Classical and Quantum Mechanical Correlation Functions of Fields in Thermal Equilibrium

Abstract

The two-point, two-time correlation functions of classical and quantum mechanical fields in thermal equilibrium in an arbitrary domain are considered. For classical fields that satisfy linear equations of motion, the correlation functions are expressed in terms of certain Green's functions of the domain. It is also shown, by evaluating the characteristic functional of the field, that a classical field is Gaussian distributed, so that all higher-order correlations can be expressed in terms of the second-order correlations. Then the second-order correlations are evaluated in various cases, and the classical and quantum mechanical results are compared. They are found to agree except within layers, about a thermal wavelength wide, near the boundaries of the domain and near the characteristic surface or light cone emanating from either of the two points. Thus the quantum mechanical correlation function can be approximated by the classical one with quantum-mechanical boundary-layer corrections.