On the functional-integral approach in quantum statistics, including mixed-mode effects and free of divergences. II. Diagram analysis and some exact relations

Abstract

For pt.I see ibid., vol.3, p.4389, 1991. In this paper the causes of the 'mathematical breakdown' of the random-phase approximation RPA' are analysed. Starting from an exact matrix formula, a diagram analysis of a functional-integral approach is developed with the following characteristics: (i) All the singularities of the integrands are cancelled completely; there is no problem of the limitation from the convergence radius of the formula used. (ii) The reality of the partition functions at every stage is guaranteed and all the benefits of the complex representation are preserved simultaneously. (iii) All the functional-integral series are transformed into two-dimensional integral series. (iv) A functional-integral approach, which can calculate the mixed mode contributions, is given for the first time. The diagrammatic rules of mixed mode contributions and some concrete examples are given. Some exact symmetry relations and expressions are suggested and proved. These symmetry relations are also preserved at every state in the author's diagram analysis. They are useful in practical calculations. A new exact relation is also derived.