Global emergent time, how does Tomita flow work?

[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  [COLOR="Red"]Jorge please invite Smerlak to talk on thermal time[/COLOR]

 

[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...

 

[marcus]

The Tomita flow, and thermal time, came up in one of the Loops parallel session talks that I think of at the moment: one by Goffredo Chirco. There may be others.
The parallel session abstracts are here:
http://www.perimeterinstitute.ca/sites/perimeter-www.pi.local/files/conferences/attachments/Parallel%20Session%20Abstracts_7.pdf

To find the videos, I can use the index I just posted in the "Loops 2013 talks" thread:
https://www.physicsforums.com/showthread.php?p=4461021#post4461021
Looking down the alphabetical list for speaker's name you see:
Goffredo Chirco, Aix-Marseille University http://pirsa.org/13070085 (0)

Conveniently, the starting time is minute zero, so we get the talk as soon as we click on the link and select flash. There is no need to wait for buffering before we start.

The KMS condition, which is essential to thermal time, also came up in this talk:
Daniele Pranzetti, Albert Einstein Institute http://pirsa.org/13070054 (0)
The paper this was based on also treats Tomita time, but that part wasn't covered in the 20-minute version.These are two outstanding talks. I wonder what others in the Loops 2013 collection deal with Tomita time. Can anybody suggest others?

The basic reason it's so interesting is that this is a global time which is observer-independent.
Instead of depending on a choice of observer, it depends on the process whose quantum state is known. That is, a vector in a boundary Hilbert space that contains information about past during and future. And more or less equivalently thanks to Israel Gelfand, a positive functional defined on the C* algebra of the process. Here "state" does not mean "state at a given instant of time". The state is a quantum description of what can be known about an entire process occurring in an enclosed spacetime region.

The state gives rise to time. It is with this tomita global time that the researchers propose to work out a general covariant QFT and a general covariant statistical mechanics. this is new because up to now these constructs have been formulated using some postulated background or observer-dependent time.

EDIT: when you click on the link for Chirco it will say that the first talk is by Bianca Dittrich, but she gave her talk in a different session and the first is actually the one you want.

 

[atyy]

I googled "tomita flow modular flow" and you are the one of the top Google finds. Would you like to comment on whether modular flow in footnote 16 on p42 of http://arxiv.org/abs/1305.3182 is Tomita flow? Bianchi is cited in ref 16 and 63. One of the authors is an "LQG guy" :p

Footnote 16 on p42, "These unitary operators implement an evolution for an internal time . This time flow is called the modular flow [29]."

Also footnote 3 on p4, "The simplest example is given by considering a global thermal state, with temperature T, and taking V to be the whole space. Then, the modular Hamiltonian is simply the ordinary (local) Hamiltonian divided by T, as is evident from eq. (1.5), and so H simply generates ordinary time translations."

A recent paper by Robert Oeckl on the Boundary Formulation of QT interestingly refers to the (M,ω) picture, suggesting that he, too, may be looking at it as a possible way to go.
http://arxiv.org/abs/1212.5571
A positive formalism for quantum theory in the general boundary formulation
Robert Oeckl (CCM-UNAM)
(Submitted on 21 Dec 2012)
We introduce a new "positive formalism" for encoding quantum theories in the general boundary formulation, somewhat analogous to the mixed state formalism of the standard formulation. This makes the probability interpretation more natural and elegant, eliminates operationally irrelevant structure and opens the general boundary formulation to quantum information theory.
28 pages

to clarify the relevance here is a quote from end of section 2 on page 4:
"...The time-evolution operator U ̃ restricted to self-adjoint operators produces self-adjoint operators. Moreover, it is positive, i.e., it maps positive operators to positive operators. It also conserves the trace so that it maps mixed states to mixed states. These considerations suggest that positivity and order structure should play a more prominent role at a foundational level than say the Hilbert space structure of H or the algebra structure of the operators on it from which they are usually derived.
Algebraic quantum field theory [7] is a great example of the fruitfulness of taking serious some of these issues. There, one abandons in fact the notion of Hilbert spaces in favor of more flexible structures built on C∗-algebras. Also, positivity plays a crucial role there in the concept of state."

From marcus's quote, Oeckl does mention algebraic quantum field theory, which is exactly http://arxiv.org/abs/1305.3182's ref [29], cited for modular flow.

Also http://www.staff.science.uu.nl/~henri105/ comments at http://mathoverflow.net/questions/5...atization-of-time-other-than-perhaps-entropy: "In algebraic quantum field theory, time evolution can be identified with the modular flow of Tomita-Takesaki theory." He says it's the Bisognano-Wichman theorem.

Incidentally, the Blanco et al paper actually does link (at linear level) the Einstein Equations with "E=TdS". So it's like a holographic version of the Jacobson derivation, as van Raamsdonk's http://arxiv.org/abs/1308.3716 "Entanglement Thermodynamics" stresses. (In the Blanco et al paper, E is the "modular energy".)

 

[marcus]

Atyy, a quote from recent Chirco et al paper nicely clears up the terminology. There are variants in terminology (just as you suggest) depending on where the speaker comes from. In mathematics, it's Tomita, in QFT it's modular.

==quote Chirco et al http://arxiv.org/abs/1309.0777 page 4== (link corrected at Atyy's suggestion)
...is called the thermal Hamiltonian. The conjugate flow parameter τ is called thermal time [6, 8]. For a non-relativistic system it is simply related to the non-relativistic time t by

τ=t/β

where β is the inverse temperature. In mathematics, the thermal time flow is called the Tomita flow [19]; the thermal hamiltonian is called the modular Hamiltonian in quantum field theory [20, 21], and the entanglement Hamiltonian in the condensed matter context [22]. For a recent discussion of this quantity in quantum gravity see also [23].

==endquote==
The reference [20] is to R. Haag, “Local quantum physics: Fields, particles, algebras”, Springer (1992)
That is, to the same standard QFT text that Blanco et al paper you found cites ([29]) in connection with what QFT people call modular flow.

I googled "tomita flow modular flow" and you are the one of the top Google finds. Would you like to comment on whether modular flow in footnote 16 on p42 of http://arxiv.org/abs/1305.3182 is Tomita flow? Bianchi is cited in ref 16 and 63. One of the authors is an "LQG guy" :p

Footnote 16 on p42, "These unitary operators implement an evolution for an internal time . This time flow is called the modular flow [29]."
...

Yes! [29] R. Haag, “Local quantum physics: Fields, particles, algebras”, Berlin, Germany: Springer (1992) (Texts and monographs in physics).

Over and beyond just noting the overlap in terminology, you are pointing out a whole bunch of interesting connections.

EDIT: Atyy, I appreciate your catching my typo error in the above Chirco paper link. I could still edit, so I corrected it.

 

[atyy]

Thanks marcus! The link for the Chirco paper is http://arxiv.org/abs/1309.0777

So does AdS/CFT instantiate Connes-Rovelli thermal time? There appears to be a derivation of the Einstein equation at linear level from "entanglement thermodynamics" dE=dS where E refers to the modular Hamiltonian that Chirco et al say is the thermal Hamiltonian in their language and the entanglement Hamiltonian in condensed matter.

The AdS/CFT papers are
http://arxiv.org/abs/1305.3182
http://arxiv.org/abs/1308.3716
http://arxiv.org/abs/1304.7100
The first two are most relevant to Tomita flow because they use the modular Hamiltonian. There's definitely the idea that these are the AdS/CFT version of Jacobson's derivation, which I think inspired Rovelli too. Edit: In fact, Connes-Rovelli thermal time preceded Jacobson's derivation.

It also makes me wonder whether Bianchi's black hole entropy is really the black hole entropy or whether he actually calculated the entangelement entropy of some other region bounded by an extremal surface - since the Ryu-Takayanagi formula says those have the same form as the BH entropy. The best way to probe the black hole interior in AdS/CFT so far seems to have been to use the Ryu-Takayanagi formula in a non-stationary spacetime since the minimal surfaces penetrate the black hole in those cases. But I believe there is no tight link between the Ryu-Takayanagi formula and the BH entropy of black holes at this time, although it seems there should be one.

==quote Chirco et al http://arxiv.org/abs/1309.0777 page 4== (link corrected at Atyy's suggestion)
...is called the thermal Hamiltonian. The conjugate flow parameter τ is called thermal time [6, 8]. For a non-relativistic system it is simply related to the non-relativistic time t by

τ=t/β

where β is the inverse temperature. In mathematics, the thermal time flow is called the Tomita flow [19]; the thermal hamiltonian is called the modular Hamiltonian in quantum field theory [20, 21], and the entanglement Hamiltonian in the condensed matter context [22]. For a recent discussion of this quantity in quantum gravity see also [23].

==endquote==

Chirco's ref 23 is Bianchi-Myers! http://arxiv.org/abs/1212.5183 which indeed has a long discussion on the modular Hamiltonian.

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?

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

Physics Monkey asked essentially the same question in post #49. Are we now in a position to answer Physics Monkey's question in more detail? Incidentally, the entanglement Hamiltonian is very common in condensed matter physics, where it's associated with the Renyi entropies. I'd never associated it with Tomita flow which I'd seen many times in marcus's posts! Duh!

For example, it's used by Swingle, McMinis and Tubman in their eq 3.8 http://arxiv.org/abs/1211.0006

 

[atyy]

http://arxiv.org/abs/1102.0440
Towards a derivation of holographic entanglement entropy
Horacio Casini, Marina Huerta, Robert C. Myers

Explicitly mentions and uses the modular flow!

The warm-up Rindler wedge example in section 2.1 is exactly (I think) the same as Connes and Rovelli's http://arxiv.org/abs/gr-qc/9406019 section 4.3.

From Casini et al's section 2.1: "One well-known example is given by Rindler space R ... In this case for any QFT, the modular Hamiltonian is just the boost generator in the X₁ direction. This result is commonly known as the Bisognano-Wichmann theorem [25]. ... Interpreted in the sense of Unruh [26], the state in R is thermal ... With this choice, the Rindler state is thermal with respect to H_τ, the Hamiltonian generating τ translations, with a temperature ... With this notation, the modular flow (2.6) on R simply corresponds to the time translation ... and the modular Hamiltonian H_R is simply related to H_τ with ..."

 

[atyy]

And here are a couple of condensed matter theorists talking about Tomita flow!

http://arxiv.org/abs/1109.1283
A geometric proof of the equality between entanglement and edge spectra
Brian Swingle, T. Senthil

"Remarkably, the entanglement Hamiltonian is the generator of a geometric flow in spacetime, and this flow may be interpreted as time evolution in Rindler space. The reduced density matrix of the half space is then a simple thermal state with respect to time evolution in Rindler space."

It looks like their main reference is to J. J. Bisognano and E. H. Wichmann, 17, 303 (1976), ISSN 00222488.

 

[Physics Monkey]

A major question in my mind is related to the issue of infinities and their role in the physics. I am accustomed to finite (but large) dimensional algebras in which the "modular flow" is really a very simple object generated by some ordinary matrices. I also have the prejudice that anything physical should be describable in a finite language like this. For example, all physical measurements with quantum fields can be described in this language.

On the other hand, my understanding is that the mathematically non-trivial aspects of modular flow in the operator algebra context are all associated with "weird" infinite algebras. So I don't see how there can be physics in such an infinite object. In fact, I have this same objection to much of old school loop gravity, which seemed way too infinite to be right.

Of course, this is mostly my vague feelings and superstitions.

 

[atyy]

http://arxiv.org/abs/1310.6335
State-Dependent Bulk-Boundary Maps and Black Hole Complementarity
Kyriakos Papadodimas, Suvrat Raju
Finally, we explore an intriguing link between our construction of interior operators and Tomita-Takesaki theory.

 

[strangerep]

I arrive late in this thread (mainly via the other "QBist" thread) and I'm trying to get a couple of things straightened out...

From Marcus's summaries about Tomita time, and also some of the references he cites, I understand that:

$S$ is an anti-linear operator.

However, then Marcus (and others) speak of an adjoint $S^*$ of an operator $S$ via a definition like:
$$
(\phi, S^* \psi) ~\equiv~ (S\phi, \psi) ~.
$$ However, in Weinberg vol1, p51, eq(2.2.7), he defines the adjoint of an anti-linear operator via:
$$
(\phi, S^* \psi) ~\equiv~ (S\phi, \psi)^* ~=~ (\psi, S\phi) ~.
$$ Weinberg's motivation for his definition is that one can then write $A^* = A^{-1}$ regardless of whether $A$ is linear or antilinear.

So what's going on with the $S$ operator used in Tomita time? Is it a typo, or an intentionally different definition of "adjoint" for antilinear operators compared to Weinberg's ? I presume it's intentionally different, else the Tomita $Q$ operator would trivially be the identity, (right?).

 

[atyy]

Papadodimas and Raju http://arxiv.org/abs/1310.6335 give the same definition of the adjoint as Weinberg in their discussion of Tomita-Takesaki theory (p57 and p86).