A robust generalization of the Legendre transform for QFT

Abstract

Although perturbative quantum field theory is highly successful, it possesses a number of well-known analytic problems, from ultraviolet and infrared divergencies to the divergence of the perturbative expansion itself. As a consequence, it has been difficult, for example, to prove with full rigor that the Legendre transform of the quantum effective action is the generating functional of connected graphs. Here, we give a rigorous proof of this central fact. To this end, we show that the Legendre transform can be re-defined purely combinatorially and that it ultimately reduces to a simple homological relation, the Euler characteristic for tree graphs. This result suggests that, similarly, also the quantum field theoretic path integral, being a Fourier transform, may be reducible to an underlying purely algebraic structure.

13 pages, 5 figures

References in corpus (3)

• Lagrange Inversion
• New Dirac Delta function based methods with applications to perturbative expansions in quantum field theory
• How to (Path-) Integrate by Differentiating

Cited by in corpus (6)

• Perturbation theory of transformed quantum fields
• Graph rules for the linked cluster expansion of the Legendre effective action
• An exact solution method for the enumeration of connected Feynman diagrams
• Legendre Transformation of the Luttinger-Ward Functional from the Bare Interaction Vertex to the Renormalized One
• Geometric Structures Induced by Deformations of the Legendre Transform
• Diagrammatics of free energies with fixed variance for high-dimensional data