Relativistic Quantum Many-Body Theory in Riemannian Space-Time

Abstract

A relativistic quantum many-body theory, which includes the strong interactions between elementary particles in curved space-time, is constructed. Using a generalized statisticaldensity operator, which incorporates the effects of gravitation as given by Einstein's field equations, as well as observables constructed from a generally covariant Lagrangian for matter fields, the definition of an $N$-point function of second-quantized matter fields is presented. The Yukawa coupling of a spinor field is then introduced. Coupled integral equations for the fermion and boson two-point functions in terms of the vertex function are given, which contain density and temperature effects in curved space-time; they are coupled to Einstein's equations through the expectation value of the energy-momentum density operator. Renormalization to effective masses and charge, as well as regularization, are discussed. The curved-space-time statistical-density operator is examined in the flat-space-time limit, and also in the nonrelativistic limit. The former agrees with previous work in relativistic statistical mechanics. The introduction of temperature and density as boundary conditions on flat-space-time $N$-point functions is carried out yielding a relativistic formalism, which may be applied to the calculation of such quantities as the equation of state of a superdense system of strongly interacting baryons. The nonrelativistic limit suggests a new approach to the statistical mechanics of Newtonian gravitation, in which such parameters as temperature become functions of coordinates. The relativistic flat-space-time limit is applicable to neutron stars at densities $\ensuremath{\rho}>{10}^{15}$ g/${\mathrm{cm}}^{3}$ consisting of strongly interacting matter.