Some Aspects of Quantum Fields in Curved Classical and Quantum Background Space-Time Using the Quantum Effective Action Formalism

Some basic questions in quantum field theory and cosmology are addressed here. We derive some formulas for the change in the canonical and anticommutation relations at an equal time for some of the well-known quantum fields in the presence of a background curved space-time metric. We then derive formulas for the approximate change in the canonical commutation relations of a field or a set of fields when a small perturbing Lagrangian is added to the unperturbed Lagrangian. We study the problem of quantizing the Klein-Gordon field interacting with the gravitational field of homogeneous and isotropic space-time of an expanding universe and also simultaneously interacting with a classical random current field. Formulas for the quantum effective action of the scale factor of the expanding universe are derived by averaging over the Klein-Gordon field, taking into account its interaction with a classical random current field. This gives us information about how the expansion rate of our universe can be affected due to quantum mechanical interaction effects. Finally, we discuss the general problem of symmetry breaking in the quantum effective action of a field when it interacts with a random Gaussian current field source. The symmetry-breaking terms are expressed in terms of the correlation field of the random current source. We also discuss how canonical commutation relations of a Bosonic field between the position and velocity fields get perturbed approximately when a small perturbing Lagrangian is added.