---
tag: 91KJT4TXG8
kind: concept
name: Maslov index
msc: [53D12]
sources:
  - url: https://en.wikipedia.org/wiki/Maslov_index
same_as:
  wikidata: Q78514551
  wikipedia: Maslov index
---

## Summary

The Maslov index is an integer attached to a loop of Lagrangian subspaces of the symplectic vector space $\R^{2n}$.
It is the class of the loop in the [[fundamental group|S1XK3HXHFX]] of the Lagrangian Grassmannian, which is isomorphic to $\Z$.
It can be applied to the tangent planes of a [[Lagrangian submanifold|H8ACPTMX08]] along a loop.
Then it counts with signs how often the loop crosses the [[Lagrangian singularities|QKPMR9MC9W]] of the projection to the base.
It enters the quantization rules of semiclassical mechanics, the index theory of Hamiltonian systems, and Floer homology.

## Explanations

### L1 human

In the plane $\R^2$, every line through the origin is a Lagrangian subspace.
Turn such a line around the origin until it comes back to itself; this takes half a turn.
The Maslov index counts how many such half turns a loop of lines makes, with a sign for the direction.
In higher dimensions the subspaces are $n$-dimensional, and the index adds up the turning of all the directions at once.
For 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.