Theta Vacua and Boundary Conditions of the Schwinger Dyson Equations

Quantum field theories and Matrix models have a far richer solution set than is normally considered, due to the many boundary conditions which must be set to specify a solution of the Schwinger-Dyson equations. The complete set of solutions of these equations is obtained by generalizing the path integral to include sums over various inequivalent contours of integration in the complex plane. We discuss the importance of these exotic solutions. While naively the complex contours seem perverse, they are relevant to the study of theta vacua and critical phenomena. Furthermore, it can be shown that within certain phases of many theories, the physical vacuum does not correspond to an integration over a real contour. We discuss the solution set for the special case of one component zero dimensional scalar field theories, and make remarks about matrix models and higher dimensional field theories that will be developed in more detail elsewhere. Even the zero dimensional examples have much structure, and show some analogues of phenomena which are usually attributed to the effects of taking a thermodynamic limit.Comment: 32 pages, Latex, 3 figure

Source Driven Solutions of Quantum Field Theories

I outline the basics of numerical solution of quantum field theory using the Source Galerkin method. This method is based on an analysis of the functional differential equations of a field theory in the presence of external sources. This approach is particularly powerful because it naturally allows the inclusion of symmetry information and avoids the fermion determinant problem associated with Monte Carlo calculations. The technique is also suitable for continuum calculations.Comment: 9 pages,Latex,talk given at 3rd AUP Workshop on QCD: Collisions, Confinement and Chaos, Paris, France, 3-8 June 199

Numerical Field Theory on the Continuum

An approach to calculating approximate solutions to the continuum Schwinger-Dyson equations is outlined, with examples for \phi^4 in D=1. This approach is based on the source Galerkin methods developed by Garcia, Guralnik and Lawson. Numerical issues and opportunities for future calculations are also discussed briefly.Comment: 13 pages, 8 figures, LaTeX 2

Where Have All the Goldstone Bosons Gone?

According to a commonly held view of spontaneously broken symmetry in gauge theories, troublesome Nambu-Goldstone bosons are effectively eliminated by turning into longitudinal modes of a massive vector meson. This note shows that this is not in fact a consistent view of the role of Nambu-Goldstone bosons in such theories. These particles necessarily appear as gauge excitations whenever they are formulated in a manifestly covariant gauge. The radiation gauge provides therefore the dual advantage of circumventing the Goldstone theorem and making evident the disappearance of these particles from the physical spectrum.Comment: 4 pages; typo correcte

From Symmetry Breaking to Topology Change I

By means of an analogy with Classical Mechanics and Geometrical Optics, we are able to reduce Lagrangians to a kinetic term only. This form enables us to examine the extended solution set of field theories by finding the geodesics of this kinetic term's metric. This new geometrical standpoint sheds light on some foundational issues of QFT and brings to the forefront core aspects of field theory.Comment: 14 pages, 18 figures; fonts corrected, for a higher definition version, check http://chep.het.brown.edu

Phase Transitions and Moduli Space Topology

By means of an appropriate re-scaling of the metric in a Lagrangian, we are able to reduce it to a kinetic term only. This form enables us to examine the extended complexified solution set (complex moduli space) of field theories by finding all possible geodesics of this metric. This new geometrical standpoint sheds light on some foundational issues of QFT and brings to the forefront non-perturbative core aspects of field theory. In this process, we show that different phases of the theory are topologically inequivalent, i.e., their moduli space has distinct topologies. Moreover, the different phases are related by "duality transformations", which are established by the modular structure of the theory. In conclusion, after the topological structure is elucidated, it is possible to use the Euler Characteristic in order to topologically quantize the theory, in resonance with the content of the Atiyah-Singer Index theorem.Comment: 24 pages, 21 figure

New Numerical Methods for Iterative or Perturbative Solution of Quantum Field Theory

A new computational idea for continuum quantum field theories is outlined. This approach is based on the lattice source Galerkin methods developed by Garcia, Guralnik and Lawson. The method has many promising features including treating fermions on a relatively symmetric footing with bosons. As a spinoff of the technology developed for ``exact'' solutions, the numerical methods used have a special case application to perturbation theory. We are in the process of developing an entirely numerical approach to evaluating graphs to high perturbative order.Comment: 10 pages, 7 figures; for inclusion in proceedings of Fourth Workshop on Quantum Chromodynamics, American University of Paris, 1 - 6 June 199

Mollifying Quantum Field Theory or Lattice QFT in Minkowski Spacetime and Symmetry Breaking

This work develops and applies the concept of mollification in order to smooth out highly oscillatory exponentials. This idea, known for quite a while in the mathematical community (mollifiers are a means to smooth distributions), is new to numerical Quantum Field Theory. It is potentially very useful for calculating phase transitions [highly oscillatory integrands in general], for computations with imaginary chemical potentials and Lattice QFT in Minkowski spacetime.Comment: 23 pages, 35 figures; check http://chep.het.brown.edu/ for high resolution figures; referees suggestions and appendix added, references streamline

The Beginnings of Spontaneous Symmetry Breaking in Particle Physics -- Derived From My on the Spot "Intellectual Battlefield Impressions"

I summarize the development of the ideas of Spontaneous symmetry breaking in the 1960s with an outline of the Guralnik, Hagen, Kibble (GHK) paper and include comments on the relationship of this paper to those of Brout, Englert and Higgs. I include some pictures from my "physics family album which might be of some amusement value".Comment: 12 pages, 9 figures, proceedings of the DPF201

A Review Of Two Novel Numerical Methods in QFT

We outline two alternative schemes to perform numerical calculations in quantum field theory. In principle, both of these approaches are better suited to study phase structure than conventional Monte Carlo. The first method, Source Galerkin, is based on a numerical analysis of the Schwinger-Dyson equations using modern computer techniques. The nature of this approach makes dealing with fermions relatively straightforward, particularly since we can work on the continuum. Its ultimate success in non-trivial dimensions will depend on the power of a propagator expansion scheme which also greatly simplifies numerical calculation of traditional perturbation graphs. The second method extends Monte Carlo approaches by introducing a procedure to deal with rapidly oscillating integrals.Comment: 17 pages, 12 figures, This document is based on a talk given by G. Guralnik at the ``Seventh Workshop on Quantum Chromodynamics'', 6-10 January 200