J. Daniel Christensen

Spin foam models of Riemannian quantum gravity

Using numerical calculations, we compare three versions of the Barrett–Crane model of four-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes given by Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spin-zero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model.

Phantom maps and homology theories

We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X, Y) of phantom maps X → Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X, HB). We show that any composite of two phantom maps is zero, and use this to reduce Margoliss axiomatisation conjecture to an extension problem. We show that a certain functor J → A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.

Numerical evidence of regularized correlations in spin foam gravity

We report on the numerical analysis of the area correlations in spin foam gravity on a single 4-simplex considered by Rovelli [C. Rovelli, Phys. Rev. Lett. 97 (2006) 151301]. We compare the asymptotics and confirm the inverse squared distance leading behaviour at large scales. This supports the recent advances on testing the semiclassical limit of the theory. Furthermore, we show that the microscopic discreteness of the theory dynamically suppresses and regularizes the correlations at the Planck scale.

Failure of Brown representability in derived categories

Abstract Let T be a triangulated category with coproducts, T c ⊂ T the full subcategory of compact objects in T . If T is the homotopy category of spectra, Adams (Topology 10 (1971) 185–198), proved the following: All homological functors { T c } op → A b are the restrictions of representable functors on T , and all natural transformations are the restrictions of morphisms in T . It has been something of a mystery, to what extent this generalises to other triangulated categories. In Neeman (Topology 36 (1997) 619–645), it was proved that Adams’ theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T . A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T ? In a recent paper, Beligiannis (Relative homological and purity in triangulated categories, 1999, preprint) made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T =D(R) of rings, and homological functors { T c } op → A b which are not restrictions of representables.

Duality and Pro-Spectra

Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spec- tra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the oppo- site of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier- Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocaliza- tion model structures generalizing known results for cofibrantly generated model categories. AMS Classification 55P42; 55P25, 18G55, 55U35, 55Q55

Dual computations of non-Abelian Yang-Mills theories on the lattice

In the past several decades there have been a number of proposals for computing with dual forms of non-Abelian Yang-Mills theories on the lattice. Motivated by the gauge-invariant, geometric picture offered by dual models and successful applications of duality in the U(1) case, we revisit the question of whether it is practical to perform numerical computation using non-Abelian dual models. Specifically, we consider three-dimensional SU(2) pure Yang-Mills as an accessible yet nontrivial case in which the gauge group is non-Abelian. Using methods developed recently in the context of spin foam quantum gravity, we derive an algorithm for efficiently computing the dual amplitude and describe Metropolis moves for sampling the dual ensemble. We relate our algorithms to prior work in non-Abelian dual computations of Hari Dass and his collaborators, addressing several problems that have been left open. We report results of spin expectation value computations over a range of lattice sizes and couplings that are in agreement with our conventional lattice computations. We conclude with an outlook on further development of dual methods and their application to problems of current interest.

Positivity of spin foam amplitudes

The amplitude for a spin foam in the Barrett–Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. As a corollary, all open spin foams going between a fixed pair of spin networks have real amplitudes of the same sign. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian Barrett–Crane model, as in statistical mechanics, even though this theory is based on a real-time (eiS) rather than imaginary-time e−S path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or half-integer. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative, which would imply similar results for the Lorentzian Barrett–Crane model.

Causal sites as quantum geometry

We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural “tangent 2-bundle,” analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well.

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C2 as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.

An efficient algorithm for the Riemannian 10j symbols

The 10j symbol is a spin network that appears in the partition function for the Barrett–Crane model of Riemannian quantum gravity. Elementary methods of calculating the 10j symbol require (j9) or more operations and (j2) or more space, where j is the average spin. We present an algorithm that computes the 10j symbol using (j5) operations and (j2) space, and a variant that uses (j6) operations and a constant amount of space. An implementation has been made available on the web.