Generating functionals for quantum field theories with random potentials

Abstract

We consider generating functionals for computing correlators in quantum field theories with random potentials. Examples of such theories include cosmological systems in context of the string theory landscape (e.g. cosmic inflation) or condensed matter systems with quenched disorder (e.g. spin glass). We use the so-called replica trick to define two different generating functionals for calculating correlators of the quantum 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. We discuss the formalism in details for a single real scalar field, but the generalization to more fields or to different types of fields is straightforward. We 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.

From wires to cosmology

“…The fact that they are independent of N s means that asymptotically the growth is purely exponential. 20 This is consistent with the asymptotical scalings we derived from the Fokker-Planck equation, but this time it does not assume that the scattering is weak.…”

“…Both effects need to be taken into account to derive the late-time correlation functions for ζ, and hence the effects on cosmological observables. It will be interesting to follow 20 It is possible that exponential particle production (localization) can be avoided when the matrices Mj are drawn from ensembles belonging to certain symmetry classes [66]. 21 We should caution the reader that the universality of our results partially relies on the assumption that the interacting fields are statistically equivalent and maximally mixed with a random but uniform distribution of non-adiabatic events.…”

“…It will be interesting to follow up on the suggestion of [25] that weak disorder during inflation could lead to additional noise in the power spectrum and bispectrum of ζ. Finally, once the effects on the curvature perturbations have been calculated, it will also be useful to compare 20 It is possible that exponential particle production (localization) can be avoided when the matrices Mj are drawn from ensembles belonging to certain symmetry classes [66]. 21 We should caution the reader that the universality of our results partially relies on the assumption that the interacting fields are statistically equivalent and maximally mixed with a random but uniform distribution of non-adiabatic events.…”

Curvature perturbations from stochastic particle production during inflation

“…Apart from the elegant mathematical correspondence, the primary drive there was that certain universal features, such as Anderson Localization [22,23] in one dimension, arise independent of the details of the systems -motivating a search for similar universality in particle production. For different approaches in the context of the early universe where simple behavior arises in spite of underlying complexity of the models, also see [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”

Quantum field-theoretic machine learning

“…The resulting φ 4 machine learning algorithm of Sections II and III retains the topology of the lattice structure and the boundary conditions, but differs from the conventional φ 4 scalar field theory due to the inhomogeneous coupling constants. To employ the tools of quantum field theory a framework involving the replica method is required, but the theories can still be formulated in terms of the functional integral with an additional averaging over the space of couplings [61]. It is noted that in our formulation the couplings are inhomogeneous but not random as they are determined during the minimization process.…”