Kastening, Boris (2000)
Recursive graphical construction of Feynman diagrams in φ^{4} theory: Asymmetric case and effective energy.
In: Physical Review E, 61 (4)
doi: 10.1103/PhysRevE.61.3501
Artikel, Bibliographie

Kurzbeschreibung (Abstract)

The free energy of a multicomponent scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys nonlinear functional differential equations which are turned into recursion relations for the connected Green’s functions in a loop expansion. These relations amount to a simple proof that W[G,J] generates only connected graphs and can be used to find all such graphs with their combinatoric weights. A Legendre transformation with respect to the external current converts the functional differential equations for the free energy into those for the effective energy G[G,φ], which is considered as a functional of the free correlation function G and the field expectation φ. These equations are turned into recursion relations for the one-particle irreducible Green’s functions. These relations amount to a simple proof that G[G,J] generates only one-particle irreducible graphs and can be used to find all such graphs with their combinatoric weights. The techniques used also allow for a systematic investigation into resummations of classes of graphs. Examples are given for resumming one-loop and multiloop tadpoles, both through all orders of perturbation theory. Since the functional differential equations derived are nonperturbative, they constitute also a convenient starting point for other expansions than those in numbers of loops or powers of coupling constants. We work with general interactions through four powers in the field.

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