Julian Barbour on does time exist

[rjbeery]

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

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

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

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

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

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

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

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

 

[Paulibus]

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

 

[marcus]

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

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

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

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

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

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

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

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

 

[Paulibus]

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

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

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

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

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

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

 

[Paulibus]

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

 

[marcus]

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

 

[marcus]

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

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

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

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

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

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

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

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

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

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

 

[marcus]

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

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

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

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

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

So there is the uniqueness of TT

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

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

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

and other good stuff that Connes and Rovelli mention.

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

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

 

[marcus]

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

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

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

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

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

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

(SA)ψ = A*ψ

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

α_t(A) = Δ^-itAΔ^it

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

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

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

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

 

[Paulibus]

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

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

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

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

 

[sheaf]

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

Did you mean to say

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

 

[HomogenousCow]

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

 

[Paulibus]

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

 

[marcus]

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

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

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

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

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

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

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

==quote==
John Baez Says:

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

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

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

 

[Paulibus]

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

 

[marcus]

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 well-informed 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 that---and 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 picture----there 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 one-parameter group α_t which acts like the passage of time on the observables---mapping each A into the corresponding observable taken a little while later---a 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 pre-bounce. 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 ten-powers of Planck density, and then subside back to low densities comparable to pre-bounce.
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 Holonomy-Diffeomorphisms & Quantum Gravity I", pages 37-39
http://arxiv.org/abs/1209.5060
G&A's reference [46] is to the paper by Connes and Rovelli:
==sample excerpt from pages 37-39==
...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 one-parameter group of automorphisms. If we consider the algebra generated by HD(M) and spectral projections of the Dirac type operator, then the semi-classical state will, provided it is separating, generate such a flow. This would imply that the dynamics of quantum gravity is state dependent¹³ - an idea already considered in [46] and [47]. Since Tomita-Takesaki 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 space-time 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.

 

[Paulibus]

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.

Could such a challenge be explored in the context of your quote from John Baez, who thinks that:

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

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

It might be an interesting experience to remember the future instead of the past, if such a switch could be arranged. Perhaps, though, that set of acausal states would be just as mysterious to us, "time"-wise, as are the present causal set. Or would switching the sign of i just be a mathematical device without any imaginable physical foundation?

 

[marcus]

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 one-by-one matrix, well that is just a single complex number so you just take the conjugate. For a two-by-two 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 with---the 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 self-adjoint. THIS IS A WAY OF TASTING THE ESSENTIAL FLAVOR OF TIME-REVERSAL. Squeezing the juice out of time-reversal.
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.

 

[marcus]

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/Repo...ew=body&id=pdf_1&handle=euclid.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 187-221 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 T-theorem. 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/math-ph/0511034
But it does not give proofs of the hard parts.

It seems that Tomita-Takesaki theory is deep, non-trivial. 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 well-defined. Rieffel and van Daele work with bounded operators and take more steps---lots 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 state---from 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 well-defined? 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 conjugate-linear and a kind of flip or involution. It is its own inverse, J²=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^itAΔ^-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 Tomita-Takesaki theorem: the statement that
Δ^it [A] Δ^-it = [A]

If you take the original star-algebra and advance each item in it by the same time-interval t, then what you get is the same star-algebra. 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/math-ph/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 so-called "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 into 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

 

[marcus]

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 Tolman-Ehrenfest 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 time-interval and dτ is the corresponding interval of Tomita time, then the Tolman-Ehrenfest 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 Tolman-Ehrenfest 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 general-relativistic thermodynamics. Here we study this notion in the restricted context of stationary spacetimes. We show that the Tolman-Ehrenfest 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 Tolman-Ehrenfest 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 = ([STRIKE]h[/STRIKE]/k) dτ/ds

 

[Paulibus]

Marcus: small queries. In one of your posts that I now can't find I'm sure you mentioned that self-adjoint 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?

 

[marcus]

Marcus: small queries. In one of your posts that I now can't find I'm sure you mentioned that self-adjoint 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? ...

A bit of miscellaneous trivia. This kind of matrices are also called "Hermitian" after Charles Hermite (born 1822) who famously studied them.
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 (self-adjoint) 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 self-adjoint matrix is where the numbers down the diagonal turn out all positive real numbers. This means the linear map is just re-scaling 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 self-adjointness 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* = x-iy)
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 19th-Century 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/Self-adjoint_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/available/etd-11202006-103150/unrestricted/dissertation.pdf
Quantum statistical mechanics, KMS states and Tomita-Takesaki theory

 

[marcus]

Paulibus and others: I hope the foregoing account of self-adjointness, 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 Tomita-Takesaki theory, and transcribed a sample excerpt

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

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

 

[Paulibus]

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!

 

[marcus]

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, involving--like everything else--uncertainty. 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 space-time 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 (10-14 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 (64:00-73:00) where he gives conclusions, outlook, and answers questions. Most of the questions were from Dittrich and (someone I think was) Ben Geloun.

 

[RUTA]

I suppose that this model (M, ω) will replace the model consisting of space-time manifold with fixed geometry and fields defined on it, in part because the "block universe" picture has philosophical shortcomings: is incompatible with quantum theory.

Of course I disagree , since there are interpretations of quantum physics which rely on blockworld (TI, two-vector, RBW, and all time-symmetric accounts). Accordingly, quantum physics isn't incompatible with BW, but rests necessarily upon it.

This last is the theme of a conference opening in Capetown in a couple of days (10-14 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

Looks like an interesting conference! I'd like to hear Ellis's talk on an "evolving spacetime." Avi tried that once and it got him nowhere (or should I say "nowhen"?). I questioned him after he presented the idea at a conference once and he admitted that a metatime was necessary and highly undesireable. That issue was the reason I entered this thread. Anyway, I believe physicists are moving outside the discipline in attempting to address the passage of time as experienced subjectively -- physics is about commonly shared experience, i.e., the objective. A person's experience of the passage of time is not a shared experience, therefore it is purely subjective. The cognitive neuroscientists can tell you all about that.

 

[marcus]

..., quantum physics isn't incompatible with BW, but rests necessarily upon it.

Looks like an interesting conference! I'd like to hear Ellis's talk on an "evolving spacetime." Avi tried that once and it got him nowhere (or should I say "nowhen"?)...

You can read Ellis' ideas about evolving space-time here
http://arxiv.org/abs/0912.0808
See also the thought experiment on page 12 of his earlier paper
http://arxiv.org/abs/gr-qc/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 space-time geometry is in principle unpredictable. As un-predetermined as are the times of radioactive decay, which conventional QM tells us are not pre-determined. 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.

 

[RUTA]

One of the points Ellis makes is that as far as we know the future space-time geometry is in principle unpredictable. As un-predetermined as as the times of radioactive decay, which conventional QM tells us are not pre-determined.

That conclusion assumes psi-ontism. Those using BW assume psi-epistemism, obviously.

 

[marcus]

That conclusion assumes psi-ontism. Those using BW assume psi-epistemism, obviously.

If anyone wants a clue as to what Ruta is talking about, some people think of the wave function "psi" in some common versions of qm as really out there, and for others it represents our knowledge.
(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/essay-contest-files/Ellis_Fqxi_essay_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 non-linear 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 large-scale 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

 

[marcus]

What George Ellis (one of the world's leading cosmologists and co-author with Stephen Hawking of The Large Scale Structure of Space-Tme) 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.