
#109
Dec512, 03:45 PM

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For math buffs fond of rigorous proof, the best paper I've found online about Tomita flow is this 1977 one by Marc Rieffel and Alfons van Daele
http://projecteuclid.org/DPubS/Repos...pjm/1102817105 Only selected parts of it are directly about Tomita flow, it delves into a bunch of related matters. The whole article is some 34 pages long. Pages 187221 of an issue of Pacific Journal of Mathematics. It would be nice if someone could point us to a more concise, say a 10 page, treatment of just the Ttheorem. Or could extract the essential line of reasoning from this paper. There is a short explanatory article commissioned by Elsevier's ENCYCLOPEDIA OF MATHEMATICAL PHYSICS, written by Stephen Summers. http://arxiv.org/abs/mathph/0511034 But it does not give proofs of the hard parts. It seems that TomitaTakesaki theory is deep, nontrivial. It is easy to say and not difficult to grasp the general idea, but drilling down to logical bedrock takes effort. The original approach involved unbounded operators, one had to wonder if and where they were welldefined. Rieffel and van Daele work with bounded operators and take more stepslots of lemmas. There's a Master's Thesis by someone named Duvenhage at Pretoria that takes essentially the same approach as Rieffel van Daele but could be helpful because it puts in more background algebra and analysis. To give an example of the kind of questions that come up, recall we have ([A], ρ) a *algebra and a statefrom which by well known means we get ([H], ψ) a hilbertspace with a cyclic separating vector which represent both the algebra and the state in a way familiar to physicists. Algebra elements A are represented as operators in customary fashion. Then a new operator S is defined by SAψ = A*ψ. How do we know this is welldefined? We are only told what SA does to the cyclic vector. And do we think of S as an operator on the hilbertspace or on the algebra? Both, but can this be done consistently? Then this operator S is resolved into two factors: S = JΔ, or in other papers S = JΔ^{1/2}. How do we know we can do this? OK as operators on the hilbertspace. The first factor is conjugatelinear and a kind of flip or involution. It is its own inverse, J^{2}=I. The second factor is positive and selfadjoint, as an operator on the hilbertspace. That means you can diagonalize it with positive real numbers down the diagonal, as learned in undergrad linear algebra class. And you can raise it to the it power to make Δ^{it}, which will be unitary. Then we define Tomita flow: α_{t}(A) = Δ^{it}AΔ^{it}. I guess that makes sense as operators on the hilbertspace, but how do we know that the flow actually stays in the original *algebra? How do we know that α_{t}(A) is still in [A]? This turns out to be a large part of the TomitaTakesaki theorem: the statement that Δ^{it} [A] Δ^{it} = [A] If you take the original staralgebra and advance each item in it by the same timeinterval t, then what you get is the same staralgebra. The time flow just shifts or shuffles or permutes the items among themselves. The fame of Tomita rests on the fact that he was able to show this, not all the stuff leading up to it, but this. So if you look at the kind of tutorial paper by Summers http://arxiv.org/abs/mathph/0511034 it is precisely this which you see as "Theorem 1.1" on page 2. This, and also a seemingly inconsequential fact about J. Namely that if you apply J front and back to every item in [A] it picks out for you all the items that commute with everything in [A], the socalled "commutant" customarily denoted with a prime, in this case [A]'. I have seen mathematicians make nimble use of this fact but its significance is not obvious, so I think of the content of "Theorem 1.1" as primarily Δ^{it} [A] Δ^{it} = [A] Delta, when turned into a unitary operator, stirs the pot without splashing any of the soup out. I supect that this Delta, which is the positive real heart of a "swap" or "reversal" operator, will eventually become part of our language because it encapsulates the intrinsic TIME inherent in a world (of observables) and a state (what we think we know about that world). And so whatever this Delta is eventually called, it will probably settle in to our collective awareness. BTW the Princeton Companion to Mathematics (page 517) points out that Δ = S*S which makes excellent sense and for economy of notation they don't bother to introduce the symbol Δ. They just use S*S, the product of the "swap" S with its adjoint S*. Minoru Tomita's work went unpublished for several years until discovered and made more presentable by Takesaki, whose name can be remembered by resolving it into "take saki". Alain Connes, in a 2010 interview, says "I am too young to have met von Neumann, but I was much more influenced at a personal level by the Japanese: Tomita and also Takesaki.” The interviewer adds: "Minoru Tomita (1924) is a Japanese mathematician who became deaf at the age of two and, according to Connes, had a mysterious and extremely original personality. His work on operator algebras in 1967 was subsequently refined and extended by Masamichi Takesaki and is known as Tomita–Takesaki Theory..." http://www.math.ru.nl/~landsman/ConnesNAW.pdf 



#110
Dec512, 09:15 PM

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Since we have this concept of a universal standard time it can be useful to compare other times with it. E.g. associated with an accelerated observer or with a location in the gravitational field.
Back in 1934 RC Tolman defined a local temperature of space associated with depth in a gravitational field, now known as the TolmanEhrenfest effect and it turns out that this temperature is the RATIO of the two rates: intrinsic Tomita time divided by proper time of a local observer. If ds is a local observer's proper timeinterval and dτ is the corresponding interval of Tomita time, then the TolmanEhrenfest temperature is proportional to dτ/ds. So the temp is a comparison of ticking rates. The local temperature is high if Tomita time is ticking a lot faster than the local observer's clock. There is a connection here to the Hawking BH temp and the Unruh temp of an accelerated observer in Minkowski space. The details are interesting and tend to validate the thermal time (i.e. Tomita time) idea. I won't go into detail at this point (supposed to help with supper) but will simply link to a relevant article: http://arxiv.org/abs/1005.2985 Thermal time and the TolmanEhrenfest effect: temperature as the "speed of time" Carlo Rovelli, Matteo Smerlak (Last revised 18 Jan 2011) The notion of thermal time has been introduced as a possible basis for a fully generalrelativistic thermodynamics. Here we study this notion in the restricted context of stationary spacetimes. We show that the TolmanEhrenfest effect (in a stationary gravitational field, temperature is not constant in space at thermal equilibrium) can be derived very simply by applying the equivalence principle to a key property of thermal time: at equilibrium, temperature is the rate of thermal time with respect to proper time  the `speed of (thermal) time'. Unlike other published derivations of the TolmanEhrenfest relation, this one is free from any further dynamical assumption, thereby illustrating the physical import of the notion of thermal time. 4 pages btw the proportionality is hbar over Boltzmann k. If T is the Tolman temperature then T = ( 



#111
Dec612, 03:07 AM

P: 165

Marcus: small queries. In one of your posts that I now can't find I'm sure you mentioned that selfadjoint matrices are the analogs of the set of real numbers. Is this (to me interesting ) statement just common knowledge, or have you a reference for it? And do you have a pointer to the "Master's Thesis by someone named Duvenhage at Pretoria" that you mentioned in post # 109?




#112
Dec612, 11:52 AM

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http://en.wikipedia.org/wiki/Charles_Hermite The photo shows him with a dour scowl (having drunk some bad wine, or found a mistake in a proof by one of his students). He was the thesis advisor of Henri Poincaré and Thomas Stieltjes. The analogy is very nice. It is undergrad math, which is the longestlasting and most beautiful kind of math. You have to know what a BASIS of a vectorspace is (a set of vectors in terms of which all the rest can be written as unique combinations). It is a CHOICE OF AXES or a choice of framework. And a matrix is a way of describing a linear transformation by saying what it does to each member of some particular basis. I don't at the moment have an online undergrad linear algebra textbook link. There might be a Kahn Academy treatment. Many years ago we used a book by Paul Halmos. You can DIAGONALIZE an hermitian (selfadjoint) matrix by finding a new basis for the vectorspace in which the same linear map is expressed by a diagonal matrix. When you do this the numbers down the main diagonal (upper L to lower R) turn out ALL REAL. A POSITIVE hermitian or selfadjoint matrix is where the numbers down the diagonal turn out all positive real numbers. This means the linear map is just rescaling along each of a fixed set of directions. No rotations no funny business. Just expanding a bit in this direction and perhaps contracting a bit in this other. There is a strong analogy between a matrix that is simply real numbers down the diagonal (and zero elsewhere) and the real numbers themselves. A selfadjoint matrix is like a bunch of real numbers applied in an assortment of specified directions. so it is the HIGHER DIMENSIONAL ANALOG of a real number. the beautiful thing is that the DEFINING CONDITION A* = A of selfadjointness is also analogous to the defining condition of realness which we can write as z* = z if you use * to mean the conjugate of a complex number (exchange i and i, if z = x+iy then z* = xiy) The only way a complex number z can have z*=z is if the imaginary part y = 0. Conjugation is flipping the complex number plane over keeping the real axis fixed, so the only way a number can have z*=z is if it is on the real axis. The business of diagonalizing matrices, or finding the right axis framework for a given linear map so that its matrix will be very simple comes under the heading of the SPECTRAL THEOREM. The "spectrum" of an operator is the list numbers down the diagonal when you put it in diagonal form. It's like analyzing some light into its different wavelengths, with a prism. You really know the beast when you know that list of numbers. I think calling it the spectrum is metaphorical, a kind of 19thCentury physicist's poetical flight of language. From a time when the most exciting thing physicists did was heat various chemical elements and separate out the colors of the light they gave off when they were hot. Determining the spectrum was the pinnacle act of analysis. We still have their word for the list of numbers down the diagonal. http://en.wikipedia.org/wiki/Spectral_theorem http://en.wikipedia.org/wiki/Selfadjoint_operator I can't recommend you go to Rocco Duvenhage's thesis. It is over a hundred of pages and you can get lost if you don't already know roughly what you are looking for, but I will put the link just in case I'm wrong and it actually is helpful to you or someone else. http://upetd.up.ac.za/thesis/availab...ssertation.pdf Quantum statistical mechanics, KMS states and TomitaTakesaki theory 



#113
Dec612, 11:23 PM

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Paulibus and others: I hope the foregoing account of selfadjointness, the analogy with real numbers, and diagonalizing was not too elementary. I tend to want to cover a range of levels: the topic is interesting enough so I think people with all different backgrounds might want to read about it. So some posts in the thread can be at a basic level, others less basic. Here's a more advanced treatment which has the merit of being very concise. It is from The Princeton Companion to Mathematics, edited by Field medalist Tim Gowers, a good math source book. I found a passage treating TomitaTakesaki theory, and transcribed a sample excerpt
http://books.google.com/books?id=ZOf...20math&f=false This is from page 517. ========quote Princeton Companion to Mathematics (2008)========== Modular theory exploits a version of the GNS construction (section 1.4). Let M be a selfadjoint 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 <T_{1}, T_{2}> = φ(T_{1}* T_{2}) 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 H_{M}. Indeed a vector V in H_{M} 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 righthand side we use multiplication in the algebra M. Because of this observation, we can think of M 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_{g1} 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 oneparameter 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== I like their expression for Δ, namely S*S. It makes sense because we know JJ = I so therefore S* S = Δ^{1/2} J J Δ^{1/2} = Δ 



#114
Dec712, 03:38 AM

P: 165

Thanks for that full and clear reply, Marcus. My interest in the spectrum of a selfadjoint (Hermitian) matrix being regarded as the "higher dimensional analog of a real number "(as you put it) was provoked by Eugene Wigner’s remark in his essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"(Communications in Pure and Applied Mathematics, vol. 13, No. 1 February 1960):
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve”. I think Wigner overstated things a bit; the supportive match between maths and physics is perhaps a bit less than a miraculous, wonderful gift. Looking at mathematics from the outside, it seems likely to me that the set of real numbers lies close to the heart of much maths. For me an interesting aspect of the real numbers is that they are commutatively symmetric under arithmetic operations like addition and subtraction; it doesn’t matter where zero is located; numbers are like a line of labels that with impunity can be translated along its length. At the heart of physics lies the symmetry of the dimensions we inhabit. As far as we know, physics is ruled by the same mathematical laws everywhere and everywhen. That’s why successful physics theories must be covariant. And why momentum and energy are conserved; because space and time have commuting translational symmetries. It seems to me that physics (an evolving description of physical reality) and mathematics (an evolving universal language used by physicists and many others) are founded on similar symmetries. Perhaps the close match between them is quite mundane and may yet come to be better understood (counting numbers were probably an abstraction invented to quantify resources, like goats. Real numbers and much else evolved from these humble roots.) Even the need for a spectrum of numbers to statistically quantify observations on a quantum scale can be understood, up to the mysterious finiteness of h. Now it seems from the work of folk like Barbour, Connes and Rovelli, that this statistical quantification promises a new understanding of time. Great stuff. An understanding of space may take longer; for practical purposes, it’s what we can swing a cat in! 



#115
Dec812, 11:56 AM

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Paulibus, one thing your post reminds me is that significant advances in physics have often been accompanied by maturing philosophical sophistication. E.g. early 20th century the role of the observer, and of measurement, no fixed prior geometry, nonexistence of the continuous trajectory, irreducible uncertainty. Taking certain philosophical (epistemic?) proposals seriously actually helped the physics develop in some cases.
So progress is not always "physics as usual". Sometimes a dialog with philosophy of science people is helpful. I suspect we will be seeing a General Covariant Quantum theory of Time emerge along lines suggested in this thread. A world (*algebra) of possible observations and a state (defined on it giving correlations and expectation values) which expresses what we think the laws are, what we know from prior observations, what we predict deduce or expect. The laws and constants of physics are after all merely correlations among actual and possible measurements, involvinglike everything elseuncertainty. They are "regularities" in the *algebra (call it M for "measurements" if you like), and are embodied in the state functional, along with what we think has been observed. The state is an elementary mathematical object, just a positive linear functional ω: M → ℂ. I suppose that this model (M, ω) will replace the model consisting of spacetime manifold with fixed geometry and fields defined on it, in part because the "block universe" picture has philosophical shortcomings: is incompatible with quantum theory. This last is the theme of a conference opening in Capetown in a couple of days (1014 December). Main theme: ideas of time and challenges to block universe idea. http://prce.hu/centre_for_time/jtf/passage.html Abstracts of scheduled talks (scroll down to get to the abstracts) http://prce.hu/centre_for_time/jtf/FullProgram.pdf BTW we should also keep an eye on tensorial group field theory "TGFT", I just watched the first 50 minutes of Sylvain Carrozza's PIRSA talk: http://pirsa.org/12120007/ It was interesting. Also the last 9 minutes (64:0073:00) where he gives conclusions, outlook, and answers questions. Most of the questions were from Dittrich and (someone I think was) Ben Geloun. 



#116
Dec812, 12:27 PM

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#117
Dec812, 01:38 PM

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http://arxiv.org/abs/0912.0808 See also the thought experiment on page 12 of his earlier paper http://arxiv.org/abs/grqc/0605049 You may have already looked at his "evolving/crystallizing" spacetime papers and would like to hear him present them in person. One of the points Ellis makes is that as far as we know the future spacetime geometry is in principle unpredictable. As unpredetermined as are the times of radioactive decay, which conventional QM tells us are not predetermined. Therefore the conventional block universe, extending into future with a predetermined spacetime geometry, cannot exist. I assume by "Avi" you mean Avshalom Elitzur, one of the other participants at this week's Capetown Time conference. 



#118
Dec812, 01:43 PM

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#119
Dec812, 03:10 PM

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(Greek roots: on = being,reality; epistem = knowledge) Anyway Ruta you said you wished you could hear Ellis' talk about the evolving block. I don't especially go for Ellis' proposed solution, but I like the clear way he describes the problem. This 2008 essay for wide audience communicates really well, and other readers of thread might enjoy it. It got the FQXi second community prize, right after Rovelli's essay. http://fqxi.org/data/essaycontestf...contest__E.pdf ==quote Ellis page 2== To motivate this, consider the following scenario: A massive object has rocket engines attached at each end to make it move either left or right. The engines are controlled by a computer that decides what firing intervals are utilised alternately by each engine, on the basis of a nonlinear time dependent transformation of signals received from a detector measuring particle arrivals due to random decays of a radioactive element. These signals at each instant determine what actually happens from the set of all possible outcomes, thus determining the actual spacetime path of the object from the set of all possible paths (Figure 1). This outcome is not determined by initial data at any previous time, because of quantum uncertainty in the radioactive decays. As the objects are massive and hence cause spacetime curvature, the spacetime structure itself is undetermined until the object’s motion is determined in this way. Instant by instant, the spacetime structure changes from indeterminate (i.e. not yet determined out of all the possible options) to definite (i.e. determined by the specific physical processes outlined above). Thus a definite spacetime structure comes into being as time evolves. It is unknown and unpredictable before it is determined. Something essentially equivalent has already occurred in the history of the universe. According to the standard inflationary model of the very early universe, we cannot predict the specific largescale structure existing in the universe today from data at the start of the inflationary expansion epoch, because density inhomogeneities at later times have grown out of random quantum fluctuations in the effective scalar field that is dominant at very early times... ...It follows that the existence of our specific Galaxy, let alone the planet Earth, was not uniquely determined by initial data in the very early universe. The quantum fluctuations that are amplified to galactic scale are unpredictable in principle. Thus spacetime evolution is not predictable even in principle in physically realisable cases. The outcome is only determined as it happens. ==endquote== An arxiv link to the same essay: http://arxiv.org/abs/0812.0240 List of the 2008 time essay contest winners: http://fqxi.org/community/essay/winners/2008.1 



#120
Dec812, 05:46 PM

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What George Ellis (one of the world's leading cosmologists and coauthor with Stephen Hawking of The Large Scale Structure of SpaceTme) says here is at once so clear and so striking that perhaps it deserves emphasis:
It follows that the existence of our specific Galaxy, let alone the planet Earth, was not uniquely determined by initial data in the very early universe. 



#121
Dec812, 10:12 PM

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I imagine that by now most people who might read this thread have figured out why the 4D block universe of General Relativity is incompatible with quantum uncertainty. The incompatibility is across the boardall common mainstream interpretations/formulations of QM for which uncertainty is an underlying bedrock principle. So perhaps I don't have to provide explanation (beyond what we already have in the quotes from George Ellis. But here's an excerpt from an essay by Carlo Rovelli that explains the point very clearly. This from page 4 of Chapter 1 of the 2009 book Approaches to Quantum Gravity, D. Oriti ed. published by Cambridge University Press ( http://arxiv.org/abs/grqc/0604045 )
==quote Chapter 1 of Approaches to Quantum Gravity== ... In classical GR, indeed, the notion of time differs strongly from the one used in the specialrelativistic context. Before special relativity, one assumed that there is a universal physical variable t, measured by clocks, such that all physical phenomena can be described in terms of evolution equations in the independent variable t. In special relativity, this notion of time is weakened. Clocks do not measure a universal time variable, but only the proper time elapsed along inertial trajectories. If we fix a Lorentz frame, nevertheless, we can still describe all physical phenomena in terms of evolution equations in the independent variable x_{0}, even though this description hides the covariance of the system. In general relativity, when we describe the dynamics of the gravitational field (not to be confused with the dynamics of matter in a given gravitational field), there is no external time variable that can play the role of observable independent evolution variable. The field equations are written in terms of an evolution parameter, which is the time coordinate x_{0}, but this coordinate, does not correspond to anything directly observable. The proper time τ along spacetime trajectories cannot be used as an independent variable either, as τ is a complicated nonlocal function of the gravitational field itself. Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable. This does not mean that GR lacks predictivity. Simply put, what GR predicts are relations between (partial) observables, which in general cannot be represented as the evolution of dependent variables on a preferred independent time variable. This weakening of the notion of time in classical GR is rarely emphasized: After all, in classical GR we may disregard the full dynamical structure of the theory and consider only individual solutions of its equations of motion. A single solution of the GR equations of motion determines “a spacetime”, where a notion of proper time is associated to each timelike worldline. But in the quantum context a single solution of the dynamical equation is like a single “trajectory” of a quantum particle: in quantum theory there are no physical individual trajectories: there are only transition probabilities between observable eigenvalues. Therefore in quantum gravity it is likely to be impossible to describe the world in terms of a spacetime, in the same sense in which the motion of a quantum electron cannot be described in terms of a single trajectory. ==endquote== Having a block universe with some definite course of geometry would be like a trajectory. A trajectory is a classical idea, it is not physical. We do not have continuous smooth trajectories, we have slits and detectors that is to say a finite number of measurements made along the way. There are an infinite number of possible observations/measurements of the path of a particle or geometry of the universe. But nature does not let herself be pinned down, we can only choose a finite number of them to make. Moreover each measurement may have a range of possible values and involve uncertainty. I suspect this is why the smooth manifoldthe continuum model of the physical world including spacetimeis apt to be replaced by something more like an algebra of observables, each one a package of uncertainty with its range of possible values. We see this replacement model being tentatively tried out by researchers. Time is then no pseudospatial "dimension" but a flow defined on the algebra. 



#122
Dec912, 02:12 PM

P: 640

The problem described by Ellis, per psiepistemism, simply reflects our inability to know the stressenergy tensor (SET) for the whole of spacetime. You need the SET to compute "a spacetime" solution (metric g) as Rovelli points out. [This is more obvious in the graphical Regge calculus version of GR where one must find a SET and metric g on every link of the graph that satisfies the graphical counterpart to Einstein's Eqns (EE).] This doesn't mean spacetime is not determined, only that we can't determine it. I don't know why that bothers him any more than the fact that we can't know the geometry of our past lightcone uniquely (per his own work in the 1980s). Does that mean the geometry of our past lightcone is not fixed? Of course not.
Rovelli's problem is averted by finding a quantum theory of gravity that doesn't follow canonical quantization. For example, one could rather seek a theory in which one computes amplitudes for spatiotemporal units ("processes" in the language of Hiley) via the path integral approach or via an algebra of process a la Hiley. In this approach, one understands a particular SET and g on the graph of Regge calculus follow as an average of many fundamental building blocks. This can be done for the Schrodinger and Dirac eqns as shown by Hiley. Of course, these formalisms assume globally flat spacetimes, so the question becomes, how do we get spacetime curvature? We postulate that can be done by modified Regge calculus whereby large graphical links are possible. We used this approach to show a flat, matterdominated GR solution (EinsteindeSitter) can match the type IA supernova data as well as the concordance model (EinsteindeSitter + lambda) without accelerating expansion (no lambda). You can read published presentations of these ideas in the following papers: “Being, Becoming and the Undivided Universe: A Dialogue between Relational Blockworld and the Implicate Order Concerning the Unification of Relativity and Quantum Theory,” Michael Silberstein, W.M. Stuckey & Timothy McDevitt. To appear in a Hiley Festschrift in Foundations of Physics. http://arxiv.org/abs/1108.2261. Appeared Online First 4 May 2012. “Modified Regge Calculus as an Explanation of Dark Energy,” W.M. Stuckey, Timothy McDevitt & Michael Silberstein, Classical & Quantum Gravity 29 055015 (2012). http://arxiv.org/abs/1110.3973. “Explaining the Supernova Data without Accelerating Expansion,” W.M. Stuckey, Timothy McDevitt & Michael Silberstein. Honorable Mention in the Gravity Research Foundation 2012 Awards for Essays on Gravitation, May 2012. International Journal of Modern Physics D 21, No. 11, 1242021 (2012) http://users.etown.edu/s/STUCKEYM/GRFessay2012.pdf So while I agree that some modification of GR is needed to accommodate quantum physics, this does not entail abandoning blockworld. On the contrary, BW is necessary in these approaches to quantum gravity. 



#123
Dec912, 02:46 PM

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I suspect you and I would definitely agree on at least one point: that how physics eventually comes to understand TIME and the problems associated with it will depend very much on what quantum theory of spacetime geometry is eventually arrived at and accepted. To be successful quantum theory of time will require a quantum theory of space AND time. At least this seems to be what you and your coauthors are striving to construct. A propos of Basil Hiley, I recently started a thread on a paper of his that just appeared, but the thread elicited little interest. You may not have noticed it. I'll get a link. http://www.physicsforums.com/showthread.php?t=651454 One of your papers with Silberstein and McDevitt was referred to in the thread, I don't recall in what connectionit may not have been clear why. 



#124
Dec1012, 03:39 AM

P: 165

And is it possible that the Hubble expansion would then vanish, because "thermal time' is somehow connected with such change? Perhaps RUTA's noting that: 



#125
Dec1012, 10:51 AM

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[11] C. Rovelli, “The Statistical state of the universe,” Class. Quant. Grav. 10 (1993) 15671578. This 12page article was published in the same issue with an 18page article: C. Rovelli, “Statistical mechanics of gravity and the thermodynamical origin of time,” Class. Quant. Grav. 10 (1993) 1549–1566. Neither seems to be on the arxiv. I have not been able to find online copies and so may have to visit the stacks at the Physics Department library here. EDIT: YAY! I found an online copy. http://siba.unipv.it/fisica/articoli....15671568.pdf 



#126
Dec1012, 12:32 PM

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It's great that earlier article is available online! Thanks to someone at University of Pavia. I printed a copy immediatelyin some cases availability is sporadic, so just to make sure. The paper is historically important and it would be nice if it were on arxiv. Here is the abstract:
http://siba.unipv.it/fisica/articoli....15671568.pdf ==quote== Class. Quantum Grav. 10 (1993) 15671578. The statistical state of the universe Carlo Rovelli Abstract. The idea that the cosmological state of the universe can be described in terms of a statistical state is discussed. A dynamical model with infinite degrees of freedom that describes a RobertsonWalker universe with nonhomogeneous electromagnetic radiation is defined. Its statistical mechanics is studied by using the covariant statistical theory developed in a companion paper. A simple statistical state that represents the cosmic background radiation is constructed. The properties of this state support the general theory; in particular, the idea, introduced in the companion paper, that a preferred time variable, denoted thermodynamical time, is singled out by the statistical state can be tested within this model. Thethermodynamical time is computed and shown to agree with the standard RobertsonWalker time. In addition, an application of the general theory to a simple special relativistic system, and a proposal for an application to full general relativity are also presented. The relevance of this application for the physics of the very early universe is discussed. ==endquote== Here's an excerpt from the introduction: In this paper, we discuss the statistical mechanics of a dynamical model that represents the universe filled with an arbitrary nonhomogeneous electromagnetic field. The purpose of this investigation is twofold. In the first place, we wish to introduce the idea of a statistical description of the state of the universe. In the second place, the model is presented as a first application of the general theory of covariant statistical thermodynamics that has been introduced in a companion paper...An excerpt from the conclusion section: In addition, we have discussed the application of the general theory to a simple special relativistic system, and we have introduced a perturhative expansion for computing an exact statistical state of the full Einstein theory that should represent a RobertsonWalker universe filled with gravitational radiation. We expect this model to be relevant for the description of the thermodynamics of the very early universe. 


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Does time exist?  General Discussion  56  
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[SOLVED] Julian Barbour's "End of Time"  Science & Math Textbook Listings  2 