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arXiv · 1612.00462

A robust generalization of the Legendre transform for QFT

David M. Jackson·Achim Kempf·Alejandro H. Morales
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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.

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David M. Jackson, Achim Kempf, Alejandro H. Morales. 2017-05-31. A robust generalization of the Legendre transform for QFT. https://doi.org/10.1088/1751-8121%2Faa6abb

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