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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)., Comment: 51 pages, 33 figures (high definition versions available upon request), partial results were presented at 1st Winter Workshop on Non-Perturbative Quantum Field Theory, http://wwnpqft.inln.cnrs.fr/, and at Miami 2010, http://cgc.physics.miami.edu/ . v2: typos fixed, references added

Symmetry Breaking is used as an "underlying principle", bringing different features of QFT to the foreground. However, the understanding of Symmetry Breaking that is used here is quite different from what is done in the mainstream: Symmetry Breaking is understood as the solution set of a given QFT, its vacuum manifold, or, more modernly, its Moduli Space. Distinct solutions correspond to different sectors, phases, of the theory, which are nothing but distinct foliations of the vacuum manifold, or points in the Moduli Space (for all possible values of the parameters of the theory). Under this framework, three different problems will be attacked: "Mollifying QFT", "Topological Transitions and Geometric Langlands Duality" and "Three-dimensional Gravity and its Phase Transitions". The first makes use of the Moduli Space of the theory in order to construct an appropriate mollification of it, rendering it viable to simulate a QFT in Lorentzian spaces, tackling the "sign problem" heads-on. The connections with Lee-Yang zeros and Stokes Phenomena will be made clear. The second will show that each different phase has its own topology which can be used as Superselection Rule; moreover, the Euler Characteristic of each phase gives it quantization condition. The mechanism via which several dualities work will also be elucidated. The last one will generalize a 0-dimensional QFT, via dimensional construction through its D-Module, and conjecture several connections between the Lie-algebra-valued extension of the Airy function and the recent Partition Function found for three-dimensional gravity with a negative cosmological constant. These three problems, put together, should exhibit a solid and robust framework for treating QFT under this new paradigm., Comment: 103 pages, 56 figures, author's Ph.D. thesis

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 figures

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/

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 streamlined

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 2003

Two new approaches to numerical QFT are presented., Comment: Lattice2002(theoretical), 3 pages

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., Comment: 3 pages, 9 figures, Lattice2002(theoretical)