John C. Baez

An Introduction to spin foam models of quantum gravity and BF theory

In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of ‘spin foam’ is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a ‘spin foam model’ we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory.In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of ‘spin foam’ is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labeled by represen- tations and vertices labeled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators. In a ‘spin foam model’ we describe states as linear combina- tions of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory.

Higher dimensional algebra and topological quantum field theory

The study of topological quantum field theories increasingly relies upon concepts from higher‐dimensional algebra such as n‐categories and n‐vector spaces. We review progress towards a definition of n‐category suited for this purpose, and outline a program in which n‐dimensional topological quantum field theories (TQFTs) are to be described as n‐category representations. First we describe a ‘‘suspension’’ operation on n‐categories, and hypothesize that the k‐fold suspension of a weak n‐category stabilizes for k≥n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n‐dimensional unitary extended TQFTs as weak n‐functors from the ‘‘free stable weak n‐category with duals on one object’’ to the n‐category of ‘‘n‐Hilbert spaces.’’ We conclude by describing n‐categorical generalizations of deformation quantization and the quantum double construction.

Introduction to Algebraic and Constructive Quantum Field Theory

The authors present a rigorous treatment of the first principles of the algebraic and analytic core of quantum field theory. Their aim is to correlate modern mathematical theory with the explanation of the observed process of particle production and of particle-wave duality that heuristic quantum field theory provides. Many topics are treated here in book form for the first time, from the origins of complex structures to the quantization of tachyons and domains of dependence for quantized wave equations. This work begins with a comprehensive analysis, in a universal format, of the structure and characterization of free fields, which is illustrated by applications to specific fields. Nonlinear local functions of both free fields (or Wick products) and interacting fields are established mathematically in a way that is consistent with the basic physical constraints and practice. Among other topics discussed are functional integration, Fourier transforms in Hilbert space, and implementability of canonical transformations. The authors address readers interested in fundamental mathematical physics and who have at least the training of an entering graduate student. A series of lexicons connects the mathematical development with the underlying physical motivation or interpretation. The examples and problems illustrate the theory and relate it to the scientific literature.Originally published in 1992.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Spin Networks in Gauge Theory

Abstract Given a real-analytic manifoldM, a compact connected Lie groupGand a principalG-bundleP→M, there is a, canonical “generalized measure” on the space A / G of smooth connections onPmodulo gauge transformations. This allows one to define a Hilbert spaceL2( A / G ). Here we construct a set of vectors spanningL2( A / G ). These vectors are described in terms of “spin networks”: graphsφembedded inM, with oriented edges labelled by irreducible unitary representations ofGand with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges. We also describe an orthonormal basis of spin network states associated to any fixed graphφ. We conclude with a discussion of spin networks in the loop representation of quantum gravity and give a category-theoretic interpretation of the spin network states.

Physics, Topology, Logic and Computation: A Rosetta Stone

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology. Namely, a linear operator behaves very much like a “cobordism”: a manifold representing spacetime, going between two manifolds representing space. This led to a burst of work on topological quantum field theory and “quantum topology”. But this was just the beginning: similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of “closed symmetric monoidal category”. We assume no prior knowledge of category theory, proof theory or computer science.

Generalized measures in gauge theory

LetP →M be a principalG-bundle. We construct well-defined analogs of Lebesgue measure on the spaceA of connections onP and Haar measure on the groupG of gauge transformations. More precisely, we define algebras of ‘cylinder functions’ on the spacesA,G, andA/G, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures onA,G, andA/G in terms of graphs embedded inM. We use this characterization to construct generalized measures onA andG whenG is compact. The ‘uniform’ generalized measure onA is invariant under the group of automorphisms ofP. It projects down to the generalized measure onA/G considered by Ashtekar and Lewandowski in the caseG = SU(n). The ‘generalized Haar measure’ onG is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure onA against generalized Haar measure gives aG-invariant generalized measure onA.

Link invariants of finite type and perturbation theory

The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra Vx containing elements gi satisfying the usual braid group relations and elements ai satisfying gi−ginfisup-1=εai, where ε is a formal variable that may be regarded as measuring the failure of ginfisup2to equal 1. Topologically, the elements ai signify intersections. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on Vx. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphisminvariant perturbation theory for quantum gravity in the loop representation.

An invitation to higher gauge theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.

From Finite Sets to Feynman Diagrams

Prediction is hard, especially when it comes to the future, but barring some unforeseen catastrophe, we can expect the amount of mathematics produced in the 21st century to dwarf that of all the centuries that came before. By the very nature of its subject matter, mathematics is capable of limitless expansion. Thanks to rapid improvements in technology, our computational power is in a phase of exponential growth. Even if this growth slows, we have barely begun to exploit our new abilities. Thus the interesting question is not whether the 21st century will see an unprecedented explosion of new mathematics. It is whether anyone will ever understand more than the tiniest fraction of this new mathematics — or even the mathematics we already have.

The Algebra of Grand Unified Theories

The Standard Model is the best tested and most widely accepted theory of elementary particles we have today. It may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three ‘grand unied theories’: theories that unify forces and particles by extending the Standard Model symmetry group U(1) SU(2) SU(3) to a larger group. These three are Georgi and Glashow’s SU(5) theory, Georgi’s theory based on the group Spin(10), and the Pati{Salam model based on the group SU(2) SU(2) SU(4). In this expository account for mathematicians, we explain only the portion of these theories that involves nite-dimensional group representations. This allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve.