In particular, how do you extract a metric tensor from a spin network? You don't. First you take general relativity and rewrite it as a gauge theory with gauge group SU(2). It's a lot like electromagnetism except the A field tells you how to parallel transport spinors (instead of charged particles) and the flux of the E field through a surface corresponds to its area (instead of the electric flux). Then you quantize this theory in a background-independent way, getting a Hilbert space of quantum states having spin networks as a basis. Spin networks of the edges correspond to flux tubes of the E field , i.e. they give area to any surface they puncture. Similarly, spin network vertices give volume to any region of space they lie in. To see this, you take various familiar geometrical observables like area and volume and turn the crank to quantize them, turning them into operators. If you work out what these operators do to spin network states, you can figure out the intuitive geometrical meaning of spin network states (which I just described). The interesting thing is that the metric tensor is NOT one of the observables you can quantize! That's one of the big surprises in loop quantum gravity: the metric is not fundamental in this theory. Instead, area plays the most fundamental role. Okay, so much for my impressionistic mini-summary of spin networks and loop quantum gravity! The following stuff fills in a few of the details... but just a few, unfortunately. For a more thorough job, try Rovelli's webpage listed below. ..................................................................... Also available at
http://math.ucr.edu/home/baez/week110.html October 4, 1997 This Week's Finds in Mathematical Physics - Week 110 John Baez Last time I sketched Wheeler's vision of spacetime foam , and his intuition that a good theory of this would require taking spin-1/2 particles very seriously. Now I want to talk about Penrose's spin networks . These were an attempt to build a purely combinatorial de_script_ion of spacetime starting from the mathematics of spin-1/2 particles. He didn't get too far with this, which is why he moved on to invent twistor theory. The problem was that spin networks gave an interesting theory of *space*, but not of spacetime. But recent work on quantum gravity shows that you can get pretty far with spin network technology. For example, you can compute the entropy of quantum black holes. So spin networks are quite a flourishing business. Okay. Building space from spin! How does it work? Penrose's original spin networks were purely combinatorial gadgets: graphs with edges labelled by numbers j = 0, 1/2, 1, 3/2,... These numbers stand for total angular momentum or spin . He required that three edges meet at each vertex, with the corresponding spins j1, j2, j3 adding up to an integer and satisfying the triangle inequalities |j1 - j2| <= j3 <= j1 + j2 These rules are motivated by the quantum mechanics of angular momentum: if we combine a system with spin j1 and a system with spin j2, the spin j3 of the combined system satisfies exactly these constraints. In Penrose's setup, a spin network represents a quantum state of the geometry of space. To justify this interpretation he did a lot of computations using a special rule for computing a number from any spin network, which is now called the Penrose evaluation or chromatic evaluation . In week22 I said how this works when all the edges have spin 1, and described how this case is related to the four-color theorem. The general case isn't much harder, but it's a real pain to describe without lots of pictures, so I'll just refer you to the original paper: 1) Angular momentum; an approach to combinatorial space time, by Roger Penrose, in Quantum Theory and Beyond, ed. T. Bastin, Cambridge University Press, Cambridge, 1971. It's easier to explain the *physical meaning* of the Penrose evaluation. Basically, the idea is this. In classical general relativity, space is described by a 3-dimensional manifold with a Riemannian metric: a recipe for measuring distances and angles. In the spin network approach to quantum gravity, the geometry of space is instead described as a superposition of spin network states . In other words, spin networks form a basis of the Hilbert space of states of quantum gravity, so we can write any state Psi as Psi = Sum c_i psi_i where psi_i ranges over all spin networks and the coefficients c_i are complex numbers. The simplest state is the one corresponding to good old flat Euclidean space. In this state, each coefficient c_i is just the Penrose evaluation of the corresponding spin network psi_i. Actually, this interpretation wasn't fully understood until later, when Rovelli and Smolin showed how spin networks arise naturally in the so-called loop representation of quantum gravity. They also came up with a clearer picture of the way a spin network state corresponds to a possible geometry of space. The basic picture is that spin network edges represent flux tubes of area: an edge labelled with spin j contributes an area proportional to sqrt(j(j+1)) to any surface it pierces. The cool thing is that Rovelli and Smolin didn't postulate this, they *derived* it. Remember, in quantum theory, observables are given by operators on the Hilbert space of states of the physical system in question. You typically get these by quantizing the formulas for the corresponding classical observables. So Rovelli and Smolin took the usual formula for the area of a surface in a 3-dimensional manifold with a Riemannian metric and quantized it. Applying this operator to a spin network state, they found the picture I just described: the area of a surface is a sum of terms proportional to sqrt(j(j+1)), one for each spin network edge poking through it. Of course, I'm oversimplifying both the physics and the history here. The tale of spin networks and loop quantum gravity is rather long. I've discussed it already in week55 and week99 , but only sketchily. If you want more details, try: 2) Carlo Rovelli, Loop quantum gravity, preprint available as gr-qc/9710008, also available as a webpage on Living Reviews in Relativity at
http://www.livingreviews.org/Articles/Volume1/1998-1rovelli/ The abstract gives a taste of what it's all about: The problem of finding the quantum theory of the gravitational field, and thus understanding what is quantum spacetime, is still open. One of the most active of the current approaches is loop quantum gravity. Loop quantum gravity is a mathematically well-defined, non-perturbative and background independent quantization of general relativity, with its conventional matter couplings. The research in loop quantum gravity forms today a vast area, ranging from mathematical foundations to physical applications. Among the most significant results obtained are: (i) The computation of the physical spectra of geometrical quantities such as area and volume; which yields quantitative predictions on Planck-scale physics. (ii) A derivation of the Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale discreteness. This discreteness emerges naturally from the quantum theory and provides a mathematically well-defined realization of Wheeler's intuition of a spacetime foam . Longstanding open problems within the approach (lack of a scalar product, overcompleteness of the loop basis, implementation of reality conditions) have been fully solved. The weak part of the approach is the treatment of the dynamics: at present there exist several proposals, which are intensely debated. Here, I provide a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature. For a nice picture of Rovelli standing in front of some spin networks, check out: 3) Carlo Rovelli's homepage,
http://www.phyast.pitt.edu/~rovelli/ which also has _link_s to many of his papers. You'll note from this abstract that the biggest
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