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 description 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 papers:

1) Roger Penrose, Angular momentum: an approach to combinatorial space-time, in Quantum Theory and Beyond, ed. T. Bastin, Cambridge U. Press, Cambridge, 1971, pp. 151-180. Also available at http://math.ucr.edu/home/baez/penrose/

Roger Penrose, Applications of negative dimensional tensors, in Combinatorial Mathematics and its Applications, ed. D. Welsh, Academic Press, New York, 1971, pp. 221-244. Also available at http://math.ucr.edu/home/baez/penrose/

Roger Penrose, On the nature of quantum geometry, in Magic Without Magic, ed. J. Klauder, Freeman, San Francisco, 1972, pp. 333-354. Also available at http://math.ucr.edu/home/baez/penrose/

Roger Penrose, Combinatorial quantum theory and quantized directions, in Advances in Twistor Theory, eds. L. Hughston and R. Ward, Pitman Advanced Publishing Program, San Francisco, 1979, pp. 301-307. Also available at http://math.ucr.edu/home/baez/penrose/

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 Ψ as

Ψ = ∑ ci ψi

where ψi ranges over all spin networks and the coefficients ci are complex numbers. The simplest state is the one corresponding to good old flat Euclidean space. In this state, each coefficient ci is just the Penrose evaluation of the corresponding spin network ψi.

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 (j(j+1))½ 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 (j(j+1))½, 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 links to many of his papers.

You'll note from this abstract that the biggest problem in loop quantum gravity is finding an adequate description of dynamics. This is partially because spin networks are better suited for describing space than spacetime. For this reason, Rovelli, Reisenberger and I have been trying to describe spacetime using "spin foams" - sort of like soap suds with all the bubbles having faces labelled by spins. Every slice of a spin foam is a spin network.

But I'm getting ahead of myself! I should note that the spin networks appearing in the loop representation are different from those Penrose considered, in two important ways.

First, they can have more than 3 edges meeting at a vertex, and the vertices must be labelled by "intertwining operators", or "intertwiners" for short. This is a concept coming from group representation theory; as described in "week109", what we've been calling "spins" are really irreducible representations of SU(2). If we orient the edges of a spin network, we should label each vertex with an intertwiner from the tensor product of representations on the "incoming" edges to the tensor product of representations labelling the "outgoing" edges. When 3 edges labelled by spins j1, j2, j3 meet at a vertex, there is at most one intertwiner

f: j1 ⊗ j2 → j3,

at least up to a scalar multiple. The conditions I wrote down - the triangle inequality and so on - are just the conditions for a nonzero intertwiner of this sort to exist. That's why Penrose didn't label his vertices with intertwiners: he considered the case where there's essentially just one way to do it! When more edges meet at a vertex, there are more intertwiners, and this extra information is physically very important. One sees this when one works out the "volume operators" in quantum gravity. Just as the spins on edges contribute area to surfaces they pierce, the intertwiners at vertices contribute volume to regions containing them!

Second, in loop quantum gravity the spin networks are embedded in some 3-dimensional manifold representing space. Penrose was being very radical and considering "abstract" spin networks as a purely combinatorial replacement for space, but in loop quantum gravity, one traditionally starts with general relativity on some fixed spacetime and quantizes that. Penrose's more radical approach may ultimately be the right one in this respect. The approach where we take classical physics and quantize it is very important, because we understand classical physics better, and we have to start somewhere. Ultimately, however, the world is quantum-mechanical, so it would be nice to have an approach to space based purely on quantum-mechanical concepts. Also, treating spin networks as fundamental seems like a better way to understand the "quantum fluctuations in topology" which I mentioned in "week109". However, right now it's probably best to hedge ones bets and work hard on both approaches.

Lately I've been very excited by a third, hybrid approach:

4) Andrea Barbieri, Quantum tetrahedra and simplicial spin networks, preprint available as gr-qc/9707010.

Barbieri considers "simplicial spin networks": spin networks living in a fixed 3-dimensional manifold chopped up into tetrahedra. He only considers spin networks dual to the triangulation, that is, spin networks having one vertex in the middle of each tetrahedron and one edge intersecting each triangular face.

In such a spin network there are 4 edges meeting at each vertex, and the vertex is labelled with an intertwiner of the form

f: j1 ⊗ j2 → j3 ⊗ j4

where j1,...,j4 are the spins on these edges. If you know about the representation theory of SU(2), you know that j1 ⊗ j2 is a direct sum of representations of spin j5, where j5 goes from |j1 - j2| up to j1 + j2 in integer steps. So we get a basis of intertwining operators:

f: j1 ⊗ j2 → j3 ⊗ j4

by picking one factoring through each representation j5:

j1 ⊗ j2 → j5 → j3 ⊗ j4


a) j1 + j2 + j5 is an integer and |j1 - j2| ≤ j5 ≤ j1 + j2

b) j3 + j4 + j5 is an integer and |j3 - j4| ≤ j5 ≤ j3 + j4.

Using this, we get a basis of simplicial spin networks by labelling all the edges and vertices by spins satisfying the above conditions. Dually, this amounts to labelling each tetrahedron and each triangle in our manifold with a spin! Let's think of it this way.

Now focus on a particular simplicial spin network and a particular tetrahedron. What do the spins j1,...,j5 say about the geometry of the tetrahedron? By what I said earlier, the spins j1,...,j4 describe the areas of the triangular faces: face number 1 has area proportional to (j1(j1+1))½, and so on. What about j5? It also describes an area. Take the tetrahedron and hold it so that faces 1 and 2 are in front, while faces 3 and 4 are in back. Viewed this way, the outline of the tetrahedron is a figure with four edges. The midpoints of these four edges are the corners of a parallelogram, and the area of this parallelogram is proportional to (j5(j5+1))½. In other words, there is an area operator corresponding to this parallelogram, and our spin network state is an eigenvector with eigenvalue proportional to (j5(j5+1))½. Finally, there is also a volume operator corresponding to the tetrahedron, whose action on our spin network state is given by a more complicated formula involving the spins j1,...,j5.

Well, that either made sense or it didn't... and I don't think either of us want to stick around to find out which! What's the bottom line, you ask? First, we're seeing how an ordinary tetrahedron is the classical limit of a "quantum tetrahedron" whose faces have quantized areas and whose volume is also quantized. Second, we're seeing how to put together a bunch of these quantum tetrahedra to form a 3-dimensional manifold equipped with a "quantum geometry" - which can dually be seen as a spin network. Third, all this stuff fits together in a truly elegant way, which suggests there is something good about it. The relationship between spin networks and tetrahedra connects the theory of spin networks with approaches to quantum gravity where one chops up space into tetrahedra - like the "Regge calculus" and "dynamical triangulations" approaches.

Next week I'll say a bit about using spin networks to study quantum black holes. Later I'll talk about dynamics and spin foams.

Meanwhile, I've been really lagging behind in describing new papers as they show up... so here are a few interesting ones:

5) C. Nash, Topology and physics - a historical essay, to appear in A History of Topology, edited by Ioan James, Elsevier-North Holland, preprint available as hep-th/9709135.

6) Luis Alvarez-Gaume and Frederic Zamora, Duality in quantum field theory (and string theory), available as hep-th/9709180.

Quoting the abstract:

"These lectures give an introduction to duality in Quantum Field Theory. We discuss the phases of gauge theories and the implications of the electric-magnetic duality transformation to describe the mechanism of confinement. We review the exact results of N=1 supersymmetric QCD and the Seiberg-Witten solution of N=2 super Yang-Mills. Some of its extensions to String Theory are also briefly discussed."

7) Richard E. Borcherds, What is a vertex algebra?, available as q-alg/9709033.

"These are the notes of an informal talk in Bonn describing how to define an analogue of vertex algebras in higher dimensions."

8) J. M. F. Labastida and Carlos Lozano, Lectures in topological quantum field theory, 62 pages in LaTeX with 5 figures in encapsulated Postscript, available as hep-th/9709192.

"In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N=2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the application of results based on duality for N=2 supersymmetric Yang-Mills theories. This involves a description of the computation of Donaldson invariants in terms of Seiberg-Witten invariants. Generalizations of Donaldson-Witten theory are reviewed, and the structure of the vacuum expectation values of their observables is analyzed in the context of duality for the simplest case."

9) Martin Markl, Simplex, associahedron, and cyclohedron, preprint available as alg-geom/9707009.

"The aim of the paper is to give an `elementary' introduction to the theory of modules over operads and discuss three prominent examples of these objects - simplex, associahedron (= the Stasheff polyhedron) and cyclohedron (= the compactification of the space of configurations of points on the circle)."

© 1997 John Baez