To bring the mathematics down to earth: the Thirring model (via Wikipedia) is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in two dimensions. A Dirac field is a fermionic field, describing for example neutrinos. The massive Thirring model would describes electrons, protons and other massive fermions. The Thirring model would imply that sub-atomic particles are solitons (standing waves), that mass is a field energy, and that particles interact as a consequence of their topology. Project 1: Finite difference simulation and asymptotic expansionThe Dirac equation known as the massive Thirring model can be analysed for Gaussian initial conditions using a finite differences scheme. It is stable for small initial conditions, but develops exponentially growing oscillatory instabilities for larger ones. Growth in what should be a conserved quantity reliably indicates the presence of instability. An asymptotic expansion of the solution to the linearised equation is a qualitatively good approximation to the system's longterm behaviour under small initial conditions. Project 2: Soliton solutionThe massive Thirring model equations have Lorentz-invariant travelling soliton solutions. Working at first with the stationary case, the Thirring model equation can be written as two complex ODE's. Using polar decomposition, the four equations in four real-valued unknowns can be reduced to one ODE and evaluated in terms of hyperbolic functions. A finite differences scheme is then used to model the behaviour of the soliton solution. Note: there already exists a 1975 paper on soliton solutions of the massive Thirring model and the original paper by Walter E. Thirring Attached are the complete projects, and the PDF write-ups |

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