AbstractsPhysics

Merging Einstein's relativity with quantum mechanics leads almost inexorably to relativistic quantum field theory (QFT). Although relativity and non-relativistic quantum mechanics have been on solid mathematical footing for nearly a century, many aspects of relativistic QFT remain elusive and poorly understood. In this thesis, we study two fundamentally important objects in quantum field theory: the S-matrix of a given QFT, which quantifies how quantum states interact and scatter off of each other, and the partition function, a quantity which defines observables of a given QFT. In chapters 2 – 6, we study generic properties of the S-matrix between on-shell massless states in four- and six-dimensions. S-matrix analysis performed with on-shell probes differ from more conventional analysis, with Feynman diagrams, in two important ways: (1) on-shell calculations are automatically gauge-invariant from start to finish, and (2) on-shell probes are inherently delocalized through all of space and time, i.e. they are ``long distance'' probes. In chapter 2, we note that one-loop scattering amplitudes have ultraviolet divergences that dictate how coupling constants in QFT ``run'' and evolve at finite distance. We show that on-shell techniques, which use exclusively long distance probes, are nevertheless sensitive to these important finite distance effects. In chapters 3 – 4, we show how the manifestly gauge-invariant on-shell S-matrix can be sensitive to what are called ``gauge anomalies'' in more conventional discussions of relativistic quantum systems, i.e. in local formulations of quantum field theory. In chapter 5 we use the basic tools of the analytic S-matrix program in an exhaustive study of the simplest non-trivial scattering processes in massless theories in four-dimensions. From the most basic incarnations of locality and unitarity, we derive many classic results, such as the Weinberg – Witten theorem, the equivalence theorem, supersymmetry, and the exclusion of ``higher spin'' S-matrices. Finally, in chapter 6, we inductively prove that the entire tree-level S-matrix of Einstein gravity in four dimensions can be recursively constructed through on-shell means. This implies, as a corollary, that the Einstein-Hilbert action may be completely excised from studies of tree-level/semi-classical scattering in General Relativity. In chapters 7 – 9, we note a surprising property of statistical mechanical partition functions for many exactly solved quantum field theories, and explore a basic corollary. In chapter 7, we note that many partition functions, $Z(T) = \sum_n e^{-E_n/T}$, that can be exactly computed and re-summed into closed-form expressions, are surprisingly self-similar under temperature reflection (T-reflection). In short, we find that $Z(+T) = e^{i \gamma} Z(-T)$, where $\gamma$ is a real number that is independent of temperature. This T-reflection symmetry only exists for a unique value of the ground state energy, often given by the naive quantization of classical potentials/Hamiltonians. In…