The Hamilton-Jacobi theory and the analogy between classical andquantum mechanics

Giuseppe Marmo; Giuseppe Morandi; Narasimhaiengar Mukunda

doi:10.3934/jgm.2009.1.317

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2009, Volume 1, Issue 3: 317-355. Doi: 10.3934/jgm.2009.1.317

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The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics

1.
Dipartimento di Scienze Fisiche and INFN, Università di Napoli "Federico II". v. Cintia. I-80126 Naples
2.
Dipartimento di Fisica and INFN, Università di Bologna. 6/2 v.le B.Pichat. I-40127 Bologna
3.
Centre for Theoretical Studies, Indian Institute of Science. Bangalore 560 012

Received: February 2009

Revised: April 2009

Published: September 2009

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Abstract

We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi ($HJ$) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the $HJ$ theory on the tangent bundle of a configuration manifold, the quantum $HJ$ theory, $HJ$ problems for general differential operators and the $HJ$ problem for Lie groups.

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Mathematics Subject Classification: 70H20, 70S05, 70FXX, 70HXX.

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