DescriptionSolitons are stable, extended objects in field theories whose energy is nonetheless localised in space. Even classically they exhibit a kind of `wave-particle' duality which is fascinating to high-energy physicists. Many integrable theories, such as the Sine-Gordon model have solitons. In such theories one can find exact classical solutions which correspond to two or more solitons scattering off one another. This project aims to study what happens when the world of solitons meets quantum mechanics. Perhaps surprisingly, one can use the symmetry of integrable models to exactly solve for the scattering of solitons even quantum mechanically. The solution relies on rich algebraic structure which provides a mathematical description of the underlying symmetry. This project has the flexibility to develop in several directions, of which I give a few examples below: One can solve exactly for the quantum soliton scattering in integrable models such as the Sine-Gordon model, where symmetry can be used to deduce algebraic equations which the scattering matrix must obey. Ultimately this can be used to provide evidence for a remarkable duality between the Sine-Gordon model at strong coupling, and a theory of massive fermions on the other, where the solitons of the Sine-Gordon theory are identified with the basic fermionic excitations of the latter!Another possible direction is to look at the scattering of so-called BPS solitons. The scattering is well-described in the low-energy limit by geodesics on a curved manifold, for example the dynamics of two monopoles is described using the Atiyah-Hitchin metric. The quantisation of this system can be analysed by studying the Schrodinger Equation on this curved manifold. A third example would be to use semi-classical methods to calculate a variety of quantities; for instance quantum corrections to soliton masses and scattering via Levinson’s theorem. PrerequisitesQuantum Mechanics III (or Physics equivalent). Having taken Solitons III would be useful but not essential, and taking Advanced Quantum Theory (or Physics course on Quantum Field Theory) will tie in nicely.Resources and referencesThere are a couple of books in the library which may be a good place to start:
There are also a number of a couple of articles it might be worth looking through |
email: Peter Bowcock