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G. S. Guralnik

Publications and source records attributed to G. S. Guralnik.

18 recordsLinked to original sources
arXiv2014

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.

hep-th↗
arXiv2013

Complex Path Integrals and the Space of Theories

The Feynman Path Integral is extended in order to capture all solutions of a quantum field theory. This is done via a choice of appropriate integration cycles, parametrized by M in SL(2,C), i.e., the space of allowed integration cycles is related to certain Dp-branes and their properties, which can be further understood in terms of the "physical states" of another theory. We also look into representations of the Feynman Path Integral in terms of a Mellin-Barnes transform, bringing the singularity structure of the theory to the foreground. This implies that, as a sum over paths, we should consider more generic paths than just Brownian ones. Finally, we are able to study the Space of Theories through our examples in terms of their Quantum Phases and associated Stokes' Phenomena (wall-crossing).

hep-th↗
arXiv2008

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.

hep-lat↗
arXiv2008

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.

hep-th↗
arXiv2006

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.

hep-th↗
arXiv2003

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.

hep-lat↗
arXiv2002

Mollified Monte Carlo

Using a common technique for approximating distributions [generalized functions], we are able to use standard Monte Carlo methods to compute QFT quantities in Minkowski spacetime, under phase transitions, or when dealing with coalescing stationary points.

hep-lat↗
arXiv2002

A Test of The Source Galerkin Method

Some results of the ongoing development of our Source Galerkin (SG) nonperturbative approach to numerically solving Quantum Field theories are presented. This technique has the potential to be much faster than Monte Carlo methods. SG uses known symmetries and theoretical properties of a theory. In order to test this approach, we applied it to phi^4 theory in zero dimensions. This model has been extensively studied and has a known set of exact solutions. This allows us to broaden the understanding of various properties of the SG method and to develop techniques necessary for the successful application of this method to more sophisticated theories.

hep-lat↗
arXiv2001

Numerical Quantum Field Theory on the Continuum and a New Look at Perturbation Theory

The Source Galerkin method finds approximate solutions to the functional differential equations of field theories in the presence of external sources. While developing this process, it was recognized that approximations of the spectral representations of the Green's functions by Sinc function expansions are an extremely powerful calculative tool. Specifically, this understanding makes it not only possible to apply the Source Galerkin method to higher dimensional field theories, but also leads to a new approach to perturbation theory calculations in scalar and fermionic field theories. This report summarizes the methodologies for solving quantum field theories with the Source Galerkin method and for performing perturbation theory calculations using Sinc approximations.

hep-lat↗
arXiv1999

New Numerical Methods for Quantum Field Theories on the Continuum

The Source Galerkin Method is a new numerical technique that is being developed to solve Quantum Field Theories on the continuum. It is not based on Monte Carlo techniques and has a measure to evaluate relative errors. It promises to increase the accuracy and speed of calculations, and takes full advantage of symmetries of the theory. The application of this method to the non-linear sigma model is outlined.

hep-lat↗
arXiv1999

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.

hep-th↗
arXiv1998

Numerical Field Theory on the Continuum

An approach to calculating approximate solutions to the continuum Schwinger-Dyson equations is outlined, with examples for ϕ^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.

hep-th↗
arXiv1997

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.

hep-th↗
arXiv1995

Source Galerkin Calculations in Scalar Field Theory

In this paper, we extend previous work on scalar $ϕ^4$ theory using the Source Galerkin method. This approach is based on finding solutions $Z[J]$ to the lattice functional equations for field theories in the presence of an external source $J$. Using polynomial expansions for the generating functional $Z$, we calculate propagators and mass-gaps for a number of systems. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. The use of polynomial expansions illustrates in a clear and simple way the ideas of the Source Galerkin method. But at the same time, this choice has serious limitations. Even after exploiting symmetries, the size of calculations become prohibitive except for small systems. The calculations in this paper were made on a workstation of modest power using a fourth order polynomial expansion for lattices of size $8^2$,$4^3$,$2^4$ in $2D$, $3D$, and $4D$. In addition, we present an alternative to the Galerkin procedure that results in sparse matrices to invert.

hep-th↗
arXiv1995

New Numerical Method for Fermion Field Theory

A new deterministic, numerical method to solve fermion field theories is presented. This approach is based on finding solutions $Z[J]$ to the lattice functional equations for field theories in the presence of an external source $J$. Using Grassmann polynomial expansions for the generating functional $Z$, we calculate propagators for systems of interacting fermions. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. Because it is not based on a statistical technique, it does not have many of the difficulties often encountered when simulating fermions. Since no determinant is ever calculated, solutions to problems with dynamical fermions are handled more easily. This approach is very flexible, and can be taylored to specific problems based on convenience and computational constraints. We present simple examples to illustrate the method; more general schemes are desirable for more complicated systems.

hep-th↗
arXiv1993

A New Approach to Numerical Quantum Field Theory

In this note we present a new numerical method for solving Lattice Quantum Field Theory. This Source Galerkin Method is fundamentally different in concept and application from Monte Carlo based methods which have been the primary mode of numerical solution in Quantum Field Theory. Source Galerkin is not probabilistic and treats fermions and bosons in an equivalent manner.

hep-ph↗