Quantum Corrections and Effective Action in Field Theory

TL;DR

This thesis explores quantum corrections in field theory using the effective action formalism, deriving nonperturbative results and applying them to gravitational potentials, with a focus on gauge-invariant observables.

Contribution

It introduces a coarse grained effective action and an exact renormalization group equation, providing new nonperturbative insights into quantum corrections in field and gravitational theories.

Findings

Derived nonperturbative effective potential for scalar fields

Established gauge-invariant test particle trajectories in quantum gravity

Obtained quantum corrections to gravitational potential at low energies

Abstract

In this Thesis we study quantum corrections to the classical dynamics for mean values in field theory. To that end we make use of the formalism of the closed time path effective action to get real and causal equations of motion. We introduce a coarse grained effective action, which is useful in the study of phase transitions in field theory. We derive an exact renormalization group equation that describes how this action varies with the coarse graining scale. We develop different approximation methods to solve that equation, and we obtain non perturbative improvements to the effective potential for a self interacting scalar field theory. We also discuss the stochastic aspects contained in this action. On the other hand, using the effective action, we find low energy and large distance quantum corrections for the gravitational potential, treating relativity as an effective low energy theory. We include the effect of scalar fields, fermions and gravitons. The inclusion of metric fluctuations causes Einstein semiclassical equations to depend on the gauge fixing parameters, and they are therefore non physical. We solve this problem identifying as a physical observable the trayectory of a test particle. We explicitly show that the geodesic equation for such particle is independent of the arbitrary parameters of the gauge fixing.