Algebraic Field Theory on Causal Sets: Local Structures and Quantization Methods

I investigate aspects of classical and quantum real scalar field theory on causal sets --- a discrete framework for space and time --- using the algebraic perspective. After reviewing and generalizing necessary notation, I consider different discretizations of the Klein-Gordon field equations to describe the dynamics of a scalar field. I generalize a recently proposed discretization method that uses a preferred past structure (which assigns a specific past element to every element of a causal set) to lattices in Minkowski spacetime of any dimension. With numerical techniques, I analyse criteria to assign a preferred past structure to more general causal sets that are generated via sprinkling --- a Poisson process on a given spacetime manifold. It turns out that there exists a method that is very successful in selecting a preferred past uniquely with high probability (for finite causal sets on Minkowski spacetime). I review quantization methods and algebraic states. For the case of a finite causal set, I show how to construct a symplectic vector space with an inner product. The given structure lets me apply the method of geometric quantization to determine a quantum algebra and define a state, which is the Sorkin-Johnston state --- commonly considered for quantum field theory on causal sets. Additionally, I discuss the relationship of the geometrically constructed quantum algebra to deformation quantization to motivate future applications like a non-perturbative construction of the quantum algebra for interacting field theories via geometric quantization.