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Dispersion relations for causal Green’s functions: Derivations using the Poincaré–Bertrand theorem and its generalizations

### Abstract

A famous theorem by Poincaré and Bertrand formally describes how to interchange the order of integration in a double integral involving two principal‐value factors. This theorem has important applications in many‐body physics, particularly in the evaluation of response functions (or ‘‘loop integrals’’) at either zero or finite temperatures. Of special interest is the loop containing an integration with respect to the energy of two causal propagators. It is shown that such a response function with two boson or two fermion lines behaves statistically like a boson, while the response function containing a boson and a fermion behaves like a fermion. Examples are given of typical loop integrals occurring in the solution of Dyson’s equations for nuclear matter in the presence of delta, nucleon, and pion interactions. A ‘‘form factor’’ that is essential for the convergence of the nucleon–pion loop integral is chosen to have little effect on the analogous nucleon–delta loop integral. The Poincaré–Bertrand (PB) theorem is then generalized to multiple integrals and higher‐order poles. From the generalization of the theorem to triple integrals, it is shown that causality is rigorously maintained, at zero temperature, for convolutions with respect to the time of products of Green’s functions and thus for Dyson’s equations. Also, for finite temperature, the three‐propagator loop integral satisfies the statistics appropriate for the loop as a whole, in direct analogy with the result for the two‐propagator loop. The intimate connection between the PB theorem and analyticity (or causality) is clearly demonstrated. Although this work considers explicitly only nuclear physics examples, the results are relevant to other fields where many‐body theory is used.

© 1990 American Institute of Physics

Received 22 August 1989
Accepted 24 January 1990

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1990-06-01

2016-02-11

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