Richard E. Norton

On the formalism of relativistic many body theory

The thermodynamic potential is constructed as an effective action functional of the various n point amplitudes (n less than or equal to 4). One of the functionals is used to obtain the equations of state as simple, convergent expressions involving the conventionally renormalized charges and masses. (auth)

Elementary particle scattering and statistical quasi-particles in quantum statistical mechanics

Abstract Previous work relating the thermodynamic potential to elementary particle S-matrix elements is generalized and rederived directly from the expressions for the diagrams of many body theory. The divergent physical region poles are shown to introduce energy derivatives of mass shell delta functions which tend to shift the energies of the scattering particles away from the elementary particle mass shell. These shifted energies are related to the statistical quasiparticle energies introduced by Balian and De Dominicis. The work of these authors is generalized to show that to all orders in the coupling strengths the many body diagrams for any system described by a relativistic or non-relativistic field theory can be summed to give: (1) the entropy and the statistical average of a non-spontaneously broken, conserved charge in terms of ideal gas-like formulae involving statistical quasi-particle energies; (2) the thermodynamic potential in terms of diagonal matrix elements of products of transition amplitudes wherein the energies of all external particles and the energy arguments of all ideal gas occupation numbers are the statistical quasi-particle energies.

On the thermodynamic potential expressed as a trace over scattering phases and occupation numbers

Abstract A formula is presented which expresses the thermodynamic potential as a trace over the product of occupation numbers and a phase operator referring to the transitions among the “excitations” associated with the absorbtive parts of the many body single particle propagators. The entropy and the statistical average of each conserved, non-spontaneously violated, charge are given simply by appropriately differentiating the occupation numbers appearing in the trace. The formula is a generalization to arbitrary temperature and density of a previously discussed formula (Ref. [2]) relating the thermodynamic potential in the regime of weak degeneracy to a trace involving the logarithm of the elementary particle S -matrix. How this latter formula arises from summing up the graphs of finite temperature field theory is also discussed.

On the equations of state in quantum statistical mechanics

Abstract By extending methods previously used to study the equations of state at low temperature, it is shown that the entropy density and the statistical average of a conserved, non-spontaneously violated, charge density can be expanded in terms of integrals over products of many body n-point amplitudes defined for real, continuous frequencies. The general structure of the expansions is described, and it is demonstrated that essentially the same spectral function determines the entropy density and the average charge densities. Certain classes of terms are worked out in detail, and the formal sum of one such class is shown to provide the contributions to the equations of state arising from composite quasiparticles associated with the poles of the n-point amplitudes. [Another term, discussed in many previous works, involves the logarithms of the elementary propagators and yields the contributions to the equations of state coming from elementary quasiparticles.] The Appendices include an extensive study of the analytic properties of many body amplitudes in the frequencies of the external and internal lines. Specialized to zero temperature, these considerations apply to the Feynman diagrams for elementary particle amplitudes.

Statistical quasi-particles as the normal component of a superfluid

It is argued from the structure of many body theory Feynman graphs that the statistical average of a spontaneously broken conserved charge density decomposes naturally into two parts: (1) a “normal” part given by an ideal gas-like expression with statistical quasi-particle emergies, and (2) a “superfluid” part which vanishes when the symmetry is restored. The momentum density of a relativistic superfluid is also discussed.