Quantum fields, vacuum fluctuations, and the Unruh effect
Dr. Nabil Iqbal
Vacuum fluctuations in Quantum Chromodynamics, image credit: D. Leinweber. We will be studying vacuum flucutations in much simpler theories than QCD.
In a relativistic quantum field theory, empty space is not actually empty: quantum fluctuations mean that virtual particles continually flicker in and out of existence. These vacuum fluctuations result in a phenomenon called the Unruh effect, where an observer who accelerates through the vacuum of a relativistic quantum field theory “feels” that she is no longer in empty space, but is instead immersed in a thermal gas with a nonzero temperature. This turns out to be a manifestation of the fact that the quantum vacuum has a high degree of quantummechanical entanglement. In addition to being interesting in its own right, this is also at the heart of Hawking radiation, which states that black holes emit thermal radiation and will thus eventually evaporate away into nothingness.
This project will develop some of the principles of relativistic quantum field theory and statistical mechanics and apply them to the physics of an accelerating observer, deriving the Unruh effect from different points of view. Some possible further topics for development then include (depending on the tastes of the student):

The entanglement of quantum fields in the vacuum

Particle detection along nontrivial worldlines

Hawking radiation from black holes
This project involves the intersection of various different concepts (quantum mechanical, geometric, etc.) and so will require considerable reading of varied references, but along the way it will also introduce a student to some of the most interesting manifestations of the quantum mechanical structure that underlies empty space itself.
Prerequisites:

MATH3111 Quantum Mechanics III (or Physics equivalent).
Corequisites:

MATH4061 Advanced Quantum Theory IV.
Exposure to concepts of geometry and curved space (e.g. through cotaking MATH4051 General Relativity) is also very helpful, particularly if you want to study Hawking radiation. Some exposure to statistical mechanical ideas (e.g. through cotaking MATH3351) also wouldn’t hurt.
Resources:
The Wikipedia pages on the Unruh effect and Hawking radiation aren’t bad places to start. More formal references include:

An introduction to black holes, information, and the string theory revolution, Susskind and Lindesay. (A grandioselynamed book that is a somewhat heuristic introduction to the basic ideas).

Spacetime and Geometry, Chapter 9, Carroll (Mostly a textbook on general relativity that nevertheless covers quantum field theory in curved space).

Quantum Fields in Curved Space, Birrell and Davies (An authoritative reference.)
General background reading on quantum field theory and geometric preliminaries will be provided as required.