Scalar quantum field theory in disordered media

A free massive scalar field in inhomogeneous random media is investigated. The coefficients of the Klein-Gordon equation are taken to be random functions of the spatial coordinates. The case of a disordered medium modeled by centered stationary and Gaussian processes is analyzed. After performing the averages over the random functions, we obtain the two-point and the four-point Green's functions of the model up to one loop. The disordered scalar quantum field theory becomes qualitatively similar to a �' 4 self-interacting theory with a frequency-dependent coupling. In the present paper we depart from the quest for analog models of quantum gravity effects and study quantum fields in a disordered environment in a more general con- text. Our study belongs to a wide program devoted to propagation of quantum matter fields in a classical back- ground (9) but with metric fluctuations. Specifically, we consider a scalar quantum field described by a Klein- Gordon-like equation, in which the parameters of the equation, the mass, and the coefficient of the second-order time derivative become random functions of the spatial coordinates. The randomness of the parameters is due to static noise sources that couple to the scalar field. In the limit of a vanishing mass, one recovers the random wave equation considered in Ref. (4). While not addressing the issue of the origin of the noise sources, we mention that one has in mind that they can be induced by a variety of phenomena. Among others, these include phenomena like metric fluctuations due to quantum creation of grav- itons in a squeezed coherent state in the presence of a black hole, and interactions with background topological defects. Similarly to the case of quantumfields in the presence of an external heat bath, the noise sources break Lorentz sym- metry since they define a preferred reference frame (10). A scalar quantum field associated with acoustic waves in a disordered medium can define a situation where sound cones fluctuate randomly. In the study of such a situation, it is important to observe that systems with disorder can be divided into two wide groups, namely, systems with quenched or annealed disorder (11-13). While in annealed systems the random field is in thermal equilibrium with the other degrees of freedom of the system, in quenched sys- tems they are not. This has implications in the way one calculates averages over the disorder. The differentiation between the two types of disorder is an important issue in the studies of the influence of impurities on phase transi- tion phenomena. Many authors have used field theory and the renormalization group to investigate systems with quenched disorder (14), with particular interest in the role played by the disorder on critical exponents. In the case of quenched random fields, by means of the replica trick it is possible to define a quenched generating func- tional of connected n-point functions. With this generating functional in hand, the influence of the disorder on phase transitions can be studied. We shall come back to these issues later in the present paper when we consider corre- lation functions. In the present paper we consider weak noise fields and implement a perturbative expansion con- trolled by small parameters that characterize the intensity of the noise correlation functions. We obtain causal two- and four-point Green's functions of the scalar field. Averaging these Green's functions over noise fluctuations, one obtains two- and four-point functions qualitatively similar to a self-interacting �' 4 theory, with a coupling