Physics and domain theory
by
Keye Martin
Naval Research Laboratory

This workshop is about a mathematical subject called "domain theory" and its importance in theoretical physics; specifically, to the areas called "general relativity" and "quantum mechanics." A 'domain' is a partially ordered set with an unusual amount of structure. In this workshop, we will say what a domain is and talk a little about why people outside of physics have been interested in them. People in computer science for example.

Domain theoretic structure has appeared in quantum mechanics, where it has been used to solve important open questions in quantum information, such as: which ensembles give rise to a given density operator? This question was originally considered by Schrodinger but only solved recently by Nielsen using a domain of finite probability distributions.

Globally hyperbolic spacetimes with their causality relations are domains, and the fact that they are means that it is possible to construct spacetime in a purely order theoretic (or causal) manner beginning from only a countable dense subset of events. This is of interest to researchers working in quantum gravity.

If you are a student reading this and found the previous two paragraphs difficult to relate to, here is something to consider before deciding not to attend the lectures: it may be possible to learn a decent amount about general relativity and quantum mechanics by simply looking at them through the lens of domain theory. Domains are simple.

In the last lecture, we will ask if domain theory has anything to teach us about physics as a whole. For instance, do these two seemingly incompatible subjects (GR and QM) have anything in common? The answer to this question may have important implications for secure communication in the not too distant future.

Date received: June 28, 2006