# Can Conne's NCG be reconciled with Verlinde's gravity as entropy?

1. May 10, 2010

### ensabah6

Verlinde has proposed to express gravity as a thermodynamic system, whereas Connes shows with a purely geometric theory, the SM lagrangian can be derived. Are the two theories reconcilable? i,e NCG with gravity as equation of state, perhaps given some type of canonical loop quantization?

http://arxiv.org/abs/1005.1169

Debye entropic force and modified Newtonian dynamics

Xin Li, Zhe Chang
(Submitted on 7 May 2010)
Verlinde has suggested that the gravity has an entropic origin, and a gravitational system could be regarded as a thermodynamical system. It is well-known that the equipartition law of energy is invalid at very low temperature. Therefore, entropic force should be modified while the temperature of the holographic screen is very low. It is shown that the modified entropic force is proportional to the square of the acceleration, while the temperature of the holographic screen is much lower than the Debye temperature $T_D$. The modified entropic force returns to the Newton's law of gravitation while the temperature of the holographic screen is much higher than the Debye temperature. The modified entropic force is connected with modified Newtonian dynamics (MOND). The constant $a_0$ involved in MOND is linear in the Debye frequency $\omega_D$, which can be regarded as the largest frequency of the bits in screen. We find that there do have a strong connection between MOND and cosmology in the framework of Verlinde's entropic force, if the holographic screen is taken to be bound of the Universe. The Debye frequency is linear in the Hubble constant $H_0$.

http://arxiv.org/abs/1005.1174

Notes on "quantum gravity" and non-commutative geometry

Jose M. Gracia-Bondia
(Submitted on 7 May 2010)
I hesitated for a long time before giving shape to these notes, originally intended for preliminary reading by the attendees to the Summer School "New paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I decide against just selling my mathematical wares, and for a survey, necessarily very selective, but taking a global phenomenological approach to its subject matter. After all, non-commutative geometry does not purport yet to solve the riddle of quantum gravity; it is more of an insurance policy against the probable failure of the other approaches. The plan is as follows: the introduction invites students to the fruitful doubts and conundrums besetting the application of even classical gravity. Next, the first experiments detecting quantum gravitational states inoculate us a healthy dose of skepticism on some of the current ideologies. In Section 3 we look at the action for general relativity as a consequence of gauge theory for quantum tensor fields. Section 4 briefly deals with the unimodular variants. Section 5 arrives at non-commutative geometry. I am convinced that, if this is to play a role in quantum gravity, commutative and non-commutative manifolds must be treated on the same footing; which justifies the place granted to the reconstruction theorem. Together with Section 3, this part constitutes the main body of the notes. Only very summarily at the end of this section we point to some approaches to gravity within the non-commutative realm. The last section delivers a last dose of skepticism. My efforts will have been rewarded if someone from the young generation learns to mistrust current mindsets.

2. May 11, 2010

### Fra

I'lll admit that I so far didn't quite understand the physical logic behind the NCG research. Like, that physical or conceptual questions suggest this way of asking questions?

The quest of trying to cast all non-commutative structures into some kind of new geometric language seems to be more like a motivated by a mathematical fascination for geometric formulations.

I guess noone has avoided that geometrical methods has been for some time a very popular "form" and representation, so given that the NCG program follows the same logic, but I personally don't think that geometry per see, is necessarily a primary concept. The sound of the word geometry, is to be quite realist minded, and I am not convinced that's what we seek.

Isn't what we seek, a deeper explanation for why a certain geometry is apparent?

Can someone maybe flesh out briefly the main PHYSICAL motivation and points of NCG? I know there are apparently non-commutative spaces etc, but isn't the question what the origin of the non-commutative structures are, rather then just interpreting them into a new geometric language - in what sense does it solve any of the problems?

I'd be interested to hear if someone can outline a bit more of the physics of NCG. I notice that several instances describes NCG more as a "mathematical discipline" rather than physics. That somehow makes more sense, but I still assume that some of you has physical reasons for liking this beyind just beeing "geoemtry geeks"? :)

I admit that I OTOH is an "information geek", and wants to cast everthing in terms of intrinsic information, but the physical and conceptual reason for this is that information, and inference are indeed very fundamental concept relating at a deep level to both inference in science and the measurement process.

I guess what I lack, is a similar "non-realist" motivation for hte NCG program - is there one?

/Fredrik

3. May 12, 2010

### paolo_th

I will try to provide some motivation for non-commutative geometry (at least from my point of view).
First of all a disclaimer: there are many different mathematical approaches to non-commutative geometry and most of what I say here applies mainly to A.Connes' version.

Geometry (in its usual sense) has always been strictly linked with classical physics: Euclidean geometry describes the physics of classical 3-space and 1-time; calculus and differential geometry of curves is at the foundations of classical Newtonian mechanics; Hamiltonian mechanics is just symplectic geometry; Minkowski geometry is at the basis of special relativity and Einstein's general relativity is essentially Lorentzian geometry ... even thermodynamics admits a description in terms of differential forms!
All the classical theory of gauge fields (a mathematically rigorous theory of "quantum" gauge fields does not exists yet!) is based on the geometry of vector bundles.
In this context, the first anomaly (i.e. a physical theory that apparently is deprived of a geometrical description) is quantum physics. Non-commutative geometry gives a "geometrical meaning" to most of the algebraic constructions in quantum physics.

Anyway, to say that non-commutative geometry provides a "geometrical interpretation for non-commutative (quantum) structures" is not wrong but a bit reductive and somehow is missing the main message !

A.Connes non-commutative geometry is the most sophisticated recent incarnation of R.Decartes' analytic geometry: instead of studying the geometry of a space ... we study algebra using the coordinate functions on that space.

The starting point is in the existence of dualities (i.e.contravariant equivalences) between categories of "classical geometrical spaces" and categories constructed out of "commutative algebras".
In practice every (say compact topological) space X is associated to a commutative C*-algebra of coordinate (continuous complex) functions C(X) and the important point is to realize that all the information on the original space X can be completely recovered by its commutative C*-algebra C(X): the space X is homeomorphic to the "spectrum" of C(X). This is the celebrated Gel'fand-Naimark duality.
Under this point of view working geometrically with (compact Hausdorff) topological spaces is perfectly equivalent to working algebraically with commutative C*-algebras (with identity): all of the geometrically meaningful properties of the space X can be algebraically codified in properties of the C*-algebra C(X).
Now the point is that although we might have usually no clear idea on how to define and deal with spaces that are more general than X ... on the other side of the duality, the codifications of geometrical properties in terms of algebraic properties still makes perfect sense if the condition of commutativity of the C*-algebra is eliminated and hence (for example) non-commutative C*-algebras can be considered as algebras of "functions" on some "generalized, but not directly defined" non-commutative spaces!

Many other dualities/equivalences are used is non-commutative geometry (Serre-Swan equivalence for example is between suitable categories of bimodules and categories of vector bundles and provides the starting point for the consideration of finite-projective bimnodules over non-commutative C*-algebras as modules of sections of vector bundles over non-commutative topological spaces).

The most effective non-commutative version of (spinorial-Riemannian) manifold is now formalized via A.Connes spectral triples (A,H,D): a pre-C*-algebra A acting as bounded operators on a Hilbert space H and a Dirac operator D that has commutators [D,a] with elements of A that can be extended to bounded operators on H.
Every compact orientable spinorial Riemannian manifold X provides a spectral triple via its algebra of smooth functions, its Hilbert space of spinor fields and its Atiyah-Singer Dirac-Pauli operator.
A recent theorem by A.Connes states that (under suitable conditions) every spectral triple (A,H,D) with a commutative algebra A is "isomorphic" to the Atiyah-Singer spectral triple of a spinorial Riemannian manifold X.

Now the main message is not to interpret non-commutative algebraic stuff geometrically ... but to
see that usual geometry is just a very small part of a much bigger landscape of geometrical non-commutative spaces. Exactly as Riemannian geometry provided A.Einstein with the suitable mathematical environment to formalize general relativity, the hope is that this much more powerful and general (non-commutative, quantum) geometries might provide the right mathematics (quantum Riemannian manifolds) for a quantum version of Einstein theory!

What are the physical motivations to consider this bigger landscape of non-commutative geometries?
Strictly related to the hopes for a mathematical theory of quantum gravity there are at least three:
1)
most of the solutions of Einstein's equations are singular ... and hence general relativity apart form trivial cases is mathematical inconsistent. The hope is that non-commutative spaces might provide "geometries" that are more suitable than usual Lorentzian manifolds to deal with singularities (black holes, big bangs etc.). Similarly apart form free fields, quantum field theory (mathematically) does not exists yet and singularities have been a disturbing feature of the theory: there is a widespread belief that modification to the short scale structure of space-time might help to resolve the problems of “ultraviolet divergences” in quantum field theory.
2)
combining quantum mechanics with general relativity we know that Minkowski space-time is not physically stable and hence it cannot describe the actual geometry of space-time ... the hope is that non-commutative geometries might do just that.
3)
in order to include the remaining physical forces (nuclear and electromagnetic) in a Kaluza-Klein “geometrization” program, going beyond the one realized for gravity by A.Einstein’s general relativity, it might be necessary to make use of non-commutative geometry ... and actually we already know that this is true because, at least at the classical level, ... this is exactly what A.Connes has been able to do with His non-commutative interpretation of the standard model: all the interactions are geometrically obtained via an (almost-commutative) spectral triple: Einstein's dream fulfilled with a mild form non-commutative geometry!

Some time ago, I did elaborate much longer on these and similar motivations in a survey paper http://arxiv.org/abs/0801.2826" [Broken] that might be useful, although a bit dated, also for references to several approaches to quantum-gravity via non-commutative geometry (a version updated to November 2009 will appear on the arXiv very soon).
[This post is already too long to add here the several interesting references].

Last edited by a moderator: May 4, 2017
4. May 12, 2010

### ensabah6

Very fascinating.

The premise of LQG is that geometry can be quantized. Evidentally since NCG has SM, the geometry should be NCG.

Do you think combining NCG with LQG is sensible based on this view? And if gravity is entropic thermodynamic, can that also be combined?

Last edited by a moderator: May 4, 2017
5. May 12, 2010

### skippy1729

commutative C*-algebra <--------------> compact hausforff topological space

non-commutative C*-algebra <---------------> Generalized NC "geometry"

This situation seems quite similar to the status of hyperbolic (Lobachevskii) geometry before there were "concrete physical" models like the Poincare Disk.

Is there a simple "concrete physical" model for NC geometry?

Skippy

6. May 12, 2010

### humanino

NCG is a continuation of Von Neumann's program. In his book (fundation of QM) Von Neumann argues that there is no need to use Dirac's ill-defined formalism. The geometrical interpretation comes along with the rigor, it is not an a-priori motivation.

Of course, it did not pick up among the physicists, because the formalism is considerably more involved and does not allow for simpler computation (on the contrary).

I fully agree with paolo_th's description of the situation, I just have a few points to add.

1) Historically we gained much insight by giving strict rigorous sense to the monstrous manipulations Fourier began to play with. I would argue, that rendered QM altogether possible through Hilbert spaces. It is a very well motivated argument that we have to take Von Neumann's message seriously.

2) More specifically, QFT already requires the general methods of type-III algebra. In practice we always perform calculation "as if" we were using type-I algebra. This type of coincidence where things work although we do not know why is the reason I refer to Fourier.

3) NCG does not need to be "reconciled with Verlinde's approach". Rovelli and Connes' work on the thermodynamics of gravitational dof (not on curved spaces) is a proposal to answer other deep questions raised by the analogies between time, thermodynamics and QM. It is a program to identify the relevant microscopic dof algebraically, and trying to use Verlinde's approach would be a catastrophic leap backward.

7. May 12, 2010

### ensabah6

Doesn't sound like you're much of a fan of gravity=entropy.

Interesting. Do you think attempting to use loop quantization in NCG is the right approach to quantization? Witten and others have writtten of string theory and NCG -- does NCG find more of a home in strings or in loops?

8. May 13, 2010

### Fra

Paolo, thank you very much for presenting your motivation! It's much appreciated!

I think understand the general idea, but from my perspective the idea is still a bit ambigous and the arguments are more "mathematical extrapolations". And with that I don't mean lack of mathematical rigour. I mean that in attempts to try to axiomatize physical theories and law, there are still choices, and I understand that in the picture you show, by starting to abstract classical physics, including GR on a particular "form" - geometry. Then there is indeed a "natural" extrapolation of thse models to more general types of geometries, that doesn't fit into classical physics.

I understand that, but this is to me, a mathematical guidance, and somehow seems to go hand in hand with the attempts to somehow axiomatize physics, and construct almost like a formal axiomatic system.

I have some issues with this way of thinking, and it is that you still have the problem of howto navigate in the set of all axiomatic systems. And I also have a feeling that Connes, like rovelli is a strong structural realist - this goes conceptually well with the axiomatic approaches.

My key objection, is that I want the axiomatic system, to be a result of a physical process, not as a result of "consistency requirements" that so to speak "lives" external to the physical interactions.

If I let my imagination run, this is the possible connection I see with entropic ideas. That the axiomatic system, seen as coded in physical systems (just beeing a "representation" of it) are evolving, by darwinian and entropic mechanism.

The so called "consistency constraints" commonly argued from the mathematical perspectives can I think be overdone, as the constraints themselvs are subject to uncertainty. I don't like when this is forgotten.

One view I've heard is that axioms are somehow "purified" extractions infered from experiment. And this is I think correct, but, then if this is to be coherent, this must be taken seriously and this "inference from experiment" is actually a physical process, and must be taken into account. I find it very incomplete to just handwave about this and refer to "science". I think that not only are physics a science, there is also a "physics of science" which is exactly what I think questioning the inference and measurement processes are about.

I think your three points are somewhat general points that supports the quest for a new framework somehow, but I don't follow that they would be specific to NCG. NCG might be a possible resolution, but maybe strategy X and Y will too.

But, I think you have answered my question, thanks. I still don't see the physical motivation but you've explained still another way of seeing it.

I think my problem is that I have a problem with the axiomatic approaches, which are mathematiclal rigourous, byt IMO physically and philosophically ambigous. And the only way I can make sense of that approach is in combination with some form of structural realism - another point where I have issues. Structural realism is just a form of realism where you consider transformations of some original information, into some more abstract form, that seems to be only "mathematical" and then considers this to be more fundamental. But it doesn't change the information. I think in a mesaurement or inference model, all information, regardless of how it's transformed or represented, must is a result of a physical process.

/Fredrik

9. May 13, 2010

### Fra

If I may try to see this in a slightly more general way, not beeing specific to NCG or verlinde, but if we instead ask this:

Is this a way to reconcile or make contact with the axiomatizing programs, that by harder utilization of various consistenct constraints and postulated axioms, tries to come up with a consistent theory of QG and unification, and the more loose "evolving law", or entropic mechanics and darwinian style programs that are admittedly less mathematically rigorous but sometimes more conceptually coherent?

I think there is, and as I see it, the connection is that it's the axiomatic system that is evolving, and each axiomatic system (and thus the corresponding model/theory) is to be seen as a STATE of an evolving picture, just like the durrent dna of a humans is simply a state of an overall evolution.

If this connection is right, then one could still make alot of exploits but it would also suggest that there exists no fixed, timeless, objective finite axiomatic system that describes reality. Instead I think what ne need to understand that this "axiomatic picture" has still a value, but it's no more cast in stone, than is the human dna. But since evolution is slow, observations and inferences made in a short timescale realtive to the overall process will still find apparent timeless law, that can be codified into axioms, in the sense that their uncertainty is not distinguishable. But this does not mean we can be sure that at somepoint observations will be made that are not described by, or are even in conflict with the current axioms.

I think of it as somewhat of a fallacy to try to seek a to hard axiomatisation, without seeing that the axiomatisation is in possibly just a physical data compression process. Let's think of the the axioms are the dna of physical law, from which the theory follows deductively and without uncertainty. If we acknowledge that the axioms are inferred, from finite and limited historical interaction they are not certain, and the rigid thinking of reality as an axiomatic system doesn't make sense.

So I think the connection between "axiomatic systems" and deductive inference and "evolving systems" and inductive inference are one of the key things yet to understand.

In this sense, I don't agree at all that raising these questions is a leap back. On the contrary do I think understanding this connection will deepen our understanding of reality. Just like the understanding of the code of life, only makes sense if you see it in the context of evolution. The dna didn't just pop out of nowhere, it has evolved. The same with physical law.

To just play mathematical bird view games and axiomatize, is somehow totally missing this point.

This is just my personal opinon as always.

/Fredrik