A new approach to lattice quantum field theories

University of Minnesota M.S. thesis. January 2018. Major: Mathematics. Advisors: Vitaly Vanchurin, Yang Li. 1 computer file (PDF); iii, 56 pages.In this thesis report, I describe an algorithm for lattice simulation of quantum/statistical fields that reduces the complexity of current techniques (Metropolis algorithm) from exponential in all the directions of space and (Euclidean-)time, to linear in (Euclidean-)time and exponential in space. This is done by building a typical field configuration spatial slice by spatial slice through an analytically obtained Markov chain from its path integral. Although the complexity still depends exponentially on the number of spatial lattice points, for quantum mechanics ($0+1$ fields) spatial slice is only a point and thus the complexity only depends linearly on the number of time lattice points and simulation becomes pretty easy. As examples, I discuss the cases of harmonic and an-harmonic oscillators along with some simulation results. The case of Gaussian fields in general (in any dimension) is trivial since in the similarity transformed space each lattice site decouples and hence there exists a random variable at each lattice site that does not interact with any other. Although the reduction of complexity from exponential in space (if possible) for higher dimensional fields in general is currently under investigation, I present a checkerboard network that we investigated along with some simulation results