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Julian Barbour on does time exist 
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#91
Nov2612, 07:35 PM

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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 nucleusswitch 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 *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. 


#92
Nov2712, 08:08 AM

P: 177

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?



#93
Nov2712, 08:40 AM

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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 noncommutativity 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 timeevolution. 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 


#94
Nov2812, 12:30 AM

P: 177

One (rather special); is because one chooses to describe the world in a quantum mechanical context, where congugate observables are (still mysteriously?) noncommutative. Two (more generally); because the world is threedimensional, 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 noncommutative change becomes possible (like successive rotations about noncolinear axes, described by adding polar vectors or tensors). Does noncommutative change only happen in three dimensions, which we seem (still mysteriously) to be endowed with? And could GR's pseudodimensional time emerge in the way Connes and Rovelli postulate just because we live in three spatial dimensions ? 


#95
Nov2812, 03:35 AM

P: 177

In checking KMS state in Wikipedia, I noticed that George Green is in the background. George was a selfeducated 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 faroff days. Just nonessential human interest stuff which guides my wary path!



#96
Nov2812, 09:20 AM

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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...rd_Sadi_Carnot Sense and Sensibility (1811), Pride and Prejudice (1813), Mansfield Park (1814), Emma (1816) Reflections on the Motive Power of Fire (1824) 


#97
Nov3012, 04:14 PM

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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...l_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 n^{2} 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 wantwe 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 algebraand 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 TomitaTakesaki business: S and it's polar decomp. etc: http://arxiv.org/abs/mathph/0511034 the first couple of pages seem enough. It is pretty basic and clearly written. It's from the Ensevelier Encyclopedia of Mathematical Physics. 


#98
Nov3012, 09:17 PM

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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. http://www.physicsforums.com/showthr...88#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/grqc/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 1621. == quote http://arxiv.org/abs/grqc/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 FriedmanRobertsonWalker 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" listI will simply quote: ==quote grqc/9406019 page 22== • Nonrelativistic 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 usestandard universe time in the standard cosmic model. and the fact that TT recovers ordinary physics time when you specialize to a NON general covariant casewith 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 ONEPARAMETER 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_{t}A. 


#99
Dec112, 11:54 AM

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The hangup 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 balancean equilibrium state.
(But people have a mental image of something "arriving" at equilibriumimagined 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 coauthor 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 selfadjoint. We need the selfadjoint part, because the Tomita flow is a oneparameter family of automorphisms given by: α_{t}(A) = Δ^{it}AΔ^{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 finitedimensional case, for instance). But in the general case this doesn’t happen, and the ConnesRovelli 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 statedependent 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 statedependent 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 nonstatedependent “god’seye 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 conjugatelinear * involution. Why should the world be a C* algebra? (I must be kidding ) See post #65 http://www.physicsforums.com/showthr...56#post4169556 


#100
Dec212, 03:02 AM

P: 177

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 nonreversible 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 noncommutative 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 etherlike 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? 


#101
Dec212, 03:13 AM

P: 203




#102
Dec212, 04:26 AM

P: 366

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



#103
Dec212, 11:11 PM

P: 177

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.



#104
Dec312, 12:16 PM

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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 "manyfingered 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.co...dtomitaflow/ 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 ‘timereversed 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 timereversed von Neumann algebra flows the other way! I know that every symplectic manifold has a ‘timereversed 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 


#105
Dec312, 09:42 PM

P: 177

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!



#106
Dec312, 10:12 PM

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I'm in general agreement about Jeff Morton's blog, especially the October 2009 post, and with the spirit of your remarks. I notice I got Jeff's location wrong, a few posts back. He was at Lisbon until recently but is now at Uni Hamburg. That's become a pretty good place for Quantum Gravity, as well as Mathematical Physics (Jeff's field). He could continue to be interested and wellinformed about QG (whatever direction his own research takes) which would be our good fortune.
The only recent question no one has responded to in this thread is from H. Cow about the current situation in QG. It's changing rapidly, and strongly affected by what's happening in Quantum Cosmology, since that is where the effects of quantum geometry are most apt to be visible, in the aftermath of the Bounce, or whatever happened around the start of the present expansion. Since that is not the main topic of this thread, I would urge H. Cow to start a thread asking about thatand also take a look at my thread about the current efforts at REFORMULATING Loop QG. ===================== Getting back to the topic of TIME. I'd be very curious to know how a cosmological bounce looks in the general covariant Heisenberg picturethere is just one timeless state, which is a positive linear functional on a C* algebra of observables. An algebra [A] and a functional ρ defined on it which gives us, among other things, expectation values ρ(A) and correlations e.g. ρ(AB)  ρ(A)ρ(B) and the like. Taken together ([A], ρ) give us a oneparameter group α_{t} which acts like the passage of time on the observablesmapping each A into the corresponding observable taken a little while latera PROCESS that mixes and morphs and stirs the observables around, a "flow" defined on the algebra [A]. So if the theory has a bounce one intuitively feels there should be an energy density observable, call it A, corresponding to a measurement made prebounce. So that we can watch the expectation value of α_{t}(A) evolve thru the bounce. In other words ρ(α_{t}(A)) should start low, as in a classical universe of the sort we're familiar with, and rise to some extremely high value within a few tenpowers of Planck density, and then subside back to low densities comparable to prebounce. Now the state ρ being timeless means that it does not change. So the challenge is to come up with an algebra of observables which undergoes a bounce, when given the appropriate timeless positive linear state functional ρ defined on it. =================== Thermal time could be starting to attract wider attention. I noticed that it comes up in the latest Grimstrup Aastrup paper, "C*algebras of HolonomyDiffeomorphisms & Quantum Gravity I", pages 3739 http://arxiv.org/abs/1209.5060 G&A's reference [46] is to the paper by Connes and Rovelli: ==sample excerpt from pages 3739== ...A more appealing possibility is to seek a dynamical principle within the mathematical machinery of noncommutative geometry. In particular, the theory of Tomita and Takesaki states that given a cyclic and separating state on a von Neumann algebra there exist a canonical time flow in the form of a oneparameter group of automorphisms. If we consider the algebra generated by HD(M) and spectral projections of the Dirac type operator, then the semiclassical state will, provided it is separating, generate such a flow. This would imply that the dynamics of quantum gravity is state dependent^{13}  an idea already considered in [46] and [47]. Since TomitaTakesaki theory deals with von Neumann algebras it will also for this purpose be important to select the correct algebra topology. ... ... Hidden within the two issues concerning of the dynamics and the complexified SU(2) connections lurks a very intriguing question. If it is possible to derive the dynamics of quantum gravity from the spectral triple construction – for instance via Tomita Takesaki theory – then it should be possible to read off the spacetime signature (Lorentzian vs. Euclidean) from the derived dynamics, for instance a moduli operator. ==endquote== I don't follow Grimstrup et al work at all closely, but note their interest. 


#107
Dec412, 02:28 AM

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#108
Dec412, 07:11 AM

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I won't be able to answer your post right away. John Baez is a teacher. I think he is using his comment to hint at how the Tomita flow of time is sensed or tasted from the algebra [A] and the function rho defined on it (that basically just gives expectation values of the observables separately and in combination).
The way the flow is constructed has very much to do with replacing i with i. That is what the * operation of a C* algebra does. On a larger scale. The slow way to digest this business intuitively begins with thinking a little about the complex conjugate operation that takes x+iy into x  iy. It flips the plane of complex numbers over along the horizontal axis. Then you think about generalizing that to matrices. For a onebyone matrix, well that is just a single complex number so you just take the conjugate. For a twobytwo that is 4 complex numbers and there is an analogous thing involving conjugates and taking the "transpose" of the matrix (exchange upper right and lower left entries. It's a swapping operation that, if repeated, gets you back what you had to start withthe mathematics term is "involution". The first thing the Tomita timeflow construction does is define a "swap" operator S that basically does the star involution in a very concrete way, analogous to simply multiplying one matrix by another: taking A > SA, and where SA turns out to have the same effect as A*. And then Tomita analyzes S into two factors, one of which is selfadjoint. THIS IS A WAY OF TASTING THE ESSENTIAL FLAVOR OF TIMEREVERSAL. Squeezing the juice out of timereversal. That is what Baez is trying to plant the idea of in the reader's mind, by making that innocentsounding observation. It is really important. The Tomita flow is based on that selfadjoint factor of S. In the usual notation this factor is called uppercase Delta Δ. This is vague and handwavy. I'll try to say it better later today. Here's a WikiP about an operation on matrices that is analogous to conjugate of complex numbers: http://en.wikipedia.org/wiki/Conjugate_transpose You may be thoroughly familiar with this already but I'll try to supply detail for other readers who might not be. 


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