On the Divergence of Perturbation Theory. Steps Towards a Convergent Series

Abstract

The mechanism underlying the divergence of perturbation theory is exposed. This is done through a detailed study of the violation of the hypothesis of the Dominated Convergence Theorem of Lebesgue using familiar techniques of Quantum Field Theory. That theorem governs the validity (or lack of it) of the formal manipulations done to generate the perturbative series in the functional integral formalism. The aspects of the perturbative series that need to be modified to obtain a convergent series are presented. Useful tools for a practical implementation of these modifications are developed. Some resummation methods are analyzed in the light of the above mentioned mechanism.

42 pages, Latex, 4 figures

References in corpus (6)

• Improved Convergence Proof of the Delta Expansion and Order Dependent Mappings
• Convergence of the Optimized Delta Expansion for the Connected Vacuum Amplitude -- Anharmonic Oscillator
• Variational Calculation of Effective Classical Potential at $T\ne 0$ to Higher Orders
• Variational Solution of the Gross-Neveu Model: finite $N$ and Renormalization
• Convergent sequences of perturbative approximations for the anharmonic oscillator I. Harmonic approach
• Scaling property of variational perturbation expansion for general anharmonic oscillator

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