Roughly speaking, instantons are solutions of the Euclidean equations of motion for the quantum particle or quantum field, which have finite action. Such solutions describe the process in which a particle in quantum theory can "propagate" from one configuration to another, which would never be allowed in classical physics. For example, if you have a potential which has two minima which are separated by a high barrier, you know that a classical particle with energy lower than the barrier can never go from one minimum to another, as it cannot tunnel through the potential barrier. In quantum mechanics, and more generally in quantum field theory, such a propagation is possible, and instantons are paths in the space of all paths which describe such a propagation.
Instantons play a very important role not just in quantum mechanics but also in quantum field theory, where they are necessary to explain a whole range of phenomena. Viewed as classical solutions of the equations of motion, instantons are also interesting as they are stable field configurations, stabilised by the topological charge which they have. In this project you will be free to choose your own path and follow a direction which you find the most interesting: to study instantons as solitons with their intriguing topological properties or their application in quantum field theory or even cosmology to explain the beginning of the Universe.
Quantum Mechanics III, Solitons III (optional) and you need to attend the Advanced Quantum Theory module in year IV. If you want to study applications of instantons in cosmology you will need to attend General Relativity IV.
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