Current Research

Recently my interests have been in the subject of string field theory. String field theory can be understood as a generalization of ordinary quantum field theory where the "fields" are function(al)s of a path, rather than local functions on spacetime. Making a Fourier decomposition of the path, it is possible to rewrite the string field as an infinite collection of local spacetime fields, which in turn give rise to "particles"---for example the graviton---corresponding to different vibrational modes of the string.

The difference between string field theory and garden variety string theory is a little bit subtle. Roughly speaking, in string theory we integrate over all possible surfaces swept out by the string in the path integral. In string field theory, we integrate over all possible string fields in the path integral. For the purpose of calculating perturbative scattering amplitudes, these two prescriptions are equivalent. The string field theory approach, however, turns out to be much more arbitrary and complicated, at least at lower orders. For this reason, string field theory has not been very popular.

So why study string field theory at all? The advantage of string field theory is that it gives a unified framework in which it is possible, at least in principle, to address questions in string theory that go beyond perturbative scattering amplitudes. Until relatively recently, this "in principle" rarely translated to "in practice," mostly because of the logistics of calculating with the theory. However, recently it was realized that it is possible to find nontrivial solutions to (open) string field theory describing an array of open string backgrounds, in particular various types of D-brane configurations. These solutions, however, are obtained numerically; we still have no analytic control of the theory. Still, these successes have again raised hope that string field theory is the long-sought-after nonperturbative definition of string theory. Much work remains to be done.

My work has mostly focused on the structure of the algebra of string fields and the role of dynamics in string field theory. Here is a list of my papers:

Level Truncation and Rolling the Tachyon in the Lightcone Basis for Open String Field Theory
In this paper I study the decay of D-branes numerically using the initial value formulation of the previous paper.

Locality, Causality, and an Initial Value Formulation of Open String Field Theory
I wrote this paper with my advisor, David Gross. We find a way of formulating the notion of "time" in open string field theory. In particular we find an initial value formulation, discuss causality, and construct a second quantized operator formalism for the interacting theory.

A Fresh Look at Midpoint Singularities in the Algebra of String Fields
In this paper I discuss some subtle issues in understanding the algebra of interacting open strings.

Moyal Formulation of Witten's Star Product in the Fermionic Ghost Sector
In this paper I explain how the algebra of interacting strings can be understood as a "matrix algebra" in the ghost sector of the theory.


Hypercomplex Numbers

In my earlier years I was very interested in Hypercomplex numbers and their applications to physics. I am a member of the The International Clifford Algebra Society and I have a webpage there. Inspired by some of the structures of Clifford Algebra, in my early days I spent a lot of time trying to construct a mathematical calculus for nonlocal functionals of curves, surfaces, etc. To my dissapointment, the formalism I developed turned out to be more or less equivalent to loop calculus, a generalization of calculus invented by Stanley Mandelstam back in the 60's for the purpose of formulating gauge theories in terms of gauge invariant objects. Sadly, I no longer think Clifford Algebras are rich enough in structure to describe the fundamental workings of the universe. Still, Clifford Algebras, hypercomplex numbers and their applications to physics are beautiful enough to be worth studying simply for the pleasure of it.

Here are some lecture notes on quaternions I wrote for my little brother a few years ago. I don't think he read them.


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