jokkon said:
Hi there thanks for all the replies. I knew that QED is a special relativistic theory but I didn't know that causality would get screwed up in a background dependent theory. Also I did read some review papers written by Carlo Rovelli and understood (I think) that a superposition of spin networks form a picture of space-time. But I am curious as to how they adapted Penrose's original idea to how LQG uses spin networks today. Not only do the spin networks look different, the quantum numbers represent something entirely different either. As far as I know the original spin networks represent total spin of the system but now they represent either area of cross section and volume.
As for the wheeler dewitt equation, i noticed that the equation itself doesn't have a time parameter so i find it weird that it would give the time evolution of spacetime.
First let me qualify something: The causality thing might be regarded as a quibble. Because a very small ripple, a tiny perturbation, wouldn't disturb things very much. I take it seriously (I think no theory based on a fixed background can be totally realistic, it can't be fundamentally right.) But not everybody thinks the same way about this.
I'm avoiding the "frozen time" issue for the moment. The issue arises already with classical GR. There is no preferred time coordinate, so the Hamiltonian or "canonical" formulation of classical GR does not give you time-evolution (like a Hamiltonian typically would in other situations.) The "canonical" formulation of GR---called ADM arnowitt, deser, misner---gives you a 3D geometry on a 3D "slice", plus a Hamiltonian constraint, which, if it is satisfied, says yes this metric together with this distribution of matter could indeed have evolved naturally, it is physical. In the "canonical" formulation of classic GR, the situation on the 3D slice somehow contains all the information about the past and the future evolution. You rightly observe that it is a little weird. This carries over to 1990s LQG which is a canonical approach, but does not carry over to the spin foam approach which is a 4D or path integral approach. For now, let's sideline this topic and I'll simply link to the Wikipedia
http://en.wikipedia.org/wiki/ADM_formalism
and avoid further discussion.
BTW compliments. Good questions! And Naty's answers are consistently on target, or seem so to me. It makes for a good thread.
I keep thinking of Lewandowski's brief description of LQG and of spinfoam, and of the relation. Lewandowski is great. There is a recent (September 2009) online audio+slide talk where he gives his account of Loop and then takes questions and comment from Carlo Rovelli, Jon Engle, Abhay Ashtekar, Laurent Freidel. I want to go back and listen to that again this afternoon. It is so concise. Maybe it could be helpful here.
About spin networks. I wouldn't stress the relation to the original Penrose invention. The use is so different. If you think about the canonical formulation, they have to have a way of describing the quantum state of the geometry of a 3D slice.
So at first they said let's represent geometry configuration space by the set of all possible "connections"* on the slice, and as functions on configuration space, in other words as functions of a connection, let's take LOOPS.
A particular loop, evaluated on a connection, will just be whatever the connection does as you run around that loop.
OK, they said, let's build a hilbertspace of quantum states out of these loops. But to make a vector space you have to add loops together. You have to stick them together, and when you do that you get a network.
A convenient basis of the Hilbertspace turned out to be these networks, which turned out to be formally similar to what Penrose had introduced. These networks either live in the 3D slice, embedded, or they exist independently as abstract labeled graphs.
About spin foams. The idea was to get away from the canonical approach, with its Hamiltonian constraint equation H=0, and working all the time on a 3D slice. The idea was to describe spacetime as an evolution from an initial 3D geometry to a final 3D geometry. To represent that path from initial to final geometry. Like a Feynman path integral. Maybe there are many paths from initial to final and each has an amplitude, and you can do a "sum over histories" or a path integral to get the overall amplitude of getting from one to the other.
So a spinfoam is the 2-complex describing how a network evolves. The trajectory of an edge becomes a 2-cell.
If a spin network is a good way to represent the quantum state of 3D geometry, then a spinfoam is a good way to represent the path along which that geometry evolves.
It seems to me that Jerzy Lewandowski describes LQG in the most concise and lucid way. Maybe in parts it is so concise as to be incomprehensible. And it is pitched at advanced seminar level. But I like how he does it---the exemplary carefulness. And I like hearing the others respond. So I'll fetch the link. Actually to get it you just have to google "ILQGS".
This stands for international lqg seminar---they do a telephone seminar a couple of times a month that links 5 or 6 locations.
http://relativity.phys.lsu.edu/ilqgs/
Jerzy's talk is the one dated 20 October 2009. Parts of the talk (such as slides 14-18) are way too technical for present purposes. But other parts provide a concentrated definitive treatment that i like very much.
*think of a "connection" as a parallel transport tool defined at every point of the hypersurface slice, that tells how a vector carried along a path thru that point will roll and yaw and come back different if you loop around and return to that point. A connection is a good way to describe the geometry and can serve as an alternative to the metric or distance-function as a way to characterize the internal experience of shape.
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EDIT TO REPLY TO NEXT POST BY ATYY
atyy said:
But what about Asymptotic Safety? Even if it turns out not to be true, the fact that it is plausible shows that the geometric nature of gravity is not sufficient to rule out quantizing gravity like electromagnetism.
Please tell me how! I am not asking ironically. Indeed I suspect that AsymSafety is right, there is a UV fixed point somewhere in theory space. But this does not mean one can write a perturbation series. AsymSafe is a non-perturbative approach. The people who do it refer to "nonperturbative renormalizability". So tell me what does "like electromagnetism" mean? What would it mean to "quantize gravity like electromagnetism"? Could you sort of sketch it out for me?
How do you represent the geometry of the universe, in your idea? How do you represent the geometry evolving dynamically along some path? Could you sketch some sort of equations to show the resemblance to QED? I am not talking about gravitons, I'm sure you realize. It's problematical to define particles on anything but a fixed flat background. The concept of particle doesn't work in dynamic geometry---you can't even say how many there are. It is just a useful approximation appropriate to a limited context. So I need a theory not of gravitons but of evolving quantum geometry.
I don't see how you would set up something analogous to QED. So I'm interested to see what you have in mind!
