Functional integral and transfer-matrix approach for 1D bosonic many-body systems with a contact potential

A 1D bosonic many-body system, related to the Bose–Einstein condensation in atomic traps and periodic optical lattices, is described by a coherent state path integral of the grand canonical partition function. Since the interaction is given by a contact potential, as commonly applied in atomic traps of BEC, the functional integral can be represented by spatial transfer matrices, ordered according to their sequential position in space. The corresponding differential transfer matrix equation is derived and the generator for space translations is given, including an approximation of slowly varying coherent state fields. The given approach of spatial transfer matrices contains large fluctuations and higher order correlation functions beyond the mean field approximation and, in analogy, can be compared to the time development operator of the Feynman path integral and the one-particle Schrödinger equation. It is also described how to obtain the spatial transfer-matrix or its corresponding generator for 2D bosonic systems and also fermions with a quartic contact interaction which leads to a differential equation with Grassmann numbers.