Stochastic quantization and Osterwalder-Schrader axioms for quantum field theory models

Source abstract

This PhD thesis constructs the $P(Φ)_2$ Euclidean quantum field theory on $\mathscr{S}'(\mathbb{R}^2)$. We first construct the finite-volume measure of the $P(Φ)_2$ model on $\mathscr{D}'(\mathbb{S}_R)$, where $\mathbb{S}_R$ is a two-dimensional sphere of radius $R\in\mathbb{N}_+$. To this end, we rely on Nelson's hypercontractivity estimate. We then use parabolic stochastic quantization and the energy method to construct the infinite-volume measure on $\mathscr{S}'(\mathbb{R}^2)$. This requires establishing the global-in-time existence of solutions to the corresponding stochastic partial differential equations with fixed cutoffs. The remainder of the thesis is devoted to verifying the Osterwalder-Schrader axioms: regularity, reflection positivity, and invariance under translations, rotations, and reflections. This combination of axioms, which is crucial for obtaining a non-trivial local relativistic quantum field theory on Minkowski spacetime, had not been established in earlier constructions based on stochastic quantization.