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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Perturbation Series At Large Orders. Summation Methods

Perturbation Series At Large Orders. Summation Methods

Quantum Field Theory and Critical Phenomena


Oxford University Press

Chapter 39 discussed the analytic structure of the ground state energy E(g) of the anharmonic oscillator. This chapter explains how the behaviour of Im E(g) for g - 0 is related to the behaviour of the coefficients Ek when the order k becomes large. It then generalizes the method to the class of potentials for which we have calculated instanton contributions. The same method can be readily applied to boson field theories, using the results of Chapter 40, while the extension to field theories involving fermions like QED requires some additional considerations. We already know that the expansion (42:1) is divergent for all values of g. This implies that, even for g small, the series does not determine the function E (g) uniquely. The chapter examines the implications of the large order behaviour for the problem of the summation of the series. Finally, it describes a few practical methods commonly used to sum divergent series of the type met in quantum mechanics and quantum field theory.

Keywords:   quantum mechanics, boson field theories, ground state energy

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