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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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Multi-Instantons In Quantum Mechanics

Multi-Instantons In Quantum Mechanics

Chapter:
(p.1020) 43 MULTI-INSTANTONS IN QUANTUM MECHANICS
Source:
Quantum Field Theory and Critical Phenomena
Author(s):

JEAN ZINN-JUSTIN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509233.003.0043

The concepts of multi-instanton configurations and contributions are non-trivial because classical equations are non-linear and thus in general a linear combination of solutions is not a solution. However, in the limit in which all instantons are largely separated, such configurations render the action almost stationary because each instanton solution differs from a constant solution only by exponentially small corrections at large distances (in a field theory this is only true if the theory is massive). This chapter examines in the context of Quantum Mechanics the significance of such multi-instanton quasi-solutions. Sections 43.1, 43.2, studies explicitly two examples which we have already considered in Chapter 41: the double-well potential and the periodic cosine potential. It then discusses general potentials with degenerate minima. It also calculates the large order behaviour in the case of the O(ν) symmetric anharmonic oscillator.

Keywords:   double-well potential, periodic cosine potential, symmetric anharmonic oscillator

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