Quantum/Statistical Field Theories with Random Potentials A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY MUDIT JAIN IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

In this thesis report (based on my paper[1]), I present a path integral formalism for treatment of Quantum/Statistical fields in flat (Minkowksi) spacetime, with random potentials. The problem is to develop a mechanism to calculate observables (correlators) for fields when there is some uncertainity/randomness in the potential for the field. Random potentials which ’drive’ the field(s) live in the field space (which we consider to be Euclidean) and for this reason, the averaging over potentials must be quenched and not annealed. Moreover, it is different from the quenched averaging done over quantum systems with random potentials (e.g. Anderson localization) where both the ’second quantized field’ and potential live in the physical space. Examples of such field theories with random potentials include cosmological systems in the context of string theory landscape (e.g. cosmic inflation) or condensed matter systems with quenched disorder (e.g. spin glass) and many more. I use the so-called replica trick to define two different generating functionals for calculating correlators of the quantum/statistical fields averaged over a given distribution of random potentials. The first generating functional is appropriate for calculating averaged (in-out) amplitudes and involves a single replica of fields, but the replica limit is taken to an (unphysical) negative one number of fields outside of the path integral. When the number of replicas is doubled the generating functional can also be used for calculating averaged probabilities (squared amplitudes) using the in-in construction. The second generating functional involves an infinite number of replicas, but can be used for calculating both in-out and in-in correlators and the replica limits are taken to only a zero number of fields (which is physically more reasonable). The formalism is presented in detail for a single real scalar field but can be naturally extended to different kinds of and/or multiple fields. As toy problems, I work out three examples: one where the mass of scalar field is treated as a random variable and two where the functional form of interactions is random, one described by a Gaussian random field and the other by a Euclidean action in the field configuration space.