Recognized as a distinct branch of mathematical physics, this book treats the subject of topological quantum field theory from the standpoint of finding invariants for knots using the pure Chern-Simons gauge theory. The Chern-Simons field theory has its beginnings in the construction of instantons in quantum field theory, but it took the work of Edward Witten and others to see the connection between it and knot theory. This connection was accomplished using the non-rigorous methods of path integrals, and these are used in this book in the initial stages to motivate the later developments. And, most importantly, the author does not hesitate to use physical arguments in place of rigorous mathematical arguments to illuminate a particular result. As an introduction, the Abelian Chern-Simons theory is considered, the (classical) action of which is metric independent and is the integral of a 3-form on a 3-manifold. Gauge invariance and general covariance are discussed as the key side-constraints for the properties of the expectation value taken over (linked) paths. In the corresponding non-Abelian theory, framings must be chosen on the knots to preserve general covariance. The goal of the book is then to calculate the expectation values of the Wilson line operators and show they define link invariants. These link invariants are polynomials, and are then shown to be associated with those defined by braid group representations on the quasi-tensor category of quasi-triangular Hopf algebras. These algebras are a "quantum" deformation of a Lie algebra, and the corresponding polynomials are called (Drinfeld) universal link polynomials. What is interesting about this connection is that with the algebraic interpretation one need not worry about doing computations in quantum field theory. This is definitely the approach taken in recent years, and mathematicians, with their insistence on rigour, are anxious to remove the subject from its quantum field theory roots. The use of quantum field theory does however have its advantages as a source of new ideas and strategies in defining link invariants, and invariants of 3-manifolds, and I believe strongly that the use of quantum field theory will lead to a resolution of the 3-dimensional Poincare conjecture. The author does not discuss this conjecture, but he does give an excellent overview of surgery on 3-manifolds via the Dehn surgery operations and some discussion of Kirby calculus. The surgery rules are then interpreted in terms of topological quantum Chern-Simons field theory. Specifically, the surgery is interpreted physically in terms of symmetries, and, via the Likorish theorem, the surgeries are thought of as being combinations of elementary surgery operations. The elementary surgery operations are then understood as twist homoeomorphisms of the complement of a tubulur neighborhood of the unknot in the standard 3-sphere. The field theory rules corresponding to the surgery operations are then formulated as symmetries in quantum field theory. The Kirby calculus then appears when the author considers the so-called "honest" surgeries, which are surgeries in which the surgery coefficients are integers. The most important part of these considerations is that one need only concentrate on Chern-Simons field theory on the standard 3-sphere.The coupling constant then only takes integer values, and hence, the author shows, the space of gauge-invariant states is then a reduced tensor algebra, which has finite dimension. The calculation of the resulting path integrals are then fairly straightforward, since one does not have to consider the difficult problem of functional integration in infinite dimensions. The case of a generic 3-manifold is handled by viewing it as the result of surgery on the standard 3-sphere. The problem in using Chern-Simons field theory in this case comes from the interpretation of the partition function, i.e. the normalization problem has to be considered since one is dealing with fields defined on different 3-manifolds (each related via surgery). But since it has been established that surgery operations are symmetry transformations, these are used to give meaning to the normalization, which, for the physicist, is nothing other than the calculation of the zero-point energy. The invariants defined by the author are not fine enough to distinguish one 3-manifold from each other, in particular, manifolds with different homotopy groups. Such invariants would be of value in resolving the 3-dimensional Poincare conjecture. A modification of the approach of topological quantum field theory, wherein genuine quantum effects, such as superposition and entanglement of states, are incorporated into the transition amplitudes between 3-manifolds, may be the way to resolve this conjecture. |