuff obviously has its background in the matrix model literature and the realization of 2d gravity in terms of dynamical triangulations (dual to matrices) was one of the successes of the 80s. But you are right, the Euclidean path integral is not only dominated by but also seems to localise in non-smooth geometries. So they try to cure this problem by changing the rules of their path integral.
[Moderator's note: Well, I understand. That's what I criticize. Every path integral in a quantum theory is dominated by non-differentiable configurations because this is necessary for the uncertainty principle. A classical configuration has sharp, well-defined values of the fields like X(t) or PHI(x,t) or g_{12}(x,t), and by the uncertainty principle, the uncertainty of the canonical momentum must therefore be infinite, which is reflected in the path integral by the fact that the |derivative| of the field is typically infinite, i.e. the non-differentiable configurations dominate. Do you agree that you could not get quantum mechanics if your path integral only summed over differentiable paths? If you succeeded to define this "truncated" integral in quantum mechanics, it would violate unitarity and the rule U(t1,t2)U(t2,t3)=U(t1,t3) because the different intervals would disagree "how much differentiable" the functions must be. LM]
It is probably fair to divide geometry into different levels of structure. One possible distinction is
It is up to discussion at which of these levels you start varying in your path integral and which parts you keep fixed.
[Moderator's note: It is fair to divide geometries, but it is never fair to "cut" some configurations from a path integral, I think. We've had a recent debate on sci.physics.strings about the overcritical electric field which was exactly about this issue - did you agree with our conclusion that you can't ever omit "unwanted" configurations? LM]
I guess, Lubos wants to vary 1-4 while keeping 0 fixed, Ambjorn and friends only vary 3-4 and I had long discussions with Hendryk Pfeiffer in which he tried to convince me that one should vary all 0--4.
[Moderator's note: I agree with him, you must vary everything. The differential structure only exists "classically" and it is a consequence of dynamics - the appearance of the derivative terms in the path integral. But the path integral over sums over all configurations of the given fields. LM]
Nobody has done a really convincing 'sum over geometries' yet, so I think it should be allowed to try all these approaches.
[Moderator's note: Nobody has found a really convincing luminiferous aether theory, so should all of us divide to different approaches how to construct aether? Actually I think that these two questions are more similar, even in details, than you might think. ;-) LM]
What Ambjorn etal find is that again in 2d you can solve this model exactly (ie compute the partition function with sources) and it agrees with expectations (whatever those would be). Second the typical configurations look much smoother (something they haven't put in, they only demand causality and global topology) than in the Euclidean case.
[Moderator's note: I am not getting this point at all. What's exactly the difference between the input and output? Typical configurations in the gravity path integral have strongly oscillating topology, both in the Minkowski and the Euclidean case, and in the Minkowski case, they have also a highly nontrivial and chaotic causal diagram. If you unphysically cut the "ugly" configurations, of course, you will end up with the "nice" ones, and because you made more constraints about the allowed configurations in the Minkowski case, you will get even nicer configurations than in the Euclidean space at the end. But that's not a result, that's your assumption. And it's an assumption that contradicts quantum mechanics. LM]
Of course, in 2D gravity is not typical, all the dynamics is in the cosmological constant and its conjugate variable, the volume, respectively. And in higher dimensions it is not possible to solve the problem analytically, you can only run in on your computer.
[Moderator's note: Right, 2D and 3D gravity don't really have gravitons as local degrees of freedom. All of us know how to compute 2D gravity as a path integral over "nice topologies" of two-dimensional spacetime: it's called the stringy worldsheet. But the conformal structure on the worldsheet is only "nice" because *any* configuration in 2D can be mapped to the "standard ones" by diff x Weyl transformations. Analogous things hold for 2D string theory - one really wants to calculate the path integral over the scalar fields in spacetime and their effects. Moreover, my arguments above that talk about the uncertainty principle for g_{12} and its time derivative can break down in d<4 because there is no such a physical degree of freedom. LM]
Another success that they claim is that they break the 'c=1 barrier'. OK, I have no idea what that really is because I try to stay away from all this old matrix model technology but Matthias Staudacher, who was around in those days, says this is quite non-trivial: In these models, you do not have to restrict yourself to pure gravity, you can couple matter to it: For example you can add an Ising spin degree of freedom to all your triangles and sum over it as well.
As, you say, in all these models you have to take the continuum limit and then you get a conformal field theory. In the old days, it was observed that whatever you did matter wise or matrix wise, you could only get models with central charge <1. But with causal triangulations coupled to matter you can break this barrier.
Finally these people claim their model has a well behaved continuum limit and I see no reason to doubt it.
[Moderator's note: in the case of the present paper, I don't have difficulties with the word "continuum limit" but rather with the word "model". You can define some set of rules that gives you a *classical* theory in some limit, but it by no means implies that your rules, before you take the limit, define a meaningful quantum theory, does it? For example, you should always ask whether your rules can lead to a unitary S-matrix, which path integrals should, and the answer will be NO in the 4D case, I think. LM]
But in the end this is only gravity if you end up in the correct universality class. That is, all your weird rules you make up to construct your discretised space-times correspond only to irrelevant operators that go away in this limit. And to show this is of course the hard part.
[Modeator's note: That may be a different way to say the same thing. You're simply not sure whether the "restricted path integral" has anything whatsoever in common with the real path integral. LM]
Robert
-- .oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oO Robert C. Helling School of Science and Engineering International University Bremen print "Just another Phone: +49 421-200 3574 stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
> It is up to discussion at which of these levels you start varying in > your path integral and which parts you keep fixed. > I guess, Lubos wants to vary 1-4 while keeping 0 fixed, Ambjorn and > friends only vary 3-4 and I had long discussions with Hendryk Pfeiffer in > which he tried to convince me that one should vary all 0--4.
What can be said about varying differentiable structures in a physical theory? As far as I know there are not many examples where several smooth structures on a given manifold are known. First and famous is the exotic S^7. Then Donaldson showed that there are exotic R^4s, but these are only implicitly known to exist.
What would a field theory on an exotic R^4 look like?
(Concerning 3)+4): Doesn't causal structure plus conformal structure imply a metric structure?)
On Wed, 17 Nov 2004, Urs Schreiber wrote: > What can be said about varying differentiable structures in a physical > theory? As far as I know there are not many examples where several smooth > structures on a given manifold are known. First and famous is the exotic > S^7. Then Donaldson showed that there are exotic R^4s, but these are only > implicitly known to exist.
These are very interesting things that a priori only look like some picky mathematical curiosities, but they potentially can even have physical consequences. However, in the path integral, it seems reasonable to guess that we don't really want to define any particular differentiable structure. And even if we do define it, we should probably sum over all of them.
This is my favorite numerology: M-theory has 11 dimensions, so let's look at exotic 11-spheres. There are 992 of them, which is twice as many as the dimension of the gauge groups in 10 dimensions (namely 496, of SO(32) or E8 x E8), and it is one half of the year in which Green and Schwarz calculated that the dimension 496 was necessary (namely 1984). ;-)
> What would a field theory on an exotic R^4 look like?
Is it really a well-posed question? Does the usual path integral focuses on the "normal" R^4 only, or does it sum over the structures?
> (Concerning 3)+4): Doesn't causal structure plus conformal structure imply a > metric structure?)
Sorry, but locally, causal structure and conformal structure of a manifold with Minkowski signature is the same thing - they determine the metric up to a local Weyl scaling, don't they? If you say what is the infinitesimal future light cone of a given point, you've defined where ds^2 vanishes, which means that assuming that ds^2 has the usual quadratic form, you've defined ds^2 up to a scaling.
This is why the Penrose diagrams, called causal diagrams, are also said to encode the conformal structure of spacetime, is not it? ___________________________________________________________________________ ___ E-mail: l...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/ eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call) Webs: http://schwinger.harvard.edu/~motl/http://motls.blogspot.com/ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^
> On Wed, 17 Nov 2004, Urs Schreiber wrote: >> What would a field theory on an exotic R^4 look like?
> Is it really a well-posed question? Does the usual path integral focuses > on the "normal" R^4 only, or does it sum over the structures?
You can imagine a path integral for gravity to be over all these structure. I was here thinking of ordinary non-gravitational field theory on a fixed exotic R^4. Just to get a feeling for how exotic exotic is.
>> (Concerning 3)+4): Doesn't causal structure plus conformal structure >> imply a >> metric structure?)
> Sorry, but locally, causal structure and conformal structure of a manifold > with Minkowski signature is the same thing
Right, I was confused. I thought Robert meant the specification of a volume element at every point by "conformal structure".
> What can be said about varying differentiable structures in a physical > theory? As far as I know there are not many examples where several smooth > structures on a given manifold are known. First and famous is the exotic > S^7. Then Donaldson showed that there are exotic R^4s, but these are only > implicitly known to exist.
I think there are plenty of examples of topological four-manifolds that admit different smooth structures. The quitic surface for example. It's really in some sense a generic phenomenon, since two four-manifolds are always homeomorphic if they have the same intersection form, but it's much harder to show that they are diffeomorphic. There are also uncountably many exotic R^4's. I think the construction of them using Casson handles are actually quite explicit, although the construction is so complicated that it's truly difficult to imagine how it could ever show up in physics.
On Wed, 17 Nov 2004 08:09:21 -0500, Robert C. Helling wrote: > 0) differentiable structure > 1) topology > 2) causal structure > 3) conformal structure > 4) metric structure
Let me reorder that...
-1) sets 0) topology 1) differentiable structure 2) causal structure + conformal structure (lets not get into details here) 4) metric structure
> [Moderator's note: It is fair to divide geometries, but it is never > fair to "cut" some configurations from a path integral, I think.
I disagree. To properly write down the path integral, the first thing you must define what you are integrating over.
Lets take some old QFT (as opposed to some fancy theory of gravity), and "fields" just some sort of function. You should really write down first which functions you want to admit, probably all smooth ones plus some more. Probably differentiable functions with suitable decay at infinity, and then the closure of that space (and extend the derivative operators to this Hilbert space).
Of course, a good physicist never does that and hopes that the details of Sobolev spaces enter the final answer just like the choice of regulator, that is not at all.
So back to gravity, I think the LModerator agrees that we should not integrate over all -1..4, since then we immediately have nasty non-separable Hilbert spaces. We'll have to extract something separable in-between that allows a reasonably well-behaved extension of the action. Of course, If I knew the specifics I'd be writing the paper right now (and then cross-list to hep/th from the functional analysis archive).
On Wed, 17 Nov 2004, Volker Braun wrote: > I disagree. To properly write down the path integral, the first thing > you must define what you are integrating over.
I agree with this sentence, but it seems to support my point. Defining what you're integrating over means choosing your fields or other degrees of freedom and their range, and you're always integrating over all of them. There can be constraints such as "you only integrate over manifolds with a spin structure". Such constraints look global in character, but they're always about our ability to define some fields - not necessarily fields with local dynamics - at every point. In other words, there cannot be any nontrivial global constraints. Once you define your fields and the set in which they take their values, you must always integrate over all of their values, I think. Making some additional constraints would break locality and unitarity.
By the way, omitting some configurations is equivalent to redefining their action to infinity, which is usually a very strange attempt to modify your dynamics.
Incidentally, Andy Neitzke tells me that :
1. Michael Green claimed that there was really a relation of the number 496 with the number of spheres, or something like that
2. There has been a paper by Edward Witten relating these exotic differential structures and global gravitational anomalies, but not necessarily involving the interesting numerology with 992
We would be very interested if someone could say something about it.
> Lets take some old QFT (as opposed to some fancy theory of gravity), > and "fields" just some sort of function. You should really write down > first which functions you want to admit, probably all smooth ones plus > some more. Probably differentiable functions with suitable decay at > infinity, and then the closure of that space (and extend the derivative > operators to this Hilbert space).
Do I understand you well? The path integral in a quantum field theory is dominated by functions that don't satisfy any of your criteria. The path integral may be *localized* on smooth functions, if you can prove it, but it is *defined* in such a way that most of the contributions come from functions that don't have derivative almost anywhere.
Path integrals are very subtle things. Mathematicians have problems to define them rigorously. Nevertheless they mathematically work for physical purposes. But they only work once you carefully follow all the rules - and one of the rules definitely is that one must integrate over all configurations of the fields that he starts with. Messing up with this rule ends up with a nonsensical theory, certainly in a generic case.
It's a sort of miracle that one can define meaningful functional integrals in quantum theories at all, but this procedure is extremely sensitive.
> Of course, a good physicist never does that and hopes that the details of > Sobolev spaces enter the final answer just like the choice of regulator, > that is not at all.
Do I understand you well that you say that it is possible to define physically meaningful functional integrals on Sobolev spaces with positive "p" i.e. only on spaces of functions whose first "p" derivatives are L2 integrable? Wow. ;-)
You can never restrict your path integral to such Sobolev spaces simply because the dominant contributions to any meaningful path integral with local physical degrees of freedom comes from non-differentiable configurations whose derivatives are definitely not L2-integrable. Quantum mechanics is jittery. Imposing some very strong differentiability criteria on the configurations in your path integral is a road to hell.
> So back to gravity, I think the LModerator agrees that we should not > integrate over all -1..4, since then we immediately have nasty > non-separable Hilbert spaces.
What does it exactly have to do with Hilbert spaces? You must integrate over -1..4 because this is how Feynman's rules work - integrate over everything. What is exactly a Hilbert space is not so transparent in the path integral formalism.
But if you're able to derive that the standard Feynman rules of path integrals end up with a non-separable Hilbert space (or another problem), then the theory simply has a non-separable Hilbert space (or the other problem) and it is a physically uninteresting theory. You will not be able to revive such a theory.
> We'll have to extract something separable in-between that allows a > reasonably well-behaved extension of the action.
Nope. The reason why the path integral seems to behave badly for pure general relativity in d>3 is that pure general relativity in d>3 is a bad quantum theory. The right way to fix it is to switch to a correct theory - add new dynamics, new fields, new interactions (such as those derivable from string theory). The wrong way to fix it is to try to modify the rules of quantum mechanics - such as the rule that Feynman's path integral is an integral over all configurations of your fields.
All the best Lubos ___________________________________________________________________________ ___ E-mail: l...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/ eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call) Webs: http://schwinger.harvard.edu/~motl/http://motls.blogspot.com/ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^
In article <pan.2004.11.17.16.25.54.200129-100...@physik.hu-berlin.de>, Volker Braun <volker.br...@physik.hu-berlin.de> wrote:
> I disagree. To properly write down the path integral, the first thing > you must define what you are integrating over.
> Lets take some old QFT (as opposed to some fancy theory of gravity), > and "fields" just some sort of function. You should really write down > first which functions you want to admit, probably all smooth ones plus > some more. Probably differentiable functions with suitable decay at > infinity, and then the closure of that space (and extend the derivative > operators to this Hilbert space).
There's a real sense in that the only relevant contributions to the path integral come from nondifferentiable functions. See Appendix 3 of Coleman's "The sses of instantons" in Aspects of Symmetry.
On Wed, 17 Nov 2004 12:34:38 -0500, Lubos Motl wrote: > [...] Such constraints look global in character, > but they're always about our ability to define some fields - not > necessarily fields with local dynamics - at every point. In other words, > there cannot be any nontrivial global constraints.
I fail to see how you distinguish constraints that "look" global from those which "are" global. On some level, they are only distinguished by your aesthetic preferences. Of course, if anything breaks locality or unitarity, it is of course bad.
> Incidentally, Andy Neitzke tells me that : [...] > 2. There has been a paper by Edward Witten relating these exotic > differential structures and global gravitational anomalies, but not > necessarily involving the interesting numerology with 992 > We would be very interested if someone could say something about it.
>> Lets take some old QFT (as opposed to some fancy theory of gravity), and >> "fields" just some sort of function. You should really write down first >> which functions you want to admit, probably all smooth ones plus some >> more. Probably differentiable functions with suitable decay at infinity, >> and then the closure of that space (and extend the derivative operators >> to this Hilbert space).
> Do I understand you well? The path integral in a quantum field theory is > dominated by functions that don't satisfy any of your criteria. The path > integral may be *localized* on smooth functions, if you can prove it, but > it is *defined* in such a way that most of the contributions come from > functions that don't have derivative almost anywhere.
Yes, I agree completely. Those things crept in where I said "closure". Sorry, should have been more explicit.
>> So back to gravity, I think the LModerator agrees that we should not >> integrate over all -1..4, since then we immediately have nasty >> non-separable Hilbert spaces.
> What does it exactly have to do with Hilbert spaces? You must integrate > over -1..4 because this is how Feynman's rules work - integrate over > everything. What is exactly a Hilbert space is not so transparent in the > path integral formalism.
Remember that point number -1) were just point sets, without any topology. There is no way you can get a separable Hilbert space out of that.
>> We'll have to extract something separable in-between that allows a >> reasonably well-behaved extension of the action.
> Nope. The reason why the path integral seems to behave badly for pure > general relativity in d>3 is that pure general relativity in d>3 is a bad > quantum theory. The right way to fix it is to switch to a correct theory - > add new dynamics, new fields, new interactions (such as those derivable > from string theory).
I agree that string theory is the best bet here. In fact, I was thinking of string theory as the "something in-between". Somewhere between point sets and complete geometry, we can try to study spaces by their loop space. Well ok, that is just words, but this is string theory to me.
On Wed, 17 Nov 2004, Volker Braun wrote: > I fail to see how you distinguish constraints that "look" global from > those which "are" global. On some level, they are only distinguished by > your aesthetic preferences. Of course, if anything breaks locality or > unitarity, it is of course bad.
Right, the criterion is, of course, whether or not you really violate unitarity or (macroscopically) locality. What I proposed was a description of the "allowed global" constraints that don't violate locality - and I said that they "look global". For example, it is OK to require that the manifolds we integrate over have a spin structure - because it is about the pointwise existence of spinorial degrees of freedom - but it is not correct to require that all noncontractible circles in your geometry must be longer than 5 meters - because such a constraint would be "really" global - it would be correlating physics in different places by a nonlocal link - and it would violate locality. I might be using inappropriate words, but you may understand me anyway.
> Yes, I agree completely. Those things crept in where I said "closure". > Sorry, should have been more explicit.
Good.
> Remember that point number -1) were just point sets, without any topology. > There is no way you can get a separable Hilbert space out of that.
I don't really know what it means to integrate over "just" sets, but if "just" sets were defining the configurations in whatever theory you would study, you would have to sum/integrate over them. If you can't get a reasonable Hilbert space in this way, then it means that your degrees of freedom should never be described just by "sets". It does not mean that you should change the rules of quantum mechanics.
One may replace the word "set" in the previous paragraph by "derived categories" or something else to get a more meaningful statement. ;-)
> I agree that string theory is the best bet here. In fact, I was thinking > of string theory as the "something in-between". Somewhere between point > sets and complete geometry, we can try to study spaces by their loop > space. Well ok, that is just words, but this is string theory to me.
Well, you have some particular sets, conformal structure and perhaps even the metric tensor on the space of possible feelings that one can get by looking on a theory. ;-) This metric may be different from others'. In my opinion, it is still more realistic to imagine string theory as being described by an even bigger path integral than the quantum gravity integrals with fluctuating topology - string field theory (closed, in this case) pictures physics as the path integral over all possible geometries but also all possible configurations of infinitely many other fields, and so on. ;-) Of course that this string field theory impression overestimates the number of degrees of freedom in quantum gravity - holography tells us that the path integral should be equivalent to a path integral of a (d-1)-dimensional non-gravitational theory - which is what works in AdS/CFT. These are subtle things, but whatever description one chooses, I think that he must simply integrate over all configurations with the pre-determined degrees of freedom. ___________________________________________________________________________ ___ E-mail: l...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/ eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call) Webs: http://schwinger.harvard.edu/~motl/http://motls.blogspot.com/ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^
In article <Pine.LNX.4.31.0411171210200.22715-100...@feynman.harvard.edu>, Lubos Motl <m...@feynman.harvard.edu> wrote:
> Incidentally, Andy Neitzke tells me that :
> 1. Michael Green claimed that there was really a relation of the number > 496 with the number of spheres, or something like that
Global gravitational anomalies have to do with pi_0(Diff^+(M)).
Similarly, you can construct exotic spheres by a clutching-type construction. In particular, the equator of a S^n is an S^{n-1}. If pi_0(Diff^+(S^{n-1})) =/= 0, glue the two hemispheres together with a diffeomorphism that isn't in the identity component.
> I think there are plenty of examples of topological four-manifolds that
> admit different smooth structures. The quitic surface for example. It's > really in some sense a generic phenomenon, since two four-manifolds are > always homeomorphic if they have the same intersection form, but it's much > harder to show that they are diffeomorphic. There are also uncountably > many exotic R^4's. I think the construction of them using Casson handles > are actually quite explicit,
Can you sketch how that works?
> although the construction is so complicated > that it's truly difficult to imagine how it could ever show up in physics.
Yes, but Lubos was imagining having a path integral where all of them are summed over, which would make them "show up" all over the place. To get a feeling for what this would mean I was wondering about the following: Break that hypothetical path integral up into the sum/integration over the smooth structures and the integration over the rest. So for each given smooth structure the rest is an "ordinary" quantum (gravitational) field theory, albeit on an exotic space. What would ordinary QFT on a fixed exotic smooth structure be like?
For instance: Can we say anything about the spectrum of the Laplace operator on an exotic sphere?
Xi Yin wrote: > > I think there are plenty of examples of topological four-manifolds that > > admit different smooth structures. The quitic surface for example. It's > > really in some sense a generic phenomenon, since two four-manifolds are > > always homeomorphic if they have the same intersection form, but it's much > > harder to show that they are diffeomorphic. There are also uncountably > > many exotic R^4's. I think the construction of them using Casson handles > > are actually quite explicit, Urs Schreiber wrote: > Can you sketch how that works?
You mean the exotic R^4?
First one needs to construct a Casson handle. The standard Casson handle is basicly an open 2-handle, namely D^2 x int(D^2), but you cut out a piece which is obtained much like the Whitehead continuum. You can find a map g: D^2xD^2 -> D2xD2 such that its restriction to D2xS1 -> D2xS1 is the map f that takes a solid torus to a thicken link inside this solid torus. By repeated acting g on D2xD2 and take the intersection of all of the images, you get the piece that you need to cut out to get the Casson handle. There are many generalizations of this procedure, which gives uncountably many Casson handles.
Now it is a hard result of Freedman that the Casson handle is actually homeomorphic to the standard open 2-handle. I don't know the proof of it. It's the key step in Freedman's classification of topological 4-manifolds. But if you think about it, the Casson handle is very exotic and it doesn't look like the ordinary 2-handle at all.
Let's call the Casson handle CH, and the ordinary open 2-handle H. Knowing that CH is homeomorphic to H, one knows that there is a TOPOLOGICALLY embedded disk. Ordinarily one can take a 4-ball union with two open 2-handles and get S^2xS^2 with a 4-ball deleted - much like that you can take a disk plus two strips and get a torus with a hole. Now instead of using the ordinary handle you can use the Casson handle CH. So you get something homeomorphic to S^2xS^2 with a 4-ball cutout, and there are two topologically embedded disks in the two CH's, which can be extended to give a topologically embedded S^2vS^2 (wedge of two spheres). You can include the missing 4-ball and get S^2xS^2, with a topologically embedded S^2vS^2 (but embedded in a very exotic way).
Finally you take the S^2xS^2, and cut out the topologically S^2vS^2, you get the exotic R^4. Let's call it X. It is clear that X is homeomorphic to R^4 (remember that to prove this we need the result of Freedman on CH).
Let's pretend that X is diffeomorphic to R^4, and try to get contradiction. If X is diffeomorphic to R^4, for any compact subset K in X, one can find a smoothly embedded 3-sphere that separates X into two parts A and B, such that A is compact and contains K. For example, we can take K to be the missing 4-ball we added to get S^2xS^2 in constructing X. So there is a compact manifold A with boundary S^3 that contains K, and another compact manifold B with boundary S^3 that contains the topologically embedded wedge of two S^2vS^2 which we cut out from S^2xS^2 to get X. Notice that B has the same homotopy type as S^2xS^2.
Previously we used the open 4-ball with two Casson handles attached. Let's call it CW. So B is contained in CW, with its boundary S^3 smoothly embedded.
It is a result of Casson and Freedman that you can embed CW topologically in any 4-manifold. In particular, you can embed 3 copies of CW's in a K3 surface. There is a B in each embedded CW, whose boundary S^3 is smoothly embedded. You can then cut out all the B's and replace them with 4-balls. The leftover of the K3 surface after this surgery would have intersection form -E8-E8, which is impossible for any smooth four-manifold by the theorem of Donaldson. This proves contradiction.
There are other examples of topological four-manifolds with more than one smooth structures that are much easier to describe. For example, the quintic surface in CP^3 is homeomorphic but not diffeomorphic to the connected sum of 9 copies of CP^2 and 44 copies of CP^2 bar (meaning CP^2 with opposite orientation). They are not diffeomorphic because they have different Seiberg-Witten invariants.
> > Xi Yin: ... although the construction is so complicated > > that it's truly difficult to imagine how it could ever show up in physics.
> Urs: Yes, but Lubos was imagining having a path integral where all of them are > summed over, which would make them "show up" all over the place. To get a > feeling for what this would mean I was wondering about the following: Break > that hypothetical path integral up into the sum/integration over the smooth > structures and the integration over the rest. So for each given smooth > structure the rest is an "ordinary" quantum (gravitational) field theory, > albeit on an exotic space. What would ordinary QFT on a fixed exotic smooth > structure be like?
> For instance: Can we say anything about the spectrum of the Laplace operator > on an exotic sphere?
The exotic spheres are not that exotic by themselves. For example, the exotic S^7 can be obtained from R^4 bundles over S^3. I don't know any results concerning the spectrum of the Laplacian on these spaces but it's not any more difficult than on other generic smooth 7-manifolds.
These Casson handles are very interesting objects, and there are some potentially interesting, but probably not quite correct, papers about it, like the following:
In this paper, Jerzy Krol uses the exotic "small" R^4 - one with the Casson Handle - to break supersymmetry in AdS/CFT. ;-) If you extract something out of this paper, it may be interesting if you describe your findings.
I'm now gonna spend some time with Casson handles and Whitehead continua. ;-) ___________________________________________________________________________ ___ E-mail: l...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/ eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call) Webs: http://schwinger.harvard.edu/~motl/http://motls.blogspot.com/ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^
Xi Yin wrote: > I think there are plenty of examples of topological four-manifolds > that admit different smooth structures. The quintic surface for > example. It's really in some sense a generic phenomenon, since two > four-manifolds are always homeomorphic if they have the same > intersection form, but it's much harder to show that they are > diffeomorphic.
You mean *simply connected* (closed smoothable oriented) four-manifolds, of course.
Urs Schreiber wrote: > What can be said about varying differentiable structures in a physical > theory? As far as I know there are not many examples where several > smooth structures on a given manifold are known.
There are certainly many examples. Kirby and Siebenmann worked out a theory which, at least in principle, provides a classification of the diffeomorphism classes of smooth structures on any topological (or piecewise linear) manifold of dimension > 4. The number is always countable, and finite on a compact manifold. There is no classification in dimension 4, but it is conjectured that the number of diffeomorphism types of smooth structures is uncountable on every noncompact topological 4-manifold. This conjecture has been proved for a large class of manifolds. Lots of closed topological 4-manifolds are known which admit infinitely many diffeomorphism types of smooth structures.
>First and famous is the exotic S^7.
... and many other exotic spheres in higher dimension follow. And once you have an exotic sphere S, you can take any smooth manifold M of the same dimension and form a connected sum M#S, which is homeomorphic but in many cases not diffeomorphic to M. The construction of exotic smooth structures on open 4-manifolds works similarly; instead of connected sums one uses "end sums" with exotic R^4s.
>Then Donaldson showed that there are exotic R^4s, but these are only >implicitly known to exist.
Some of them can be described very explicitly: as handlebodys (with infinitely many handles). This has been done by Bi\v{z}aca and Gompf in the 1990s.
By the way: Donaldson provided an essential ingredient for the existence proof. But it was Freedman who first proved that exotic R^4s exist (by putting together his own results and Donaldson's).
>What would a field theory on an exotic R^4 look like?
What kind of field theory? One that has only the smooth structure as background structure? If the theory really uses the smooth structure (as all interesting theories I know of do), then its set of solutions on an exotic R^4 would probably look very different from the set of solutions on standard R^4. For instance, Donaldson's invariants of a closed smooth 4-manifold M are derived from the moduli space of solutions of a certain Yang/Mills theory on M. These invariants can distinguish nondiffeomorphic smooth structures on the same closed topological 4-manifold, so the solution spaces of the Yang/Mills theory look in general quite different for homeomorphic but nondiffeomorphic manifolds. It's similar for the Seiberg/Witten invariants.
But I think that the discussion of exotic smoothness is beside the point of this thread anyway. In a physical theory, one would have to use path integrals over the set of smooth structures, not over the set of *diffeomorphism classes* of smooth structures, on a given topological manifold. (Whatever the integration measures might be. Any suggestions?) The set of all smooth structures (= maximal C^infty atlases) on a nonempty topological manifold of dimension > 0 is *always* uncountable. I am pretty sure that it is irrelevant for this physical discussion which of these smooth structures are homeomorphic. The fact that the forgetful functor
> Does the usual path integral focuses on the "normal" R4 only, or does > it sum over the structures?
Urs Schreiber replied:
> You can imagine a path integral for gravity to be over all these > structure.
My imagination seems to be too limited. Could you write down an explicit (toy) example and explain its intuitive meaning? What sort of theory does it describe? What would the integration measure (which has probably no rigorous definition, or has it?) mean on a heuristic level?
(Maybe these questions are too hard. Then let's try to figure out what happens in a discretized version: consider triangulations of space(time) instead of smooth structures. We integrate over all (not necessarily homeomorphic) triangulations, right? What's the integrand? What's the measure? What kind of physics is the theory supposed to describe?)
>>What would a field theory on an exotic R^4 look like?
> What kind of field theory?
Well, morally if you want to vary the smooth structure you will even more want to vary the metric. So the answer is: gravitational field theories. But I guess it would help to see much simpler toy examples.
> In a physical theory, one would have to use path > integrals over the set of smooth structures, not over the set of > *diffeomorphism classes* of smooth structures, on a given topological > manifold.
I am not sure why you think so.
> Lubos Motl asked:
>> Does the usual path integral focuses on the "normal" R4 only, or does >> it sum over the structures?
> Urs Schreiber replied:
>> You can imagine a path integral for gravity to be over all these >> structure.
Just for the record, I never said that in the usual path integral quantization prescription for gravity one sums over smooth structures. The discussion here is about what would happen if one would do it - and how.
> My imagination seems to be too limited. Could you write down an explicit > (toy) example and explain its intuitive meaning?
I have never thought about this stuff before, but what comes to mind is the following:
Pick any manifold which admits a finite number N of smooth structures {S_n}_{n=1}^N.
(Probably what I mean here is what you call diffeomorphism classes of smooth structures.)
Pick any one of these S_i and with respect to it compute the path integral Z(S_i) for a given field theory on the given manifold with that smooth structure.
Sum up the result to get Z = sum_i=1^N Z(S_i)
> (Maybe these questions are too hard. Then let's try to figure out what > happens in a discretized version: consider triangulations of space(time) > instead of smooth structures. We integrate over all (not necessarily > homeomorphic) triangulations, right? What's the integrand? What's the > measure? > What kind of physics is the theory supposed to describe?)
You would do that if you think that this is a way to deal with gravity.
Maybe it is better to turn the question around: Does a sum over smooth structure arise anywhere by itself?
For instance in AdS/CFT: Is there any indication that the path integral on the AdS side should involve a sum over smooth structures?
Urs Schreiber wrote: > I have never thought about this stuff before, but what comes to mind is the > following:
> Pick any manifold which admits a finite number N of smooth structures > {S_n}_{n=1}^N.
> (Probably what I mean here is what you call diffeomorphism classes of smooth > structures.)
> Pick any one of these S_i and with respect to it compute the path integral > Z(S_i) for a given field theory on the given manifold with that smooth > structure.
> Sum up the result to get Z = sum_i=1^N Z(S_i)
This sounds interesting, although I don't see any reason to compute the partition sum with fixed underlying topological manifold. In physics my feeling is that manifolds that are not diffeomorphic should be treated as totally distinct objects. If you want to sum over manifolds that are not diffeomorphic, you might as well sum over all topologies.
For example, consider M-theory on AdS_4xS^7, which is dual to the low energy limit of M2-brane world volume theories. You can replace the S^7 by exotic S^7, which correspond to putting the M2-brane at the singularity of some local Calabi-Yau 4-fold (hep-th/9810201). I don't have any understanding of these 3D CFTs. It would be interesting to see what you get by summing up these 28 different CFTs.
> Urs Schreiber wrote: >> Sum up the result to get Z = sum_i=1^N Z(S_i)
> This sounds interesting, although I don't see any reason to compute the > partition sum with fixed underlying topological manifold. In physics my > feeling is that manifolds that are not diffeomorphic should be treated as > totally distinct objects. If you want to sum over manifolds that are not > diffeomorphic, you might as well sum over all topologies.
True. That's natural considering the hierarchy of geometrical notions
> -1) sets > 0) topology > 1) differentiable structure > 2) causal structure + conformal structure (lets not get into details > here) > 4) metric structure
mentioned by Volker Braun previously.
(BTW, how about including a point -2) fixing some category that one wants to work in. There are other possible choices apart from the category Set... ;-)
> For example, consider M-theory on AdS_4xS^7, which is dual to the low > energy limit of M2-brane world volume theories. You can replace the S^7 by > exotic S^7, which correspond to putting the M2-brane at the singularity of > some local Calabi-Yau 4-fold (hep-th/9810201). I don't have any > understanding of these 3D CFTs. It would be interesting to see what you > get by summing up these 28 different CFTs.
Hm, this seems to be dual to M-theory with given asymptotic *topology* specified, somehow. (?)
Urs Schreiber wrote: >"Marc Nardmann" <Marc.Nardm...@bigfoot.de> schrieb [...]
>>>What would a field theory on an exotic R^4 look like?
>>What kind of field theory?
>Well, morally if you want to vary the smooth structure you will even more >want to vary the metric. So the answer is: gravitational field theories. But >I guess it would help to see much simpler toy examples.
>>In a physical theory, one would have to use path >>integrals over the set of smooth structures, not over the set of >>*diffeomorphism classes* of smooth structures, on a given topological >>manifold.
>I am not sure why you think so.
I was thinking about the general case of a theory which might contain background structures (e.g. an integration measure). For a background-free (gravitational) theory the integration over the set of diffeomorphism classes might be well-defined. Let's see:
My interpretation of the situation we are discussing is as follows. We have a topological manifold M. Let Sm(M) denote the set of all smooth structures on M. For each element A of M, we have a well-defined smooth fibre bundle E(A) [e.g. the bundle of symmetric bilinear forms of index 1 on the tangent bundle of A] and can consider smooth sections in E(A) [smooth Lorentzian metrics on A]. (Maybe we want to allow sections of lower regularity as well, say C^2, but let's take smooth sections for simplicity.) Let's denote the set of all these sections by F(A).
We form the disjoint union of all the sets F(A) and denote it by F(M). (There might be situations where it would be meaningful to take the union instead of the disjoint union: e.g. when E(A) is the trivial bundle A x R --> A [where R denotes the real line], so F(A) is the set of smooth functions from A to R. Then the constant functions would be contained in F(A) for every A. But even in such situations, let's take the disjoint union.) So we have a set-theoretic fibre bundle p: F(M) --> Sm(M) whose fibre over A is F(A).
Now we consider an action S which is defined on the domain F(M):
S: F(M) --> R .
[For instance, in our example from above, we could take S(g) = integral over M of the scalar curvature of g with respect to the measure defined by g (assuming that the manifold M is compact), i.e. the Einstein/Hilbert action.]
The path integral should now have the form
integral over F(M) of exp(iS) with respect to some measure on F(M)
(or a Wick-rotated version thereof), where the measure on F(M) has to be specified. That's my interpretation of our situation.
Let's denote the set of diffeomorphism classes of smooth structures on M by DiffSm(M).
What would it mean in the present context to take "the path integral over DiffSm(M)" instead of "the path integral over Sm(M)"? I guess it would mean:
"Take an appropriate measure on DiffSm(M) and, for each A in Sm(M), an appropriate measure dX_A on F(A). For each equivalence class in DiffSm(M), choose one representative A. Compute the integral of S over the fibre F(A) with respect to dX_A. *Prove that the result does not depend on the chosen representative A.* Now do the integration over DiffSm(M)."
That the result be independent of the choice of A means that whenever there is a diffeomorphism phi: A --> A', we have
integral over F(A) of S dX_A = integral over F(A') of S dX_{A'} .
I think that to prove such a property, we have to assume that our bundle E(A) and the measure dX_A "depend naturally on A" (words used in a non-standard way), i.e., for every diffeomorphism phi: A --> A', we have a measure-preserving map T(phi): F(A) --> F(A') such that (preferably) functoriality holds: T(phi ° psi) = T(phi) ° T(psi) and T(id_A) = id_{F(A)}. [In our favourite example, we can take T(phi)(g) to be the pullback of the metric g by phi^{-1} and hope that the measures make this measure-preserving.]
Moreover, we have to assume that S satisfies, for all diffeomorphisms phi, the condition
S ° T(phi) = S .
[This holds in our favourite example: the integral over M of the scalar curvature of g with respect to the measure defined by g is equal to the integral over M of the scalar curvature of (phi^{-1})^*(g) with respect to the measure defined by (phi^{-1})^*(g).]
Then clearly
integral over F(A') of S with respect to dX_{A'} = integral over F(A) of S ° T(phi) with respect to dX_A = integral over F(A) of S with respect to dX_A ,
as desired.
However, if there is any background structure in our theory, then the condition S ° T(phi) = S will almost certainly fail. That was the problem I was thinking about in the paragraph you cited above. Then the integration over DiffSm(M) would simply make no sense, while integration over Sm(M) would still be meaningful.
Urs wrote: > <>Just for the record, I never said that in the usual path integral > quantization prescription for gravity one sums over smooth structures. The > discussion here is about what would happen if one would do it - and how.
That's how I interpreted your words. A propos what would happen: In our Einstein/Hilbert example on the topological manifold R^4, which questions would seem interesting to you?
I wrote: > <>My imagination seems to be too limited. Could you write down an explicit > (toy) example and explain its intuitive meaning?
Urs replied:
>I have never thought about this stuff before, but what comes to mind is the >following:
>Pick any manifold which admits a finite number N of smooth structures >{S_n}_{n=1}^N.
>(Probably what I mean here is what you call diffeomorphism classes of smooth >structures.)
>Pick any one of these S_i and with respect to it compute the path integral >Z(S_i) for a given field theory on the given manifold with that smooth >structure.
If the theory satisfies the requirements I described above (being background-free in particular), this should work. But I am still not sure what the interpretation resp. intuitive meaning of such a theory is. What is the path integral supposed to tell us in the Einstein/Hilbert case, for instance?
>Sum up the result to get Z = sum_i=1^N Z(S_i)
Okay, in such a finite case, a simple counting measure is a reasonable choice. But which measure could we choose on R^4, for example? Or on the larger set of all (diffeomorphism classes of) 4-dimensional smooth structures (not necessarily representing the same homeomorphism type)? I have some vague ideas, but this seems to be a subtle issue.
> Urs Schreiber wrote: >> Marc Nardman originally wrote: >>>In a physical theory, one would have to use path >>>integrals over the set of smooth structures, not over the set of >>>*diffeomorphism classes* of smooth structures, on a given topological >>>manifold.
>>I am not sure why you think so.
> I was thinking about the general case of a theory which might contain > background structures (e.g. an integration measure). For a > background-free (gravitational) theory the integration over the set of > diffeomorphism classes might be well-defined. Let's see:
> My interpretation of the situation we are discussing is as follows. We > have a topological manifold M. Let Sm(M) denote the set of all smooth > structures on M. For each element A of M, we have a well-defined smooth
Just to make sure: I guess in that last sentence you really meant "For each element A of Sm(M)", where A is a smooth atlas on M, right?
(This at least seems to be what you have in mind further below.)
I follow the construction of the path integral that you outline. This is, in very mathy language, what I had in mind, too:
[...]
> Now we consider an action S which is defined on the domain F(M):
> S: F(M) --> R .
[...]
> integral over F(A) of S dX_A = integral over F(A') of S dX_{A'} .
[...]
> Moreover, we have to assume that S satisfies, for all diffeomorphisms > phi, the condition
> S ° T(phi) = S .
(= diffeo invariance of the action)
> However, if there is any background structure in our theory, then the > condition S ° T(phi) = S will almost certainly fail.
Hm, ok, so this is the point that we need to sort out. Probably it depends a lot on what kind of "background structure" you have in mind.
To me it seems that the usual cases of interest in physics don't have a problem here, but let me know if you think I am missing something.
For instance if we want to consider YM theory our action is given by the integral of
tr F \wedge * F
over M. The metric encoded in "*" is fixed (up to diffeos) and not supposed to be integrated over. Still, the corresponding action is diffeo invariant.
> If the theory satisfies the requirements I described above (being > background-free in particular), this should work. But I am still not > sure what the interpretation resp. intuitive meaning of such a theory > is. What is the path integral supposed to tell us in the > Einstein/Hilbert case, for instance?
It would give us a theory of quantum gravity in which the smooth structure of spacetime is allowed to "fluctuate" (at least if we enhance the pure EH action to something which has a well defined path integral). As Lubos and Xi Yin have pointed out, if we are willing to allow fluctuating topologies we should also allow such fluctuating smooth structures. So from this point of view it seems like the most natural thing to do. On the other hand, I have never seen it discussed before and as you point out the cases where there are uncountably many smooth structures on a fixed topology raises a couple of problems.
>>Sum up the result to get Z = sum_i=1^N Z(S_i)
> Okay, in such a finite case, a simple counting measure is a reasonable > choice. But which measure could we choose on R^4, for example? Or on the > larger set of all (diffeomorphism classes of) 4-dimensional smooth > structures (not necessarily representing the same homeomorphism type)? I > have some vague ideas, but this seems to be a subtle issue.
I have no idea which measure to choose in such nontrivial cases. Please let us know about your ideas here.
> A propos what would happen: In our > Einstein/Hilbert example on the topological manifold R^4, which > questions would seem interesting to you?
Thinking about it, a central question would probably be related to observability and suppression of fluctuations of smooth structures.
What would an exotic R^4 "look like" for those living in it? Would they still see plane waves passing by (approximately)? Can we have a flat metric on an exotic R^4? In the paper hep-th/9810201 that Xi Yin mentioned it says that it is not known if one can have Einstein metrics on exotic spheres.
If an exotic R^4 really would look exotic to its inhabitants, then the next thing I would like to know is how much it is suppressed in the path integral, compared to ordinary R^4. Since the path integral weight exp(-S) depends on the smooth structure only very indirectly, this is probably very hard to see.
Urs Schreiber wrote: >I forgot to reply to the following interesting question that Marc Nardmann >posed:
>>A propos what would happen: In our >>Einstein/Hilbert example on the topological manifold R^4, which >>questions would seem interesting to you?
>Thinking about it, a central question would probably be related to >observability and suppression of fluctuations of smooth structures.
Okay, so the question is how to verify fluctuations of the smooth structure experimentally.
>What would an exotic R^4 "look like" for those living in it? Would they >still see plane waves passing by (approximately)?
I guess everything depends on the length scale. Leaving fluctuations aside, is the universe we live in diffeomorphic to R^4? Or to S^3 x R? Or to an exotic R^4? If (differential-)topological complications show up only on very large (or very small) scales, would we notice? (Regarding S^3 x R, the WMAP data might help. I assume you have heard of http://arxiv.org/abs/astro-ph/0310253.) On a small/medium scale, there can be plane waves, of course.
Since there are no exotic R^3s, an exotic R^4 is not diffeomorphic to a product of a 3-manifold with R. So if it's equipped with a Lorentzian metric, there occurs "change of spatial topology". This might be observable on a long enough time scale.
I admit that I'm a bit confused by "fluctuations of the smooth structure". We need some kind of interference phenomenon to detect those, don't we? I.e. an observable that can be measured repeatedly by an "experimenter" such that the resulting probability distribution shows a "destructive interference" effect, not compatible with the assumption that each experiment E involved only one smooth structure A(E)...
>Can we have a flat metric on an exotic R^4?
Every exotic R^4 admits a flat Riemannian and a flat Lorentzian metric (but not a geodesically complete one). The reason is that every contractible smooth n-manifold can be immersed into R^n. The immersion can be used to pull back structures (e.g. flat metrics) from R^n to the manifold.
>In the paper hep-th/9810201 that Xi Yin mentioned it says >that it is not known if one can have Einstein metrics on exotic spheres.
I don't know, but if this is unknown, I'm not surprised: constructing Einstein metrics is not easy. However, it is well-known that some exotic spheres do not admit any Riemannian metric of positive scalar curvature, whereas some others admit metrics of positive Ricci curvature.
>If an exotic R^4 really would look exotic to its inhabitants, then the next >thing I would like to know is how much it is suppressed in the path >integral, compared to ordinary R^4. Since the path integral weight exp(-S) >depends on the smooth structure only very indirectly, this is probably very >hard to see.
Indeed. Now we should really worry about the measure. But unless it behaves very strange, the measure of the standard R^4 (a one-point subset of an uncountable measure space) should be zero.
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