October 16, 1999

This Week's Finds in Mathematical Physics (Week 140)

John Baez

Let's start with something fun: biographies!

1) Norman Macrae, John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence and Much More, American Mathematical Society, Providence, Rhode Island, 1999.

2) Steve Batterson, Stephen Smale: The Mathematician Who Broke the Dimension Barrier, American Mathematical Society, Providence, Rhode Island, 2000.

Von Neumann might be my candidate for the best mathematical physicist of the 20th century. His work ranged from the ultra-pure to the ultra-applied. At one end: his work on axiomatic set theory. At the other: designing and building some of the first computers to help design the hydrogen bomb - which was so applied, it got him in trouble at the Institute for Advanced Studies! But there's so much stuff in between: the mathematical foundations of quantum mechanics (von Neumann algebras, the Stone-Von Neumann theorem and so on), ergodic theory, his work on Hilbert's fifth problem, the Manhattan project, game theory, the theory of self-reproducing cellular automata.... You may or may not like him, but you can't help being awed. Hans Bethe, no dope himself, said of von Neumann that "I always thought his brain indicated that he belonged to a new species, an evolution beyond man". The mathematician Polya said "Johnny was the only student I was ever afraid of." Definitely an interesting guy.

While von Neumann is one of those titans that dominated the first half of the 20th century, Smale is more typical of the latter half - he protested the Vietnam war, and now he even has his own web page!

3) Stephen Smale's web page, http://www.math.berkeley.edu/~smale/

He won the Fields medal in 1966 for his work on differential topology. Some of his work is what you might call "pure": figuring out how to turn a sphere inside out without any crinkles, proving the Poincare conjecture in dimensions 5 and above, stuff like that. But a lot of it concerns dynamical systems: cooking up strange attactors using the horseshoe map, proving that structural stability is not generic, and so on - long before the recent hype about chaos theory began! More recently he's also been working on economics, game theory, and the relation of computational complexity to algebraic geometry.

Now for some papers on spin networks and spin foams:

4) Roberto De Pietri, Laurent Freidel, Kirill Krasnov, and Carlo Rovelli, Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, preprint available as hep-th/9907154.

The Barrett-Crane model is a very interesting theory of quantum gravity. I've described it already in "week113", "week120" and "week128", so I won't go into much detail about it - I'll just plunge in....

The original Barrett-Crane model involved a fixed triangulation of spacetime. One can also cook up versions where you sum over triangulations. In some ways the most natural is to sum over all ways of taking a bunch of 4-simplices and gluing them face-to-face until no faces are left free. Some of these ways give you manifolds; others don't. In this paper, the authors show that this "sum over triangulations" version of the Barrett-Crane model can be thought of as a quantum field theory on a product of 4 copies of the 3-sphere. Weird, huh?

But it's actually not so weird. The space of complex functions on the (n-1)-sphere is naturally a representation of SO(n). But there's another way to think of this representation. Consider an triangle in Rn. We can associate vectors to two of its edges, say v and w, and form the wedge product of these vectors to get a bivector v ^ w. This bivector describes the area element associated to the triangle. If we pick an orientation for the triangle, this bivector is uniquely determined. Now, a bivector of the form v ^ w is called "simple". The space of simple bivectors is naturally a Poisson manifold - i.e., we can define Poisson brackets of functions on this space - so we can think of it as a "classical phase space". Using geometric quantization, we can quantize this classical phase space and get a Hilbert space. Since rotations act on the classical phase space, they act on this Hilbert space, so it becomes a representation of SO(n). And this representation is isomorphic to the space of complex functions on the (n-1)-sphere!

Thus, we can think of a complex function on the (n-1)-sphere as a "quantum triangle" in Rn, as long as we really just care about the area element associated to the triangle. One can develop this analogy in detail and make it really precise. In particular, one can describe a "quantum tetrahedron" in Rn as a collection of 4 quantum triangles satisfying some constraints that say the fit together into a tetrahedron. These quantum tetrahedra act almost like ordinary tetrahedra when they are large, but when the areas of their faces becomes small compared to the square of the Planck length, the uncertainty principle becomes important: you can't simultaneously know everything about their geometry with perfect precision.

Let me digress for a minute and sketch the history of this stuff. The quantum tetrahedron in 3 dimensions was invented by Barbieri - see "week110". It quickly became important in the study of spin foam models. Then Barrett and I systematically worked out how to construct the quantum tetrahedron in 3 and 4 dimensions using geometric quantization - see "week134". Subsequently, Freidel and Krasnov figured out how to generalize this stuff to higher dimensions:

5) Laurent Freidel, Kirill Krasnov and Raymond Puzio, BF description of higher-dimensional gravity, preprint available as hep-th/9901069.

6) Laurent Freidel and Kirill Krasnov, Simple spin networks as Feynman graphs, preprint available as hep-th/9903192.

So much for history - now back to business. So far I've told you that the state of a "quantum triangle" in 4 dimensions is given by a complex function on the 3-sphere. And I've told you that a "quantum tetrahedron" is a collection of 4 quantum triangles satisfying some constraints. More precisely, let

H = L2(S3)

be the Hilbert space for a quantum triangle in 4 dimensions. Then the Hilbert space for a quantum tetrahedron is a certain subspace T of H x H x H x H, where "x" denotes the tensor product of Hilbert spaces. Concretely, we can think of states in T as complex functions on the product of 4 copies of S3. These complex functions need to satisfy some constraints, but let's not worry about those....

Now let's "second quantize" the Hilbert space T. This is physics jargon for making a Hilbert space out of the algebra of polynomials on T - usually called the "Fock space" on T. As usual, there are two pictures of states in this Fock space: the "field" picture and the "particle" picture. On the one hand, they are states of a quantum field theory on the product of 4 copies of S3. But on the other hand, they are states of an arbitrary collection of quantum tetrahedra in 4 dimensions. In other words, we've got ourselves a quantum field theory whose "elementary particles" are quantum tetrahedra!

The idea of the de Pietri-Freidel-Krasnov-Rovelli paper is to play these two pictures off each other and develop a new way of thinking about the Barrett-Crane model. The main thing these guys do is write down a Lagrangian with some nice properties. Throughout quantum field theory, one of the big ideas is to start with a Lagrangian and use it to compute the amplitudes of Feynman diagrams. A Feynman diagram is a graph with edges corresponding to "particles" and vertices corresponding to "interactions" where a bunch of particles turns into another bunch of particles.

But in the present context, the so-called "particles" are really quantum tetrahedra! Thus the trick is to write down a Lagrangian giving Feynman diagrams with 5-valent vertices. If you do it right, these 5-valent vertices correspond exactly to ways that 5 quantum tetrahedra can fit together as the 5 faces of a 4-simplex! Let's call such a thing a "quantum 4-simplex". Then your Feynman diagrams correspond exactly to ways of gluing together a bunch of quantum 4-simplices face-to-face. Better yet, if you set things up right, the amplitude for such a Feynman diagram exactly matches the amplitude that you'd compute for a triangulated manifold using the Barrett-Crane model!

In short, what we've got here is a quantum field theory whose Feynman diagrams describe "quantum geometries of spacetime" - where spacetime is not just a fixed triangulated manifold, but any possible way of gluing together a bunch of 4-simplices face-to-face.

Sounds great, eh? So what are the problems? I'll just list three. First, we don't know that the "sum over Feynman diagrams" converges in this theory. In fact, it probably does not - but there are probably ways to deal with this. Second, the model is Riemannian rather than Lorentzian: we are using the rotation group when we should be using the Lorentz group. Luckily this is addressed in a new paper by Barrett and Crane. Third, we aren't very good at computing things with this sort of model - short of massive computer simulations, it's tough to see what it actually says about physics. In my opinion this is the most serious problem: we should either start doing computer simulations of spin foam models, or develop new analytical techniques for handling them - or both!

Now, this "new paper by Barrett and Crane" is actually not brand new. It's a revised version of something they'd already put on the net:

7) John Barrett and Louis Crane, A Lorentzian signature model for quantum general relativity, preprint available as gr-qc/9904025.

However, it's much improved. When I went up to Nottingham at the end of the summer, I had Barrett explain it to me. I learned all sorts of cool stuff about representations of the Lorentz group. Unfortunately, I don't now have the energy to explain all that stuff. I'll just say this: everything I said above generalizes to the Lorentzian case. The main difference is that we use the 3-dimensional hyperboloid

H3 = {t2 - x2 - y2 - z2 = 1}

wherever we'd been using the 3-sphere

S3 = {t2 + x2 + y2 + z2 = 1}

It's sort of obvious in retrospect, but it's nice that it works out so neatly!

Okay, here are some more papers on spin networks and spin foams. Since I'm getting lazy, I'll just quote the abstracts:

8) Sameer Gupta, Causality in spin foam models, preprint available as gr-qc/9908018.

We compute Teitelboim's causal propagator in the context of canonical loop quantum gravity. For the Lorentzian signature, we find that the resultant power series can be expressed as a sum over branched, colored two-surfaces with an intrinsic causal structure. This leads us to define a general structure which we call a "causal spin foam". We also demonstrate that the causal evolution models for spin networks fall in the general class of causal spin foams.

9) Matthias Arnsdorf and Sameer Gupta, Loop quantum gravity on non-compact spaces, preprint available as gr-qc/9909053.

We present a general procedure for constructing new Hilbert spaces for loop quantum gravity on non-compact spatial manifolds. Given any fixed background state representing a non-compact spatial geometry, we use the Gel'fand-Naimark-Segal construction to obtain a representation of the algebra of observables. The resulting Hilbert space can be interpreted as describing fluctuation of compact support around this background state. We also give an example of a state which approximates classical flat space and can be used as a background state for our construction.

10) Seth A. Major, Quasilocal energy for spin-net gravity, preprint available as gr-qc/9906052.

The Hamiltonian of a gravitational system defined in a region with boundary is quantized. The classical Hamiltonian, and starting point for the regularization, is required by functional differentiablity of the Hamiltonian constraint. The boundary term is the quasilocal energy of the system and becomes the ADM mass in asymptopia. The quantization is carried out within the framework of canonical quantization using spin networks. The result is a gauge invariant, well-defined operator on the Hilbert space induced from the state space on the whole spatial manifold. The spectrum is computed. An alternate form of the operator, with the correct naive classical limit, but requiring a restriction on the Hilbert space, is also defined. Comparison with earlier work and several consequences are briefly explored.

11) C. Di Bartolo, R. Gambini, J. Griego, J. Pullin, Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants, preprint available as gr-qc/9909063.

We present a quantization of the Hamiltonian and diffeomorphism constraint of canonical quantum gravity in the spin network representation. The novelty consists in considering a space of wavefunctions based on the Vassiliev knot invariants. The constraints are finite, well defined, and reproduce at the level of quantum commutators the Poisson algebra of constraints of the classical theory. A similar construction can be carried out in 2+1 dimensions leading to the correct quantum theory.

12) John Baez, Spin foam perturbation theory, preprint available as gr-qc/9910050 or at http://math.ucr.edu/home/baez/foam3.ps

We study perturbation theory for spin foam models on triangulated manifolds. Starting with any model of this sort, we consider an arbitrary perturbation of the vertex amplitudes, and write the evolution operators of the perturbed model as convergent power series in the coupling constant governing the perturbation. The terms in the power series can be efficiently computed when the unperturbed model is a topological quantum field theory. Moreover, in this case we can explicitly sum the whole power series in the limit where the number of top-dimensional simplices goes to infinity while the coupling constant is suitably renormalized. This `dilute gas limit' gives spin foam models that are triangulation-independent but not topological quantum field theories. However, we show that models of this sort are rather trivial except in dimension 2.

© 1999 John Baez