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Global emergent time, how does Tomita flow work? 
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#37
Jan713, 12:56 AM

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PF Gold
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...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 selfadjoint. 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 observerindependent 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. Ttime is the logarithm of change to the base Q. When specific cases are considered and the arithmetic is done, the units of Tflow 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 "oneparameter group of automorphisms" defined on M. A flow for short. 


#38
Jan713, 08:00 AM

P: 5,632

Marcus: Last question on 'Ashtekar time'.....
From the November paper: 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.... 


#39
Jan713, 10:41 AM

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PF Gold
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Also I'm no great authority on this Ttime 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 definedthe 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 picturehis "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 sofar Ashtekar is not using to do cosmology. But when you make the setups correspondmake enough assumptions to bridge between the different models of the worldthen apparently you get agreement! I haven't gone thru all the steps so I have to take this partly on faith. Ttime 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. 


#40
Jan713, 10:47 AM

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Marcus:
great explanations in the prior post.....thanks! 


#41
Jan813, 04:50 PM

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Physics "TRASH TALK' :
Marcus I wondered what you thought about this: from: http://arxiv.org/pdf/1007.4094v2.pdf pg 21 Modular noncommutative geometry in physics /////////////////////////// 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... 


#42
Jan813, 05:40 PM

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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 wrongthis is just the message I get. ===quote Bertozzini Conti Lewkee.. page 21=== 5.3 Modular noncommutative geometry in physics Despite the fact that most, of the literature on noncommutative 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 noncommutative geometry for the foundations of physics looks still quite weak and disputable. In this subsection rather than discussing the vast panorama of applications of noncommutative 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 noncommutative 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 noncommutative ideas (such as semifinite and modular spectral triples, phasespaces etc.) to physics. ===endquote=== 


#43
Jan913, 08:41 AM

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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/EhrenfestTolman_effect 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 covariant theories, the notion of time depends on the state [of] the system in a general covariant context, extending time ﬂow to generally covariant theories depends on the thermal state of the system, time ﬂow is determined by the thermal state. 


#44
Jan1413, 05:38 PM

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PF Gold
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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 Ttime himself but maybe he wants to the next generation theorists to be in the limelightI don't understand any of that, really. But one way or another, Tomita flow time is a really important idea. It is the only observerindependent time that we have in full GR, the quantum version. I don't mean when there is a prior fixed curved spacetime, 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



#45
Jan1513, 09:07 AM

P: 175

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 "observerindependent", as we now think clocktime 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 "observerindependent". 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 "observerindependent"? 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 "observerindependent", then is "Thermal" a good choice for qualifying something that is claimed to be truly invariant? (Just quibbling.) 


#46
Jan1513, 12:55 PM

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prey to observer dependency.... 


#47
Jan1513, 09:15 PM

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PF Gold
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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 "observerindependent". It would be wise to call Ttime 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 Ttime) is precisely that it is observer independent. 


#48
Jan1513, 09:30 PM

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PF Gold
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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 worldcorrelations 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 flowTtime. ===================== What I have been wondering about lately is how LQG will be formulated in staralgebra 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 coauthor 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 almostcommutative 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 wellknown 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.) 


#49
Jan1613, 11:58 AM

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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? 


#50
Jan1613, 12:13 PM

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PF Gold
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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 Ttime to local observer time can be given a physical meaning, which is kind of interestinga 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 


#51
Jan1713, 08:21 PM

P: 482

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... .



#52
Jan1713, 11:42 PM

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PF Gold
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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/ 


#53
Jan2013, 12:44 PM

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PF Gold
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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 coauthor) 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 observerindependent time that we have in full GR. I don't mean when there is a prior fixed curved spacetime, 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. 


#54
Jan2113, 03:15 AM

P: 175

I'm still struggling with the concept of an observerindependent time. Before relativity, physics theories and the measured quantities that they interrelate were assumed to be observerindependent. 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 observerindependence to the theories that rule the interrelation of measurements. Measurements of time, distance and temperature were revealed to be observerdependent features of observerindependent 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 observerdependent measured quantities, like clock time, distance and temperature  by observerdependent solutions to observerindependent equations? 


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