Maslov indices, Poisson brackets, and singular differential forms

¹ Perimeter Institute for Theoretical Physics - 31 Caroline Street North Waterloo, Ontario Canada N2L 2Y5
² Aix Marseille Université, CNRS, CPT, UMR 7332 - 13288 Marseille, France
³ Université de Toulon, CNRS, CPT, UMR 7332 - 83957 La Garde, France
⁴ Department of Physics, University of California - Berkeley, CA 94720-7300, USA

Received: 11 March 2014
Accepted: 12 May 2014

Abstract

Maslov indices are integers that appear in semiclassical wave functions and quantization conditions. They are often notoriously difficult to compute. We present methods of computing the Maslov index that rely only on typically elementary Poisson brackets and simple linear algebra. We also present a singular differential form, whose integral along a curve gives the Maslov index of that curve. The form is closed but not exact, and transforms by an exact differential under canonical transformations. We illustrate the method with the 6j-symbol, which is important in angular-momentum theory and in quantum gravity.