Global emergent time, how does Tomita flow work?

[marcus]

Tomita time is an intrinsic observer-independent time variable available to us for fully general relativistic analysis. As far as I know it is the only such time available for things like
general covariant statistical mechanics
GC statistical quantum mechanics
GC quantum field theory

Here I want to discuss some nuts&bolts of the Tomita flow construction.

In GR, "coordinate time" is not really a time---it's neither observable nor physically meaningful. On the other hand, observer time is not defined until one has already fixed on a particular curved space-time geometry. So observer time cannot be used if the geometry itself is included as part of the dynamics. Furthermore in a quantum theoretical treatment the problem is worse since a space-time cannot be determined any more than can a continuous particle trajectory.

But to do certain kinds of analysis we need an independent time variable. The Tomita flow is a one-parameter group of transformations of the observables algebra M that arises naturally as powers U^t of a distinguished unitary operator U. I want to discuss the natural way you get that operator U. Tomita time is the real number t that appears as the exponent. It is the parameter of the one-parameter group of changes that operate on the observable algebra M, mapping one element X of M to another.

X → U^tXU^-t

The construction is not all that complicated. It is described on page 517 of the Princeton Companion to Mathematics.
http://books.google.com/books?id=ZO...6AEwAw#v=onepage&q=minoru tomita math&f=false

Thinking of M as the C* algebra generated by all possible measurements, this one-parameter group shifts and shuffles the observables among themselves. Obviously a key question is how do you construct the basic operator U whose powers U^t serve to represent the passage of time?

To start with, we represent what we think we know about the world (statistical correlations among measurements, levels of confidence, uncertainty, variances, expectation values) by a positive state function ω: M → ℂ.

Positive here means that for any X in the algebra ω(X* X) > 0, and equals zero only in case X itself is zero. The state ω defines an inner product on M which allows us to treat M as a hilbert space M.

Now the star mapping X → X* can be viewed as an operator S on the hilbert space M. The map S:M → M is almost linear--the term for it is conjugate linear, because scalar multiplication carries over using the complex conjugate of the scalar.

I suppose the letter S is used to denote this operator version of the * because S stands for "star". Whatever the reason, in every treatment of Tomita flow I've seen, this notation has been used.

Now the first cleverness occurs. Since M is now a hilbert space we have a well-defined notion of the ADJOINT of an operator. That is also always denoted by a star. In matrix representations it's the conjugate transpose--you flip the matrix over and take conjugates when the entries are complex. And it's defined more generally using the inner product:
<T*X, Y> = <X, TY>.
So we can take the star of the star mapping S. It gets slightly non-trivial here simply because we are using * in two senses: the original C* algebra operation and the new adjoint available now that M is a hilbert space with an inner product. Now we can multiply S together with its adjoint S*.

That's it. S* S is a well-defined operator on the observables algebra M viewed as a hilbert space. It's easy to see that it is self-adjoint. Self adjoint operators are the analogs of real numbers, just as unitary operators are analogs of the numbers on the unit circle. To see that it is self-adjoint you just have to verify that
<S*S X, Y> = <X, S*S Y>

The unitary operator U, mentioned earlier, which is the seed of the Tomita flow, the seed of change, the seed of the passage of time in the world described by (M, ω) is simply given by
U = (S*S)ⁱ

It's a basic fact about hilbertspace that you can raise a self-adjoint operator to an imaginary power like i, and get a unitary. This is analog of raising a real number x to imaginary power and getting a number xⁱ on the unit circle. Like those numbers e^2πit we are always seeing.

So that is how U, the seed of the Tomita flow, is obtained.
The flow simply consists of bracketing an observable X with that unitary U raised to powers t and -t. That is what CHANGE is, how the passage of time works, in the star-algebra world defined by (M, ω)

Time is the logarithm of change, to the base U. And it is observer-independent.

Time is not a pseudo-spatial dimension, in this view of the world. It is the parameter of a flow on the observables.

X → U^t X U^-t

 

[marcus]

We need to keep handy a few links to discussions of Tomita flow, as a global emergent time, in the literature:
Here's a Vimeo video of part of a talk on Tomita time by Matteo Smerlak:
http://vimeo.com/33363491
from a March 2011 workshop at Nice.

Here's the article by Alain Connes and Carlo Rovelli:
http://arxiv.org/abs/gr-qc/9406019

and a seminal 1993 paper, The Statistical State of the Universe
http://siba.unipv.it/fisica/articoli/C/Class%20Quantum%20Grav_vol.10_1993_pp.1567-1568.pdf
which shows how T-time recovers usual conventional time in several interesting contexts, notably standard Friedmann-model cosmology.

Wherever a local observer's time is defined, one can compare it with the global emergent T-time. The ratio between the two times is physically meaningful as shown in this paper by Smerlak and Rovelli.
http://arxiv.org/abs/1005.2985

Here's a recent paper showing how T-time could be used in approaches to general relativistic statistical mechanics and general covariant statistical QM.
http://arxiv.org/abs/1209.0065
===================

It is unlikely that QG can be formulated in terms of a 4D spacetime geometry, for the same reason that in quantumtheoretical treatments a particle does not have a definite trajectory. One can make a finite number of observations about where it goes, and they are correlated, but one cannot say that a continuous trajectory exists. Same with a 4D spacetime which is a trajectory of the geometry. This is explained in Chapter 1 of Approaches to Quantum Gravity (D. Oriti ed.)
http://arxiv.org/abs/gr-qc/0604045 (see page 4 of preprint)

This indicates that the set of all possible measurements M must replace the 4D manifold space-time, as a basis for GC QFT (general covariant quantum field theory). Each of the observables in M has its own uncertainty built in. Since the correlations are statistical we have no difficulty obtaining an indefinite causal structure, or an uncertain geometric evolution. Features that may be difficult to obtain with a 4D manifold seem to come automatically with the C* algebra M.

The state ω defined on M, represents what Bohr referred to as "what we can SAY" about nature or the world, as opposed to what IS.

Jeff Morton's blog on T-time (with John Baez comment):
http://theoreticalatlas.wordpress.c...time-hamiltonians-kms-states-and-tomita-flow/

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."

 

[marcus]

To get a little intuition about the form of the Tomita flow:
X → U^t X U^-t
let's imagine a concrete example. Mozart takes time off from writing Don Giovanni and for his amusement he uses a crude spectrograph to measure the color of the sun. That is measurement X

Now we want to see what that measurement turns into when advanced by t = 200 years.
Ronald Reagan takes time off from golf, napping at staff meetings, and his busy schedule of public appearances to go out and use the same crude spectrograph to record the mix of wavelengths of the sun.

The recipe is UNDO 200 years of change. Do Mozart's measurement of sunlight. Then RESTORE 200 years of change

X → U^{+200 years} X U^{-200 years}

This is how we map the measurement Moz. made into the one Ron made.
I have to go, no time to edit back soon.

 

[Drakkith]

Marcus, you always manage to make my brain explode! I think that's a good thing!

 

[marcus]

Marcus, you always manage to make my brain explode! I think that's a good thing!

I'm delighted to hear this! I'm excited by this development in quantum gravity too. It must be what it feels like when a possibly major development comes along.

 

[MTd2]

Hi Marcus! Is this related to my question in the other thread?

 

[marcus]

Hi Marcus! Is this related to my question in the other thread?

I replied in the other thread.
https://www.physicsforums.com/showthread.php?p=4209977#post4209977

EDIT to respond to following post:

Hi Arivero, KMS states are referred to quite a lot in the Connes-Rovelli paper, as I recall. Also in Rovelli's recent paper (on general relativistic statistical mechanics).
http://arxiv.org/abs/gr-qc/9406019
http://arxiv.org/abs/1209.0065

To give context for any readers not familiar with it, I'll simply quote the lead paragraph at WikiP.
http://en.wikipedia.org/wiki/KMS_state
"KMS state
In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo-Martin-Schwinger state or, more commonly, a KMS state: a state satisfying the KMS condition. Kubo (1957) introduced the condition, Martin & Schwinger (1959) used it to define thermodynamic Greens functions, and Rudolf Haag, M. Winnink, and N. M. Hugenholtz (1967) used the condition to define equilibrium states and called it the KMS condition."

 

[arivero]

First time I saw it, it was for some study of states (KMS states??) in C*-algebras.

 

[marcus]

Hi Alejandro, I replied to you earlier in postscript to post#7, but wanted to ask what role you think the KMS condition plays? It gets mentioned (I gave some references earlier) but does not seem to be needed in defining Tomita flow or the associated time.

========quote Princeton Companion to Mathematics (2008) page 517 ==========
Modular theory exploits a version of the GNS construction (section 1.4). Let M be a self-adjoint algebra of operators. A linear functional φ: M → C is called a state if it is positive in the sense that φ(T*T) ≥ 0 for every T in M (this terminology is derived from the connection described earlier between Hilbert space theory and quantum mechanics). for the purposes of modular theory we restrict attention to faithful states, those for which φ(T*T) = 0 implies T = 0. If φ is a state, then the formula

<T₁, T₂> = φ(T₁* T₂)

defines an inner product on the vector space M. Applying the GNS procedure, we obtain a Hilbert space H_M. The first important fact about H_M is that every operator T in M determines an operator on HM. Indeed a vector V in HM is a limit V = lim_n→∞ V_n of elements in M, and we can apply an operator T in M to the vector V using the formula

TV = lim TV_n

where on the right-hand side we use multiplication in the algebra M. Because of this observation, we can think of M as an algebra of operators on H_M, rather than as an algebra of operators on whatever Hilbert space we began with.

Next, the adjoint operation equips the Hilbert space H_M wtih a natural "anti linear" operator
S: H_M → H_M by the formula [see footnote]

S(V) = V*.

Since U*_g = U_g^-1 for the regular representations, this is indeed analogous to the operator S we encountered in our discussion of continuous groups. The important theorem of Minoru Tomita and Masamichi Takesaki asserts that, as long as the original state φ satisfies a continuity condition, the complex powers

U_t = (S*S)^it

have the property that

U_t M U_-t = M for all t.

The transformations of M given by the formula T → U_t T U_-t are called the modular automorphisms of M.
Alain Connes proved that they depend only in a rather inessential way on the original faithful state φ. To be precise, changing φ changes the modular automorphisms only by inner automorphisms, that is, transformations of the form T → UTU^-1 where U is a unitary operator in M itself. The remarkable conclusion is that every von Neumann algebra M has a canonical one-parameter group of "outer automorphisms," which is determined by M alone and not by the state φ that is used to define it.[footnote] The interpretation of this formula on the completion H_M of M is a delicate matter.

==endquote==
Nowhere here is the KMS condition invoked. I think KMS plays a role in one or more significant APPLICATIONS of the Tomita flow idea, but is not essential to its definition.

http://books.google.com/books?id=ZO...6AEwAw#v=onepage&q=minoru tomita math&f=false

BTW the chapter on operator algebras this is quoted from is by Nigel Higson and John Roe (the PCM is edited by Timothy Gowers)

 

[marcus]

I think the Tomita flow ("T-flow") and Tomita time ("T-time") are going to be important because physics will probably come to be formulated on a quantum geometric basis to handle certain extreme situations where the geometry is an important part of the dynamics. And QG is unlikely to involve spacetimes (= continuous trajectories). T-time is the only option I know of being proposed as an independent time variable for general covariant statistical mechanics or GC quantum theory. And it is intrinsic--arises naturally when we have a C*algebra M and a state ω representing the world and what we think we know about it.

Given that it is going to be of major importance, I would like to see if there's a good notation that will make it easy to grasp. What do you call (S* S)ⁱ a certain self-adjoint operator raised to the power i? It's a unitary operator on H_M, the algebra M competed as hilbertspace. Earlier I denoted it simply by the letter U. Maybe that's not distinctive enough, what about Q? And let me try using A as a generic element of the algebra M.

Let's try that. S is the star operation on M carried over to H_M. S* is the adjoint.
The product S* S is a positive self-adjoint operator on H_M. A positive real number raised to the power i is just a number on the unit circle, and we can pick a basis that diagonalizes S* S and raise every eigenvalue to the power i. We get a unitary operator.
Let's try denoting that Q = (S* S)ⁱ

Then Tomita's theorem says we have a flow on M that is simply given by
A → Q^t A Q^-t
where t is a real number parameter of the flow, and corresponds to time (measured in Planck units in cases where it has been checked.)

So advancing a measurement or observation A, by some 200 years,
to get another measurement Q^{+200 years} A Q^{-200 years}
can be thought of intuitively as "undo 200 years of change, perform A, and restore the 200 years." Q is the basic Tomita unitary operator, Q^t is how change is represented. Intuitively "T-time is the logarithm of change, to the base Q. And it is observer independent. And it is not a pseudospatial fake dimension"

 

[marcus]

First time I saw it, it was for some study of states (KMS states??) in C*-algebras.

This is rather technical, readers may wish to skip this post. I found a source
http://arxiv.org/pdf/math-ph/0511034v1.pdf
that summarizes stuff about KMS (kubo martin schwinger) that Arivero was asking about.
This is from page 2, section 2.1:
==quote==
The modular automorphism group satisfies a condition which had already been used in mathematical physics to characterize equilibrium temperature states of quantum systems in statistical mechanics and field theory — the Kubo–Martin–Schwinger (KMS) condition. If M is a von Neumann algebra and {α_t | t ∈ ℝ} is a σ-weakly continuous one-parameter group of automorphisms of M, then the state φ on M satisfies the KMS-condition at (inverse temperature) β (0 < β < ∞) with respect to {α_t} if for any A,B ∈ M there exists a complex function F_A,B(z) which is analytic on the strip {z ∈ C | 0 < Imz < β} and continuous on the closure of this strip such that
F_A,B(t) = φ(α_t(A)B) and F_A,B(t + iβ) = φ(Bα_t(A)) ,
for all t ∈ ℝ . In this case, φ(α_iβ(A)B) = φ(BA), for all A,B in a σ-weakly dense, α- invariant *-subalgebra of M. Such KMS-states are α-invariant, i.e. φ(α_t(A)) = φ(A), for all A ∈ M, t ∈ ℝ, and are stable and passive (cf. Chapter 5 in [3] and [5]).
Every faithful normal state satisfies the KMS-condition at value β = 1 (henceforth called the modular condition) with respect to the corresponding modular automorphism group.
...
...
The modular automorphism group is therefore endowed with the analyticity associated with the KMS-condition, and this is a powerful tool in many applications of the modular theory to mathematical physics. In addition, the physical properties and interpretations of KMS-states are often invoked when applying modular theory to quantum physics.
Note that while the non-triviality of the modular automorphism group gives a measure of the non-tracial nature of the state, the KMS-condition for the modular automorphism group provides the missing link between the values ω(AB) and ω(BA), for all A, B ∈ M (hence the use of the term “modular”, as in the theory of integration on locally compact groups).
==endquote==
This article was written by Stephen Summers for the Elsevier Encyclopedia of Mathematical Physics.
So FWIW this summarizes the fact that the modular automorphism group (the Tomita flow we were talking about) has to do with the non-commutativity of the observables algebra.
And it also points out the usefulness in applications that comes from the modular group satisfying the KMS-condition.
Here's another possibly useful source: an open access article by Marc Rieffel and Alfons van Daele about Tomita's theorem and related, also called Tomita-Takesaki theory.
http://projecteuclid.org/DPubS?verb...e=UI&handle=euclid.pjm/1102817105&page=record

 

[John86]

Marcus,
Not directly related links but Aspect and Connes talking about time in these talks !
Hope it is of your tatse..

http://noncommutativegeometry.blogspot.nl/

http://math.univ-lille1.fr/~cempi/conf/Inaugurale/programme.html

 

[marcus]

... Connes talking about time ... !
Hope it is of your taste..

http://math.univ-lille1.fr/~cempi/conf/Inaugurale/programme.html

Connes in the first 5 minutes of the video:
"the origin of time is quantum mechanical and it is really coming from the non-commutativity of the variables.."

Very much to my taste! I hope others will watch the beginning of the Connes video. Thanks!

 

[marcus]

One reason I think Tomita time is important is that it is so strongly motivated. It's not clear to me that the motivation to use the parameter of the Tomita flow as time, in the study of physical systems, was appreciated by Tomita, or by the younger mathematician Takesaki who helped bring his research to light in 1970. Tomita's interest was in abstract operator algebras. It was, I think, Connes and Rovelli who first realized that the Tomita theory applies to the "problem of time" in GR and the even more severe problems with time that people were faced with in QG. To review the motivation for T-time, I'll quote from page 2 of their 1994 paper.
== http://arxiv.org/abs/gr-qc/9406019 page 2 ==
In a general covariant theory there is no preferred time flow, and the dynamics of the theory cannot be formulated in terms of an evolution in a single external time parameter. One can still recover weaker notions of physical time: in GR, for instance, on any given solution of the Einstein equations one can distinguish timelike from spacelike directions and define proper time along timelike world lines. This notion of time is weaker in the sense that the full dynamics of the theory cannot be formulated as evolution in such a time. In particular, notice that this notion of time is state dependent.

Furthermore, this weaker notion of time is lost as soon as one tries to include either thermodynamics or quantum mechanics into the physical picture, because, in the presence of thermal or quantum “superpositions” of geometries, the spacetime causal structure is lost. This embarrassing situation of not knowing “what is time” in the context of quantum gravity has generated the debated issue of time of quantum gravity. As emphasized in [4], the very same problem appears already at the level of the classical statistical mechanics of gravity, namely as soon as we take into account the thermal fluctuations of the gravitational field. Thus, a basic open problem is to understand how the physical time flow that characterizes the world in which we live may emerge from the fundamental “timeless” general covariant quantum field theory [9].

In this paper, we consider a radical solution to this problem. This is based on the idea that one can extend the notion of time flow to general covariant theories, ...
==endquote==

 

[martinbn]

Mathematically this is well known and straightforward given the algebra, but what exactly is M? Generated by all possible measurements is not clear enough (to me). What are observables in general relativity?

p.s. Sorry for the, I guess, stupid questions.

 

[marcus]

Mathematically this is well known and straightforward given the algebra, but what exactly is M? Generated by all possible measurements is not clear enough (to me). What are observables in general relativity?

p.s. Sorry for the, I guess, stupid questions.

Certainly not stupid questions! I can try to give my opinion. There has been a lot of discussion of this very thing: What are observables in general relativity? I think the question is intimately related to the challenge of combining GR with QM, and also with problems of time in GR. My opinion will necessarily be partly speculative.

You probably know that QM has evolved to a higher level of abstraction by throwing away the hilbert space (which can be recovered but is not essential to the analysis once one has the algebra).

Now what I think needs to happen and that is beginning to happen is that we can throw away the manifold that used to be necessary in GR.

Once GR has shed its manifold, and QM has discarded its Hilbertspace, the two will find they are together, they won't even have to TRY to unite---they will already be joined. This is how I think things are going.

So I think AdS/CFT is something of a distraction from the main evolutionary course, it is too elaborately manifold-based for it to be squarely in the right direction.
===here's a paraphrase something I wrote earlier which might help, or might be redundant===

People differ as to their interpretation of QM but what they are talking out mathematically is what John von Neumann presented in 1932 (Foundations of Quantum Mechanics) and subsequently in what is called von Neumann algebras. Originally these were algebras of operators on a Hilbertspace, but later they were axiomatized. It is the axiomatic stucture of the algebra, saying how its elements behave (the "operators" which no longer operate on anything) which gives meaning to the elements.
This is a familiar way mathematics develops.
The elements of a set in axiomatic set theory can be anything or nothing in particular, the structure gives them meaning. They are the primitives--one cannot say what they are. The "points" in a differential manifold are the primitives in diff geom. One cannot say what a "point" is, but the axiomatic structure saying how they behave gives them their meaning.

Originally von Neumann defined algebras of operators on a specific Hilbertspace. then that was axiomatized. what were operators on something are now primitives with no special meaning in and of themselves---their behavior and what we can do with them determined by the axioms and the C* algebra structure.

But we can THINK of M as an infinite menu of possible acts of measurement, whose results can be added, subtracted, multiplied with each other, having the * operation and the normed topology that von Neumann thought up etc.

 

[marcus]

Martin, you might be interested in taking a look at the WikiP on John von Neumann:
http://en.wikipedia.org/wiki/John_von_Neumann
the man was phenomenal. Probably the greatest pure/applied mathematician born in the 20th century bar none.
Basic contributions in so many areas of mathematics and physics (including computer architecture and game theory). 1903-1957.
It helped me to get some perspective when I saw the comments by other mathematicians, about von Neumann and then saw what he stated he thought was his most essential accomplishment. This is from a short list of facts he submitted to the National Academy of Sciences: "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."

He was the person who put quantum theory in the mathematical form it is today, and also axiomatized the operator *-algebra that evolved into the C* algebra.
Interpretation is another business---there can be lots of different interpretations of the same basic mathematical structures.

 

[martinbn]

marcus (#16), yes, that helps me, and I think I understand your point. I do like the idea, as you said history shows that was the way in many areas, including algebraic geometry, Grothendieck's point of view, which can be paralleled to Gelfand's. The geometric object is determined by a certain algebra, so the properties of the algebra can be abstracted and the theory can be built without the geometric object and in greater generality. To come back to my question if we start with a Lorentzian manifold, what is the algebra? I suppose the generators have to correspond to measurements, but this needs a more formal description, and what are the relations? Now that I said this, I seem to remember seeing a paper by Geroch, where, if I am not mistaken, he was defining an algebra of observables, but I have to find it and look at it again.

#17 Oh, yes, I have looked into his life, he is indeed quite remarkable.

 

[marcus]

Martin, this caught my attention and I'm not sure how relevant it is:
http://arxiv.org/abs/1109.0036
William Donnelly is a PhD student advised by Ted Jacobson at U. Maryland, and he expects to finish his thesis in 2013. This paper was published in PRD 2012.
Decomposition of entanglement entropy in lattice gauge theory
William Donnelly
(Submitted on 31 Aug 2011 (v1), last revised 26 Apr 2012 (this version, v2))
We consider entanglement entropy between regions of space in lattice gauge theory. ...
====
One thing that got my attention was on page 1:
"Closely related to lattice gauge theory is loop quantum gravity, which is formulated as an SU(2) lattice gauge theory on a superposition of lattices. Although this paper will not discuss loop quantum gravity, entanglement entropy in loop quantum gravity was discussed in Refs. [20, 21], and we expect the techniques of this paper to generalize easily to a superposition of lattices. We note also that the Hilbert space of edge states in SU(2) lattice gauge theory is closely related..."

Donnelly's earliest paper, published PRD 2008, when he was a Master's student at Waterloo advised by Achim Kempf, is:
http://arxiv.org/abs/0802.0880
Entanglement Entropy in Loop Quantum Gravity
William Donnelly
(Submitted on 6 Feb 2008)
The entanglement entropy between quantum fields inside and outside a black hole horizon is a promising candidate for the microscopic origin of black hole entropy. We show that the entanglement entropy may be defined in loop quantum gravity, and compute its value for spin network states. The entanglement entropy for an arbitrary region of space is expressed as a sum over punctures where the spin network intersects the region's boundary. Our result agrees asymptotically with results previously obtained from the isolated horizon framework, and we give a justification for this agreement. We conclude by proposing a new method for studying corrections to the area law and its implications for quantum corrections to the gravitational action.
4 pages

I will try to explain why this interests me. If I take seriously what Rovelli and Connes say about the problems with time in both GR and QG, then I suspect there will be a future development of LQG in the C* format. That is, using (M,ω). Then one gets a global emergent time, "T-time", that depends only on the state ω, not on the observer. Looking ahead, how will a classical manifold geometry be recovered?

So that is the question that's on my mind, at the moment. If they go that way, with (M, ω), how will they get some conventional geometric stuff back out? I can see how they could get the equivalent of a Cauchy surface. The state ω might enforce a bounce, and that is a place to start counting T-time---a reference marker for the Tomita flow. So one gets to specify a subset of the algebra M, a particular moment in effect. Now one can define a REGION of that "cauchy surface", and its complement. Subsets A and A^c.

This is the sort of thing discussed in the Bianchi Myers paper. The entanglement entropy between a region and its complement. And Bianchi Myers cite Donnelly's 2008 paper about entanglement entropy between regions in LQG. I suspect something is brewing here. I'll be interested to learn what Donnelly's thesis turns out to be about.

 

[marcus]

Since we're on a new page, I'll summarize. Links will come later, after the summary. Anyone who persists in trying to represent the world as a 4D manifold with fields plastered on it will have trouble with time. The alternative is to represent the world as a star-algebra M and a state ω representing what we think we know about it (i.e. correlations amongst observable etc.).

Technically a C*-algebra is an abstract generalization of a von Neumann operator algebra. Axiomatizing the observable algebra allows getting rid of the Hilbert space and a quantum state becomes a positive functional ω:M→ℂ, on the abstract algebra M. von Neumann would approve

Given a state functional ω on M, Gelfand and friends tell us how to construe M as a Hilbert space H_M. This is really great! We were not given a hilbertspace to start with, but anytime we want we can recover one that M ACTS ON as operators.

The abstract star operation on M becomes a conjugate linear transformation S: H_M → H_M defined on Gelfand's hilbertspace. This is something new, so things begin to happen.

Because H_M has an inner product, we know what the ADJOINT of S is. The inner product tells us, see earlier post. Call the adjoint S*. The operator product of S* with S is positive and self-adjoint. Such an operator can be raised to complex powers (think of diagonalizing a matrix and raising the eigenvalues.) In particular the operator S*S can be raised to the power i.

Tomita now defines a UNITARY operator Q = (S* S)ⁱ on the (Gelfand) hilbertspace H_M.
Real powers of Tomita's unitary Q correspond to the passage of an observer-independent world time. viewed as shifting measurements around amongst themselves.

The Tomita flow can be considered as a map M → M from earlier measurements to later ones, defined by
A → Q^t A Q^-t
This can be thought of as taking a measurement A in M to a corresponding measurement made t units of time later. To take an example, we can think of the Tomita flow converting a measurement A into one made, say, 200 years later (i.e. Q^{200 years} A Q^{-200 years}) this way:
"The later measurement is what you get if you undo 200 years of change, perform the earlier measurement, and then restore 200 years of change."

Here "change" means Q^t, the Tomita unitary raised to a real number power. The exponent t would be 200 years expressed in natural (Planck) time units.
T-time is the logarithm of change to the base Q.

When specific cases are considered and the arithmetic is done, the units of T-flow time turn out to be Planckian natural time units. Technically this is called an "automorphism" of the algebra M, and letting t range along the real line ℝ we get a "one-parameter group of automorphisms" defined on M. A flow for short.

 

[marcus]

There's a fuller listing of relevant links in post #2. I've tried to boil that down to essentials.
Video of part of a talk on Tomita time by Matteo Smerlak:
http://vimeo.com/33363491

Article by Alain Connes and Carlo Rovelli:
http://arxiv.org/abs/gr-qc/9406019

Seminal 1993 paper, The Statistical State of the Universe
http://siba.unipv.it/fisica/articoli/C/Class%20Quantum%20Grav_vol.10_1993_pp.1567-1568.pdf showing how [what was later realized to be Tomita flow time] recovers usual conventional time in several interesting cases including standard cosmology.

Ratios between a local observer's time and the global emergent T-time can be physically meaningful as shown in the paper by Smerlak and Rovelli http://arxiv.org/abs/1005.2985

T-time fills a need in formulating general covariant statistical mechanics and general covariant statistical QM http://arxiv.org/abs/1209.0065

Section in Princeton Companion to Mathematics on the Tomita flow:
http://books.google.com/books?id=ZO...6AEwAw#v=onepage&q=minoru tomita math&f=false
===================
Post #2 has more discussion along these lines.
===================
Personally I think that the recent paper by Bianchi and Myers is potentially applicable in this context, as a way of revealing the regional structure implicit in the algebra and state (M, ω). Independent of any particular observer.

 

[TrickyDicky]

Since we're on a new page, I'll summarize. Links will come later, after the summary. Anyone who persists in trying to represent the world as a 4D manifold with fields plastered on it will have trouble with time. The alternative is to represent the world as a star-algebra M and a state ω representing what we think we know about it (i.e. correlations amongst observable etc.).

Technically a C*-algebra is an abstract generalization of a von Neumann operator algebra. Axiomatizing the observable algebra allows getting rid of the Hilbert space and a quantum state becomes a positive functional ω:M→ℂ, on the abstract algebra M. von Neumann would approve

Thanks. It's good to have someone bringing to PF and explaining this completely new perspective(to me at least, I know it's been around for a few years now, and that some of the ideas date back from von Neumann early work). It is really a fascinating venue to explore.
I'm concerned about some points though, this is a complete change in paradigm and one has to get used to it, I mean, people found weird that theories in physics might come in different number of dimension (4, 9, 10, 11, 27...), but this is a deeper switch to maybe no manifold at all. I guess I'm still really fond of the old manifold, and I would like to see that this idea get some empirical backing soon.
I listened to the Connes lecture linked here and found it full of ideas, and amusing(not only for his strong accent ). It is really great that his noncommutative algebraic geometry model is able to recover most of phenomenology of the standard model of particle physics, but since the aim of quantum gravity is to bridge the gap between QM and GR and ultimately make them compatible, I missed the part where these star-algebras give any prediction or explanation about the gravitational part.

 

[marcus]

T.D., I will have to respond to your post piecemeal--a bit at a time. Thanks for the stimulus. Just now, wondering how to reply to your questions, I found what looks like an excellent review paper (July 2010). Published in the Russian online journal SIGMA special issue on "noncommutative spaces and fields."
http://arxiv.org/abs/1007.4094
Modular Theory, Non-Commutative Geometry and Quantum Gravity
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul
(Submitted on 23 Jul 2010 (v1), last revised 19 Aug 2010 (this version, v2))
This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
47 pages. (cites 260 items)

I like it's being comparatively self-contained. Where possible the level is pedagogical--they define what a C* is, in basic terms, instead of assuming the reader already knows stuff like that. From what I read, the writing is clear.
Ideologically they are strongly on the side of making what we ordinarily think of as differential geometry realities "take shape" in algebra context. They put this better than I just did. We have to learn ways to somehow "recover" familiar manifold geometry out of a C* setup. Again, I'm tempted to go fetch quotes instead of saying it in my own words.

I didn't know of Roberto Conti before. Here are his 31 papers on arxiv:
http://arxiv.org/find/grp_math,grp_physics/1/au:+Conti_R/0/1/0/all/0/1
They seem mostly in the "math.OA" branch of arxiv---the "operator algebras" part of the math arxiv.
So you only get a subset of his work if you just search in physics. Snapshot: http://owpdb.mfo.de/detail?photo_id=10625 http://owpdb.mfo.de/detail?photo_id=16368

 

[marcus]

About this paper I mentioned:
http://arxiv.org/abs/1007.4094
Modular Theory, Non-Commutative Geometry and Quantum Gravity
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul

T.D., the part of it relevant to your question starts on page 23. That is, digestible portions of section 6:Perspectives on Modular Algebraic Quantum Gravity pages 23-38.

Especially some excerpts from 6.1 (Construction of modular spectral geometries)
6.2 (Physical meaning of modular spectral geometries)
6.4 (Finding the macroscopic geometry) very short because either just beginning or tentative work in progress, only about a dozen relevant papers are cited here.
6.5 (Connection with other approaches to quantum geometry)
6.6 (Quantum physics)

My sense is that Paolo Bertozzini, the junior author was the main motivator of this work. To get an idea of who Bertozzini is, one can watch a portion of the YouTube where he gives a lecture at Imperial College in London about it. Unfortunately in the part I watched the camera stays pointed at the lectern and does not follow the speaker to the blackboard or slidescreen. This is frustrating, so overall the YouTube seems useless. But a few minutes watch gives a sense of the person.

In footnote 14 on page 23, it says Bertozzini, Conti, Lewkee have a work in progress called
Modular algebraic quantum gravity. It is somewhat discouraging to me that this work has not appeared.

My intuitive feeling is that the key to getting geometry out is by way of a C* setup that imitates deSitter---a single bounce. Or one of the other one-bounce cosmic models. That may provide a reference time. A time zero for the Tomita flow.

This would in effect "slice" the C* algebra according to T-time slices. In that case one would have a chance to define and study the subsets of the algebra corresponding to 3D regions, and their boundaries.

 

[TrickyDicky]

About this paper I mentioned:
http://arxiv.org/abs/1007.4094
Modular Theory, Non-Commutative Geometry and Quantum Gravity
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul

T.D., the part of it relevant to your question starts on page 23. That is section 6:
Perspectives on Modular Algebraic Quantum Gravity

Especially 6.1 (Construction of modular spectral geometries)
6.2 (Physical meaning of modular spectral geometries)
6.4 (Finding the macroscopic geometry) very short because either just beginning or tentative work in progress

Thanks Marcus, I'll read it and maybe comment afterwards.

 

[marcus]

I overlooked something that actually makes the Bertozzini YouTube potentially quite useful! His SLIDES are available at the Oxford University site.
http://www.cs.ox.ac.uk/quantum/slides/clap2-paolobertozzini.pdf
I didn't realize that when I posted #24 earlier. (mistake highlighted in red)

...from 6.1 (Construction of modular spectral geometries)
6.2 (Physical meaning of modular spectral geometries)
6.4 (Finding the macroscopic geometry) very short because either just beginning or tentative work in progress, only about a dozen relevant papers are cited here.
6.5 (Connection with other approaches to quantum geometry)
6.6 (Quantum physics)

My sense is that Paolo Bertozzini, the junior author was the main motivator of this work. To get an idea of who Bertozzini is, one can watch a portion of the YouTube where he gives a lecture at Imperial College in London about it. Unfortunately in the part I watched the camera stays pointed at the lectern and does not follow the speaker to the blackboard or slidescreen. This is frustrating, so overall the YouTube seems useless. But a few minutes watch gives a sense of the person.
...
This would in effect "slice" the C* algebra according to T-time slices. In that case one would have a chance to define and study the subsets of the algebra corresponding to 3D regions, and their boundaries.

Here is the abstract for the seminar talk:
==quote==
Published on May 1, 2012
Speaker: Paolo Bertozzini (Thammasat University)
Title: Categories of spectral geometries
Event: Categories, Logic and Foundations of Physics II (May 2008, Imperial College London)
Slides: http://www.cs.ox.ac.uk/quantum/slides/clap2-paolobertozzini.pdf

Abstract: In A. Connes' non-commutative geometry, "spaces" are described "dually" as spectral triples. We provide an overview of some of the notions that we deem necessary for the development of a categorical framework in the context of spectral geometry, namely: (a) several notions of morphism of spectral geometries, (b) a spectral theory for commutative full C*-categories, (c) a tentative definition of strict-n-C*-categories, (d) spectral geometries over C*-categories. If time will allow, we will speculate on possible applications to foundational issues in quantum physics: categorical covariance, spectral quantum space-time and modular quantum gravity.
==endquote==
Much of the first 3/4 of the slide set is rather technical abstract math. BUT the last quarter or so addresses the problem of how you do QG in star-algebra context, and recover geometry. Efforts to do this have been made by various people, along various lines. I found it interesting to see the different things that have been tried, going back even to around 1972. The slide set (last 1/4) has many references, including to a 1999 paper by Rovelli that I didn't know about:
http://arxiv.org/abs/gr-qc/9904029.

 

[Paulibus]

Marcus: your summary post #20 is very helpful. But I still find the concept of our very complicated "world" described (or represented mathematically) "as a star-algebra M and a state ω representing what we think we know about it " very abstract, but intensely interesting, even for a non-mathematician.

I believe that examples often help comprehension. It might be useful to describe the nuts and bolts of how some small toy quantum "world" could be described along these abstract lines. An example I can think of is a small finite world of one-dimensional simple harmonic oscillators confined by potential-barrier walls. Would Tomita time in such a world connect to anything we think we know about this world? Such as the half-life of radioactive particles it might crudely model?

 

[marcus]

...It might be useful to describe the nuts and bolts of how some small toy quantum "world" could be described along these abstract lines...

I'll keep an eye out for an example like that, it would be extremely valuable as a conceptual aid! And pedagogically too, I think. Maybe some more adept person than myself will construct one.

What i picture, at first, is something so elementary and finite that many things about it would be trivial. Tomita time, for instance, would probably not flow. I picture an example based on a tetrahedron. Perhaps four area observables generate the algebra.

Two tetrahedra can be joined to form a 3D hypersphere---not embeddable in our 3D space but still interesting as a simple compact boundaryless space.

Or even simpler, as you suggested, something in lower dimension. start by thinking of a triangle---say equilateral at first. Three length observables (analogous to the tet's four area observables.) then think of gluing two equilateral triangles together to form a 2D sphere. And let them, after all, not be equilateral (that was only to help me imagine them for starters.)

Then think of adding some kind of field observables that live on this simple 2D sphere world, and describe the observable algebra. It might be a lovely algebra! But it's not even 9AM in morning here and already I feel like a dummy, completely inadequate

It's a really good idea to get a pedagogical example of this kind of thing. Maybe there is someone I can ask, for whom it would be easy. I'm very new at this M,ω stuff (and old in years, which doesn't help either.)

 

[marcus]

A toy model (M, ω) would correspond to having only a finite number of sites where measurements are made.
This could cause M to be finite dimensional. That is why I mentioned a tetrahedron.

You know there are finite groups, even finite fields (analogs of ℝ and ℂ, but actually with only finitely many elements.) things like that can be good for toy models.

From the different vertices of the tet one might measure angles, distances. Area operators might be defined. I suppose a matter field could be defined on that minimalist picture. Some labels at the vertices, or along the edges.

So that would add more operators to M, some having to do with matter, as well as the geometric measurements.
================
I think the "spin networks" that Penrose introduced, and which are the basis of Loop gravity, would serve as a basis for this kind of finite-dimensional (M,ω). In usual LQG they can carry matter fields as well as geometric information. sometimes the labels on the vertices and edges of the graph are Lie group elements, sometimes finite group elements, something group representations e.g. spins, sometimes (with Lewandowski) the labels can even be operators, as I recall. There are different styles.

But essentially a spin network is a finite combinatorial/algebraic structure. So it would be natural devise a way to transform such a thing (a labeled graph) into a finite dimensional star algebra M and state ω.

Then one might imagine that by making the spin network more and more complicated one might get a more realistic picture, not so "toy". My intuitive feeling is that is not satisfactory, some more creative math has to happen to get up out of this toy level. But at least it seems to offer a way to construct toy models of increasing size and complexity---to sort of ramp things up. Just speculating.

I still haven't described enough structure to allow for time-evolution, really need an infinite dimensional star algebra for that, I suspect. I'll keep an eye out for the kind of thing you mentioned---a toy model that can illustrate Tomita flow. Wish I knew of one that we could examine right now, but I don't.

 

[Paulibus]

Thanks for these thoughts, Marcus. The idea of making a toy model out of glued-together triangles (or tetrahedra joined to form a 3D hypersphere) in order to provide observables is rather abstract to me, especially since I've (probably mistakenly) considered spheres (2- or hyper-) as geometrical constructs with a center-like item equidistant from some suitable n-dimensional perimeter; i.e. circle related. I'm hoping for a more physics-based model -- a structure based on triangles or tetrahedra sounds to me related to a Buckminster-Fuller dome. Probably my limited knowledge.

Let's hope that somebody can attenuate the abstract nature of Tomita flow sufficiently for us to connect it smoothly with the time that passes for us all. At a rate inversely proportional to the years that remain for each of us!

 

[atyy]

So why doesn't LQC seem to use thermal time?

Does thermal time give rise to a preferred foliation, since it is global and observer independent?

I noticed one Bohmian in a one particular World advocating a preferred foliation (maybe that's not so accurate - he says "proper foliation"): http://arxiv.org/abs/1205.4102

Can Bohmians use thermal time, but instead of thermal equilibrium, use the Bohmian quantum equilibrium?

 

[marcus]

So why doesn't LQC seem to use thermal time?

I thought it did! That was one of the points Rovelli made in 1993 paper, the first or one of the first T-time papers. Also referred to in 1994 Connes Rovelli, towards the end. I quoted and gave link earlier in thread. Recovers Friedmann time if you make the usual Friedmann assumptions.

And LQC uses F. time (agrees with classical expansion after first few Planck seconds. So LQC runs on T-time.

Does thermal time give rise to a preferred foliation, since it is global and observer independent?

YES! if you have a preferred time zero to use as base point. Of course you need one starting slice of measurements to start with. Like e.g. the "bounce". Then with that as reference you can advance everything by one time unit and get another slice.

Cosmology, as you know, has a global preferred time that is independent of observer. And has a starting place. So cosmology has a foliation. So since T-time recovers Friedmann time it would have to have a foliation too, given the same helpful assumptions.

About Bohmians, I wouldn't know. Maybe someone else here. My hunch is that (M,ω) formulation obviates or weakens motives for B.

 

[Naty1]

Marcus:

Anyone who persists in trying to represent the world as a 4D manifold with fields plastered on it will have trouble with time.

Is this a reference to problems in LQG which LQC is trying to circumvent??

such as: from one of Ashtekar's papers...we discussed here:

How has LCG resolved the Big Bang Singularity?

https://www.physicsforums.com/showthread.php?t=662565

...because LQG does yet offer the quantum version of full Einstein’s equations which one can linearize around a quantum FLRW spacetime.

Which is a better set of papers, in your opinion, to mull over, those of Ashtekar in the other thread or the papers in this one?? [I sure like the Matteo video...that's as far as I have gotten here...]

 

[marcus]

Marcus:
...Is this a reference to problems in LQG which LQC is trying to circumvent??

No, not a reference to LQG or LQC in particular. Problem with time in GR was identified before LQG or LQC. Quantizing GR, however you try to do it, whatever approach, exacerbates the problem of time in GR.

A good half-page explanation is in rovelli's essay "unfinished revolution" Google "rovelli revolution". From like page 3 or 4 , gave link earlier.
You've got to understand the problem. It is serious and across-the-board.

Which is a better set of papers, in your opinion, to mull over, those of Ashtekar in the other thread or the papers in this one?? [I sure like the Matteo video...that's as far as I have gotten here...]

MATTEO SMERLAK's talk! Yes! I'm glad you like it. I'm not sure what the relevance of Ashtekar paper is. His work is always relevant and worth looking over.But it is not specifically about time---the topic here.

Maybe the relevance is to understanding the narrowing separation betw. LQC and LQG. People who don't get this treat them as static and don't realize how much overlap is growing. Have to watch over time and get a sense of momentum, rates of change.
Full LQG already has a simple case of bounce cosmology, recovered deSitter. Full LQG has the cosmological constant. These are developments in the past 3 years or so on that side.
Meanwhile LQC has made remarkable progress in past 3 years with increasing the complexity of the models to include more degrees of freedom---so more realistic, more fluctuations, more matter, just last year bringing in Fock space.

These are fast moving research programs in the process of merging.

That's an important perception to understanding and knowing what to expect. I guess one reason to read Ashtekar papers is to get a sense of that---a feel for how it's going on the LQC side. But I don't see his papers so relevant to the time issue itself. Maybe indirectly.

One of Ashtekar's best PhD students is now a postdoc at Marseille. He has written a great paper (several actually). I would almost say THAT is what one should study to keep up with LQC.

 

[atyy]

[I sure like the Matteo video...that's as far as I have gotten here...]

MATTEO SMERLAK's talk! Yes! I'm glad you like it.

Yes, me too. Listening to his talk was the first time I understood anything about thermal time. Before that I knew of Connes-Rovelli but it was indigestible to me. Now that you've linked the Rovelli 1993, I think I shall see if I can make any headway with that.