Global emergent time, how does Tomita flow work?

marcus

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

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/ea...cience.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:
http://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/201...ells-time.html

torsten

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 refering 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...0.1.1.141.4325
and
B. Schroer , H. -w. Wiesbrock: Modular Theory and Geometry
http://citeseerx.ist.psu.edu/viewdoc...=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=d1Vjsm9pQlc
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 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

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, gonna read those now...