Maths projects →Project IV topics
Project IV 2015-2016
Is the Universe stable?
Description
One of the most interesting possibilities allowed by quantum field theory is for the "vacuum" to be in an unstable state, with the prospect that the Universe will ultimately undergo catastrophic decay. The papers by Coleman and Coleman and de Luccia that introduced this idea are widely regarded as among the most beautiful, and beautifully written, papers in all of mathematical physics:
"This is disheartening. The possibility that we are living in a false vacuum has never been a cheering one to contemplate. Vacuum decay is the ultimate ecological catastrophe; in a new vacuum there are new constants of nature; after vacuum decay, not only is life as we know it impossible, so is chemistry as we know it. However, one could always draw stoic comfort from the possibility that perhaps in the course of time the new vacuum would sustain, if not life as we know it, at least some structures capable of knowing joy. This possibility has now been eliminated."
In this project you will investigate the mathematical ideas behind quantum instability and phase transitions. The project is quite challenging: you will learn the rudiments of quantum mechanics and quantum field theory with the aim of understanding the computation by Coleman and de Luccia. From there the project is quite open ended with the possibility of learning more about instantons and other non-perturbative effects, and similar phase transitions that are thought to have taken place in the early Universe, including the famous Higgs mechanism.
Prerequisites
Analysis in many variables II, Mathematical Physics II, Quantum Mechanics III and a co-requisite is Advanced Quantum Theory IV (Some Physics modules may possibly count as a substitute)
Resources and references
There is a large number of papers and some books discussing the mathematics; some good starting points are the original papers
- Coleman and de Luccia's paper
- The false vacuum in Wikipedia with other original refs.
- Coleman, S. 1988, Aspects of Symmetry, by Sidney Coleman, pp. 416. ISBN 0521318270. Cambridge, UK: Cambridge University Press, February 1988.
- A good overviews of quantum mechanics is contained in Hey and Walters and also in The Feynman lectures on physics / Feynman, Leighton, Sands. Then try the introduction in Quantum mechanics / Leonard I. Schiff. Also the QMIII course notes.
- A basic familiarity with quantum field theory will be required. Introductory texts on quantum field theory such as Quantum Field Theory in a Nutshell by A.Zee, Princeton University Press, Quantum Field Theory by Itzykson, C. and Zuber, J. B. New York: McGraw-Hill, 1980. Another good introduction to quantum field theory is Aitchison and Hey