Fermions in Infinite Dimensions with Hard Sphere Interactions.

When examining the Many Body problem with interactions, it is useful to study a concrete model which is simple enough so that the validity of a given method can be assessed, such as hard-core systems, since hard cores can be used as a reference system for a perturbative expansion for potentials with repulsive cores. The partition function for a hard-core fluid, when expanded in terms of the Mayer f-functions, results in integrals that can be interpreted diagrammatically, and the number of diagrams increases rapidly. A way of organizing these diagrams is then of interest; one approach is to do so in terms of the dimensionality. Going to the limit of infinite dimensions shows which diagrams are more important from a dimensional standpoint. Classically, it has been found that betaP = rho + (1/2)vrho^2. The current work extends the treatment to fermions. Because the Partition Function is a trace in the quantum mechanical case, it corresponds to particles being created at a given momentum, scattered by other particles, and then annihiliated at the points they were created. Because of the trace, all of the diagrams are closed, and are analogous to vacuum field fluctuations. When the Grand Partition Function is interpreted diagrammatically in this way, the effects of the quantum extensions appear as diagrammatic additions to the classical fugacity expansion diagrams of the equation of state. Since the potential energy terms are singular, the ladder diagrams are summed over to achieve an integral equation in which the magnitude of the potential interaction is absent. When written out in detail, the equation resembles the integral equation for the wavefunction of the two-body scattering problem, and so the effective interactions between torons may be expressed in terms of the solutions to the two-body problem. The calculations at present indicate that there is reason to expect that the self-interaction of a two -toron and the mutual interaction of one torons may become the dominant additions from the quantum side to the diagrammatic expansion.