the Maslov index
/ MAH-slov /
Picture a loop of Lagrangian planes — the half-dimensional 'omega-blind' subspaces — turning continuously and eventually returning to where it started. As it turns, it can wind around the space of all such planes some whole number of times. The Maslov index is that integer winding number. It is the symplectic-geometry counterpart of the winding number of a closed curve around the origin, and it shows up as a topological correction term whenever you push a wave or a quantum state along a classical trajectory.
Precisely, because the Lagrangian Grassmannian Lambda(n) has fundamental group pi_1(Lambda(n)) = Z, every loop of Lagrangian subspaces gamma: S^1 -> Lambda(n) has a well-defined homotopy class, an integer mu(gamma) = [gamma] in Z. This integer is the Maslov index of the loop. There are several refinements built on the same idea: the Maslov index of a path of Lagrangians relative to a fixed reference Lagrangian (counting signed crossings of the 'train' where the moving plane meets the reference plane nontrivially), and the Maslov class of a Lagrangian submanifold L in (M, omega), an element of H^1(L; Z) obtained by trivializing the bundle of tangent Lagrangian planes against a chosen background. In each version the count is intrinsically about how Lagrangian planes rotate.
Why it matters: the Maslov index is the integer that fixes phases in semiclassical (WKB) approximation and supplies the famous +1/2 corrections to Bohr-Sommerfeld quantization — the reason the harmonic oscillator's energy levels are (n + 1/2) hbar omega and not n hbar omega. In modern symplectic topology it provides the grading of Floer homology and controls the dimension (the expected Fredholm index) of moduli spaces of pseudoholomorphic curves with Lagrangian boundary. A common confusion is to treat it as a single number for a Lagrangian: the relevant object depends on what you measure — a loop gives an integer, a path gives a (possibly half-integer in some conventions) relative index, and a submanifold gives a cohomology class — so always specify which Maslov index is meant.
For the harmonic oscillator in (R^2, dp ^ dq), one period of the motion rotates every tangent line through 360 degrees, so the loop of tangent Lagrangian lines (here Lambda(1) = S^1) winds around exactly twice, giving Maslov index 2. Fed into the Bohr-Sommerfeld rule, this Maslov 2 produces the +1/2 zero-point shift, recovering the exact quantum levels E_n = (n + 1/2) hbar omega.
The oscillator loop has Maslov index 2; this is the source of the +1/2 in (n + 1/2) hbar omega.
There is no single 'the' Maslov index — loop, path-relative, and submanifold (cohomology-class) versions all carry the name and differ. Sign and normalization conventions also vary across authors (some count crossings with half-integer weights), so cite the convention.