|Dec30-12, 07:57 AM||#18|
Global emergent time, how does Tomita flow work?
marcus (#16), yes, that helps me, and I think I understand your point. I do like the idea, as you said history shows that was the way in many areas, including algebraic geometry, Grothendieck's point of view, which can be paralleled to Gelfand's. The geometric object is determined by a certain algebra, so the properties of the algebra can be abstracted and the theory can be built without the geometric object and in greater generality. To come back to my question if we start with a Lorentzian manifold, what is the algebra? I suppose the generators have to correspond to measurements, but this needs a more formal description, and what are the relations? Now that I said this, I seem to remember seeing a paper by Geroch, where, if I am not mistaken, he was defining an algebra of observables, but I have to find it and look at it again.
#17 Oh, yes, I have looked into his life, he is indeed quite remarkable.
|Dec30-12, 11:24 PM||#19|
Martin, this caught my attention and I'm not sure how relevant it is:
William Donnelly is a PhD student advised by Ted Jacobson at U. Maryland, and he expects to finish his thesis in 2013. This paper was published in PRD 2012.
Decomposition of entanglement entropy in lattice gauge theory
(Submitted on 31 Aug 2011 (v1), last revised 26 Apr 2012 (this version, v2))
We consider entanglement entropy between regions of space in lattice gauge theory. ...
One thing that got my attention was on page 1:
"Closely related to lattice gauge theory is loop quantum gravity, which is formulated as an SU(2) lattice gauge theory on a superposition of lattices. Although this paper will not discuss loop quantum gravity, entanglement entropy in loop quantum gravity was discussed in Refs. [20, 21], and we expect the techniques of this paper to generalize easily to a superposition of lattices. We note also that the Hilbert space of edge states in SU(2) lattice gauge theory is closely related..."
Donnelly's earliest paper, published PRD 2008, when he was a Master's student at Waterloo advised by Achim Kempf, is:
Entanglement Entropy in Loop Quantum Gravity
(Submitted on 6 Feb 2008)
The entanglement entropy between quantum fields inside and outside a black hole horizon is a promising candidate for the microscopic origin of black hole entropy. We show that the entanglement entropy may be defined in loop quantum gravity, and compute its value for spin network states. The entanglement entropy for an arbitrary region of space is expressed as a sum over punctures where the spin network intersects the region's boundary. Our result agrees asymptotically with results previously obtained from the isolated horizon framework, and we give a justification for this agreement. We conclude by proposing a new method for studying corrections to the area law and its implications for quantum corrections to the gravitational action.
I will try to explain why this interests me. If I take seriously what Rovelli and Connes say about the problems with time in both GR and QG, then I suspect there will be a future development of LQG in the C* format. That is, using (M,ω). Then one gets a global emergent time, "T-time", that depends only on the state ω, not on the observer. Looking ahead, how will a classical manifold geometry be recovered?
So that is the question that's on my mind, at the moment. If they go that way, with (M, ω), how will they get some conventional geometric stuff back out? I can see how they could get the equivalent of a Cauchy surface. The state ω might enforce a bounce, and that is a place to start counting T-time---a reference marker for the Tomita flow. So one gets to specify a subset of the algebra M, a particular moment in effect. Now one can define a REGION of that "cauchy surface", and its complement. Subsets A and Ac.
This is the sort of thing discussed in the Bianchi Myers paper. The entanglement entropy between a region and its complement. And Bianchi Myers cite Donnelly's 2008 paper about entanglement entropy between regions in LQG. I suspect something is brewing here. I'll be interested to learn what Donnelly's thesis turns out to be about.
|Jan1-13, 03:37 PM||#20|
Since we're on a new page, I'll summarize. Links will come later, after the summary. Anyone who persists in trying to represent the world as a 4D manifold with fields plastered on it will have trouble with time. The alternative is to represent the world as a star-algebra M and a state ω representing what we think we know about it (i.e. correlations amongst observable etc.).
Technically a C*-algebra is an abstract generalization of a von Neumann operator algebra. Axiomatizing the observable algebra allows getting rid of the Hilbert space and a quantum state becomes a positive functional ω:M→ℂ, on the abstract algebra M. von Neumann would approve
Given a state functional ω on M, Gelfand and friends tell us how to construe M as a Hilbert space HM. This is really great! We were not given a hilbertspace to start with, but anytime we want we can recover one that M ACTS ON as operators.
The abstract star operation on M becomes a conjugate linear transformation S: HM → HM defined on Gelfand's hilbertspace. This is something new, so things begin to happen.
Because HM has an inner product, we know what the ADJOINT of S is. The inner product tells us, see earlier post. Call the adjoint S*. The operator product of S* with S is positive and self-adjoint. Such an operator can be raised to complex powers (think of diagonalizing a matrix and raising the eigenvalues.) In particular the operator S*S can be raised to the power i.
Tomita now defines a UNITARY operator Q = (S* S)i on the (Gelfand) hilbertspace HM.
Real powers of Tomita's unitary Q correspond to the passage of an observer-independent world time. viewed as shifting measurements around amongst themselves.
The Tomita flow can be considered as a map M → M from earlier measurements to later ones, defined by
A → Qt A Q-t
This can be thought of as taking a measurement A in M to a corresponding measurement made t units of time later. To take an example, we can think of the Tomita flow converting a measurement A into one made, say, 200 years later (i.e. Q200 years A Q-200 years) this way:
"The later measurement is what you get if you undo 200 years of change, perform the earlier measurement, and then restore 200 years of change."
Here "change" means Qt, the Tomita unitary raised to a real number power. The exponent t would be 200 years expressed in natural (Planck) time units.
T-time is the logarithm of change to the base Q.
When specific cases are considered and the arithmetic is done, the units of T-flow time turn out to be Planckian natural time units. Technically this is called an "automorphism" of the algebra M, and letting t range along the real line ℝ we get a "one-parameter group of automorphisms" defined on M. A flow for short.
|Jan1-13, 04:04 PM||#21|
There's a fuller listing of relevant links in post #2. I've tried to boil that down to essentials.
Video of part of a talk on Tomita time by Matteo Smerlak:
Article by Alain Connes and Carlo Rovelli:
Seminal 1993 paper, The Statistical State of the Universe
http://siba.unipv.it/fisica/articoli....1567-1568.pdf showing how [what was later realized to be Tomita flow time] recovers usual conventional time in several interesting cases including standard cosmology.
Ratios between a local observer's time and the global emergent T-time can be physically meaningful as shown in the paper by Smerlak and Rovelli http://arxiv.org/abs/1005.2985
T-time fills a need in formulating general covariant statistical mechanics and general covariant statistical QM http://arxiv.org/abs/1209.0065
Section in Princeton Companion to Mathematics on the Tomita flow:
Post #2 has more discussion along these lines.
Personally I think that the recent paper by Bianchi and Myers is potentially applicable in this context, as a way of revealing the regional structure implicit in the algebra and state (M, ω). Independent of any particular observer.
|Jan2-13, 12:13 PM||#22|
I'm concerned about some points though, this is a complete change in paradigm and one has to get used to it, I mean, people found weird that theories in physics might come in different number of dimension (4, 9, 10, 11, 27....), but this is a deeper switch to maybe no manifold at all. I guess I'm still really fond of the old manifold, and I would like to see that this idea get some empirical backing soon.
I listened to the Connes lecture linked here and found it full of ideas, and amusing(not only for his strong accent ). It is really great that his noncommutative algebraic geometry model is able to recover most of phenomenology of the standard model of particle physics, but since the aim of quantum gravity is to bridge the gap between QM and GR and ultimately make them compatible, I missed the part where these star-algebras give any prediction or explanation about the gravitational part.
|Jan2-13, 02:48 PM||#23|
T.D., I will have to respond to your post piecemeal--a bit at a time. Thanks for the stimulus. Just now, wondering how to reply to your questions, I found what looks like an excellent review paper (July 2010). Published in the Russian online journal SIGMA special issue on "noncommutative spaces and fields."
Modular Theory, Non-Commutative Geometry and Quantum Gravity
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul
(Submitted on 23 Jul 2010 (v1), last revised 19 Aug 2010 (this version, v2))
This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
47 pages. (cites 260 items)
I like it's being comparatively self-contained. Where possible the level is pedagogical--they define what a C* is, in basic terms, instead of assuming the reader already knows stuff like that. From what I read, the writing is clear.
Ideologically they are strongly on the side of making what we ordinarily think of as differential geometry realities "take shape" in algebra context. They put this better than I just did. We have to learn ways to somehow "recover" familiar manifold geometry out of a C* setup. Again, I'm tempted to go fetch quotes instead of saying it in my own words.
I didn't know of Roberto Conti before. Here are his 31 papers on arxiv:
They seem mostly in the "math.OA" branch of arxiv---the "operator algebras" part of the math arxiv.
So you only get a subset of his work if you just search in physics. Snapshot: http://owpdb.mfo.de/detail?photo_id=10625 http://owpdb.mfo.de/detail?photo_id=16368
|Jan3-13, 02:56 PM||#24|
About this paper I mentioned:
Modular Theory, Non-Commutative Geometry and Quantum Gravity
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul
T.D., the part of it relevant to your question starts on page 23. That is, digestible portions of section 6:Perspectives on Modular Algebraic Quantum Gravity pages 23-38.
Especially some excerpts from 6.1 (Construction of modular spectral geometries)
6.2 (Physical meaning of modular spectral geometries)
6.4 (Finding the macroscopic geometry) very short because either just beginning or tentative work in progress, only about a dozen relevant papers are cited here.
6.5 (Connection with other approaches to quantum geometry)
6.6 (Quantum physics)
My sense is that Paolo Bertozzini, the junior author was the main motivator of this work. To get an idea of who Bertozzini is, one can watch a portion of the YouTube where he gives a lecture at Imperial College in London about it. Unfortunately in the part I watched the camera stays pointed at the lectern and does not follow the speaker to the blackboard or slidescreen. This is frustrating, so overall the YouTube seems useless. But a few minutes watch gives a sense of the person.
In footnote 14 on page 23, it says Bertozzini, Conti, Lewkee have a work in progress called
Modular algebraic quantum gravity. It is somewhat discouraging to me that this work has not appeared.
My intuitive feeling is that the key to getting geometry out is by way of a C* setup that imitates deSitter---a single bounce. Or one of the other one-bounce cosmic models. That may provide a reference time. A time zero for the Tomita flow.
This would in effect "slice" the C* algebra according to T-time slices. In that case one would have a chance to define and study the subsets of the algebra corresponding to 3D regions, and their boundaries.
|Jan3-13, 03:08 PM||#25|
|Jan3-13, 03:58 PM||#26|
I overlooked something that actually makes the Bertozzini YouTube potentially quite useful! His SLIDES are available at the Oxford University site.
I didn't realize that when I posted #24 earlier. (mistake highlighted in red)
Published on May 1, 2012
Speaker: Paolo Bertozzini (Thammasat University)
Title: Categories of spectral geometries
Event: Categories, Logic and Foundations of Physics II (May 2008, Imperial College London)
Abstract: In A. Connes' non-commutative geometry, "spaces" are described "dually" as spectral triples. We provide an overview of some of the notions that we deem necessary for the development of a categorical framework in the context of spectral geometry, namely: (a) several notions of morphism of spectral geometries, (b) a spectral theory for commutative full C*-categories, (c) a tentative definition of strict-n-C*-categories, (d) spectral geometries over C*-categories. If time will allow, we will speculate on possible applications to foundational issues in quantum physics: categorical covariance, spectral quantum space-time and modular quantum gravity.
Much of the first 3/4 of the slide set is rather technical abstract math. BUT the last quarter or so addresses the problem of how you do QG in star-algebra context, and recover geometry. Efforts to do this have been made by various people, along various lines. I found it interesting to see the different things that have been tried, going back even to around 1972. The slide set (last 1/4) has many references, including to a 1999 paper by Rovelli that I didn't know about:
|Jan5-13, 12:16 AM||#27|
Marcus: your summary post #20 is very helpful. But I still find the concept of our very complicated "world" described (or represented mathematically) "as a star-algebra M and a state ω representing what we think we know about it " very abstract, but intensely interesting, even for a non-mathematician.
I believe that examples often help comprehension. It might be useful to describe the nuts and bolts of how some small toy quantum "world" could be described along these abstract lines. An example I can think of is a small finite world of one-dimensional simple harmonic oscillators confined by potential-barrier walls. Would Tomita time in such a world connect to anything we think we know about this world? Such as the half-life of radioactive particles it might crudely model?
|Jan5-13, 10:54 AM||#28|
What i picture, at first, is something so elementary and finite that many things about it would be trivial. Tomita time, for instance, would probably not flow. I picture an example based on a tetrahedron. Perhaps four area observables generate the algebra.
Two tetrahedra can be joined to form a 3D hypersphere---not embeddable in our 3D space but still interesting as a simple compact boundaryless space.
Or even simpler, as you suggested, something in lower dimension. start by thinking of a triangle---say equilateral at first. Three length observables (analogous to the tet's four area observables.) then think of gluing two equilateral triangles together to form a 2D sphere. And let them, after all, not be equilateral (that was only to help me imagine them for starters.)
Then think of adding some kind of field observables that live on this simple 2D sphere world, and describe the observable algebra. It might be a lovely algebra! But it's not even 9AM in morning here and already I feel like a dummy, completely inadequate
It's a really good idea to get a pedagogical example of this kind of thing. Maybe there is someone I can ask, for whom it would be easy. I'm very new at this M,ω stuff (and old in years, which doesn't help either.)
|Jan5-13, 10:36 PM||#29|
A toy model (M, ω) would correspond to having only a finite number of sites where measurements are made.
This could cause M to be finite dimensional. That is why I mentioned a tetrahedron.
You know there are finite groups, even finite fields (analogs of ℝ and ℂ, but actually with only finitely many elements.) things like that can be good for toy models.
From the different vertices of the tet one might measure angles, distances. Area operators might be defined. I suppose a matter field could be defined on that minimalist picture. Some labels at the vertices, or along the edges.
So that would add more operators to M, some having to do with matter, as well as the geometric measurements.
I think the "spin networks" that Penrose introduced, and which are the basis of Loop gravity, would serve as a basis for this kind of finite-dimensional (M,ω). In usual LQG they can carry matter fields as well as geometric information. sometimes the labels on the vertices and edges of the graph are Lie group elements, sometimes finite group elements, something group representations e.g. spins, sometimes (with Lewandowski) the labels can even be operators, as I recall. There are different styles.
But essentially a spin network is a finite combinatorial/algebraic structure. So it would be natural devise a way to transform such a thing (a labeled graph) into a finite dimensional star algebra M and state ω.
Then one might imagine that by making the spin network more and more complicated one might get a more realistic picture, not so "toy". My intuitive feeling is that is not satisfactory, some more creative math has to happen to get up out of this toy level. But at least it seems to offer a way to construct toy models of increasing size and complexity---to sort of ramp things up. Just speculating.
I still haven't described enough structure to allow for time-evolution, really need an infinite dimensional star algebra for that, I suspect. I'll keep an eye out for the kind of thing you mentioned---a toy model that can illustrate Tomita flow. Wish I knew of one that we could examine right now, but I don't.
|Jan6-13, 08:07 AM||#30|
Thanks for these thoughts, Marcus. The idea of making a toy model out of glued-together triangles (or tetrahedra joined to form a 3D hypersphere) in order to provide observables is rather abstract to me, especially since I've (probably mistakenly) considered spheres (2- or hyper-) as geometrical constructs with a center-like item equidistant from some suitable n-dimensional perimeter; i.e. circle related. I'm hoping for a more physics-based model -- a structure based on triangles or tetrahedra sounds to me related to a Buckminster-Fuller dome. Probably my limited knowledge.
Let's hope that somebody can attenuate the abstract nature of Tomita flow sufficiently for us to connect it smoothly with the time that passes for us all. At a rate inversely proportional to the years that remain for each of us!
|Jan6-13, 02:17 PM||#31|
So why doesn't LQC seem to use thermal time?
Does thermal time give rise to a preferred foliation, since it is global and observer independent?
I noticed one Bohmian in a one particular World advocating a preferred foliation (maybe that's not so accurate - he says "proper foliation"): http://arxiv.org/abs/1205.4102
Can Bohmians use thermal time, but instead of thermal equilibrium, use the Bohmian quantum equilibrium?
|Jan6-13, 03:07 PM||#32|
And LQC uses F. time (agrees with classical expansion after first few planck seconds. So LQC runs on T-time.
Cosmology, as you know, has a global preferred time that is independent of observer. And has a starting place. So cosmology has a foliation. So since T-time recovers Friedmann time it would have to have a foliation too, given the same helpful assumptions.
About Bohmians, I wouldn't know. Maybe someone else here. My hunch is that (M,ω) formulation obviates or weakens motives for B.
|Jan6-13, 03:39 PM||#33|
such as: from one of Ashtekar's papers.....we discussed here:
How has LCG resolved the Big Bang Singularity?
Which is a better set of papers, in your opinion, to mull over, those of Ashtekar in the other thread or the papers in this one?? [I sure like the Matteo video...that's as far as I have gotten here...]
|Jan6-13, 04:34 PM||#34|
A good half-page explanation is in rovelli's essay "unfinished revolution" Google "rovelli revolution". From like page 3 or 4 , gave link earlier.
You've got to understand the problem. It is serious and across-the-board.
Maybe the relevance is to understanding the narrowing separation betw. LQC and LQG. People who don't get this treat them as static and don't realize how much overlap is growing. Have to watch over time and get a sense of momentum, rates of change.
Full LQG already has a simple case of bounce cosmology, recovered deSitter. Full LQG has the cosmological constant. These are developments in the past 3 years or so on that side.
Meanwhile LQC has made remarkable progress in past 3 years with increasing the complexity of the models to include more degrees of freedom---so more realistic, more fluctuations, more matter, just last year bringing in Fock space.
These are fast moving research programs in the process of merging.
That's an important perception to understanding and knowing what to expect. I guess one reason to read Ashtekar papers is to get a sense of that---a feel for how it's going on the LQC side. But I don't see his papers so relevant to the time issue itself. Maybe indirectly.
One of Ashtekar's best PhD students is now a postdoc at Marseille. He has written a great paper (several actually). I would almost say THAT is what one should study to keep up with LQC.
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