Auxiliary-field quantum Monte Carlo calculations for the relativistic electron gas

We have written a version of the Hubbard-Stratonovich transformation for the case of the relativistic jellium, and have developed an auxiliary-field method for performing a quantum Monte Carlo calculation on such a system. We have successfully applied such a technique to a system with two electrons in a repeated box, and report on the results here. Also, we comment on the relationship between the functional form of the above theory and Feynman's formulation of quantum electrodynamics, namely the relationship of the auxiliary fields to the scalar and vector potentials in quantum electrodynamics. We find that gauge freedom enables us to find an auxiliary-field representation in which all fields are real, thus in principle removing the need to use imaginary auxiliary fields.