Eur. Phys. J. C (2009) 64: 495-527
https://doi.org/10.1140/epjc/s10052-009-1155-3

Regular Article - Theoretical Physics

Diagonal representation for a generic matrix valued quantum Hamiltonian

¹ Institut Fourier, UMR 5582 CNRS-UJF UFR de Mathématiques, Université Grenoble I, BP74, 38402, Saint Martin d’Hères Cedex, France
² Laboratoire de Physique Moléculaire et des Collisions, ICPMB-FR CNRS 2843, Université Paul Verlaine-Metz, 57078, Metz Cedex 3, France

^* e-mail: herve.mohrbach@univ-metz.fr

Received: 22 June 2009
Revised: 28 July 2009
Published online: 30 September 2009

Abstract

A general method to derive the diagonal representation for a generic matrix valued quantum Hamiltonian is proposed. In this approach new mathematical objects like non-commuting operators evolving with the Planck constant promoted as a running variable are introduced. This method leads to a formal compact expression for the diagonal Hamiltonian which can be expanded in a power series of the Planck constant. In particular, we provide an explicit expression for the diagonal representation of a generic Hamiltonian to the second order in the Planck constant. This result is applied, as a physical illustration, to Dirac electrons and neutrinos in external fields.