The Bern-Kosower Rules and Their Relation to Quantum Field Theory

For forty years, Feynman diagrams have been the method of choice for computing scattering amplitudes in quantum field theory. However, in gauge theories they are troublesome; the rules break the gauge symmetry, and only in the final answer is the symmetry restored. The Bern-Kosower rules are a new technique for calculating scattering amplitudes in gauge theories. Their applicability is as yet limited, but they are more efficient than Feynman diagrams. What is especially unusual about this method is that it was derived not from field theory but from string theory. When a physical process in string theory has energy small relative to the Planck mass, the theory behaves as a simple non-abelian gauge theory. Consequently, one may compute field theory scattering amplitudes by taking the low energy limit of the corresponding string amplitudes. Bern and Kosower studied gluon scattering in string theory, and found that the low energy limit can be calculated using a set of simple rules bearing little resemblance to Feynman rules and exceeding them in efficiency. The power of the new rules has recently been demonstrated in a computation never previously performed. It is embarrassing that theories as well studied as gauge theories should harbor a mystery of this magnitude. On the one hand, the Bern-Kosower rules exist; on the other, no derivation from gauge theory of these rules has been found. In this thesis, I have attempted to bridge this gap. My main result is that the Bern-Kosower rules for quantum electrodynamics, and portions of the rules for quantum chromodynamics, may be derived from first-quantized field theory. After an introduction to first-quantized scalar field theories, I present a derivation of the Bern -Kosower rules requiring no prior knowledge of string theory. Next, I present my main result, that one-loop effective actions of particles in background gauge fields can be used to derive the starting point for the Bern-Kosower rules and many of its other features. In the last part of the thesis I explore the special relationship between gauge invariance and the Bern-Kosower rules.