Intuitive content of Loop Gravity--Rovelli's program
|Oct15-03, 04:57 PM||#1|
Intuitive content of Loop Gravity--Rovelli's program
In the "perche i nostri discorsi" thread, selfAdjoint gave a concise sketch of the direction that Carlo Rovelli sees in loop gravity
Probably what we need most now is an intuitive sketch of what makes Loop Gravity different from your typical field theory----how is the backgroundlessness implemented?
Things are coming to a head. Rovelli's new book "Quantum Gravity" is major and has IMO material for a bunch of PhD dissertations just expanding on details. It also contains the "Dialog" as a final chapter. You can connect the points made about loop gravity in the Dialog to chapters and sections in the main part of the book. Also Smolin's April 2003 paper lays out what has been accomplished and what remains to do and what the prospects are for getting loop gravity tested---it is a thorough review and comparison: "How far are we from a quantum theory of gravity?"
Plus we have good accounts of loop cosmology by Bojowald
like the recent paper "Quantum Gravity and the Big Bang", and in some of Ashtekar's papers. It appears progress in cosmology has been dramatic of late. New researchers have been getting into LQG at the level of cosmology.
Plus there is this month's Berlin symposium "Strings meets Loops" which will probably generate a series of overview talks
aimed at wider audience----e.g. another cosmology overview by Bojowald, another full theory overview by Ashtekar, a spin foam overview by Rovelli, and so on.
So there is more and more accessible information than there was a year ago, about loop gravity. It looks to me as if new research possibilities are coming into focus. For example, these days I keep seeing papers about the "low energy limit" or "semi-classical limit", another place where newcomers are getting in (like those Argentine people this month---Kozameh, Gleiser, Parisi)
It seems to me to be a good time to try to say what loop gravity is about, in the simplest possible way. I am apt to make several false starts on this. If someone else has been thinking about it and wants to try, go ahead....
|Oct17-03, 06:38 PM||#2|
was practically laughing just now thinking of my trying to give an intuitive explanation of loop gravity. Meteor has a copy of Three Roads, which probably has a perfectly good one, and I am too lazy to go down to the library and get a copy---never seen it.
so meteor or selfAdjoint are probably better equiped to do a sketch of loop gravity
BTW I believe that to give a good conceptual description of something often requires deeper understanding, paradoxically, than to descrbe it technically, so it is a place where one's shortcomings become evident HOWever here goes
I told you before I would make several false starts, it is inevitable, so lets get started
in loop gravity the excitations of space (or geometry or gravity allee same bizness weiss da) are polymers----essentially ball-and-stick models like of very large protein molecules
so the first analogy is a drumhead with sand sprinkled on it. you know that every different way it can vibrate is shown by the lines that appear in the sand when you excite---I hope you did this at the science museum as a kid and know what I mean: nice diagrams of lines appear on the surface and these diagrams CATALOG all the modes of vibration or the excitation modes.
the next analogy is a sink full of soapsuds---you had to wash the dishes as a kid and you can imagine the whole sink or the whole universe (whats the difference) full of soapsuds. Now in the middle of every bubble put a dot, and if one bubble contacts another bubble then connect those two dots with a line.
Now you have a network (a ball-and-stick molecule, a polymer) that fills the whole sink or universe. And we are going to give a number Q to each point (or ball, or dot, or vertex) in the network.
And a number P to each connecting line.
Whatever for? Why label each point and line with a number? (Roger Penrose thought of doing it, he is the one to ask about it) Well, using that additional information the network can tell us the VOLUME of any region----just add up the numbers attached to each point in the region, somehow-----and the AREA of any surface---just add up the numbers, in some fashion, that are attached to every line that punctures the surface.
Furthermore, if this labeled network's sticks become flexy the whole thing can be squashed flat and stuffed into your dresser drawer where you ordinarily keep undershirts.
So here is a thing which you can squash any shape and store anywhere which nevertheless tells you everything about the geometry of the universe, or the kitchen sink I forget which.
|Oct17-03, 06:58 PM||#3|
Yes, I have a copy of the book, and in it a spin network is defined as a directed graph, jointly with a series of rules that guide the evolution of the graph. To each edge of the graph is assigned a number, and the area of a surface depends on the value of the numbers of the edges that punctures the surface.Btw, the area of a surface can be computed with a formula that includes the Immirzi parameter. The volume of an object is proportional to the number of nodes of the graph thatr are inside the object.
However, in various documents on Arxiv, i've found that the edges of the graph are labeled with group representations of Lie groups, and the vertices with intertwining operators(damned if i now what's an intertwining operator!)I don't know if these spin networks are the same that the cited in the book of Smolin
Spin networks are used in the canonical approach to quantum gravity, but another approach, the sum-over-histories approach, has adopted a particular version of them, called spin foams, that are cell complexes.
|Oct18-03, 09:20 AM||#4|
Intuitive content of Loop Gravity--Rovelli's program
|Oct18-03, 09:42 AM||#5|
|Oct18-03, 01:22 PM||#6|
here's another piece of the jigsaw puzzle
in classical mechanics things move along trajectories---curved paths parametrized by time---and when you quantize the trajectories go away.
the curved paths things travel along dont exist any more, you have to erase the trajectories (or in Feynmann sum over histories you "integrate" all possible ways of getting from here to there---in any case the clear picture of a path loses reality and dissipates)
in GR, the 4D manifold is not a real thing (individual points are not events and have no physical meaning) because of gauge invariance any point will do----to define an event you need matter, like the event that two particles cross paths (at some point, but which point doesnt make any difference it is an arbitrary choice.
arbitrary choices needed to express something mathematically are called "gauge"----extra physically meaningless information that gets unavoidably mixed in as part of keeping the books.
in GR the 4D manifold is there so that you can write down the trajectory of the gravitational field. GR does not suggest that 4D spacetime exists, it is a mathematical amenity used for defining evolution of the gravitational field and the matter that goes into shaping the field.
but when GR is quantized, the trajectory goes away and one simply has a 3Dmanifold, with a space of all possible geometries defined on it
again, as in classical case, the points of the 3D manifold have no physical meaning---they are just "gauge". One can define surfaces and volumes only using matter---Rovelli and Ashtekar both use examples like by a surface I mean for instance the top of this desk.
The quantum states are functions defined on the space of all possible geometries that the 3D manifold can have. Analogous to quantizing a particle moving on the line by "wave functions" defined on the line.
The curious thing is that no one started out thinking of the "wave functions" defined on the space of geometries as spinnetworks! Nobody was looking for spin networks or asking for them! It just turned out that they appeared as the best way to CATALOG the functions defined on the space of all possible gravitational fields or all possible geometries.
At first they tried defining these functions using loops and they got a hilbertspace of loop-functions, but they couldnt get an orthogonal basis: the loops were too redundant. So they eventually borrowed spinnetworks from Penrose and they turned out to give an orthogonal basis for the space.
Also I even believe that the basis is countable and the hilbertspace is separable----technical conveniences to be sure.
So when I mentioned this polymer network thing that describes the geometry of the whole universe, but that you can stuff into the dresser drawer, it is a quantum state of geometry (a functional like a wavefunction defined on the collection of all classical geometries) and all quantum states look like this or combinations of things like this.
and matter fields must be defined on things like this
and the quantum state can evolve! At noon by some clock it can be this one in the top drawer and then at one o'clock it can be like this other one in the bottom drawer.
But it is a disconnected hopping, and there is no absolute clock you just have to choose some physical PROCESS (essentially something involving matter) to serve as a clock. This clock is part of the world and there is a correlation between what you observe the clock says and where you observe the pendulum is, or how far away the galaxies are, or whatever else. There is no one absolute time that drives the rest only correlations between different processes
processes which include, among other things, the change in the state of the 3D geometry of space.
So, when you check things out using cosmology, the evolution equation is a finite DIFFERENCE equation! It is not a differential equation. when you do loop cosmology the Friedmann equation that all cosmologists depend on becomes a step by step difference equation----e.g. Bojowald uses the scalefactor as his clock, there must be some physical process to use as clock, and correlates other stuff like curvature and inflation and density with the scalefactor. And so does everybody else that has been doing loop cosmology that I have seen. For example "Quantum Gravity and the Big Bang" the talk Bojowald gave recently.
it is interesting how the concept of time changes.
In Rovelli's textbook "Quantum Gravity" there is a philosophical section on time which I found really interesting---he finds that different branches of physics use ideas of time that are actually different from each other and also from everyday vernacular time.
he is able to distinguish around 8 or 9 different ideas of time.
Quantizing General Relativity seems to exert a strong influence on the ideas of time because both QM and GR bring insights about time which, if you try to put them together, produce something that seems radically new.
(of course one can avoid having to think about it if one throws out GR and replaces it with a lobotomized form or if one is very careful to only use GR and QM in separate situations and never together on the same problem)
|Oct19-03, 04:53 AM||#7|
Marcus, there are a vast number of papers that continue to reveal a discrete direction, I am really glad that you take the time to post the most interesting idea's from many fields.
You may have this link allready?..but if so others may find it interesting:http://uk.arxiv.org/abs/gr-qc?0306059
Rovelli for me seems to be an architect of new thinking.
|Oct19-03, 10:53 AM||#8|
I will quote the last 5 sentences in (the conclusions part of) this paper and try to say why I think it is interesting
"...We have studied the propagator of our model in detail. We have shown that in the semiclassical limit it has a simple relation with the Hamilton function of the classical theory, but this relation is not a simple exponential, as one might have expected.
Instead, the propagator is real. It is the sum of two exponential
terms complex conjugate to each other, that propagate backward and forward, respectively, along the motions. Accordingly, the physical Hilbert space splits between forward and backward propagating states.
We expect this structure to be the same in quantum general relativity."
this is the kind of simple example (a system like a springbob with only a couple of degrees of freedom) that physics teachers love to use when introducing a new method---try the new approach out on the simplest thing in sight: an harmonic oscillator, a single particle in a potential well, whatever. Then the maths do not obscure the ideas.
So this is Daniele Colosi (a grad student at Marseille) and Rovelli having fun with a toy that moves in a simple ellipse. Is this your reading too? I just looked at the paper. I like it. Maybe we should make a thread about this paper or just look at it in this thread
|Oct20-03, 11:39 AM||#9|
A Simple Background Independent Hamiltonian Quantum Model (Colosi/Rovelli)
Minkowski Vacuum in Background Independent Quantum Gravity
(Conrady, Doplicher, Oeckl, Rovelli, Testa)
and several related 2003 spin foam papers
recent work involving the hamiltonian seems to connect (in ways I didnt anticipate and dont fully understand) to recent spin foam work
there is the fact that in August 2002 John Baez and some others posted computer results that some spin foam numbers were not what some people expected them to be----this seems to have lead to increased interest in spin foams: something new to understand about them----several new papers with new ideas
then there is the fact that at this months symposium it is Rovelli who is talking about spin foams (and he and his associates have recently, in late 2002 and in 2003) put out several papers on spin foams
then there is the fact that several of these recent papers link up the hamiltonian and spin foam approaches----they are or seem to be trying to cure problems in both the hamiltonian and discover how to use spin foam models properly in a way that suggests some underground connection between the two
i had till now not paid attention much to spin foam quantum gravity but now because of these little hints Ive been reading in the past day or so, and because Rovelli has chosen to do the spinfoam presentation, I am beginning to pay more attention and trying to understand a little better.
BTW at the symposium the loop lineup looks like this
Ashtekar: quantum geometry and applications (this means overview and application to big bang, inflation, black holes...)
Bojowald: loop quantum cosmology (a strong run of results by him and about 10 other researchers over past 3 years, giving guidance to development of the full theory by testdriving in the cosmology case)
Rovelli: spin foams (this is the one that I cannot antipate, it will have unexpected things)
Lewandowski: the hamiltonian (this presumably will be profoundly analytic/algebraic as is the way with people from Warsaw. maybe selfAdjoint will help us understand this one [:)],
it has now been 5 years since Lewandowski found fault with Thiemann's hamiltonian and there has been a great deal of work involving hamiltonians since then! Perhaps L will summarize some of this. As befits a growing theory, the main issue here remains unresoved and people are still discovering how it should look, as for example in the paper you gave the link to )
This is merely by way of saying thanks for the link to that paper. It has given me something to do during spare moments for the past day or so
|Oct20-03, 12:03 PM||#10|
In this paper spin networks are defined like trivalent graphs with edges labeled by positive integers.A trivalent graph is a regular graph with 3 edges arriving at each node. The spin network has to follow 2 rules:
-The sum of the 3 edges that converge at a given node has to be an even number
-Each of these 3 edges can't be superior to the sum of the other 2
Do somebody know the paper where the labels passed from being numbers to group representations? Do the Lie groups have to be some specific Lie group? Are actually spin networks continued to be defined as trivalents graphs? Must the group representations be irreducible representations?
Ok, Ok, very much questions but this is interesting stuff
|Oct20-03, 01:31 PM||#11|
I'm not sure, but I think gr-qc/9707010 is early. See also Baez's TWF #110.
|Oct20-03, 01:32 PM||#12|
to say irreducible representation of SU2 is sort of like saying "spin" because there is one for each dimension and so roughly speaking one for each integer (or half integer if you divide each integer by two according to the quaint ancient custom of physicists)
the papers where Penrose made up "spin network" idea are not online!
however I have read about these papers and my understanding is that ALREADY AT THE BEGINNING penrose thought of the graph as labeled by "spins" that is to say either halfintegers or, what is the same, irreducible reps of SU2
as an interesting insight into human, or at least Penrose, nature, he regularly FLIPFLOPPED at the beginning between having the labels be integers which he called "colours" and dividing them all by two and calling them "spins". As a civilized mathematician he wanted to call them colours but as a savage physicist driven by brute instinct and prejudice he needed to divide them by two---as is the custom---and call them spins.
so this ambiguity of labeling has been there from the start
remember also that as children, while others are taught to skip rope and play hopscotch, physicists are taught that SU2 is the "double cover" of the rotation group, which is why an electron can turn around 720 degrees before it looks normal again. the first time it turns around it appears to have pointed teeth and is wearing a Count Dracula costume but then it turns around another 360 degrees and is its old self. But doubtless you know all this already!
|Oct20-03, 02:18 PM||#13|
I like it!
The inside of Fort knox has a safe where the rotation of the combination/wheel-number will dictate if one opens the safe or not?
The inside cogs and wheels are dictated by the outside combination wheel, one false turn and CPT kicks in and you are forever going to be turning..and turning..the dials. Yet if one were on the inside and the back of the door had a seethrough covering, one can guide a way through any amount of infinite combinations with ease!
|Oct21-03, 09:41 AM||#14|
I'm trying know to fathom what's the meaning of the Poisson algebra. I will post something about it
|Oct21-03, 10:23 AM||#15|
Good for you if you post on the Poisson algebra. This is a missing piece in our discussions here.
|Oct22-03, 03:58 PM||#17|
I told you I would make several false starts. Eventually there should be a non-technical description of loop gravity in only one to ten pages. Let's keep this thread going until we have one, or find one in the literature.
the basic picture of any quantum theory is you have a space of possibilities (configurations, might be simply a set of possible positions and momentums) and then you define a kind of "(not)probability" function or wave-function on that space of possibilities.
If the space of all possibilities, of whatever it is (one particle, N particles, a field, a geometry of the universe) is called A, then the the wavefunctions or quantum states or "(not)probability" functions are just complex valued functions on
usually there is a measure defined on called A so you can integrate these functions and they are "square integrable" which means they dont run off to infinity too much and have finite integrals
and I have to say that in mathematics this is, in a certain way, as basic as things ever get----a space, some complex-number-valued functions defined on that space---and being able to integrate or sum each of them, so each one has a finite size.
a loaf of bread, a jug of wine, and hilbert space---this is all we ask and it does not seem like a lot----the rest is trimmings.
so in a certain sense if I could just tell you how to build the configuration space called A of loop gravity and then, if I could just explain how to define a function on that space----and get a hilbertspace consisting of all the complex-number-valued functions on called A then I would be done explaining. All the rest----the selfadjoint operators on the hilbertspace, their evolution, their spectra, and all----that all "hatches" from how the original hilberspace of wavefunctions, or quantum states or whatever you call them, is defined.
So here we are down in the basement and there are not even any "spin networks" or "loops" around. I have to tell you the space called A of loop gravity.
Psssst! It is the space of "connections". A connection is one way to clothe a bare manifold with geometry if it has no geometry. The whole destiny of loop gravity, win or lose, succeed or fail, is in this one choice----the geometry of the universe shall be represented by the possible "connections" on a 3D continuum, a 3D manifold.
Until 1986 the guys like John Wheeler who were trying to quantize GR used the space of "metrics" for their called A and it gave them headaches. After 1986 almost everybody switched over to representing the geometries by connections.
Hey, the whole thing could go into another iteration if some yet other set of variables for GR were found---something that captured the essence of the shape of the world that was not a metric and not a connection---then you could have a new configuration space called A and a new hilbertspace of complex-valued functions defined on it.
It took 50 years to get from a space of metrics to a space of connections---people have tried to quantize GR for a long time. I am not betting they come up with something to replace connections, but they might.
So we are looking at the most basic question---how do you describe the shape of the world, what is a connection, how do you arrange all the possibilites to make a configuration-space, a space of all possible connections, how do you define functions on that space, that have their values in the complex number plane?
what is a connection?
|loop quantum gravity|
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