# Global emergent time, how does Tomita flow work?

by marcus
Tags: emergent, flow, global, time, tomita, work
Astronomy
PF Gold
P: 23,227
 Quote by 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?
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 HM.

When you have a conventional quantum theory you always have this! But now the star of M itself becomes an operator defined on HM. 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
αt A = QtAQ-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.
P: 74
 Quote by marcus 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 HM. When you have a conventional quantum theory you always have this! But now the star of M itself becomes an operator defined on HM. 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 at A = QtAQ-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.
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 IN 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, IIIlambda 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
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.
Astronomy
PF Gold
P: 23,227
 Quote by 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 ...
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
 P: 74 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.
P: 483
 Quote by marcus 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, gonna read those now...
 Astronomy Sci Advisor PF Gold P: 23,227 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, 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.
P: 8,613
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."

 Quote by marcus 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 André Henriques comments at http://mathoverflow.net/questions/57...rhaps-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".)
Astronomy
PF Gold
P: 23,227
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.
 Quote by 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]." ...
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.
P: 8,613
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.

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 by marcus ==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.

 Quote by 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?
 Quote by marcus It gives the right limit in several interesting cases. I don't know about asymptotic AdS. Confirming cases are listed and discussed in the 1994 paper by Alain Connes and Carlo Rovelli. I gave the link in post #2 of this thread. The RATIO of T-time to local observer time can be given a physical meaning, which is kind of interesting---a general relativistic temperature identified by Tolman around 1930. There's a link to the Smerlak Rovelli paper about that also in post #2, I think. http://www.physicsforums.com/showthr...23#post4209223
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
 Sci Advisor P: 8,613 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 Hτ, 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 HR is simply related to Hτ with ..."
 Sci Advisor P: 8,613 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.
 Sci Advisor HW Helper P: 1,334 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.
 Sci Advisor P: 8,613 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.
P: 1,924
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?).
 Sci Advisor P: 8,613 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).
 Astronomy Sci Advisor PF Gold P: 23,227 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/gr-qc/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 in-thread summary and new backup source links.
 Astronomy Sci Advisor PF Gold P: 23,227 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 Wise-Gielen idea. So I want to keep that in sight. Maybe * algebra can be used to build a quantum version of Wise-Gielen observer space. Correct me if I am mistaken (Rep and Atyy too!) but I think so far Wise-Gielen 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 time-flow 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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