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Global emergent time, how does Tomita flow work? 
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#55
Jan2113, 05:27 PM

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PF Gold
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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 mnemonica 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^{t}AQ^{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 Ttime 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. 


#56
Jan2113, 05:43 PM

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PF Gold
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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 ShauYu Lan, PeiChen 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 ω_{0} = mc^{2}/ ħ, where c is the speed of light and ħ is the reduced Planck constant. A clock referenced to ω_{0} would enable highprecision mass measurements and a fundamental definition of the second. We demonstrate such a clock using an optical frequency comb to selfreference a RamseyBordé atom interferometer and synchronize an oscillator at a subharmonic of ω_{0}. 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 measurethe 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...ellstime.html 


#57
Jan2313, 04:22 AM

P: 77

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 measuretheoretic 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 II_{1} and II_{\infinity} used in statistical physics (spin glas models, TemperleyLieb algebra, Jones knot polynomial) type III divided into III_{0}, III_{lambda} and III_{1} connected to QFT Especially the last factor III_{1} 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 4359 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 III_{1}. To say it again: Tomitas theory is interesting and relevant to understand time but t is not the time but the probability. 


#58
Jan2313, 02:08 PM

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http://arxiv.org/abs/mathph/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 itfolks not close to a university library. My hunch (as a nonexpert) 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 


#59
Jan2313, 02:46 PM

P: 77

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. 


#60
Jan2513, 07:23 PM

P: 483

In the clip Matteo Smerlak suggests the following: Two notions of time;
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 spacedependent. 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... 


#61
Aug113, 02:22 PM

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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/sit...bstracts_7.pdf To find the videos, I can use the index I just posted in the "Loops 2013 talks" thread: http://www.physicsforums.com/showthr...21#post4461021 Looking down the alphabetical list for speaker's name you see: Goffredo Chirco, AixMarseille 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 20minute 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 observerindependent. 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 observerdependent 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. 


#62
Sep313, 07:39 PM

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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." Also André Henriques comments at http://mathoverflow.net/questions/57...rhapsentropy: "In algebraic quantum field theory, time evolution can be identified with the modular flow of TomitaTakesaki theory." He says it's the BisognanoWichman 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".) 


#63
Sep413, 12:03 PM

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PF Gold
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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 nonrelativistic system it is simply related to the nonrelativistic 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. 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. 


#64
Sep413, 02:29 PM

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Thanks marcus! The link for the Chirco paper is http://arxiv.org/abs/1309.0777
So does AdS/CFT instantiate ConnesRovelli 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, ConnesRovelli 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 RyuTakayanagi 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 RyuTakayanagi formula in a nonstationary spacetime since the minimal surfaces penetrate the black hole in those cases. But I believe there is no tight link between the RyuTakayanagi formula and the BH entropy of black holes at this time, although it seems there should be one. For example, it's used by Swingle, McMinis and Tubman in their eq 3.8 http://arxiv.org/abs/1211.0006 


#65
Sep513, 02:39 PM

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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 warmup Rindler wedge example in section 2.1 is exactly (I think) the same as Connes and Rovelli's http://arxiv.org/abs/grqc/9406019 section 4.3. From Casini et al's section 2.1: "One wellknown example is given by Rindler space R ... In this case for any QFT, the modular Hamiltonian is just the boost generator in the X_{1} direction. This result is commonly known as the BisognanoWichmann 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 ..." 


#66
Sep513, 11:59 PM

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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. 


#67
Sep1013, 09:10 PM

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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 nontrivial 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. 


#68
Oct2313, 09:30 PM

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http://arxiv.org/abs/1310.6335
StateDependent BulkBoundary Maps and Black Hole Complementarity Kyriakos Papadodimas, Suvrat Raju Finally, we explore an intriguing link between our construction of interior operators and TomitaTakesaki theory. 


#69
Jan514, 02:40 AM

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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: $$ (\phi, S^* \psi) ~\equiv~ (S\phi, \psi) ~. $$ However, in Weinberg vol1, p51, eq(2.2.7), he defines the adjoint of an antilinear 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?). 


#70
Jan514, 06:54 AM

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



#71
Jan514, 12:54 PM

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Could simply be a matter of mathematical conventions. If someone would like to write a brief summary of the Tomita flow in context of C* quantum formalism, and use Weinberg's conventions, I'd be delighted to go along with their notation. There's nothing sacred about the particular way that Alain Connes and Carlo Rovelli did it in their 1994 paper http://arxiv.org/abs/grqc/9406019 , or that section of the Princeton Companion to Mathematics that I linked to. Nice to have some explicit summary posted here in thread as well as backup sources available online though.
I'm not sure how useful that section of the Princeton Companion actually is, but here's the link FWIW: http://books.google.com/books?id=ZOf...20math&f=false Maybe we need a fresh new one page inthread summary and new backup source links. 


#72
Jan514, 02:14 PM

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PF Gold
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Rep, I really like your bringing up Derek Wise' thing about lifting to "observer space" and the general idea of reality with no "official" spacetime still being real and representable.
So the question now for me, about the C* formulation, is could it be useful in implementing that WiseGielen idea. So I want to keep that in sight. Maybe * algebra can be used to build a quantum version of WiseGielen observer space. Correct me if I am mistaken (Rep and Atyy too!) but I think so far WiseGielen is purely classical. Also I think Rovelli C* picture has a shortcoming in the following sense: I do not see a way in that context to realize any one particular observer's construction of space or of spacetime, or, say, of his past lightcone. There may BE an obvious way, but I don't see it. Suppose the C* picture needs further elaboration so that it contains something that is not an official spacetime but which verges on looking like a bundle of observers. Could we give the C* picture something extra that sort of looks like it is enough like spacetime to allow us to work with it and do spacetime things. As it is, the C* picture is just a normed algebra of measurements with maybe a timeflow defined on them. It is pretty vague. I'm worried. Maybe we should go back to the QB thread. Maybe there is no immediately obvious way to apply the C* picture to the idea in the other thread. 


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