Maslov indices, Poisson brackets, and singular differential forms

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.

5 pages

References in corpus (3)

• Discreteness of the volume of space from Bohr-Sommerfeld quantization
• The Ponzano-Regge asymptotic of the 6j symbol: an elementary proof
• Closed-Orbit Theory of Spatial Density Oscillations in Finite Fermion Systems

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• Four-dimensional Quantum Gravity with a Cosmological Constant from Three-dimensional Holomorphic Blocks
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Exact relations between homoclinic and periodic orbit actions in chaotic systems
• Geometric determination of heteroclinic and unstable periodic orbit classical actions