Global emergent time, how does Tomita flow work?

[marcus]

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=ZO...6AEwAw#v=onepage&q=minoru tomita math&f=false
Maybe we need a fresh new one page in-thread summary and new backup source links.

 

[marcus]

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 anyone 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.

 

[strangerep]

Rep,

Evo sometimes abbreviates my name to "Strange", but that can create ambiguities in different contexts.

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.

That's the impression I got too.

Also I think Rovelli C* picture has a shortcoming in the following sense: I do not see a way in that context to realize anyone 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. [...]

I had envisaged a (representation of) an algebra of observables $A$ associated to an inertial observer. That observer constructs a spacetime as a homogeneous space for the associated group. For multiple observers, we construct tensor products like $A\otimes A$, and hence products of their respective spacetimes . (Think: products of symplectic phase spaces in classical mechanics.) But there must be more to it than that if we are to accommodate mutual accelerations associated with interactions, etc.

So yes, for this part of the discussion we should go back to the other (QBist) thread. I'll resume that later. I only came over here to clarify the Tomita-time construction, so I need to study the references that you and atyy mentioned above.

 

[strangerep]

[...] 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.[...]

Now that I've read and pondered some more on Tomita-time, and thermal time, I begin to think that the antiunitary $J$ part of the operator $S = J \Delta^{1/2}$ is a mere red herring for these purposes.

The generic idea behind the thermal time construction starts with an arbitrarily-chosen fiducial (aka cyclic) state operator $\omega$. As a state operator, it satisfies $\omega^* = \omega$ (hence all its eigenvalues are real). Also, its eigenvalues are all nonnegative.

The operator $\Delta$ is simply $\omega$. The operator $\omega^{1/2}$ makes sense (simply take the square roots of the eigenvalues). Similarly, the operator $U(t) := \omega^{it}$ also makes sense by similarly raising the eigenvalues to that power.

The original state operator $\omega$ is obviously invariant under conjugation by $U(t)$, i.e., $U(t) \, \omega \, U(-t) \;=\; \omega$, etc.

However, if $\omega$ is a pure state then one and only one of its eigenvalues is 1 while the others are zero (cf. Ballentine p52). In that case $\omega^n = \omega$, where $n$ is any complex number. Therefore, the "flow" represented by the $U(t)$ is trivial (the identity) if $\omega$ is pure, but can be nontrivial if $\omega$ is nonpure -- which is the case for the usual thermal (Gibbs) state $\omega = \exp (-\beta H)$ .

The Tomita construction seems to start from any antilinear operator $S$, performs a "polar" decomposition of it, obtaining a corresponding $\Delta$ thereby which can be used as the fiducial state operator. But so what? For a given algebra, how is $S$ picked out? And is this essentially equivalent to picking out a fiducial state operator $\omega$, as is usually done? I don't see what the fuss is all about. What am I missing?

 

[naima]

Usual QM is hamiltonian dependent. Once you have H you know how operators evolve with time in the Heisenberg picture.
To study KMS condition we compute the average value $\langle \alpha_t(A) B \rangle$ in the state exp(-H).

With Tomita machinery no Hamiltonian to begin with but things are state dependent.
given a density matrix $\rho$ we send $A\rho$ to $S(A \rho) =A^*\rho$ then Tomita theorem associates $\rho$ to a KMS flow $\sigma_{\rho}(s)$ (we compute $\langle \sigma_{\rho}(s) (A) B \rangle$ in the state $\rho$
You see that S is not picked among others. One $\rho$, one Tomita flow.
Could you tell me why in $S = J \Delta$ Connes calls J the phase of S and $\Delta$ the modulus of S?

 

[strangerep]

[...]
You see that S is not picked among others. One $\rho$, one Tomita flow.

Yes, that's the impression I got. But I find it quite weird to rely on such a thing for time-flow, since the flow becomes trivial if the state is pure.

Could you tell me why in $S = J \Delta$ Connes calls J the phase of S and $\Delta$ the modulus of S?

I haven't read much of Connes, but I guess the terminology just follows the standard terminology in linear algebra and functional analysis. E.g., in Lax's textbook on Linear Algebra, there's a "polar decomposition" theorem 22 on p139, which says:

Let $Z$ be a linear mapping of a complex Euclidean space into itself. Then $Z$ can be factored as $$Z = RU ~,$$where $R$ is a nonnnegative self-adjoint mapping, and $U$ is unitary.

So the terminology of calling $R$ a "modulus" and $U$ a "phase" is just a generalization of terms used in polar decomposition of a complex number: $z = r e^{i\theta}$.

Looking at Lax's proof, I think it goes through similarly, if $Z$ is anti-linear instead. In that case the decomposition is of the form $Z = RJ$ where now $J$ is anti-unitary. So calling it a "phase" is perhaps an abuse of terminology, but mathematicans love to call different things by the same name.

 

[Chronos]

Too many assumptions for my comfort zone.

 

[strangerep]

Too many assumptions for my comfort zone.

Can you elaborate? (I'm kinda struggling with this stuff.)

 

[naima]

Yes, that's the impression I got. But I find it quite weird to rely on such a thing for time-flow, since the flow becomes trivial if the state is pure.

When the state is almost pure his entropy is very small. As the one-parameter s grows the operators seem to be frozen.
Rovelli wrote once that the time flow is a product of our ignorance (entropy). the sentence is mysterious but in this point of view it may be taken into account.

I think that Marcus was wrong with:

Tomita time is an intrinsic observer-independent time variable available to us for fully general relativistic analysis.

Being state dependent how could it be observer-independent?

It would be interesting to see how it is an emerging time.

 

[strangerep]

When the state is almost pure his entropy is very small. As the one-parameter s grows the operators seem to be frozen.
Rovelli wrote once that the time flow is a product of our ignorance (entropy). [...]

Can you recall the reference? I'd like to read the context of Rovelli's remark.

Indeed, I have trouble making sense of it. Does it suggest that the less one knows about the global state, the faster time seems to flow?? And does this even sit consistently with Lorentz boosts and relative time dilation between different observers? It also seems to be in contradiction with known gravitational time dilation in which a clock in a stronger gravitational field runs slower.

Hmmm,... let's see,... an observer accelerating strongly knows less(?) than a weakly-accelerating observer. (I mean in terms of entanglement entropy associated with their Rindler horizons). So...
[Oops! Brain crash -- core dumped. I'll have to think about that further after I reboot.]

I think that Marcus was wrong with:

Tomita time is an intrinsic observer-independent time variable available to us for fully general relativistic analysis.

Being state dependent how could it be observer-independent?

I was under the impression that the state $\omega$ which generates Tomita time flow is analogous to the "fiducial vacuum state" from which Fock spaces are built (except that the latter is pure but the former is nonpure). So it's a "given". Then, just as inequivalent Fock spaces can arise from different choices of vacuum state, so different universes arise from different choices of $\omega$.

But,... as you see,... I struggle with all this...

It would be interesting to see how it is an emerging time.

Yes -- emerging from what? Spin network states?

 

[naima]

Can you recall the reference? I'd like to read the context of Rovelli's remark.

Indeed, I have trouble making sense of it. Does it suggest that the less one knows about the global state, the faster time seems to flow?? And does this even sit consistently with Lorentz boosts and relative time dilation between different observers? It also seems to be in contradiction with known gravitational time dilation in which a clock in a stronger gravitational field runs slower.

It is a quotation from Rovelli's small book "what is time? What is space?"

He writes that he worked on a timeless theory that had no success around him unless he met Alain Connes (1982 Fields medal). he realized that his theory was a soecial case of Connes's theory. They wrote a paper together.

He says that the flow of time is an emerging effect of our ignorance. If we had a perfect knowledge of things they would seem to be frozen (i use here my proper words).
Take the zeno effect if you observe continuously a up spin it will freeze. nothing will happen to it. You must accept not to observe it for a while to see becoming down.
This may be in relation to the frozen flow associated to a pure state. Only decohered observers would see things moving. A finite time observer has only access to events inside the future cone of his birthday and the past cone of his death (the diamond) but his wave function may be nor null outside this region. This is the reason while entropy and temperature is associated to this diamond.

You will see here answers from Rovelli.
It is hard to imagine that what we (decohered observers) see is not what really happens but only WE see.

I do not know what is the physical meaning of the state which generate Tomita flow. is it a vacuum seen by the observer or the state of the observer?

 

[strangerep]

You will see here answers from Rovelli.

Thanks. Normally, I don't read much of the FQXi comments (too much waffle), but Rovelli's responses are interesting (and also a lesson in how to remain polite).

I'll quote a few of Rovelli's remarks from your link which seem relevant to my other confusions above...

[...] it is important to recall that the thermal time hypothesis does not REPLACE dynamics. My entire point is that dynamics can be expressed as correlations between variables, and does not NEED a time to be specified. The thermal time is only the one needed to make sense of our sense of flowing time, it is not a time needed to compute how a simple physical system behaves. The last can be expressed in terms of correlations between a variable and a clock hand, without having to say which one is the time variable. Therefore the question about the flow of time defined by bodies at different temperature is a question about thermodynamics out of equilibrium.

[...]

[...] all temporal "effects" that are captured by ordinary mechanics have nothing to do with thermal time. They just have to do with the fact that there are laws that govern the relations among variables. The additional peculiar "flowing" of time is an "effect" which is not the same thing as temperature, but (if we believe the thermal time hypothesis) it emerges in a thermodynamcal/statistical situation only.

I do not know what is the physical meaning of the state which generate Tomita flow. is it a vacuum seen by the observer or the state of the observer?

I'm not sure about that, but this extract from another of Rovelli's responses seems relevant (emboldening is mine):

[...] In a timeless world, a small subsystem (us) whose interaction with the rest of the universe is limited to a very small number of variables, and therefore who has no access to the exact state of the rest of the universe (that is, it has the same state for many different states of the universe), can be correlated with the rest of the world in such a way to have an imprecise information about the rest of the system (a way to express these notions precisely using Shannon information theory is in my work on relational quantum theory); then with respect to this subsystem a Tomita flow is defined; and this flow itself is the physical underpinning of the perception of the flow of time, whatever this perception is.

Finally, Marcus: if you're still reading this thread... Does the following Rovelli quote remind you of anything we were discussing recently?

[...] getting rid of space at the fundamental level is not very new. I think that what general relativity does is precisely so. It is the realization that the Newtonian "space" is nothing else that one of the physical fields that make up reality. Reality is not a space inside which things moves, but rather an ensemble of fields in interaction. So, my answer is that we must forget space and forget time. Forgetting space is easy; we have centuries of traditions that give us examples about how to think the world without a fundamental space. Forgetting time is more difficult, [...]

 

[julcab12]

I remember my first account here a couple of yrs ago- PF 'Existence of time'. My understanding is raw at that time(and still does today^^) and i have the sense that time can also be expressed not in the usual experience. Way back when i was a kid I used to play with clocks comparing and manipulating each one to a point of breaking. I'm very curious. Instead of imaging godzillas and voltrons. I'm often confined to such unusual questions. Crazy stuff like 'what if' time clocks were never invented and all we can see are movements and have a sense of measurement through uniformity of events e.g changes in location of the sun and passing of seasons, etc etc. In my attempt. I imagined myself as an outside observer. I was trying to deduced everything as a variable of movements/dynamics(my version of thinking of what is thermal today) confined from a reference of absolute stillness. I always thought that physicality of movement/vibration/thermal creates the experience of passing moment--- time in a sense that it is emergent, consequential and the uniformity of such events are what is measured.

I was surprised when i read the links. Brought back old memories.

----CR reply "is that dynamics can be expressed as correlations between variables, and does not NEED a time to be specified. The thermal time is only the one needed to make sense of our sense of flowing time, it is not a time needed to compute how a simple physical system behaves. The last can be expressed in terms of correlations between a variable and a clock hand, without having to say which one is the time variable. Therefore the question about the flow of time defined by bodies at different temperature is a question about thermodynamics out of equilibrium... ALL dynamical systems (classical) can be simply reformulated in a way that puts time on the same ground as the other variables, an din this case the dynamics is expressed by a "Wheeler-DeWitt-like" constraint.

 

[torsten]

Indeed, I have trouble making sense of it. Does it suggest that the less one knows about the global state, the faster time seems to flow?? And does this even sit consistently with Lorentz boosts and relative time dilation between different observers? It also seems to be in contradiction with known gravitational time dilation in which a clock in a stronger gravitational field runs slower.

I also worked with Tomitas flow in the context of foliations on spacetime. Here it seems there is an interesting perspective. Connes used the flow to change from factor II foliations (as described by factor II von Neumann algebras) to factor III foliations. Both foliations are related by a postive, measurable function as density. But from the physical point of view, this function can be interpreted as probability function. Then your quote above will make sense.

 

[naima]

But from the physical point of view, this function can be interpreted as probability function. Then your quote above will make sense.

Could you elaborate for almost laymen?

 

[torsten]

Could you elaborate for almost laymen?

The space of leafs of a foliation is a complicated space. Consider for instance a curve covering the torus, the so-called Kronecker foliation of the torus. A continuous function over the leaf space of this foliation can be only the constant function (otherwise the function is not continuous).
Connes had now the brilliant idea to associate a von Neumann algebra of operators to the leaf space of a foliation. Then from the structure of this algebra, one can recover the properties of the foliation. The leaf space was the first example of a non-commutative space (and a motivation for the following constructions). In case of the Kronecker foliation of the torus, one obtains a factor II algebra for the leaf space.
But there are physically more interesting foliations, mainly foliations of hyperbolic manifolds having a factor III as leaf space. By using Tomitas theory, Connes constructed a new foliation (with factor II leaf space) from the factor III leaf space. It is a total space of the bundle of positive densities or equivalently the space of probability functions (after integration). The probability function is defined over the transverse bundle of the foliation, i.e. the space who labelled the leafs.
I hope it is now more understandable