A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain
U functions defined for the same system with Boltzmann statistics. It is then shown that these
U functions can be evaluated in successive approximations in terms of a binary kernel
B which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.
We present a full many-body analysis of the problem of a single ↓ atom resonantly interacting with a Fermi sea of ↑ atoms. A series of successive approximations permits us to clarify the quite mysterious agreement between Monte Carlo results and approximate calculations taking only into account single particle-hole excitations. We show that it results from a nearly perfect destructive interference of the contributions of states with more than one particle-hole pair. Our treatment provides, at the same time, an essentially exact solution to this problem.