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  "body": "\n## Summary\n\nThe Maslov index is an integer attached to a loop of Lagrangian subspaces of the symplectic vector space $\\R^{2n}$.\nIt is the class of the loop in the [[fundamental group|S1XK3HXHFX]] of the Lagrangian Grassmannian, which is isomorphic to $\\Z$.\nIt can be applied to the tangent planes of a [[Lagrangian submanifold|H8ACPTMX08]] along a loop.\nThen it counts with signs how often the loop crosses the [[Lagrangian singularities|QKPMR9MC9W]] of the projection to the base.\nIt enters the quantization rules of semiclassical mechanics, the index theory of Hamiltonian systems, and Floer homology.\n\n## Explanations\n\n### L1 human\n\nIn the plane $\\R^2$, every line through the origin is a Lagrangian subspace.\nTurn such a line around the origin until it comes back to itself; this takes half a turn.\nThe Maslov index counts how many such half turns a loop of lines makes, with a sign for the direction.\nIn higher dimensions the subspaces are $n$-dimensional, and the index adds up the turning of all the directions at once.\nFor a light beam or a quantum wave, each unit of Maslov index along a closed path shifts the phase by a fixed amount, which corrects the naive quantization rule.\n",
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