Julian Barbour on does time exist

[marcus]

Hmmm, that paper is almost two decades old, but I guess the concept hasn't changed much from then since you are linking it...

Yes! I do think the Connes Rovelli paper is very well written. What they say there can probably not be said much better by anybody. But the idea has developed and the most recent paper is, as you may know, Rovelli's September 2012 "General relativistic statistical mechanics".

I think the point is this is a major outstanding problem that may be nearing the time when it is ripe to work on. In a general covariant theory there is no preferred idea of time, and so one cannot do thermodynamics or stat mech as we ordinarily think of it.

One can do these things on an arbitrary fixed curved spacetime, but that is not the full GR treatment. So eventually humans HAVE to do thermo and stat mech in full GR context. Or the quantum version of that. But researchers must use their efforts wisely and not work on problems which are not ready to be addressed. For a while they only slowly chip away, or prepare some ideas to start with. that is how i see it.

I think one should not immediately think of a 4D lorentzian manifold (just my private opinion) I think one should think of the observable algebra, possibly abstractly as a C*-algebra. And the state embodies what we think we know and expect about all the observations. The fine thing is that this state itself uniquely specifies a one-parameter flow on the observables---the modular group of automorphisms of the algebra---uniquely up to some equivalence relation.
that is very abstract, but then one can in various cases make it specific using the familiar tools of the Hilbertspace, the 4D manifold, the fields written on the manifold, and so on. Or (I don't know) maybe LQG tools and Hilbertspace. At the moment I do not see any suggestion of a connection with LQG, it seems like an entirely separate development. (Except for sharing the general covariant GR perspective.)

 

[marcus]

I must repeatedly stress that this is only a hypothesis put forward to be tested, but C&R just could have hit on the way to handle time in a generally covariant quantum system. Remember that all we actually have is an algebra of observables. A 4D differential manifold is sheer mathematical fiction, as far as anyone knows. All we really have are our observations, a finite number of them, of which we can multiply and add together some to predict others (because they form an algebra).

==quote http://arxiv.org/abs/gr-qc/9406019 page 14==
Let us now return to generally covariant quantum theories. The theory is now given by an algebra A of generally covariant physical operators, a set of states ω, over A, and no additional dynamical information. When we consider a concrete physical system, as the physical fields that surround us, we can make a (relatively small) number of physical observation, and therefore determine a (generically impure) state ω in which the system is. Our problem is to understand the origin of the physical time flow, and our working hypothesis is that this origin is thermodynamical. The set of considerations above, and in particular the observation that in a generally covariant theory notions of time tend to be state dependent, lead us to make the following hypothesis.

The physical time depends on the state. When the system is in a state ω, the physical time is given by the modular group αt of ω.

The modular group of a state was defined in eq.(8) above. We call the time flow defined on the algebra of the observables by the modular group as the thermal time, and we denote the hypothesis above as the thermal time hypothesis.

The fact that the time is determined by the state, and therefore the system is always in an equilibrium state with respect to the thermal time flow, does not imply that evolution is frozen, and we cannot detect any dynamical change. In a quantum system with an infinite number of degrees of freedom, what we generally measure is the effect of small perturbations around a thermal state. In conventional quantum field theory we can extract all the information in terms of vacuum expectation values of products of fields operators, namely by means of a single quantum state |0⟩. This was emphasized by Wightman...

...Given the quantum algebra of observables A, and a quantum state ω, the modular group of ω gives us a time flow α_t. Then, the theory describes physical evolution in the thermal time in terms of amplitudes of the form
F_A,B(t) = ω(α_t(B)A) (26)
where A and B are in A. Physically, this quantity is related to the amplitude for detecting a quantum excitation of B if we prepare A and we wait a time t – “time” being the thermal time determined by the state of the system.

In a general covariant situation, the thermal time is the only definition of time available. However, in a theory in which a geometrical definition of time independent from the thermal time can be given, for instance in a theory defined on a Minkowski manifold, we have the problem of relating geometrical time and thermal time. As we shall see in the examples of the following section, the Gibbs states are the states for which the two time flows are proportional. The constant of proportionality is the temperature. Thus, within the present scheme the temperature is interpreted as the ratio between thermal time and geometrical time, defined only when the second is meaningful.⁶

We believe that the support to the thermal time hypothesis comes from analyzing its consequences and the way this hypothesis brings disconnected parts of physics together. In the following section, we explore some of these consequences. We will summarize the arguments in support the thermal time hypothesis in the conclusion...
==endquote==

C&R are telling us that in a fully generally covariant system without making some additional arbitrary choices, *the modular group flow is the only definition of time we've got*.
Further, it gives us transition amplitudes.
Further, if we go ahead and arbitrarily make a choice of geometry (e.g. Minkowski) then we can compare that time with the inherent modular group time, and the ratio can have a physical meaning.

 

[Paulibus]

I think that the central idea of Rovelli’s essay “Forget Time”; his proposed “thermal time hypothesis” ; is a “timely”, important and thought provoking reminder of an uncomfortable truth, namely that the way physicists describe reality (which is their job description!) is dominated by our anthro’centric perspective. We are a species distinguished by our peculiarly elaborate communication skills.

Rovelli persuasively argues that:

... what we call “time” is the thermal time of the statistical state in which the world happens to be, when described in terms of the macroscopic parameters we have chosen.

(My emphasis.) Thermal time is taken as the variable that the system is “in equilibrium” with respect to. In the case of say, a gas, his thermal time, I gather, reduces to our ordinary time (to within a proportionality factor). Since our macroscopic-scale description of equilibrium hinges on the statistically and thermally defined concept of temperature, in this case “thermal” is a very appropriate label.

What about situations where we are as yet unable to quantify entropy, but just trust the Second Law implicitly? As Rovelli says: “Time is ... the expression of our ignorance of the full microstate”. Is Rovelli suggesting that our concept of time is an statistical artefact of the scale we human beings inhabit? Just a tool; a parameter that physics uses to describe quantitatively our human circumstances, with thermodynamics as a sort of catch-all background?

Roll on the next chapter in this story.

 

[marcus]

... In the case of say, a gas, his thermal time, I gather, reduces to our ordinary time (to within a proportionality factor)...
What about situations where we are as yet unable to quantify entropy, ...?
...
Roll on the next chapter in this story.

Your post raises several interesting issues, I'm focussing on one right now---the cases where thermal time "reduces to our ordinary time". It seems important to list those offered in the Connes Rovelli paper, since the good consequences of the thermal time hypothesis (TTH) support one's suspicion that it is possibly right and worth investigating.

I've moved over to and am working from the Connes Rovelli paper, since it is the main source and considerably more complete than any of the other papers (including the wider-audience FQXi essay.) The C&R paper has 77 cites, over a third of which are in the past 4 years. It is the root paper that other thermal time papers (including by Rovelli) refer to for detailed explanation.

So what I would propose as a "next chapter in the story" is to make sure we get the main points that C&R are making. I'll run down the main corroborative cases they give on page 22, in their conclusions. These are explained in the preceding section, pages 16-21.
== http://arxiv.org/abs/gr-qc/9406019 ==
...
• Classical limit; Gibbs states. The Hamilton equations, and the Gibbs postulate follow immediately from the modular flow relation (8).
• Classical limit; Cosmology. We refer to [11], where it was shown that (the classical limit of) the thermodynamical time hypothesis implies that the thermal time defined by the cosmic background radiation is precisely the conventional Friedman-Robertson-Walker time.
• Unruh and Hawking effects. Certain puzzling aspects of the relation between quantum field theory, accelerated coordinates and thermodynamics, as the Unruh and Hawking effects, find a natural justification within the scheme presented here.
...
==endquote==

They also include three other supporting points. One that is not discussed in the paper and they simply mention in passing is the widely shared notion that time seems bound up with thermodynamics and there are indeed hundreds of papers exploring that general idea in various ways (far too numerous to list). Their idea instantiates this widely shared intuition among physicists.

Another supporting point is that the thermal time formalism provides a framework for doing general relativistic statistical mechanics. Working in full GR, where one does not fix a prior spacetime geometry, how can one do stat mech? A way is provided here (and see http://arxiv.org/abs/1209.0065 )

The sixth point is the one they give first in their "conclusions" list---I will simply quote:
==quote gr-qc/9406019 page 22==
• Non-relativistic limit. In the regime in which we may disregard the effect of the relativistic gravitational field, and thus the general covariance of the fundamental theory, physics is well described by small excitations of a quantum field theory around a thermal state |ω⟩. Since |ω⟩ is a KMS state of the conventional hamiltonian time evolution, it follows that the thermodynamical time defined by the modular flow of |ω⟩ is precisely the physical time of non relativistic physics.
==endquote==

There is one other supporting bit of evidence which I find cogent and which they do not even include in their list. This is the uniqueness. Have to go, back shortly.

 

[marcus]

The way I see the uniqueness point is that once you have a C*-algebra A of all your observables, and a (positive trace-class) state functional ρ representing what you think you know about the world, then there is only one time evolution that you can define from the given (A,ρ) without making any further choices.

It is the natural canonical flow of time, given the world as we know it. We know the world as a bunch A of observables/measurements that are interrelated by adding subtracting multiplying etc. that is what an algebra is. And as a probabilistic functional ρ defined on that algebra, representing our information about what values those observables take. Given those two things (A,ρ) there is a unique flow defined on the algebra, taking each observable along to subsequent versions of itself.

I'm not entirely clear or comfortable with this, but it seems reasonable to try thinking along those lines. GR is timeless, QM says what counts are the measurements, we take those hints seriously and we say that the world exists (timelessly) as an algebra of observations A. Specifically a C* algebra (abstract form of von Neumann algebra) and such an algebra has a natural idea of state defined on it representing what we think we know and expect. So this pair (A, ρ) is the world. And that pair gives you a unique time flow. The one-parameter group of automorphisms on the algebra that takes any observable to the next, to the next, to the next. There is a natural built-in way to make the observables flow. That's time. Or one idea of it.

 

[marcus]

I think perhaps the essential thing about time-ordering is it makes a difference which measurement you do first. All these differences are encoded in the non-commutativity of the algebra of observations, so time-orderings are already latent in the algebra. We shouldn't be too surprised that an algebra of observables, helped by a timeless state function to define the world, would have a preferred flow.

 

[Paulibus]

...it seems reasonable to try thinking along those lines. GR is timeless, QM says what counts are the measurements, we take those hints seriously and we say that the world exists (timelessly) as an algebra of observations A. Specifically a C* algebra (abstract form of von Neumann algebra) and such an algebra has a natural idea of state defined on it representing what we think we know and expect. So this pair (A, ρ) is the world. And that pair gives you a unique time flow. The one-parameter group of automorphisms on the algebra that takes any observable to the next, to the next, to the next. There is a natural built-in way to make the observables flow. That's time. Or one idea of it.

Sounds sensible to me, put this way (barring C* algebra; new to me). But for a long time I've thought of time as a "dimension", one of four absolutely mysterious and fundamental such items in the "Universe Lucky Packet" that when unwrapped, started stuff off with a singular bang, or a softer bounce, neither of which we understand properly yet.

What are simple folk like me to think if Connes and Rovelli's approach turns out to be right?

Time is part of the flexible spacetime geometry responsible for gravity. Time curves as one of the four dimensions described by the Riemann tensor. So I've understood. Or is it ct, which dimensionally is space-like, that curves? Or perhaps just c that changes from place to place, so bending light around galaxies? Strange thoughts pass by.

 

[marcus]

... it seems reasonable to try thinking along those lines. GR is timeless, QM says what counts are the measurements, we take those hints seriously and we say that the world exists (timelessly) as an algebra of observations A. Specifically a C* algebra (abstract form of von Neumann algebra) and such an algebra has a natural idea of state defined on it representing what we think we know and expect. So this pair (A, ρ) is the world. And that pair gives you a unique time flow. The one-parameter group of automorphisms on the algebra that takes any observable to the next, to the next, to the next. There is a natural built-in way to make the observables flow. That's time. Or one idea of it.

Sounds sensible to me, put this way (barring C* algebra; new to me). But for a long time I've thought of time as a "dimension", one of four absolutely mysterious and fundamental such items in the "Universe Lucky Packet" that when unwrapped, started stuff off with a singular bang, or a softer bounce, neither of which we understand properly yet.

What are simple folk like [us all] to think if Connes and Rovelli's approach turns out to be right?

Time is part of the flexible spacetime geometry responsible for gravity. Time curves as one of the four dimensions described by the Riemann tensor. So I've understood. Or is it ct, which dimensionally is space-like, that curves? Or perhaps just c that changes from place to place, so bending light around galaxies? Strange thoughts pass by.

I imagine we're all rather much in the same fix as you describe, or at least I am. Geometrizing time as a pseudo-spatial dimension works so well! It's become part of how we think.

And it may be right! This approach proposed by Connes and Rovelli may be wrong. It is just an hypothesis which they argue should be thought through.

You put the mental dilemma very precisely---and the business of light bending around galaxies and clusters of galaxies is very beautiful. As well as being essential to observational cosmology nowadays---they depend on the magnification produced by lensing. The whole business of 4D geometry is compellingly beautiful...

It's a challenge to hold two contradictory ways of thinking, at least for a while, in one's head. I can't say it any better than you just did.

 

[marcus]

Paulibus, I don't want to raise false hopes. But I am beginning to find thermal time understandable and (for me) it comes of reading pages 16 and 17 of the Connes Rovelli paper. I'm comfortable with ordinary operator algebra on ordinary hilbertspace. this is undergrad math major level. there are many steps of algebra but you just have to go thru them patiently. IFF you also are comfortable it might work for you. Then you wouldn't have to feel mystified by it. Maybe in a day or two I will try to make INTUITIVE sense of the 20 or so steps of algebra on those pages. In case you don't like vectors and matrices and would find it tedious to work thru.

What it does is go thru the NON relativistic case where there is already a hamiltonian and it shows that the jazzy new thermal time flow RECOVERS the conventional hamiltonian time evolution. IOW the jazzy new idea of time specializes down to the right thing---it is a valid generalization of what we already think of as time-evolution flow.

So for me, the pages 16 and 17 are the core of the Connes Rovelli paper and at least for now the core and the whole business. It's not so unintuitive now.

 

[marcus]

here ( gr-qc/9406019 pages 16,17) we have a conventional situation with hilbertspace H and hamiltonian H. Of course we have the algebra A of observables, the operators on the HAnd the quantum state ω is a density matrix: that's what we want to study and finesse a time evolution from. And of course we have the algebra A of observables, the operators on the H

Imagine it in positive diagonal form, we'll need its square root k_o = ω^1/2.
Now the trick is the "GNS construction" which is like obviously a bunch of matrices can themselves form a vectorspace! You can add two and get another matrix. You can multiply by a scalar.

If we want to think of k_o as an operator we write it k_o. If we want to think of it as a vector in a vectorspace where the vectors are actually matrices we write it | k_o>

This (which appears kind of dumb at first sight) is actually the cleverest thing on the whole two pages. I've seen this in math before, something that looks utterly pointless turns out not to be. It is so pointless that it takes clever people like Gelfand Naimark Segal to think of it. We can make a mixed state (a matrix) into a pure state (a vector) in a "higher" hilbertspace this way.

Now all the operators A which used to act on H can act on vectors like k_o, call a generic such "vector" k. The key analytic condition is that k k* have finite trace (equation 30).
Define the new action of any operator A by
A |k> = |Ak>
It's obvious. k WAS an operator, so A by k is another operator so |Ak> is a vector. It is the vector which |k> gets mapped to.

So now we can do something a little interesting. We can define the set
{ A |k_o> for all A}
I think I've seen that called the "folium" of |k_o>. Anyway the set of all vectors that |k_o> gets mapped to, using all possible operators in the algebra. that is a vectorspace and it has |k_o> as a "cyclic" vector. I don't like the term "cyclic" for this but it has historical roots and is conventional. Call it seed or generator if you want. It generates the whole vectorspace when operated on by the algebra A. Above all algebra requires patience, now we are at the top of page 17, where things begin to happen. I'll continue later.

|k_o> is going to play the role of a "thermal vacuum state". the vanilla state from which the thermal time arises. the authors say a little about it at the top of page 17 that could provide extra intuition. But I'll continue this later.

 

[rjbeery]

Here's my two cents;

IF:
1) All observers share a reality
and
2) There is some definition of "state" or "now" describing physical existence which extends beyond those observers (which utilizes the concept of Time in any way)
and
3) We consider what Relativity does to our usual definitions of Time

THEN:
Applying the definition of #2 for all observers in #1, taking #3 into consideration, we conclude that the entirety of the history/future of the Universe eternally exists as a physical representation in what is literally a static 4D Block Universe; the flow of time and the concept of becoming are emergent properties of being sentient.

 

[marcus]

Hi RJ, block universe was discussed some earlier in posts #50 and 52 of this thread. Here is a link to post #52
https://www.physicsforums.com/showthread.php?p=4140332#post4140332

 

[rjbeery]

Hi RJ, block universe was discussed some earlier in posts #50 and 52 of this thread. Here is a link to post #52
https://www.physicsforums.com/showthread.php?p=4140332#post4140332

Ahh, thanks marcus. I was too lazy to work through 4 pages of comments. Also, there wasn't much mention of Block Universe that I could see (even in your referenced posts).

A block universalist might say I can't make a decision as the future is already present in some sense. But he/she seems unable to say how many essays that eventually will have been read by me

This is because an observer in the "now" has access to information which has been stored in some manner accessible to the observer in the current state; this information gets stored via entropic processes. What this means is that entropy doesn't increase with time, but rather information is available for storage as entropy is increased. Any observer in the "now" might naturally conclude that states to which he has information (i.e. the past) are of a different character than those states which reveal how many essays you have or will read, but that isn't the case (IMHO).

 

[marcus]

If one accepts GR then the decay of a radioactive nucleus will affect the geometry of the universe (as the distribution of mass always does, in GR) and according to QM the time when the nucleus will decay has not yet been determined (unless you postulate "hidden variables") and cannot in principle be predicted. Thus the geometry (the metric) of the universe is not predetermined.
So if one accepts ordinary physics (GR and QM) there can be no block universe.

George Ellis put this amusingly in his FQXi essay that I linked to above. He described a massive rocket powered sled zooming back and forth along a track under the control of a radioactive decay (Schrödinger Cat) mechanism that tells it when to go east and when to go west.

In ordinary GR, the "coordinate time" is not physically meaningful. Not measurable. One needs to break general covariance by introducing an observer, or e.g. a uniformly distributed gas of particles, as is done in cosmology.

A good discussion of the status of time in modern physics is provided in a few pages of the Rovelli essay that Naty linked to earlier in this thread. It is called "Unfinished Revolution" and was posted on arxiv in 2006 or 2007. Google "rovelli revolution" and you should get it. It is wide audience. I don't personally know of any working physicist who takes the traditional block U idea seriously. The prevailing question is where do we go from here.
=========================

What I've been gradually working thru, in the past few posts, is the idea that there IS an intrinsic time-flow on the space of observables, which arises from specifying a STATE ω of the universe. This is akin to what Barbour has been saying: time is certainly real but not as a pseudo-spatial dimension or as something fundamental. It arises from more basic stuff. In this case it arises as a one-parameter group of transformations of the space of observables. What I'm trying to understand better is how this socalled "modular group" α_t or "flow" arises from specifying the algebra of observables A and the state ω. The formalism we are working with here is compatible with BOTH QM and GR, it is used to delve into "general covariant statistical quantum mechanics" which definitely seems interesting.

 

[marcus]

Sources
http://arxiv.org/abs/gr-qc/9406019 pages 16,17
WikiP: "Gelfand-Naimark-Segal construction"
WikiP: "KMS state"
WikiP: "Tomita-Takesaki theory" (not so good I think, but at least article exists)
WikiP: "Polar decomposition" (article exists, I haven't used or evaluated it)

The basic situation that general covariant quantum physics deals with is an algebra A of observables. That's the world. After all QM is about making measurements/observations. And a temporal flow α_t is a oneparameter group of automorphisms of that algebra.

automorphism means it maps an observable A onto another observable α_tA which you can think of as making the same observation but "t timeunits later".
oneparameter group means that doing α_s and then doing α_t has the same flow effect as doing α_s+t. the parameter t is a real number.
And automorphism means it preserves the algebra operations, it is linear etc etc.

Observables are in fact an algebra because you can add and multiply observables together to predict other observables or to find how they correlate with each other.

The statistical quantum state of the world is represented by a positive functional on the algebra which we can think of as a density matrix ω and its value on an observable A can be written either as ω(A) or as trace(Aω). The state ω is what gives the observables their expectation values and their correlations.

A nice thing about a density matrix ω is that it has a square root ω^1/2. Think of writing it as a diagonal matrix with all positive entries down the diagonal, and taking the square root of each entry.

More about this later. From an algebra A and a state of the world ω it is possible to derive a unique flow α_t on the algebra. Taking each observable A into a progression of "later" evolved observables α_tA, for every timeparameter number t.

 

[rjbeery]

If one accepts GR then the decay of a radioactive nucleus will affect the geometry of the universe (as the distribution of mass always does, in GR) and according to QM the time when the nucleus will decay has not yet been determined (unless you postulate "hidden variables") and cannot in principle be predicted. Thus the geometry (the metric) of the universe is not predetermined.
So if one accepts ordinary physics (GR and QM) there can be no block universe.

George Ellis put this amusingly in his FQXi essay that I linked to above. He described a massive rocket powered sled zooming back and forth along a track under the control of a radioactive decay (Schrödinger Cat) mechanism that tells it when to go east and when to go west.

The hidden variable problem goes away in a Block Universe; nothing remains to be determined because it already exists. The unknown variables are hidden from us locally but reside local to the respective particles in the future. "When" a nucleus decays relative to an observer is a problem of information availability, not some intrinsic Universal randomness.

I emboldened the words in your post which show that we have problems thinking without a "flow of time". Technically, neither GR nor QM have any mechanism for a FLOW of time whatsoever. They are completely time-symmetric theories, yet you are suggesting that one direction is preferred over the other. The time parameter is only a marker along the 4D Block Universe in my view.

 

[marcus]

nothing remains to be determined because it already exists...

You are on your own, RJ. Working physicists assume QM. Your picture is incompatible with QM. I've tried explaining this to you but it doesn't seem to get across.

How about you read a few pages of Rovelli's wide-audience essay Unfinished Revolution, that I suggested you look at earlier?
Section 1.2 "Time" is less than a page long. It starts at the bottom of page 3 and covers part of page 4.

Google "rovelli unfinished revolution" and you get the arxiv version: http://arxiv.org/abs/gr-qc/0604045

 

[rjbeery]

You are on your own, RJ. Working physicists assume QM. Your picture is incompatible with QM. I've tried explaining this to you but it doesn't seem to get across.

How about you read a few pages of Rovelli's wide-audience essay Unfinished Revolution, that I suggested you look at earlier?
Section 1.2 "Time" is less than a page long. It starts at the bottom of page 3 and covers part of page 4.

Google "rovelli unfinished revolution" and you get the arxiv version: http://arxiv.org/abs/gr-qc/0604045

I will, and I will also read George Ellis' FQXi essay but I could not find your link to it. I assume I can Google it without much problem. Regardless, I'm not speaking from a position of naivete; QM is not incompatible with Block Time and I'd be happy to discuss specifically why you think this (other than referencing others' papers).

 

[marcus]

I will, and I will also read George Ellis' FQXi essay but I could not find your link to it. I assume I can Google it without much problem. Regardless, I'm not speaking from a position of naivete; QM is not incompatible with Block Time and I'd be happy to discuss specifically why you think this (other than referencing others' papers).

I gave the link in the post I pointed you to:
http://fqxi.org/community/essay/winners/2008.1
go there, scroll down to "second community prize", there is Ellis's abstract and a link to the PDF.

I already explained the incompatibility using the same example Ellis did, radioactive decay changes the distribution of mass---Ellis's rocket sled just makes it more colorful.

 

[marcus]

I need to plug ahead with how time (as a flow on the observable algebra) emerges. For continuity, here are the essentials of the last post:

Given an algebra A of observables and a state of the world ω it is possible to derive a unique flow α_t on the algebra. Taking each observable A into a progression of "later" evolved observables α_tA, for every timeparameter number t.

A nice thing is that this "thermal time" construct RECOVERS ordinary time when we start with a conventional Hamiltonian H and hilbertspace H. this is what Connes Rovelli show on pages 16 and 17 of their paper. See link:
Sources
http://arxiv.org/abs/gr-qc/9406019 pages 16,17
WikiP: "Gelfand-Naimark-Segal construction"
WikiP: "KMS state"
WikiP: "Tomita-Takesaki theory" (not so good I think, but at least article exists)
WikiP: "Polar decomposition" (article exists, I haven't used or evaluated it)

The basic situation that general covariant quantum physics deals with is an algebra A of observables. That's the world. After all QM is about making measurements/observations. And a temporal flow α_t is a oneparameter group of automorphisms of that algebra.

automorphism means it maps an observable A onto another observable α_tA which you can think of as making the same observation but "t timeunits later".
oneparameter group means that doing α_s and then doing α_t has the same flow effect as doing α_s+t. the parameter t is a real number.
And automorphism means it preserves the algebra operations, it is linear etc etc.

Observables are in fact an algebra because you can add and multiply observables together to predict other observables or to find how they correlate with each other.

The statistical quantum state of the world is represented by a positive functional on the algebra which we can think of as a density matrix ω and its value on an observable A can be written either as ω(A) or as trace(Aω). The state ω is what gives the observables their expectation values and their correlations.

A nice thing about a density matrix ω is that it has a square root ω^1/2. Think of writing it as a diagonal matrix with all positive entries down the diagonal, and taking the square root of each entry.

The observable algebra (think matrices) IS a vector space. You can add matrices entry-wise and so on. The celebrated GNS construction makes a vectorspace out of |ω^1/2⟩ together with all the other density matrices and their like which you can get by applying elements A of the algebra to that root vector. that is called the FOLIUM of ω
|Aω^1/2⟩ for all A in A
It is a hilbertspace. The special good things about this hilbertspace (they give it a name, K) is that the algebra acts on it, after all it was MADE by having the algebra act on the single root vector |ω^1/2⟩ and seeing what you get, and the other thing is just that: it has what is called a "cyclic vector", a root or generator: the whole hilbertspace is made by having the algebra of operators act on that one |ω^1/2⟩, as we have seen.

ω(A) = ⟨ω^1/2|A|ω^1/2⟩

Now what C&R do is they construct an operator, by giving its polar decomposition. This is what happens on page 17. And the operator obtained by putting the polar decomp. together has the effect of doing a matrix transpose, or mapping A → A*. They call this operator S.

SA |ω^1/2⟩ = A* |ω^1/2⟩

There is some intuition behind this (there is already something about it on page 7 but I'm looking at page 17). It is like swapping creation and annihilation operators. Undoing whatever an operator does. Partly it is like getting your hands on what is implicitly an infinitesimal time-step, except there is no time yet. More importantly, transpose is tantamount to commuting
(AB)* = B*A*
So if we can just take the anti-unitary part out of the picture it's almost like swapping order: AB → BA. Yes very handwavy, but there is some underlying intuition, will get back to this.

We are going to build from that swapping or reversal operator S. In particular we will use the positive self-adjoint part of the polar decomposition. More about this later.

 

[rjbeery]

I gave the link in the post I pointed you to:
http://fqxi.org/community/essay/winners/2008.1
go there, scroll down to "second community prize", there is Ellis's abstract and a link to the PDF.

I already explained the incompatibility using the same example Ellis did, radioactive decay changes the distribution of mass---Ellis's rocket sled just makes it more colorful.

OK, I read Ellis' paper and I'm not seeing his point or the problem with the rocket sled. Any macro-scale process which is dependent upon an apparently random quantum process can be time-reversed in the same way that thermodynamic systems are: an extraordinarily unlikely series of physically plausible events "conspire" to make it happen.

Did I drop the glass on the floor to watch it shatter, or did the heat in the floor molecules synchronize at precisely the right moment to make the shards jump into the air, coalesce and fuse into a proper glass shape, flying up onto the table only to be stopped by my hand? Equivalently, did ongoing radioactive decay make the sled change directions, or did rogue alpha particles bombarding our nucleus-switch cause the direction changes?

Ellis' arguments are basically all Epistemological in nature. He is trying to tie a preferred direction of time to the fact that we apparently only know things about the past. He says

A closely related feature is the crucial question of time irreversibility: the laws of physics, chemistry, and biology are irreversible at the macro scale, as evidenced inter alia by the Second Law of Thermodynamics, even though the laws of fundamental physics (the Dirac equation, Schroedinger’s equation, Maxwell’s equations, Einstein’s field equations of gravity, Feynman diagrams) are time reversible. This irreversibility is a key aspect of the flow of time: if things were reversible at the macro scale, there would be no genuine difference between the past and the future, and the physical evolution could go either way with no change of outcome; both developments would be equally determined by the present. The apparent passage of time would have no real consequence, and things would be equally predictable to the past and the future.

He claims that the macro scale events are irreversible* via the Second Law of Thermodynamics, therefore time flow exists in one direction. The problem is that entropy only increases until equilibrium is reached! What would Ellis say about time flow direction in a theoretical Universe in systemic thermal equilibrium?

*As I'm sure you are aware, the Second Law of Thermodynamics is a tendency or likelihood, not a law. Is Ellis suggesting that the preferred direction of the flow of time is also a mere likelihood? This is a spurious argument.

 

[Paulibus]

Marcus, Your sequence of posts 80, 85,90 are really helpful in following what Connes and Rovelli are doing. Hope you continue with them. One point I'm not clear on. In Rovelli's recent paper (1209.0065), he remarks that "The root of the temporal structure is thus coded in the non commutativity of the Poisson or quantum algebra” (near the end of p.1). The "thus" puzzles me. Is it indeed obvious?

 

[marcus]

Marcus, Your sequence of posts 80, 85,90 are really helpful in following what Connes and Rovelli are doing. Hope you continue with them. One point I'm not clear on. In Rovelli's recent paper (1209.0065), he remarks that "The root of the temporal structure is thus coded in the non commutativity of the Poisson or quantum algebra” (near the end of p.1). The "thus" puzzles me. Is it indeed obvious?

No it's not obvious, to me at least. Instead it strongly piqued my curiosity.
I am beginning to understand now (but keep in mind that I am not an expert or an active researcher, I just watch developments and hopefully comprehend a bit of it.)

When the world is an algebra of observables then it HAS to be that the temporal structure is coded in the non-commutativity because it is coded in the fact that it matters which observation you make first.

And when we look at the Tomita construction and the KMS condition what we see is a mathematical struggle involving the study of AB versus BA. Like combing the flow out of a tangled head of noncommutative hair. I will get a page reference to Connes Rovelli that illustrates.

I'm so glad you are interested in this too! We'll certainly continue working thru it, as you suggest. My intuition now is that noncommutativity of measurements or observation has within it the essence of timeflow, but that I just need to study the stuff some more to see how.

A page reference. Try Connes Rovelli page 13 equation 23. The important thing is not to get bogged at the start by trying to grasp every little math detail but to see the main thing they are doing. They are invoking the "KMS condition". The state ω is a functional on the observables and it has the property that ω(AB) is almost the same as ω(BA). In fact it would give exactly the same number if you apply the TIME EVOLUTION flow gamma to A slightly differently. You can compensate for swapping the order if you "skootch" A by a little in the time-evolution.

This equation 23 is the KMS condition which you also see as the last equation in the WikiP article "KMS State" where they say that a KMS state is one satisfying a certain "KMS condition" which is verbatim the same as equation 23.

Intuitively IMHO, KMS condition gives a way of defining a steady state which is somehow more generally applicable than older ways, but which reduces back to, say, Gibbs idea of equilibrium where that is applicable. The people who showed you could recover the older idea from KMS have names like Haag Hugenholtz Winnink. The S in KMS stands for Julian Schwinger, who shared the QED prize with Feynman. KMS dates back to late 1950s. this is just nonessential human interest stuff but it sometimes helps

 

[Paulibus]

When the world is an algebra of observables then it HAS to be that the temporal structure is coded in the non-commutativity because it is coded in the fact that it matters which observation you make first.

This is what I had concluded also, sketched in a doggy sort of way below. But must an algebra of observables be non-commutative? and if so, why? Two reasons seem possible to me.

One (rather special); is because one chooses to describe the world in a quantum mechanical context, where congugate observables are (still mysteriously?) non-commutative.

Two (more generally); because the world is three-dimensional, for any context we find useful to quantify and describe change in, such as QM and GR.

I speculate that scalar changes of physical quantities in one dimension are perforce commutative, and that in two dimensions the same is true; the order in which like quantities (say vectors) are added doesn't matter (parallelogram law). But in three dimensions non-commutative change becomes possible (like successive rotations about non-colinear axes, described by adding polar vectors or tensors).

Does non-commutative change only happen in three dimensions, which we seem (still mysteriously) to be endowed with? And could GR's pseudo-dimensional time emerge in the way Connes and Rovelli postulate just because we live in three spatial dimensions ?

 

[Paulibus]

In checking KMS state in Wikipedia, I noticed that George Green is in the background. George was a self-educated younger contemporary of Jane Austen's. I'm fearful whenever his functions are involved in something; there was genius in the water they drank in those far-off days. Just non-essential human interest stuff which guides my wary path!

 

[marcus]

I wish she could have met Sadi Carnot b. 1796, whose book Réflexions sur la puissance motrice du feu was published while he was still in his twenties. He was another younger contemporary. Here is a portrait:
http://en.wikipedia.org/wiki/Nicolas_Léonard_Sadi_CarnotSense and Sensibility (1811), Pride and Prejudice (1813), Mansfield Park (1814), Emma (1816)
Reflections on the Motive Power of Fire (1824)

 

[marcus]

I should keep on developing the thermal time idea, as in posts #90. 92, 93... I got distracted elsewhere and left the job half done. Thanks to Paulibus for help and encouragement!

I will include plenty of links to source and background articles e.g. from WikiP
http://en.wikipedia.org/wiki/Gelfand–Naimark–Segal_construction
I'm not an expert and can't be completely sure my take on every point is correct. But it seems to me that the GNS construction is the key thing.

Observables form an abstract normed algebra of the C* type. Most basically an algebra is something with addition and multiplication. Think of a bunch of n x n matrices over ℂ. The matrices themselves form a n² dimensional vectorspace.
Starting with an algebra you can CONSTRUCT a vectorspace that the algebra acts on.

GNS is a slightly more refined version. You start with an algebra [A] with a specified positive linear functional ρ defined on it. Think of a density matrix, a generalized "state".
ρ(A) is the complex conjugate of ρ(A*).

GNS construction gives you a hilbertspace [H] with the algebra ACTING on it and a CYCLIC VECTOR ψ in [H] such that
ρ(A) = ⟨ψAψ⟩
and I'll explain what a cyclic vector is in a moment. That one vector can generate the whole hilbertspace.

Two things to stress: The construction gives us a REPRESENTATION of the abstract [A] as a bunch of operators acting on the constructed [H]. It is just as if the algebra were not abstract but all along consisted of ("matrices" i.e.) operators on the hilbertspace. GNS tells algebras they don't have to be abstract if they don't want---we can always build a good hilbertspace for them to act on where they'll feel completely at home, as operators.

The other thing to stress is what a cyclic vector is. Essentially it means that the whole hilbertspace can be gotten just by acting on that one vector ψ by elements A of the algebra---and taking limits if necessary, the set [A]ψ is dense in [H].
====================

Intuitively GNS works this way: you make the hilbertspace by considering [A] itself (the "matrices") as a vectorspace and factoring out stuff as needed. So any "matrix" can be considered both as a vector or an operator on the vectors. And the original state functional, the "density matrix" ρ, well intuitively we can take its square root and that will be a square matrix and therefore can be viewed as the vector ψ. That's basically where the cyclic vector ψ comes from and why ρ(A) = ⟨ψAψ⟩.

====================

So far we are just using the GNS construction. Thanks to Mssrs Gelfand Naimark Segal for the goodies. Now the next key step is to define an operator S on [H] using the cyclic vector. For every A in [A] we consider the vector Aψ and we say what S does to that.
SAψ = A*ψ
That defines SA adequately because the vectors Aψ are dense in the hilbertspace.
It's called "anti linear" or "conjugate linear" because in multiplying the source by a scalar converts into multiplying the target by its complex conjugate. The * operation is conjugate linear in that sense and it carries over to S.
Next we take the polar decomposition of S.
http://en.wikipedia.org/wiki/Polar_decomposition
It is a piece of that POLAR DECOMPOSITION that gives us thermal time.
(This is how thermal time arises, from nothing but an abstract algebra and a statistical state defined on that algebra.)
There's a encyclopedia article on Tomita-Takesaki business: S and it's polar decomp. etc:
http://arxiv.org/abs/math-ph/0511034
the first couple of pages seem enough. It is pretty basic and clearly written.
It's from the Ensevelier Encyclopedia of Mathematical Physics.

 

[marcus]

Before proceeding to derive the thermal time flow from that operator S mentioned at the end of the preceding post, I should review some of the motivation. TT is general covariant which other kinds of physical time are not. And yet it agrees with regular physical time in several specialized cases.
I'll quote from post #74 earlier where these were mentioned.
https://www.physicsforums.com/showthread.php?p=4171588#post4171588
This is paraphrasing the Connes Rovelli paper which has 77 cites, over a third of which are in the past 4 years. So it is fairly well known and still probably the best source on TT definition and basics.
http://arxiv.org/abs/gr-qc/9406019
==quote post #74==
... I'll run down the main corroborative cases they give on page 22, in their conclusions. These are explained in the preceding section, pages 16-21.
== quote http://arxiv.org/abs/gr-qc/9406019 ==
...
• Classical limit; Gibbs states. The Hamilton equations, and the Gibbs postulate follow immediately from the modular flow relation (8).
• Classical limit; Cosmology. We refer to [11], where it was shown that (the classical limit of) the thermodynamical time hypothesis implies that the thermal time defined by the cosmic background radiation is precisely the conventional Friedman-Robertson-Walker time.
• Unruh and Hawking effects. Certain puzzling aspects of the relation between quantum field theory, accelerated coordinates and thermodynamics, as the Unruh and Hawking effects, find a natural justification within the scheme presented here.
...
==endquote==

They also include three other supporting points. One that is not discussed in the paper and they simply mention in passing is the widely shared notion that time seems bound up with thermodynamics and there are indeed hundreds of papers exploring that general idea in various ways (far too numerous to list). Their idea instantiates this widely shared intuition among physicists.

Another supporting point is that the thermal time formalism provides a framework for doing general relativistic statistical mechanics. Working in full GR, where one does not fix a prior spacetime geometry, how can one do stat mech? A way is provided here (and see http://arxiv.org/abs/1209.0065 )

The sixth point is the one they give first in their "conclusions" list---I will simply quote:
==quote gr-qc/9406019 page 22==
• Non-relativistic limit. In the regime in which we may disregard the effect of the relativistic gravitational field, and thus the general covariance of the fundamental theory, physics is well described by small excitations of a quantum field theory around a thermal state |ω⟩. Since |ω⟩ is a KMS state of the conventional hamiltonian time evolution, it follows that the thermodynamical time defined by the modular flow of |ω⟩ is precisely the physical time of non relativistic physics.
==endquote==

There is one other supporting bit of evidence which I find cogent and which they do not even include in their list. This is the uniqueness.
==endquote==

So there is the uniqueness of TT

and the fact TT independent of arbitrary choices, all you need is the algebra of observables (the world) and a positive linear functional defined on it (the state of the world: our information about it.) You don't have to choose a particular observer or fixed geometry

and the fact that TT recovers the time that cosmologists use--standard universe time in the standard cosmic model.

and the fact that TT recovers ordinary physics time when you specialize to a NON general covariant case---with a Hamiltonian and the Hilbert space of usual QM.

and other good stuff that Connes and Rovelli mention.

That all makes me tend to think that this is a good way to get your basic time. It is set up as a ONE-PARAMETER FLOW operating on the OBSERVABLES ALGEBRA.

The flow is denoted α_t where t is time, and it carries any given A in [A] into subsequent observables α_tA.

 

[marcus]

The hang-up some people say they have about the TT hypothesis centers on the word "equilibrium". The root meaning here is "balanced" but the STATE that we are talking about is "4D" or timeless. It represents how we think the world is. Period. Including all physical reality past present and future. So naturally it does not COME into equilibrium. Ideally it simply IS how it is. Our idea of how the world is must not change with time and therefore it is in balance---an equilibrium state.

(But people have a mental image of something "arriving" at equilibrium---imagined as a state at a certain time. That's the wrong way to think about a timeless state.)

I think the way to understand TT is as the logical completion of the Heisenberg picture. You could call it "general covariant Heisenberg time". In the Heisenberg picture the world is an algebra of observables and there is just one state. The hilbertspace is not essential, you only use one state in it and you can throw away the rest. The hilbertspace was used, historically, to construct the algebra, but once you have the algebra you can discard it and you will always be able to recover that sort of representation (by GNS) from the algebra itself. That one state vector that you keep is really just a positive linear functional on the algebra. Something that assigns expectation values to observables.

And once we have specified [A] and the state functional ρ we automatically get a flow α_t on the algebra, by Tomita. The idea of global time is given automatically independent of any observer or any assumption about background geometry.

The best independent critical commentary on TT which I have seen is by the mathematician Jeff Morton (Baez PhD and Baez co-author now at Lisbon). You can see that he gets hung up on what I believe is the wrong "equilibrium" notion. But he has otherwise a very clear assessment. His insight helped me when I was confused earlier about the TT. This is from his blog "Theoretical Atlas" October 2009. I've added an exponent 1/2 to align his notation with other sources used in this thread. He uses ω, instead of ρ, for the state.

==quote Jeff Morton==
First, get the algebra [A] acting on a Hilbert space [H], with a cyclic vector ψ (i.e. such that [A]ψ is dense in [H] – one way to get this is by the GNS representation, so that the state ω just acts on an operator A by the expectation value at ψ, as above, so that the vector ψ is standing in, in the Hilbert space picture, for the state ω). Then one can define an operator S by the fact that, for any A in [A], one has

(SA)ψ = A*ψ

That is, S acts like the conjugation operation on operators at ψ, which is enough to define since ψ is cyclic. This S has a polar decomposition (analogous for operators to the polar form for complex numbers) of JΔ^1/2, where J is antiunitary (this is conjugation, after all) and Δ^1/2 is self-adjoint. We need the self-adjoint part, because the Tomita flow is a one-parameter family of automorphisms given by:

α_t(A) = Δ^-itAΔ^it

An important fact for Connes’ classification of von Neumann algebras is that the Tomita flow is basically unique – that is, it’s unique up to an inner automorphism (i.e. a conjugation by some unitary operator – so in particular, if we’re talking about a relativistic physical theory, a change of coordinates giving a different t parameter would be an example). So while there are different flows, they’re all “essentially” the same. There’s a unique notion of time flow if we reduce the algebra [A] to its cosets modulo inner automorphism. Now, in some cases, the Tomita flow consists entirely of inner automorphisms, and this reduction makes it disappear entirely (this happens in the finite-dimensional case, for instance). But in the general case this doesn’t happen, and the Connes-Rovelli paper summarizes this by saying that von Neumann algebras are “intrinsically dynamic objects”. So this is one interesting thing about the quantum view of states: there is a somewhat canonical notion of dynamics present just by virtue of the way states are described. In the classical world, this isn’t the case.

Now, Rovelli’s “Thermal Time” hypothesis is, basically, that the notion of time is a state-dependent one: instead of an independent variable, with respect to which other variables change, quantum mechanics (per Rovelli) makes predictions about correlations between different observed variables. More precisely, the hypothesis is that, given that we observe the world in some state, the right notion of time should just be the Tomita flow for that state. They claim that checking this for certain cosmological models, like the Friedman model, they get the usual notion of time flow. I have to admit, I have trouble grokking this idea as fundamental physics, because it seems like it’s implying that the universe (or any system in it we look at) is always, a priori, in thermal equilibrium, which seems wrong to me since it evidently isn’t. The Friedman model does assume an expanding universe in thermal equilibrium, but clearly we’re not in exactly that world. On the other hand, the Tomita flow is definitely there in the von Neumann algebra view of quantum mechanics and states, so possibly I’m misinterpreting the nature of the claim. Also, as applied to quantum gravity, a “state” perhaps should be read as a state for the whole spacetime geometry of the universe – which is presumably static – and then the apparent “time change” would then be a result of the Tomita flow on operators describing actual physical observables. But on this view, I’m not sure how to understand “thermal equilibrium”. So in the end, I don’t really know how to take the “Thermal Time Hypothesis” as physics.

In any case, the idea that the right notion of time should be state-dependent does make some intuitive sense. The only physically, empirically accessible referent for time is “what a clock measures”: in other words, there is some chosen system which we refer to whenever we say we’re “measuring time”. Different choices of system (that is, different clocks) will give different readings even if they happen to be moving together in an inertial frame – atomic clocks sitting side by side will still gradually drift out of sync. Even if “the system” means the whole universe, or just the gravitational field, clearly the notion of time even in General Relativity depends on the state of this system. If there is a non-state-dependent “god’s-eye view” of which variable is time, we don’t have empirical access to it. So while I can’t really assess this idea confidently, it does seem to be getting at something important.
==endquote==
Jeff Morton's blog: http://theoreticalatlas.wordpress.com

The state (a linear functional on the observables) is what we believe to be timelessly true about the world.
The world is the algebra of observations.
So far this is more or less what Wittgenstein said in chapter 1 of Tractatus. I wonder why the algebra of observables should be normed, and over the complex numbers, and equipped with a conjugate-linear * involution. Why should the world be a C* algebra? (I must be kidding )
See post #65 https://www.physicsforums.com/showthread.php?p=4169556#post4169556

 

[Paulibus]

Your series of posts describing how time can be described has been most illuminating for me, Marcus, and it does indeed seem consistent with Wittgenstein's philosophical take that you quoted. Thanks for explaining an abstract perspective that Heisenberg would have appreciated in a way that I could actually make a lot of sense of.

But to be really convincing, even if the world is, as you say, "an algebra of observations", I guess that folk like Barbour, Connes and Rovelli may have to formulate some kind of predictive description, with an aspect that can be verified by physical evidence.

Time, that non-reversible Moving Finger, is a slippery concept to handle, even by mathematically inclined folk with plenty of Wit. I have neither Piety nor sufficient Wit and find myself wondering about really elementary "why" questions to add to your list, like why does Planck's constant exist at all, so making ODTAA a non-commutative process?

I suspect it is because no Operation (One Damn Thing) that happens After Another, does so on a virgin playing field, even if the operation is algebraic and the playing field is as tenuous and ether-like as the cosmic microwave background. Perhaps both the operations and the algebra are only descriptive shadows of reality cast on the cave walls of our minds?

 

[sheaf]

Time, that non-reversible Moving Finger,...

Did you mean to say

Time, those non-reversible Moving Many Fingers,...

 

[HomogenousCow]

Just a quick question, what is the current state of quantum gravity??

 

[Paulibus]

Sheaf: No, I wouldn't dare to tamper with the words of the great Muslim philosopher Omar Khayyam! See Rubaiyat of Omar Khayyam, quatrain No. 51.

 

[marcus]

...
Time, that non-reversible Moving Finger, is a slippery concept to handle, even by mathematically inclined folk with plenty of Wit. I have neither Piety nor sufficient Wit and find myself wondering about really elementary "why" questions to add to your list, like why does Planck's constant exist at all, so making ODTAA a non-commutative process?
...

I got your reference at once--it was put so perfectly that I couldn't think of any appropriate response!
The words came to mind without their original punctuation and are so transcribed.

The moving finger writes, and having writ
moves on––nor all your piety nor wit
shall lure it back to cancel half a line,
nor all your tears wash out a word of it.

One can well ask "why" the apparent connection between AFTERNESS, as in odtAa, and algebraic unswitchability. John Baez put in a related comment at Jeff Morton's blog (of the "I think this is cool..." sort) when they were discussing Rovelli thermal time idea.
I'll get a link.

The joking reference to "many-fingered time" was sly of Sheaf and a bit arcane. It is a modern hypothesis that a few people have explored. (Including Demystifier among others.) I think it comes in different versions. One picture (not Demy's) might be of a block universe past that grows forwards in time from many different points, in a sort of uncoordinated way. Sheaf must know a lot more about it than I. The idea would have baffled Mssrs Fitzgerald and Khayyam, I imagine. We don't really need to consider it here, since the thermal time construction gives us one unique universal time (which we can compare local and observer times to.)

Here's link to Jeff Morton's blog post about TT.
http://theoreticalatlas.wordpress.c...time-hamiltonians-kms-states-and-tomita-flow/

Here's the Baez quote from "Theoretical Atlas":

==quote==
John Baez Says:

October 30, 2009 at 12:08 am
I think every von Neumann algebra has a ‘time-reversed version’, namely the conjugate vector space (where multiplication by i is now defined to be multiplication by -i) turned into a C*-algebra in the hopefully obvious way. And I think the Tomita flow of this time-reversed von Neumann algebra flows the other way!

I know that every symplectic manifold has a ‘time-reversed version’ where the symplectic structure is multiplied by -1. This is equivalent to switching the sign of time in Hamilton’s equations.

I think it’s cool how time reversal is built into these mathematical gadgets.
==endquote

 

[Paulibus]

Thanks for the pointer to Jeff Morton's blog. It's a gem. And for translating Sheaf's post -- I didn't know about that particular gadget; all mathematical gadgets are most definitely cool, like Omar's stuff. The wonder for me is how so many of them are practical, as well!