Description

Soon after Einstein formulated his General Theory of Relativity, treating gravity as an expression of the curvature of spacetime, Kaluza and Klein proposed that electromagnetism could also be included in this description if spacetime had additional dimensions. This idea went out of fashion after the discovery of the nuclear forces, but today, the idea of extra dimensions of spacetime plays a central role in string theory, which predicts that spacetime is fundamentally ten-dimensional. 

This project will explore how this mechanism operates. You will see how the wave equation on a higher-dimensional spacetime can be decomposed to give a spectrum of fields in the four-dimensional universe. You can study both traditional compactifications, where the additional dimensions are curled up small, and more recent ideas of `large' extra dimensions. There is also the possibility of exploring the history of the idea of extra dimensions. 

Prerequisites

Analysis in Many Variables II (or PHYS 2611 Mathematical methods in Physics).

Mathematical Physics II or some equivalent prior exposure to modern physics will be very helpful; please see me before signing up for the project if you don't have this. 

No knowledge of general relativity is expected, and the project has plenty of scope for development without needing to master the dynamics of spacetime.

Resources

Brian Greene's book "The Elegant Universe" includes a nice popular account of compactification in string theory. There are also many discussions of varying quality on the web: search on "extra dimensions" or "Kaluza-Klein theory". 

Zweibach, "A First Course in String Theory" (CUP), is the most accessible introduction to string theory. Chapter 2 provides a useful discussion of compactification and extra dimensions.