DeFacio
 
Driver
 
small product photo
  Jefferies
Elementary Introduction to the Feynman Operator Calculus, and Unification of Hyperbolic and Parabolic Evolution Equations
 

Brian DeFacio
Department of Physics and
Astronomy, University of Missouri
Columbia

Title

Abstract

Wars have been a  common activity among our species for longer than written records ; and even beyond oral traditions.While consulting  for the LMI Corp. in McLean , VA  for the US Navy it became clear to me that the Feynman Functional gave a formulation of  the matters our panel was discussing. Some of these issues will be presented to this workshop.
Geometric Path Integrals
Feynman Functionals  for old problems: Understanding some complex systems in modern warfare
 

Bruce Driver
Department of Mathematics,
University of California, San
Diego
Title

Abstract

 

Tepper Gill
Department of Electrical
and Computer Engineering
and Department of Mathematics,
Howard University
Title

Abstract

I In this talk, we present an elementary introduction to the Feynman operator calculus by directly extending the notion of strong continuity to the time-ordered setting. We then provide generalizations of a few generation theorems (including the Hille Yosida theorem) and use these results to unify the two distinct approaches to existence and uniqueness theorems for general hyperbolic and parabolic (time-dependent) evolution equations. We impose no domain restrictions and only assume strong continuity for the given family of operator-valued functions.
  Brian Jefferies
School of Mathematics,
University of New South
Wales

Title
Feynman's Operational Calculus with Brownian Time-Ordering

Abstract
The idea of replacing a time-ordering measure in Feynman's operational calculus by Brownian motion is quite natural and has been considered by G. W. Johnson and G. Kallianpur, who use the term "stochastic Dyson series". Solutions to the Schrodinger equation with a stochastic perturbation are represented as a stochastic Dyson series. A similar representation of the solutions of linear stochastic partial differential equations seems to be closely related to problems in harmonic analysis concerning maximal regularity of operators acting in a Banach space.
 
 
kallianpur
 
Kauffman
 
Lapidus
 
Rajeev
Sengupta

Gopinath Kallianpur
Department of Statistics,
University of North Carolina,
Chapel Hill
Titl
e

A Stochastic Feynman Integral
  Louis Kauffman
Department of Mathematics,
Statistics,
and Computer Science
University of Illinois Chicago
Title
Feynman Integrals and the Theory of Knots

Abstract
Witten showed how to formulate the Jones polynomial and its generalizations in terms of functional integrals and quantum field theory. This talk will review the heuristics of these constructions and how they are related to fully rigorous combinatorial models for the topological invariants. We then raise the question of new and as yet undiscovered physical interpretations for the categorifications of these invariants, particularly for Khovanov homology that generalizes the bracket polynomial
state summation for the Jones polynomial.
 

Michel Lapidus
Department of Mathematics,
University of California,
Riverside

Title

Abstract

In this talk, we will discuss some of the analogies  and differences between the Feynman integral approach to quantum physics and the Feynman  operational calculus approach. In particular, by using perturbation expansions, one can see that the Feynman integral approach  naturally leads to the basic heuristic  rules of the operational calculus.  After having briefly recalled  the heuristic underlying the two approaches, we will show how, in turn,  the latter can lead to a precise definition (under suitable hypotheses) of the former. In fact, under suitable assumptions, a mathematical realization of the  operational calculus can also be viewed as an appropriate extension of and substitute for the Feynman integral in situations when the ‘integral’ itself is difficult to define directly. Conversely, the latter is a global object whose definition, in principle, does not depend on the existence of  any particular perturbation expansion. Much of the usefulness of the heuristic Feynman integral in contemporary mathematics and physics relies on the tension and cross-fertilization between these two apparently very different settings and approaches.
From the Feynman Integral to Feynman’s Operational Calculus, and Back
  Sarada Rajeev
Department of Physics
and Astronomy,
University of Rochester
Quantization of Contact Geometry and Quantum Thermodynamics

Title

Abstract

In classical thermodynamics, like in classical mechanics, there is a notion of canonically conjugate observables; pressure is conjugate to volume, temperature to entropy, magnetic field to magnetization etc. I will argue that if quantum effects are included in thermodynamics, the observables do not commute and even satisfy a kind of thermodynamic uncertainty principle. This algebra of thermodynamic observables can be determined in most cases of physical interest by analogy to the passage from classical mechanics to quantum mechanics. I will also review the geometric formulation of classical mechanics in terms of contact geometry ( the work of J. W. Gibbs). This is then generalized to the quantum theory to get a non-commutative contact geometry. A reformulation in terms of path integrals will be discussed briefly as well.
Ambar Sengupta
Department of Mathematics,
Louisiana State University
Title

Abstract
Two-dimensional Yang-Mills gauge theory and three dimensional Chern-Simons theory have produced mathematical problems ranging from probability theory to topology of the moduli space of flat connections. This talk will present an overview of some of these mathematical problems and results.
Functional Integrals in Low-Dimensional Gauge Theories