On-shell scattering and temperature-reflections

Abstract

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 chapter 8, we note that a certain limit of quantum chromodynamics (QCD) is symmetric under T-reflections only when its vacuum energy vanishes. We verify that this is indeed the correct vacuum energy of this theory through two calculations that are independent of each other and independent of the presence or absence of T-reflection symmetry. In chapter 9, we use this T-reflection symmetry to obtain a detailed understanding of the presence/absence of Hagedorn phase transitions in a certain calculable limit of QCD.