Archive | 2019
Quantum fluctuations in non-globally hyperbolic spacetimes : Flutuações quânticas em espaços-tempos não-globalmente hiperbólicos
Abstract
Quantization of fields on backgrounds provided by General Relativity is the primary cornerstone of Quantum field theory in curved spaces. Such spacetimes are usually globally hyperbolic, i.e., spacetimes where the Cauchy problem associated with hyperbolic equations in particular, the Klein-Gordon equation is well-posed. Conversely, in non-globally hyperbolic spacetimes, lost of predictability jeopardizes the quantization procedure. Although, Wald argues that one recovers a sensible dynamical evolution of fields by finding the positive self-adjoint extensions of the spatial component of the differential wave operator. Moreover, Ishibashi and Wald show that this prescription for dynamics is the only whose outcomes are physically consonant. According to their prescription, in spacetimes such as the Global Monopole or anti-de Sitter where global hyperbolicity is absent -, a sensible dynamics is obtained through the imposition of a set of boundary conditions at the naked singularity or the conformal boundary, respectively. In our work, we examine the effects that these non-Dirichlet boundary conditions have on physically relevant quantities, e.g., the expectation value of the stress-energy tensor. Our first two toy models consist of examining the scattering of scalar fields on the three-dimensional conical spacetime and the Global Monopole. We obtain contributions to the scattering amplitude depending on the boundary condition parameter exclusively and independent of topological features of the respective manifolds. Additionally, in the Global Monopole, we find analytical contributions to the fluctuations of the stress tensor and the field squared. Once again, our contributions depend solely on the boundary condition parameter. Besides, our results resemble those of scalar fields in Minkowski spacetime with a point removed, which confirm that any contributions from boundary conditions are unrelated to the topology of the spacetime. Finally, we consider the propagation of scalar fields on the anti-de Sitter spacetime. The imposition of Robin boundary conditions at infinity for one of the modes of the wave equation results in expectation values that do not respect all symmetries of adS. We pay the price for adopting a prescription for sensible dynamics in adS, namely the violation of energy conditions and the breakdown of spacetime invariance. Ultimately, in this dissertation, we exploit the development of a physically consistent Quantum Field Theory on non-globally hyperbolic spacetimes. Nevertheless, such prescription gives rise to non-trivial outcomes. Amongst the infinitely many dynamical evolutions that fields may admit, we are not aware of any constraints in nature that force us to choose a specific one. In any case, our results reveal that, if somehow nature determines a particular boundary condition, such a choice shall influence physically relevant quantities.