Diagram Expansions in Quantum Statistics

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Abstract
The Montroll‐Ward‐Lee‐Yang approach to quantum statistics is generalized to multicomponent systems. It is also generalized so as to include external fields. The formalism is constructed in a volume‐dependent manner, and includes internal coordinates, such as spin, from the beginning. It is rigorously proved that the quantum‐mechanical volume‐dependent cluster integral may be expressed in terms of connected diagrams only. The rules for drawing these diagrams are given. By simply generalizing the meaning of the word ``determinant,'' all arguments are made to apply to both Fermi‐Dirac and Bose‐Einstein statistics simultaneously. A statistics factor, γ = ±1, for bosons (fermions) is introduced, in terms of which single formulas apply to both statistics. Rules are stated, by means of which the γ dependence of the contribution to the pressure for any diagram is given in terms an elementary topological property of the diagram.

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