Geometrical properties of Maslov indices in periodic-orbit theory

Maslov indices in periodic-orbit theory are investigated using phase space path integral. Based on the observation that the Maslov index is the multi-valued function of the monodromy matrix, we introduce a generalized monodromy matrix in the universal covering space of the symplectic group and show that this index is uniquely determined in this space. The stability of the orbit is shown to determine the parity of the index, and a formula for the index of the n-repetition of the orbit is derived. PACS numbers: 05.45.Mt, 02.40.Ma Over the past few decades a large number of studies have been made on periodic-orbit theory [1]. However, the Maslov index, which is an additional phase factor appearing in periodic orbit theory, doesn’t seem to be thoroughly understood. For hyperbolic orbits, Robbins [2] showed that these indices are the winding numbers which are defined by the invariant Lagrangian manifolds around these orbits. Moreover, it was conjectured that the same argument could be extended to elliptic orbits and more general orbits which have mixed stability. However, this is not the case since general orbits don’t necessarily have such invariant manifolds around them. Furthermore, Brack et al. [3] investigated periodic orbits in anisotropic harmonic oscillator (these orbits are elliptic), and showed the Maslov index of the n-repetition of the orbit μn is not equal to nμ1. This result contradicts Robbins’s conjecture, which leads to μn = nμ1. In this letter, we propose a new approach to the problem which can be applied to all periodic orbits, irrespective of the type of the stability. Our method [email protected]