Maslov Index

The Maslov index is an integer phase correction in semiclassical mechanics. It records signed phase changes from caustic crossings, including their multiplicities; in the simplest one-dimensional setting each ordinary crossing contributes one. A raw count of geometric intersections is not the general definition. In the convention used here, each unit of Maslov index contributes the phase factor

$$\exp\left( - i\frac{\pi}{2} \right)$$

to a transported semiclassical branch.

The index is the bookkeeping device behind the $1/2$ in one-dimensional WKB quantization and the phase factor in semiclassical propagators. It is not an extra empirical rule. It is what remains when a local WKB branch is continued through places where its naive amplitude would become singular.

Required background. Turning-point matching supplies the local Airy phase, and Bohr–Sommerfeld quantization combines the two boundary phases.

Caustic phase corrections

The leading WKB form in an allowed region is

$$\psi(x) \approx \frac{C_+}{\sqrt{p(x)}} e^{(i/\hbar)\int^x p(x')\,dx'} + \frac{C_-}{\sqrt{p(x)}} e^{-(i/\hbar)\int^x p(x')\,dx'}.$$

At a turning point $p(x)=0$, this expression fails. The exact local equation is repaired by Airy matching, as described on Turning Points and Connection Formulas. The result is a $\pi/4$ offset for the standing wave attached to a decaying tail. In traveling-wave language reflection at that turning point has amplitude $r=-i$, a phase loss of $\pi/2$. These are compatible descriptions, not two shifts to add separately.

When the wave is carried around a complete closed orbit, these local phase losses accumulate. The Maslov index packages the accumulated caustic phase into a single integer.

One-Dimensional Smooth Well

For a smooth one-dimensional bound state, the classical closed orbit has two turning points. The EBK phase condition can be written

$$\frac{1}{\hbar} \oint p\,dx - \frac{\pi}{2}\mu = 2\pi n.$$

The two ordinary turning points give

$$\mu=2.$$

Therefore

$$\oint p\,dx = 2\pi\hbar \left( n+\frac12 \right).$$

This is the modern WKB version of the Bohr-Sommerfeld rule. The $1/2$ is the two-turning-point Maslov correction.

Caustics and Branch Changes

A caustic is a place where the projection from a Lagrangian manifold in phase space down to configuration space becomes singular. In one-dimensional WKB this is visible as a turning point. In multidimensional semiclassics, caustics occur when nearby classical trajectories focus, so the configuration-space amplitude predicted by a single branch diverges.

The divergence is not physical. It signals that the chosen coordinate representation is a bad local description of the semiclassical wave. Passing through the caustic changes the branch of the square root in the semiclassical amplitude, and that branch change is tracked by the Maslov phase.

This is closely analogous to geometrical optics. Ray amplitudes diverge at focal caustics, but the wave field remains finite when described by an appropriate uniform approximation.

EBK Convention

For an integrable system, the EBK rule around a cycle $\gamma_j$ is

$$J_j = \oint_{\gamma_j}p\cdot dq = 2\pi\hbar \left( n_j+\frac{\mu_j}{4} \right).$$

Equivalently,

$$\frac{J_j}{\hbar} - \frac{\pi}{2}\mu_j = 2\pi n_j.$$

This equation is often the least ambiguous way to remember the convention. If another source defines the Maslov index with the opposite sign, the final quantization rule can still agree after translating conventions.

Propagator Convention

For the semiclassical propagator, the Maslov index appears as a phase attached to each classical trajectory:

$$K_{\mathrm{sc}}(q_b,t_b;q_a,t_a) \sim \sum_\gamma A_\gamma \exp\left[ \frac{i}{\hbar}S_\gamma - i\frac{\pi}{2}\nu_\gamma \right].$$

For a positive-time scalar Schrödinger propagator with positive kinetic energy, take $A_\gamma=(2\pi\hbar)^{-d/2}|D_\gamma|^{1/2}e^{-i\pi d/4}$ and $\nu_\gamma=0$ on a sufficiently short nonfocal branch. Subsequent conjugate points increment $\nu_\gamma$ with their multiplicity. This separates the initial free Gaussian phase from the later caustic phases. The same geometric phenomenon is being measured, but notation varies across EBK, WKB, Morse theory, and trace-formula literature. The safe habit is to state the phase convention along with the index.

Harmonic-Oscillator Example

The one-dimensional harmonic oscillator propagator has caustics at times

$$T = \frac{k\pi}{\omega}, \qquad k\in\mathbb Z,$$

because all trajectories refocus after half periods. For $T>0$ away from these times, $\nu=\lfloor\omega T/\pi\rfloor$ in the convention above. At a focal time the kernel is a distribution, not a finite function:

$$K(x_b,k\pi/\omega;x_a,0) =e^{-ik\pi/2}\delta\bigl(x_b-(-1)^kx_a\bigr).$$

This follows directly from $E_n=\hbar\omega(n+1/2)$ and parity $(-1)^n$. Acting on a regular wavefunction gives its reflected or restored profile with a finite phase; an infinite isolated prefactor is not a formula for the kernel at the focus.

For bound-state quantization, the same oscillator has two turning points on each full orbit, so $\mu=2$. The WKB rule gives

$$E_n = \hbar\omega \left( n+\frac12 \right),$$

which is exact for the oscillator.

Exercises

1. What is the one-dimensional oscillator evolution operator after one full classical period? Why would omitting the Maslov phases give the wrong answer?

Solution

At $T=2\pi/\omega$, every eigenstate acquires $e^{-i2\pi(n+1/2)}=-1$, so $U(T)=-I$. Two focal crossings contribute $(-i)^2=-1$. Classical phase-space points return to themselves, but the quantum propagator retains this phase, which matters when compared coherently with another evolution.

2. An isotropic oscillator in $d$ dimensions focuses all $d$ coordinates at a half period. What is the focal phase?

Solution

The conjugate point has multiplicity $d$ and contributes $e^{-i\pi d/2}$. The evolution is that factor times full spatial inversion. Counting it as one ordinary one-dimensional crossing would miss the multiplicity.

References

• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315-397, 1972.
• M. Brack and R. K. Bhaduri, Semiclassical Physics, Westview Press, 2003.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Reidel, 1981.