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Exact and approximate dynamics of the quantum mechanical O ( N ) model

Exact and approximate dynamics of the quantum mechanical O ( N ) model We study the dynamics of the quantum mechanical O ( N ) model as a specific example to investigate the systematics of a 1 / N expansion. The closed time path formalism melded with an expansion in 1 / N is used to derive time evolution equations valid to order 1 / N (next-to-leading order). The effective potential is also obtained to this order and its properties are elucidated. In order to compare theoretical predictions against numerical solutions of the time-dependent Schrödinger equation, we consider two initial conditions consistent with O ( N ) symmetry, one of them a quantum roll, the other a wave packet initially to one side of the potential minimum, whose center has all coordinates equal. For the case of the quantum roll we map out the domain of validity of the large- N expansion. We also discuss the existence of unitarity violation in this expansion, a well-known problem faced by moment truncation techniques. The 1 / N results, both static and dynamic, are contrasted with those given by a Hartree variational ansatz at given values of N . A comparison against numerical results leads us to conclude that late-time dynamical behavior, where nonlinear effects are significant, is not well described by either approximation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review D American Physical Society (APS)

Exact and approximate dynamics of the quantum mechanical O ( N ) model

16 pages

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References (1)

Publisher
American Physical Society (APS)
Copyright
Copyright © 2000 The American Physical Society
ISSN
1089-4918
DOI
10.1103/PhysRevD.62.125015
Publisher site
See Article on Publisher Site

Abstract

We study the dynamics of the quantum mechanical O ( N ) model as a specific example to investigate the systematics of a 1 / N expansion. The closed time path formalism melded with an expansion in 1 / N is used to derive time evolution equations valid to order 1 / N (next-to-leading order). The effective potential is also obtained to this order and its properties are elucidated. In order to compare theoretical predictions against numerical solutions of the time-dependent Schrödinger equation, we consider two initial conditions consistent with O ( N ) symmetry, one of them a quantum roll, the other a wave packet initially to one side of the potential minimum, whose center has all coordinates equal. For the case of the quantum roll we map out the domain of validity of the large- N expansion. We also discuss the existence of unitarity violation in this expansion, a well-known problem faced by moment truncation techniques. The 1 / N results, both static and dynamic, are contrasted with those given by a Hartree variational ansatz at given values of N . A comparison against numerical results leads us to conclude that late-time dynamical behavior, where nonlinear effects are significant, is not well described by either approximation.

Journal

Physical Review DAmerican Physical Society (APS)

Published: Dec 15, 2000

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