A robust generalization of the Legendre transform for QFT

作者：Jackson D M; Kempf A; Morales A H^*

来源：Journal of Physics A-Mathematical and Theoretical, 2017, 50(22): 225201.

DOI：10.1088/1751-8121/aa6abb

• Legendre transform
• quantum effective action
• Lagrange inversion
• tree enumeration

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, the quantum field theoretic path integral, being a Fourier transform, may also be reducible to an underlying purely algebraic structure.

• 出版日期2017-6-2
• 单位UCLA