James Cook's Homepage

James Cook's Homepage

Teaching :

Interests and Current Research:
My interests generally fall under the category of theoretical or mathematical physics. Lately I have been investigating the mathematical basis for supersymmetric field theory. Hopefully, I will establish a better mathematical foundation for some of my work from last year. Ultimately, we'd like to better understand the geometry underlying the superfield construction. Superfields are functions with rather unusual domains. The domain of a superfield contains abstract numbers called Grassmans. Grassmans have the strange property that they can be non-zero and yet when you square them you get zero. The domain of a superfield is parametrized by variable constructed from infinite real or complex linear sums of products of grassmans. Probably sounds worse than it is. The reason for studying such unusual spaces is that these superspaces admit a very natural action of the super Poincaire group. The super Poincaire group is the group formed by the super Poicaire algebra which is the only known physically reasonable extension of the Poicaire algebra. In addition to the usual translations, boosts, and rotations the super Poincaire group has what are called supersymmetry transformations. These supersymmetry transformations become supertranslations when realized on a particular superspace. This particular superspace is known as N=1 rigid superspace and is one of the central objects of interest in supersymmetric field theory. Supersmooth functions from rigid superspace form representations of the super Poicaire group. As such it is easy to construct models which have supersymmetry ( invariance under the super symmetry transformations ) with the help of superfields. This invariance falls out very naturally in the construction. What is beautiful about it is that what seems simple at the level of superfields can entail very messy things at the level of usual relativistic field theory. Superfield models reduce to ordinary relativistic field models (mappings from Minkowski space as opposed to superspace) at what is known as the "component field level". Component fields are the typical fields discussed in relativistic field theory ( scalar fields, Weyl spinors, vector fields,... ), a whole ensemble are hidden within a particular superfield. Without the superfield construction one would have to find how to couple the ensemble together under the supersymmetry transformations. These couplings are somewhat messy, yet fall out almost for free within the superfield construction. Anyway, there is still much left to do in understanding what exactly a "superfield" is, especially in the context of what is called gauge theory.

(Old news)My previous research was on a gauge field theory in noncommutative Minkowski spacetime. I wanted to understand if it was physically sensible to write a supersymmetric (SUSY) field theory for which the underlying spactime is noncommutative (actually in the space I study there is no supersymmetry,but when the noncommutative deformation is set to zero we recover the usual SUSY field theory). In a nutshell, SUSY says that particles and the forces that govern them should be balanced (there should be equal numbers of bosons (force carriers) and fermions (matter) ). There is no experimental verification of SUSY at this time, but many physicsists hope it may be detected about 2010. Noncommutative geometry places constraints on space itself, so that space is no longer continuous below some certain scale. Essentially what this means is that we have done away with the point, hence Noncommutative geometry is pointless. The theory I worked out naturally extended ideas from both SUSY and Noncommutative geometry to encorporate them simultaneously. If your interested I've posted a preprint of my work below.

my paper

Last year Dr. Ronald Fulp and I worked through the theory of super Lie groups. Our paper is currently under review,

our paper

You can get some of the big ideas from my poster,

the poster

we hope to apply our theory of super Lie groups to treat Super Yang Mills in terms of supersmooth geometry. That work is currently in progress(2007).

Fun Links:
Warren Siegel's webpage. I especially enjoy his university disclaimer.
World of Mathematics
Wikipedia Math Dictionary
MacTutor History of Math Archive
Marilyn Daily's Parametric Playground (Maple Worksheet)
Homestar Runner or better yet TROGDOR! (the Burninator)
Lots and Lots of Math History

My talk on representation theory in quantum mechanics

This talk outlines the story of isospin
Many mathematical and physical details are avoided, the goal is to eulicidate the main algebraic motivation for using representation theory in physics. Along the way we learn what quarks are (naively) and what they explain (the eightfold way to begin). Mostly, we follow Greiner's symmetries in QM.

Magnetic Monopoles

Notes on relativistic electromagnetism and the Dirac monopole and the associated mathematics of principal fiber bundles, particularly Hopf bundles. Mostly finished,but contain some minor errors. Should add more on Yang-Mills theory later...( pgs 1-24 from fall 2005)
magnetic monopoles

Notes from some past and present courses

North Carolina State University

Church

Other Schools I've Attended

Notes on representation theory in quantum mechanics

My talk on representation theory in quantum mechanics

Magnetic Monopoles

Notes from some past and present courses