Global emergent time, how does Tomita flow work?

atyy

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 any one 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'.....

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:

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

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.

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:
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

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...


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

marcus

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*-module


Hilbert C* Module
http://en.wikipedia.org/wiki/Hilbert_C*-module


On 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


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

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

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

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

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

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.
http://www.physicsforums.com/showthr...23#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... .