Theory of Strongly Coupled Many-Fermion Systems. I. Convergence of Linked-Cluster Expansions

A strongly coupled system—the limiting case of a highly degenerate many-fermion system for which the variation of the kinetic energy is neglected, and the interaction restricted to a region of momentum space neighboring the Fermi surface—has been analyzed in a manner not dependent upon assumptions about the convergence of power series expansions or on partial summations of infinite series. The vacuum expectation value of the resolvent operator, ${〈\frac{1}{\mathrm{}(H-z)\mathrm{}}〉}_0$, is expressed as the Laplace transform of the exponential of a function linearly dependent on the volume of the system. It is shown that the linked-cluster expansion of the vacuum expectation value of the resolvent operator has a zero radius of convergence as a power series in the coupling constant. The most serious physical consequence of this is that a nontrivial interaction never results in a "normal" system.