An approach for the calculation of one-loop effective actions and vacuum energies: scattering methods

An approach for calculating one-loop effective actions and vacuum energies is suggested. Spectral functions are functions of the eigenvalue spectrum of an operator. The starting point is to regard one-loop effective actions and vacuum energies in quantum field theory and scattering phase shifts and amplitudes in quantum mechanics as spectral functions of an operator. Different spectral functions of the same operator may belong to different fields of physics. Methods for calculating various spectral functions can be interconverted through transform relations among spectral functions. In this paper, we convert the method for calculating scattering phase shifts and amplitudes to a method for calculating one-loop effective actions and vacuum energies. In principle, all methods for the calculation of scattering phase shifts and amplitudes can be converted to methods for the calculation of the one-loop effective actions and vacuum energies. As an example, we convert the Born approximation in quantum-mechanical scattering theory to a method for calculating one-loop effective actions and vacuum energies. In appendices, we provide integral representations of the Bessel function for calculating the sum encountered.