Galilei-covariant path-integral quantization of non-relativistic complex scalar fields.
In order to study Galilei-invariant systems, the functional quantization of non-relativistic complex scalar fields is investigated by exploiting Galilean covariance, a geometrical structure defined in a \((4+1)\)-dimensional Minkowski space. Thereby a new approach to treat the quantization of the Schrödinger field dynamics, including nonlinear interactions, is developed. The physical motivation is twofold: from a theoretical point of view, this formalism allows one to express hard nonrelativistic models within a rather simple covariant form. Second, it may be applied to nuclear and low-temperature physics, many-body theory, superfluidity, Bose-Einstein condensation, etc. The aim of the authors is to expand the use of quantum field theoretical methods in statistical physics by borrowing the concept of Lorentz-covariance in order to build Galilei-invariant models of the condensed state. In this paper, as well as in the authors' previous articles, those field models are considered whose form is inspired by Lorentz-like expressions. However, their Galilean counterparts should not be seen necessarily as their corresponding non-relativistic limits and might apply to entirely different physical systems. Such concepts as creation and annihilation of pairs should not be understood in the present context in the relativistic sense, but, for instance, in terms of elementary excitations, quasi-particles and holes with respect to the Fermi sphere. Despite the fact that the interpretation of some aspects of Galilean covariance is still needed, technical aspects of the theory are developed in the paper. Specifically, the path-integral quantization of Galilei-invariant fields is investigated, because its well-known relativistic version provides a perturbation theory in which the \(S\)-matrix is manifestly Lorentz-invariant. The generating functionals and the Green's functions are defined by utilizing the five-dimensional prescription of Galilean covariance. As an example, the self-interacting scalar field with coupling term \(\lambda| \Phi| ^4\) in two and three dimensions, as well as the nonlinear Schrödinger equation are considered. The calculations of scattering amplitudes in two dimensions agree with results found in the literature.
scientific article; zbMATH DE number 2012727
scientific article