Global emergent time, how does Tomita flow work?

[atyy]

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?

 

[marcus]

So why doesn't LQC seem to use thermal time?

I thought it did! That was one of the points Rovelli made in 1993 paper, the first or one of the first T-time papers. Also referred to in 1994 Connes Rovelli, towards the end. I quoted and gave link earlier in thread. Recovers Friedmann time if you make the usual Friedmann assumptions.

And LQC uses F. time (agrees with classical expansion after first few Planck seconds. So LQC runs on T-time.

Does thermal time give rise to a preferred foliation, since it is global and observer independent?

YES! if you have a preferred time zero to use as base point. Of course you need one starting slice of measurements to start with. Like e.g. the "bounce". Then with that as reference you can advance everything by one time unit and get another slice.

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.

 

[Naty1]

Marcus:

Anyone who persists in trying to represent the world as a 4D manifold with fields plastered on it will have trouble with time.

Is this a reference to problems in LQG which LQC is trying to circumvent??

such as: from one of Ashtekar's papers...we discussed here:

How has LCG resolved the Big Bang Singularity?

https://www.physicsforums.com/showthread.php?t=662565

...because LQG does yet offer the quantum version of full Einstein’s equations which one can linearize around a quantum FLRW spacetime.

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...]

 

[marcus]

Marcus:
...Is this a reference to problems in LQG which LQC is trying to circumvent??

No, not a reference to LQG or LQC in particular. Problem with time in GR was identified before LQG or LQC. Quantizing GR, however you try to do it, whatever approach, exacerbates the problem of time in GR.

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.

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...]

MATTEO SMERLAK's talk! Yes! I'm glad you like it. I'm not sure what the relevance of Ashtekar paper is. His work is always relevant and worth looking over.But it is not specifically about time---the topic here.

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.

 

[atyy]

[I sure like the Matteo video...that's as far as I have gotten here...]

MATTEO SMERLAK's talk! Yes! I'm glad you like it.

Yes, me too. Listening to his talk was the first time I understood anything about thermal time. Before that I knew of Connes-Rovelli but it was indigestible to me. Now that you've linked the Rovelli 1993, I think I shall see if I can make any headway with that.

 

[marcus]

I should recall this link from post #2 http://arxiv.org/abs/gr-qc/0604045 and the simple statement it gives on page 4 of the MOTIVATION for getting away from 4D manifold in any approach to QG, and why T-time is so interesting.
==quote page 4 of 0604045==
In general relativity, when we describe the dynamics of the gravitational field (not to be confused with the dynamics of matter in a given gravitational field), there is no external time variable that can play the role of observable independent evolution variable. The field equations are written in terms of an evolution parameter, which is the time coordinate x⁰, but this coordinate, does not correspond to anything directly observable. The proper time τ along spacetime trajectories cannot be used as an independent variable either, as τ is a complicated non-local function of the gravitational field itself. Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable. This does not mean that GR lacks predictivity. Simply put, what GR predicts are relations between (partial) observables, which in general cannot be represented as the evolution of dependent variables on a preferred independent time variable.

This weakening of the notion of time in classical GR is rarely emphasized: After all, in classical GR we may disregard the full dynamical structure of the theory and consider only individual solutions of its equations of motion. A single solution of the GR equations of motion determines “a spacetime”, where a notion of proper time is associated to each timelike worldline.

But in the quantum context a single solution of the dynamical equation is like a single “trajectory” of a quantum particle: in quantum theory there are no physical individual trajectories: there are only transition probabilities between observable eigenvalues. Therefore in quantum gravity it is likely to be impossible to describe the world in terms of a spacetime, in the same sense in which the motion of a quantum electron cannot be described in terms of a single trajectory.
==endquote==
It's an interesting article, if you ever want the whole thing and don't have the link handy, just google "revolution rovelli" without the quotes.

In classical GR, there is no physically meaningful (or observable) time variable that can be used to study a fully general covariant system. The proper time of an observer depends on already having a fixed geometry, a fixed spacetime. As a rule, coordinate time is not physically meaningful. In the quantum case the problem is still more severe. Quantum theory does not allow a 4d manifold spacetime to exist, any more than does the "continuous trajectory" of a particle. There are only correlations amongst observations made along the way. General remark, not limited to anyone particular QG approach.

So to achieve a fully general covariant (GC) analysis we need a world comprised of those observations, and the correlations amongst them. This motivates picturing the world as (M, ω), not as a 4d manifold with fields.

And then, as a bonus in a number of interesting cases, you get Tomita flow.

 

[marcus]

...Since we're on a new page, I'll bring forward part of post #20 giving a general summary of Tomita flow. Links will come later.

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.

Given a state functional ω on M, Gelfand and friends tell us how to construe M as a hilbertspace H_M. We were not given a hilbertspace to start with, M was given to us as an abstract algebra. But anytime we need to we can call up a hilbertspace that M ACTS ON as operators.

The abstract star operation on M becomes a conjugate linear transformation S: H_M → H_M defined on Gelfand's hilbertspace. This is something new, so things begin to happen.

Because H_M has an inner product, we know what the ADJOINT of S is. Call the adjoint S*, defined using the inner product, by ⟨S*X, Y⟩ = ⟨X, SY⟩. 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 H_M.
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 → Q^t 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. Q^{200 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 Q^t, 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.

 

[Naty1]

Marcus: Last question on 'Ashtekar time'...

I'm not sure what the relevance of Ashtekar paper is. His work is always relevant and worth looking over.But it is not specifically about time---the topic here.

When you answered an atyy question in this thread and said LQC of Ashtekar et al DOES use thermal time, I thought it likely WAS significant...Ashtekar seems to think so:

From the November paper:

...one can regard the background scalar field φ as a relational time variable with respect to which physical observables evolve. This is a new conceptual element, made necessary by quantum gravity considerations.

This was the first time I have ever read such a statement and am still trying to digest it. Is this significant or more a cop out?? Is this field even an observable??

Also, thanks for posting the Unfinished Revolution link...we discussed that sometime ago and I am even more impressed rereading it now when I understand a [little] bit more...thanks to you and others here...

 

[marcus]

Classical/semiclassical corroboration--chaos, volume gap

...When you answered an atyy question in this thread and said LQC of Ashtekar et al DOES use thermal time, I thought it likely WAS significant...

I think you drew the correct conclusion that T-time IS significant. But what you read from me probably wasn't clearly worded enough.
Also I'm no great authority on this T-time business, just beginning to get used to it. You realize that thermal time is NOT THE SAME as classical Friedmann model cosmology time. THEY JUST HAPPEN TO AGREE, which is very nice, but they are DEFINED DIFFERENTLY.

Likewise in conventional LQC going back on the order of 10 years Ashtekar and others have been using a scalar field as a relational clock. I regret to say I can't explain how this works, maybe later today, but that is NOT THE SAME as thermal time! It just happens to agree with classical Friedmann model time within a few Planck time units after the bounce, when the classical model becomes reliable.

And therefore it also agrees with thermal time.

You can have three things which mathematically are defined differently arising from different mathematical setups but which nevertheless give the same numbers where they overlap.

There will be places where they don't overlap because one or the other is not defined---the formal setup that it is based on and depends on for its definition fails for some reason. Like Friedmann time is real good back to near the start of expansion but then it blows up.
So Ashtekar has to patch it by putting some simple matter into the picture---his "scalar field". In a vague hand wavy way that is like putting an observer in who is somehow able to survive the bounce.

So they graft different ideas of time together to make a workable continuous one. And the fact that there is agreement on the overlap makes one confident that the definitions are right.

Thermal time, or Tomita time, is a comparatively new one for me. It is not the same. You need a different setup (the star algebra) which so-far Ashtekar is not using to do cosmology. But when you make the setups correspond---make enough assumptions to bridge between the different models of the world---then apparently you get agreement! I haven't gone thru all the steps so I have to take this partly on faith.

T-time is much more general. It is not limited to Friedmann cosmology or LQC. But when you make it apply to them as a special case then apparently it checks out. Which is really good, otherwise I probably wouldn't be so interested. It's nice to have a better machine but you want that machine to give the same answers you are used to in the old familiar situations, so you think maybe you can trust what it says in some new unfamiliar ones.

Also, thanks for posting the Unfinished Revolution link...we discussed that sometime ago and I am even more impressed rereading it now when I understand a [little] bit more..

Great! I had the same experience! Several times I've come back to that brief argument on page 4 of http://arxiv.org/abs/gr-qc/0604045 , just those 3 paragraphs, and each time it has meant more to me. Like you, each time I understand more of the context and it means more. Also as you said, thanks in part to others contributing to this thread and similar ones.

 

[Naty1]

Marcus:

Likewise in conventional LQC going back on the order of 10 years Ashtekar and others have been using a scalar field as a relational clock. I regret to say I can't explain how this works, maybe later today, but that is NOT THE SAME as thermal time!

AHA! No wonder I could not figure out how they were related...

great explanations in the prior post...thanks!

 

[Naty1]

Physics "TRASH TALK' :
Marcus I wondered what you thought about this:

from: http://arxiv.org/pdf/1007.4094v2.pdf pg 21

Modular non-commutative geometry in physics

Despite the fact that most, of the literature on non-commutative geometry is actually heavily motivated or directly inspired by physics (Heisenberg quantum mechanics, standard model, renormalization in perturbative quantum field theory, deformation quantization, just to mention a few) and the strong interest shared by theoretical physics for this mathematical subject, when compared to the outstanding structural achievements of Tomita–Takesaki modular theory in quantum statistical mechanics and algebraic quantum field theory, the fundamental relevance of non-commutative geometry for the foundations of physics looks still quite weak and disputable.

Whoaaaaaa, dudes...a little feud here? "still" ? A bit overstated maybe?? or not??

///////////////////////////
But man oh man was I relieved when I read the following...because so far reading on Tomita Time I have been thinking to myself, "How did these guys EVER tie all this together" ...and as you can see, it took a number of people, a number of approaches, and some time. It's not as if someone started with a grand vision...

The first indirect indications of the existence of a deep connection between (equilibrium) statistical mechanics (and hence modular theory [and Tomita time#]), quantum field theory and gravity (that, after A. Einstein’s theory of general relativity, essentially means geometry of four-dimensional Lorentzian manifolds) came, after J. Bardeen, B. Carter, S. Hawking results on black hole laws, from the discovery of entropy of black holes by J. Bekenstein [20, 21], black holes’ thermal radiation by S. Hawking and the vacuum thermalization ef fect by W. Unruh .

# My comment in parenthesis []

.. The point here is the existence of a correspondence between modular theory and von Neumann algebras on one side and Poisson geometry of classical systems on the other. The existence of an interplay between general relativity, gravitation and thermodynamics, has been reinforced by the important work of T. Jacobson that obtained for the first time a thermodynamical derivation of Einstein equations from the equivalence principle. This work has been further expanded, among several authors, by T. Padmanaban. This line of thoughts, has recently been exploited in order to infer that, being of thermodynamical origin, gravitation (contrary to electromagnetism and other subnuclear forces) cannot possibly be a fundamental force of nature and hence should not be subjected to quantization, but explained as a macroscopic phenomenon emergent from a different theory of fundamental degrees of freedom (usually strings) and after the recent appearence of E. Verlinde e-print on the interpretation of Newtonian gravity as an entropic force has led to a fantastic proliferation of research papers

So now it appears Gravity [relativity] AND time find their origins in thermodynamics!

 

[marcus]

Physics "TRASH TALK' :
Marcus I wondered what you thought about this:

from: http://arxiv.org/pdf/1007.4094v2.pdf pg 21

Modular non-commutative geometry in physics

I'll requote the passage manually without using the quote "button" so it does not go away so easily--and try to interpret. I'm actually still trying to understand this. Modular theory (Tomita flow etc) does use non-commutative algebra but not necessarily Connes NCG. It uses the C* format (M,ω) where for example the star algebra can be about a quantum field on a manifold space-time geometry. What Bertozzini and Conti are pointing out is that non-commutative geometry of the kind Connes has developed seems less utilized in physics than one might expect given the circumstances they point to. I think that's what they're saying.

As Bertozzini Conti Lewkee.. describe it, C* algebra approach is comparatively widely used and successful. But the specific Alain Connes approach which involves a "spectral triple" and dirac operator and special extra axioms has NOT lived up to expectations so far, so to speak. This could simply be because it is wrong (a possible C* approach to geometry but not the right one) or because the physicists have been slow to take to it, or it might need some more time to grow on them.

I don't see them as assigning blame for Connes NCG underutilization, but more simply pointing out an opportunity, some research for somebody to do. I could easily be wrong--this is just the message I get.

===quote Bertozzini Conti Lewkee.. page 21===
5.3 Modular non-commutative geometry in physics
Despite the fact that most, of the literature on non-commutative geometry is actually heavily motivated or directly inspired by physics (Heisenberg quantum mechanics, standard model, renormalization in perturbative quantum field theory, deformation quantization, just to mention a few) and the strong interest shared by theoretical physics for this mathematical subject, when compared to the outstanding structural achievements of Tomita–Takesaki modular theory in quantum statistical mechanics and algebraic quantum field theory, the fundamental relevance of non-commutative geometry for the foundations of physics looks still quite weak and disputable. In this subsection rather than discussing the vast panorama of applications of non-commutative geometry to physics and model building (see the book by A. Connes, M. Marcolli [87] for a recent very complete coverage of the physics applications of non-commutative geometry and, for a really pedestrian list of references, our companion survey paper [30]), we proceed to describe the very few available instances and hints of a direct applicability of modular non-commutative ideas (such as semi-finite and modular spectral triples, phase-spaces etc.) to physics.
===endquote===

 

[Naty1]

Marcus, your reply was surprising, because I read the except very differently...glad I asked...
so I tried to do some checking...without success...

The three Wikipedia links which follow have a lot of mathematical information for anyone interested, but too technical for me...

Too technical for me to be able to figure out how,for example, C* [Tomita] and W* [von Neumann] are different as applied to physical systems, and the mathematical terminology is also not so easy to understand ...Von Neumann algebra
http://en.wikipedia.org/wiki/Von_Neumann_algebra

C* Algebra
http://en.wikipedia.org/wiki/Hilbert_C*-moduleHilbert C* Module
http://en.wikipedia.org/wiki/Hilbert_C*-moduleOn a more basic level, here are some notes I made which may help introduce some of the concepts of thermal time to those, who like me, are new to the subject:


Thermal Time [Wikipedia]
http://en.wikipedia.org/wiki/Ehrenfest-Tolman_effect

The Ehrenfest–Tolman effect (also known as the Tolman–Ehrenfest effect), created by Richard C. Tolman and Paul Ehrenfest, argues that temperature in space at thermal equilibrium varies with the spacetime curvature. Specifically, it depends on the spacetime metric…….This relationship leads to the concept of thermal time which has been considered as a possible basis for a fully general-relativistic thermodynamics. It has been shown that the Tolman–Ehrenfest effect can be derived by applying the equivalence principle to the concept that temperature is the rate of thermal time with respect to proper time.

Talk by Matteo Smerlak:

The Tolman effect shows that in the presence of a stationary gravitational field temperature is not homogeneous at equilibrium…
T[x] is proportional to 1/[root{g(oo)[x]}] in stationary coordinates.

Is proportional to 1/c sqd so is a very small effect……the meaning of ‘thermal equilibrium’ is not obvious….typical descriptions... thermodynamically, information, KMS or stochastically IGNORE GRAVITY….non relativistically, at equilibrium,

thermal time = B x mechanical time…

Where B is temperature….and mechanical time is local time in GR, metric dependent….proper time…thermal time is the flow in phase space….
Tomita–Takesaki modular theory
Tomita Time…… Connes and Rovelli

descriptive comments
a global emergent time, [Marcus]
Hypothesis: the origin of physical time flow is thermodynamical,

Our postulate: that thermal time state defines the physical time,

an evolution of an external time parameter in generally co-variant theories,

the notion of time depends on the state [of] the system in a general co-variant context,

extending time flow to generally co-variant theories depends on the thermal state of the system,

time flow is determined by the thermal state.

 

[marcus]

Naty, thanks for posting your notes on the (first 15 minutes) of Matteo Smerlak's talk
http://vimeo.com/33363491
It's an outstanding talk! and it is helpful to have some notes that one can glance at as a reminder of what he was covering.

I would be really happy if Jorge Pullin, who organizes the online seminar ILQGS would give the final timeslot (7 May) of this semester to a thermal time talk! Perhaps Matteo Smerlak could give the talk.

Rovelli seems to be stepping back these days to give center stage to younger researchers. He could of course present the seminar on T-time himself but maybe he wants to the next generation theorists to be in the limelight---I don't understand any of that, really.

But one way or another, Tomita flow time is a really important idea. It is the only observer-independent time that we have in full GR, the quantum version. I don't mean when there is a prior fixed curved space-time, I mean the full dynamical geometry and matter. There is no other way to do fully general covariant statistical mechanics, which requires some kind of time. Or so I think anyway. So it is clear to me how I wish Jorge should allocate that last timeslot.
http://relativity.phys.lsu.edu/ilqgs/schedulesp13.html

DATE	Seminar Title	                    Speaker 	     Institution
Jan 29 [B]Entanglement in loop quantum gravity[/B] Eugenio Bianchi  Perimeter Institute
Feb 12 [B]Dynamical chaos and the volume gap [/B]  Hal Haggard	     CPT Marseille
Feb 26 [B]Gravity electroweak unification[/B]	    Stephon Alexander  Haverford College
Mar 12 [B]Quantum reduced loop gravity[/B]	    E.Alesci/F.Cianfrani Univ. Erlangen	 
Mar 26 [B]Bianchi I LQC[/B]	                    Brajesh Gupt     LSU
Apr  9 TBA	                            Karim Noui	     Univ Tours
Apr 23 TBA                                  Martin Bojowald  Penn State	 
May  7  Jorge please invite Smerlak to talk on thermal time

 

[Paulibus]

I agree that the topic of "Thermal Time" is important. Anything that clarifies the pedigree of old Father Time is fascinating. Especially when it is claimed to be "observer-independent", as we now think clock-time isn't; a surprise that was in the end forced on us by observation, rather than just the beauty of relativistic reasoning. I suppose that the measured speed of light is a good accepted example of something that is truly "observer-independent". The local laws of physics may provide a better, more general example.

In the absence of observed examples of an invariant time, consider a common macroscopic attribute which, like Time, we have in a sense created out of the whole cloth of ordinary experience, namely Temperature. To me it looks like the quintessential "thermal" attribute of a thermodynamic system. It is emergent in a statistical sense and can be simply defined for a classical system (from the distribution of velocities of gas atoms) or for a quantum mechanical system (from the distribution of energy among its states).

Is measured temperature "observer-independent"? I suspect not. Temperature might vary from one observer to another; like the black body radiation escaping from a cavity in a moving body, which I think would be measured differently by relatively moving observers. Or by observers situated at different gravitational potentials.

If temperature is not "observer-independent", then is "Thermal" a good choice for qualifying something that is claimed to be truly invariant? (Just quibbling.)

 

[Naty1]

Is measured temperature "observer-independent"? I suspect not. Temperature might vary from one observer to another; like the black body radiation escaping from a cavity in a moving body, which I think would be measured differently by relatively moving observers. Or by observers situated at different gravitational potentials.

According to Bill Unruh, you are correct! [and, analogously, Hawking relative to HAwking radiation].

If temperature is not "observer-independent", then is "Thermal" a good choice for qualifying something that is claimed to be truly invariant? (Just quibbling.)

I don't think it's a quibble...but I do not think the scalar field utilized as a clock falls
prey to observer dependency...

 

[marcus]

I agree that the topic of "Thermal Time" is important. Anything that clarifies the pedigree of old Father Time is fascinating. Especially when it is claimed to be "observer-independent",...

Is measured temperature "observer-independent"? I suspect not. ...

If temperature is not "observer-independent", then is "Thermal" a good choice for qualifying something that is claimed to be truly invariant? (Just quibbling.)

Paulibus and Naty, thanks for interesting comments! I think the above is a valid point about NOMENCLATURE (not necessarily physics).

The idea of a heat bath breaks Lorentz invariance. There is a preferred frame in which the bath is not moving. I think in fact temperature is not "observer-independent". It would be wise to call T-time by some other name than "thermal". After all it is really TOMITA FLOW TIME. And the interesting thing about Tomita flow time (which is what I mean by T-time) is precisely that it is observer independent.

 

[marcus]

It is really important for us to understand the construction of the Tomita flow, based on the pair (M,ω).
You start with a star algebra and a state, the state representing what we think we know about the world---correlations among measurements based on our guesses as to the physics equations that govern them and on our accumulated data. The amazing thing is that this gives a time flow---T-time.
=====================

What I have been wondering about lately is how LQG will be formulated in star-algebra terms. We have to have some way to get from spin networks to C*algebras.
there is a recognized way to get from directed graphs to C*algebras, which could inspire ideas, but by itself this is not enough. Directed graphs are not enough.

Today there appeared a nice paper by Matilde Marcolli (the dazzling Caltech math professor and co-author with Alain Connes). It claims to GENERALIZE the "spin network" to the "gauge network" and then to present a procedure to get from "gauge network" to star algebra. I have the highest regard for Marcolli so am ready to take a good bit on faith here.
Notice that whereas in LQG it is the spin networks that form an orthonormal basis for the LQG Hilbert space, here, in Marcolli's generalization it is the gauge networks that form an orthonormal basis for the Hilbert space. They play an analogous role.

http://arxiv.org/abs/1301.3480
Gauge networks in noncommutative geometry
Matilde Marcolli, Walter D. van Suijlekom
(Submitted on 15 Jan 2013)
We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C*-algebra of gauge network correspondences.
...

http://www.its.caltech.edu/~matilde/
http://www.math.ru.nl/~waltervs/index.php?page=home
(Walter Daniel van Suijlekom b. 1978, dual career as professional musician, interesting. PhD 2005 at SISSA Trieste. Since 2007 postdoc at Nijmegen, same place as Renate Loll. Has taught some interesting courses at Nijmegen including NCG, i.e. spectral geometry.)

 

[Physics Monkey]

Presumably if Tomita time is going to be useful, it should agree with usual notions in the right limit. Has it been checked that this time gives the usual global time evolution in, for example, asymptotically AdS spaces?

Also, we know that the Cauchy problem is not well posed in AdS because of the need for boundary conditions. Is this freedom apparent in Tomita time? Are there other freedoms?

 

[marcus]

Presumably if Tomita time is going to be useful, it should agree with usual notions in the right limit. Has it been checked that this time gives the usual global time evolution in, for example, asymptotically AdS spaces?
...

It gives the right limit in several interesting cases. I don't know about asymptotic AdS.

Confirming cases are listed and discussed in the 1994 paper by Alain Connes and Carlo Rovelli.

I gave the link in post #2 of this thread.

The RATIO of T-time to local observer time can be given a physical meaning, which is kind of interesting---a general relativistic temperature identified by Tolman around 1930. There's a link to the Smerlak Rovelli paper about that also in post #2, I think.
https://www.physicsforums.com/showthread.php?p=4209223#post4209223

 

[DennisN]

Still taking time to follow the threads here about time :). I also enjoyed the Matteo Smerlak clip but I got deeply disappointed that it was only 15 min :(. If anyone knows any link to the full clip, I would be very happy. This topic is very interesting; although I find it quite abstract and difficult, I have a feeling that I am slowly understanding more about it as I go along. I hope... .

 

[marcus]

I also enjoyed the Matteo Smerlak clip but I got deeply disappointed that it was only 15 min :(. If anyone knows any link to the full clip, I would be very happy...

I was disappointed too, by it not being complete. I'm glad you enjoyed what we have of the talk however! There is a complete 54 minute video of a 2010 talk by Smerlak about related topics, that he gave at Perimeter Institute.

A typical way to get talks at the Perimeter Institute Recorded Seminar Archive is to simply google "pirsa smerlak" or pirsa with the name of the speaker.

If you try it please let me know if the video works, usually they do. I get it fine on my computer. It is available in several formats from PIRSA but I use the "flash" format.

The talk is November 2010 so over 2 years old and that makes a difference. But it is still pretty good, I think. When you google "pirsa smerlak" you get
http://pirsa.org/10110071/

 

[marcus]

As a reminder: http://vimeo.com/33363491
It was reported earlier this month that Matteo Smerlak has accepted a postdoc at Perimeter Institute starting 2013.

I think there is a shift of attention towards the relation between QG (quantum geometry) and THERMODYNAMICS. For example Stefano Liberati has been working on that angle all along as has Ted Jacobson of course, and Goffredo Chirco (a Liberati PhD and co-author) has just accepted a Marseille postdoc position, starting 2013.

Because of the growing interest in QG+Thermo, Tomita flow time is a key idea. It is the only observer-independent time that we have in full GR. I don't mean when there is a prior fixed curved space-time, but rather the full dynamical geometry and matter. There is no other way to do fully general covariant statistical mechanics, which requires some kind of time.

 

[Paulibus]

I'm still struggling with the concept of an observer-independent time. Before relativity, physics theories and the measured quantities that they inter-relate were assumed to be observer-independent. Neither time, distance or temperature were thought to depend on the state of motion or the mass/energy environment of the observer. But the advent of general relativity confined observer-independence to the theories that rule the inter-relation of measurements. Measurements of time, distance and temperature were revealed to be observer-dependent features of observer-independent theory. This dichotomy has amply been confirmed by prediction and observation.

Where does thermal/Tomita time fit into this scheme? For instance, defining thermal time as the logarithm of change (Marcus' post #20) seems to imply that in a perfectly symmetric invariant world (no change whatever; ratio of repeated measurements always 1), thermal time would not flow (log 1 = 0). But we do live in a world of fundamental theory assumed to be perfectly symmetric, the same everywhere and everywhen, invariant and involving invariant constants like c and h.

Is the flow of thermal time engendered only by the changes of observer-dependent measured quantities, like clock time, distance and temperature --- by observer-dependent solutions to observer-independent equations?

 

[marcus]

I'm still struggling with the concept of an observer-independent time. Before relativity, physics theories and the measured quantities that they inter-relate were assumed to be observer-independent. Neither time, distance or temperature were thought to depend on the state of motion or the mass/energy environment of the observer. But the advent of general relativity confined observer-independence to the theories that rule the inter-relation of measurements. Measurements of time, distance and temperature were revealed to be observer-dependent features of observer-independent theory. This dichotomy has amply been confirmed by prediction and observation.

Where does thermal/Tomita time fit into this scheme? For instance, defining thermal time as the logarithm of change (Marcus' post #20) seems to imply that in a perfectly symmetric invariant world (no change whatever; ratio of repeated measurements always 1), thermal time would not flow (log 1 = 0). But we do live in a world of fundamental theory assumed to be perfectly symmetric, the same everywhere and everywhen, invariant and involving invariant constants like c and h.

Is the flow of thermal time engendered only by the changes of observer-dependent measured quantities, like clock time, distance and temperature --- by observer-dependent solutions to observer-independent equations?

Hi Paulibus, your post encourages me to look critically at the idea of Tomita time and try to saywhere it comes from. How did a time-like flow get into the picture in the first place.

We tend to think of a quantum theory as something with a Hilbertspace (assumed over ℂ), and operators corresponding to measurements. COULD THIS BE WRONG? Because it sneaks a time idea in there, makes it implicit. Maybe that should not be allowed, in which case we should have a different formulation of quantum theory, say involving only probabilities and no complex numbers, no Hilbertspace (or C* generalization). Maybe it is our prejudice about what a quantum theory should look like that dooms us to the disease of a preferred time. This sounds kind of silly but I'm saying it to try to open up the box and get it all on the table.

What I said about "logarithm of change to the base Q" is just a way to think of the construction which helps me as a sort of mnemonic---a memory aid.

The real definition is: you assume you are given a quantum theory as (M,ω) a star algebra with a state. Basically this goes back to von Neumann, it is a format which subsumes the usual hilbertspace one. Maybe it's wrong. But assuming that, you can recover a hilbertspace and have the algebra M act on it. So M is no longer abstract, it is realized or represented as operators acting on H_M.

When you have a conventional quantum theory you always have this! But now the star of M itself becomes an operator defined on H_M. Call this operator S.

Now we can define a unitary Q = (S* S)ⁱ

Tomita showed that for every real number t, we have an automorphism of the algebra M that they always denote by the letter alpha
α_t A = Q^tAQ^-t

I'm not sure about this but I think that on the face of it there is no reason to suppose that this parametrized group of automorphisms corresponds to "time" in any sense. I think it may have been Connes and Rovelli who realized that it does in fact agree with usual time concepts in several interesting cases. So that was a surprise for everybody: the Tomita flow (which we get simply because the M algebra has a star) correpsonds to physical time in interesting cases!

These authors did not claim to KNOW for certain that it was the right form of time for doing, say, general covariant statistical mechanics. Rather they cautiously proposed it and conjectured that it could be right. I don't think they have done enough with T-time to know, yet.

It's curious because it seems unique and because it comes merely from having a star operation in the conventional observables algebra. Should the star operation be made illegal so that we don't risk finding ourselves with a preferred time? I realize I can't answer your question in a satisfactory way. I'm still wondering about this and hoping to hear more from one or another directions.

 

[marcus]

This may or may not be relevant. There is a "clock" that just tracks the rotations of a particle's wave function which depends on its mass. So mass and time seem linked. Maybe we can't get away from some version of time.

This was published online 10 January 2013 by Science journal.
http://www.sciencemag.org/content/early/2013/01/09/science.1230767

A Clock Directly Linking Time to a Particle's Mass
Shau-Yu Lan, Pei-Chen Kuan, Brian Estey, Damon English, Justin M. Brown, Michael A. Hohensee, Holger Müller

Department of Physics, 366 Le Conte Hall MS7300, University of California, Berkeley.
Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley.
ABSTRACT
Historically, time measurements have been based on oscillation frequencies in systems of particles, from the motion of celestial bodies to atomic transitions. Relativity and quantum mechanics show that even a single particle of mass m determines a Compton frequency ω₀ = mc²/ ħ, where c is the speed of light and ħ is the reduced Planck constant. A clock referenced to ω₀ would enable high-precision mass measurements and a fundamental definition of the second. We demonstrate such a clock using an optical frequency comb to self-reference a Ramsey-Bordé atom interferometer and synchronize an oscillator at a subharmonic of ω₀. This directly demonstrates the connection between time and mass. It allows measurement of microscopic masses with 4 × 10⁻⁹ accuracy in the proposed revision to SI units. Together with the Avogadro project, it yields calibrated kilograms.
===================

I realize you could say what of it, because you can think of the particle, or the Cesium atom, as an observer. So this is the observer's time. But it makes me think. It is quantum mechanics that does this. Mass has an intrinsic frequency which it appears people are able to measure---the Compton frequency. Classical mass did not used to have an innate frequency.

I started a thread about this in Quantum Mechanics forum:
https://www.physicsforums.com/showthread.php?t=664870
Bee Hossenfelder blogged about it a few days ago on the 17th Jan., I urge reading her post:
http://backreaction.blogspot.com/2013/01/how-particle-tells-time.html

 

[torsten]

The real definition is: you assume you are given a quantum theory as (M,?) a star algebra with a state. Basically this goes back to von Neumann, it is a format which subsumes the usual hilbertspace one. Maybe it's wrong. But assuming that, you can recover a hilbertspace and have the algebra M act on it. So M is no longer abstract, it is realized or represented as operators acting on H_M.

When you have a conventional quantum theory you always have this! But now the star of M itself becomes an operator defined on H_M. Call this operator S.

Now we can define a unitary Q = (S* S)ⁱ

Tomita showed that for every real number t, we have an automorphism of the algebra M that they always denote by the letter alpha
a_t A = Q^tAQ^-t

I'm not sure about this but I think that on the face of it there is no reason to suppose that this parametrized group of automorphisms corresponds to "time" in any sense. I think it may have been Connes and Rovelli who realized that it does in fact agree with usual time concepts in several interesting cases. So that was a surprise for everybody: the Tomita flow (which we get simply because the M algebra has a star) corresponds to physical time in interesting cases!

These authors did not claim to KNOW for certain that it was the right form of time for doing, say, general covariant statistical mechanics. Rather they cautiously proposed it and conjectured that it could be right. I don't think they have done enough with T-time to know, yet.

It's curious because it seems unique and because it comes merely from having a star operation in the conventional observables algebra. Should the star operation be made illegal so that we don't risk finding ourselves with a preferred time? I realize I can't answer your question in a satisfactory way. I'm still wondering about this and hoping to hear more from one or another directions.

Hi marcus, interesting thread.
I'm also occupied by Tomitas modular theory for a long time (four years). At first I suggest two references (you mentioned the Bertozzi paper also referring to my work):
Borcherts: On Revolutionizing of Quantum Field Theory with Tomita's Modular Theory (published in J. Math. Phys.)
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.141.4325
and
B. Schroer , H. -w. Wiesbrock: Modular Theory and Geometry
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.497

Tomitas modular theory was also noticed in QFT and therefore should have an impact for quantum gravity. If I interpreted the tone of the discussion right then you ask for a geometric interpretation of this theory.
Interestingly, Connes gave one. In the first work of noncommutative geometry, he considered the leaf space of a foliation. In simple cases like fibrations, the leaf space has the structure of a Banach manifold or something similar. But for the Kronecker foliation (a single infinite curve who twist around the torus) of the torus, the usual leaf space is even no Haussdorff space.
Motivated by measure-theoretic considerations of foliations by Hurder etc., Connes began to realize that a noncommutative von Neumann algebra can be a suitable object for foliations.
In case of the Kronecker foliation he obtained the factor II_\infinity von Neumann algebra. The classification of von Neumann algebras (= observablen algebra) with center the complex numbers (called factors) are given by three types:
type I divided into I_N and I_\infinity covers the usual quantum mechanics
type II divided into II₁ and II_\infinity used in statistical physics (spin glas models, Temperley-Lieb algebra, Jones knot polynomial)
type III divided into III₀, III_lambda and III₁ connected to QFT
Especially the last factor III₁ represents the observablen algebra of a QFT with one vacuum vector.
For every factor there is a foliation where Connes leaf space model is this factor (see the pages 43-59 in Connes book "Noncommutative geometry" 1995)
In this geometric model there is also an interpretation of Tomitas theory which uncovers also the role of the paramter t:
Every factor III foliation (i.e. a foliation having a factor III von Neumann algebra as leaf space) can be obtained from a type II foliation (see Proposition 8 and 9 on pages 57,58).
Let (V, F) be a codimension q type III foliation with the transverse bundle N=TV/F defined at every x\in V. Now one considers associated principal R+*-bundle of positive densities defined on page 57. The total space V' of this bundle over V defines a new foliation (V',F') of type II.
But then Tomitas modular operator Q is dual to the density. I found only one possible interpretation:
Tomitas parameter t is the probability and not the time.

For another geometric interpretation of the factor III I refer to my own paper:
http://arxiv.org/abs/1211.3012
"Quantum Geometry and Wild embeddings as quantum states"
We considered a wild embedding. Remember an embedding is a map i:K->M so that i(K) is homeomorphic to K. If i(K) can be reperesented by a finite polyhedron (or a finite triangulation) then one calls i a tame embedding otherwise it is a wild embedding.
Examples of wild embeddings are Alexanders horned sphere or Antoise necklace, check out youtube for movies

http://www.youtube.com/watch?v=Pe2mnrLUYFU&NR=1
In the paper we constructed a von Neumann algebra associated to a wild embedding. In particular we show that the deformation quantization of a tame embedding leads to a wild embedding so that its von Neumann algebra is a factor III₁.

To say it again: Tomitas theory is interesting and relevant to understand time but t is not the time but the probability.

 

[marcus]

Hi marcus, interesting thread.
I'm also occupied by Tomitas modular theory for a long time (four years). At first I suggest two references (you mentioned the Bertozzi paper also referring to my work):
Borcherts: On Revolutionizing of Quantum Field Theory with Tomita's Modular Theory (published in J. Math. Phys.)
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.141.4325
and
B. Schroer , H. -w. Wiesbrock: Modular Theory and Geometry
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.497
...

Hi Torsten, thanks for commenting! I was able to find the Schroer Wiesbrock paper at arxiv:
http://arxiv.org/abs/math-ph/9809003
It looks "over my head" at the moment but I'm glad to have it as a reference.

You noticed that the Bertozzini et al paper (http://arxiv.org/abs/1007.4094) that I mentioned earlier cites your paper with Krol (http://arxiv.org/abs/1001.0882) . I didn't mention it but it also cites several papers by Jesper Grimstrup about grafting Connes NCG standard model onto LQG geometric basis. (http://arxiv.org/refs/1007.4094)

Borcherts' papers from around 1998 and 1999 are much cited. I could not find online the one you mentioned ("On Revolutionizing...") but if and when curiosity overcomes my natural sloth I can always walk over onto campus and find it in J. Math. Phys.

If you have another choice source which is online, other people might appreciate it---folks not close to a university library.

My hunch (as a non-expert) is that the Tomita flow can be SEVERAL DIFFERENT things. It can be trivial (no flow at all) on certain C* algebras. It can agree with the passage of TIME as Rovelli and Alain Connes found in some interesting cases, and you have found that in other cases it corresponds to PROBABILITY.

Much of what you say is fascinating and of immediate interest. I have been thinking about the FOLIATION which the Tomita flow might induce, especially in the case of C* Loop cosmology where there is a certain subset of the algebra corresponding to observables at the moment of the bounce. If you take that as a simultaneous set and advance from there you would get a spacelike foliation of the algebra.

You say that in a Tomita flow foliation each individual leaf is a Banach manifold. (I recall years ago there was a saying among the students "Whenever anything nice happens, it happens in a Banach space." ) Presumably Banach manifold would be one whose tangent space at every point is Banach. This sounds nice. But a bit overwhelming. It may take me a while to assimilate some of what you say in the above post

 

[torsten]

Sorry, there was a misunderstanding: the leaf space or the space of all leafs has the structure of Banach manifold but only for simple foliations. In case of Tomita flows, the leaf space is an ugly space: there is only one possible continuous function over this space, the constant function. That is the reason why Connes changed to operators.
I'm not shure that the TIME of Rovelli and Connes is a real time. They found a unitary dynamics driven by t but that describes only the possibilities not what happens actual.

 

[DennisN]

If you try it please let me know if the video works, usually they do. I get it fine on my computer. It is available in several formats from PIRSA but I use the "flash" format. The talk is November 2010 so over 2 years old and that makes a difference. But it is still pretty good, I think. When you google "pirsa smerlak" you get
http://pirsa.org/10110071/

Yes, the clip worked and I enjoyed it, thank you very much, Marcus! The details are a couple of notches above my education/understanding, but I think I at least understand the essence of it better now, thanks to the clip. What I got out of it was the following; (For others: the Matteo Smerlak clip is about time, temperature, Tolman effect, thermal equilibrium in stationary spacetimes (in the presence of gravity)).

In the clip Matteo Smerlak suggests the following:

Two notions of time;

• mechanical time t (in time-reversible mechanical equations)
• thermal time τ (in time-irreversible thermodynamic equations)

Mechanical time is proper time along stationary worldlines. Thermal time is associated to the ignorance of the microscopic dynamics, represented by statistical states; the thermal time flow is induced by a statistical state. The quantum version of this is the "Tomita modular flow".

Temperature as "the speed of time"; the (inverse) temperature β is (the scale of) thermal time with respect to mechanical time; roughly speaking

thermal time = β * mechanical time

at thermal equilibrium in stationary spacetimes. Temperature is space-dependent.

I am regretfully not qualified to further evaluate the arguments/equations in detail (and as Marcus said, the clip is two years old), but I must say I am intrigued by this particular kind of bridging between thermodynamics, relativity and quantum mechanics - it is very interesting! I saw there are other interesting posts above, going to read those now...