Numerical methods in non-perturbative quantum field theory

This thesis applies techniques of non-perturbative quantum field theory for solving both bosonic and fermionic systems dynamically on a lattice. The methods are first implemented in a bosonic system to examine the quantum decay of a scalar field oscillon in 2+1D. These configurations are a class of very long-lived, quasi-periodic, non-topological soliton. Classically, they last much longer than the natural timescales in the system, but gradually emit energy to eventually decay. Taking the oscillon to be the inhomogeneous, (quantum) mean field of a self-interacting scalar field enables an examination of the changes to the classical evolution in the presence of quantum fluctuations. The evolution is implemented through applying the Hartree approximation to the quantum dynamics. A statistical ensemble of fields replaces the quantum mode functions to calculate the quantum correlators in the dynamics. This offers the possibility for a reduction in the computational resources required to numerically evolve the system. The application of this method in determining the oscillon lifetimes, though, provides only a negligible gain in computational efficiency: likely due to the lack of any space or time averaging in measuring the lifetimes, and the low dimensionality. Evolving a Gaussian parameter-space of initial conditions enables comparing the classical and quantum evolution. The quantum fluctuations significantly reduce the lifetime compared to the classical case. Examining the evolution in the oscillatory frequency demonstrates the decay in the quantum system occurs gradually. This markedly contrasts the classical evolution where the oscillon frequency has been demonstrated to evolve to a critical frequency when the structure abruptly collapses. Despite the distinctly different evolution and lifetime, a similar range of the Gaussian initial conditions in both cases generates oscillons. This indicates the classical effects dominate the early evolution, and the quantum fluctuations most significantly alter the later decay. The methods are next implemented in a fermionic system to examine "tunnelling of the 3rd kind". This phenomenon is examined in the case where a uniform magnetic field propagates through a classical barrier by pair creation of fermions: these cross unimpeded through the barrier and annihilate to (re-)create the magnetic field in the classically shielded region. A statistical ensemble of fields, similarly to the oscillon simulations, is initially constructed for evaluating the fermionic contribution in the gauge field dynamics. This ensemble, importantly and in contrast to the bosonic case, involves two sets of fields to reproduce the anti-commuting nature of the fermion operator. The ensemble method, again, offers the possibility for a reduction in the computational resources required to evolve the system numerically. A test case indicates the method for the tunnelling system, though, requires impracticable computational resources. Using the symmetries in the system to construct an ansatz for the fields provides an alternative method to evolve the dynamics on a lattice. This procedure effectively reduces the system to a 1+1 dimensional problem with the fermion mode functions summed over the three-dimensional momentum space. The significant decrease in the real-time for the evolution (and quite attainable computational resources) on applying the ansatz provides a practical technique to examine the tunnelling. Measuring the magnetic field in the classically shielded region confirms the analytic estimates. These (qualitatively) reproduced the exponential decrease estimated in the classical transmission on varying the interaction strength between the barrier and the magnetic field. The observed tunnelling signal, moreover, matches the perturbative, analytic estimate within the expected correction in the lattice configuration. These bosonic and fermionic quantum, dynamical simulations demonstrate limitations to the benefits in applying the ensemble method. The highly practical and successful tunnelling computations, in contrast, indicate the potential power of a suitable ansatz to significantly reduce the computational times in simulations on a lattice