MT01 — Maslov Dequantisation

Field Value
Domain Mathematical Methods
System WKB / stationary phase as ℏ → 0
Group Sp(2n, ℝ) (symplectomorphisms of phase space)
H^k tier H⁰
ISA Origami (β → ∞)
Status Validated
Opcodes ORBIT · TWIST · LABEL
Papers Paper 543, Paper 201, Paper 477

Physical system

Maslov dequantisation is the passage ℏ → 0 from quantum wave mechanics to classical geometric optics, made algebraically precise: complex amplitudes become tropical quantities (piecewise-linear phase functions), quantum superposition becomes classical max, and interference becomes argmax. The algebraic transition is:

ℏ → 0:  (ℂ, +, ×)  →  (ℝ ∪ {−∞}, max, +)

The complex semiring of amplitudes deforms into the tropical semiring of actions. This is the MGE in the β → ∞ limit. The MGE Gibbs weight exp(−βE) tropicalises to argmax(−E) as β → ∞, selecting the minimum-energy orbit — the classical ground state.

In field theory: Maslov dequantisation explains why QFT path integrals reduce to classical field equations in the limit ℏ → 0 (saddle-point / stationary-phase approximation). The classical trajectory is the saddle point of the action; the quantum amplitude is the sum over all paths weighted by exp(iS/ℏ); as ℏ → 0, the sum is dominated by the saddle — the tropical maximum.

Target category

Symp(2n, ℝ) — the symplectic category whose objects are cotangent bundles T*Q (phase spaces) and whose morphisms are canonical transformations (Lagrangian correspondences). The semiclassical limit maps:

  • Quantum states (wavefunctions) → Lagrangian submanifolds of T*Q
  • Quantum observables → classical Hamiltonians on T*Q
  • Quantum time evolution → Hamiltonian flow (symplectomorphism)

Interpretation functor

F: C → Symp(2n, ℝ) defined by:

Opcode F(opcode)
ORBIT Classical trajectory: (q(t), p(t)) = flow of Hamilton’s equations; the tropical fixed point of the quantum ORBIT
TWIST Maslov index correction μ ∈ ℤ at caustics: the half-integer phase shift ψ → ψ · e^{iμπ/2} that restores the correct WKB phase when characteristics cross; the H¹ correction to H⁰
LABEL Action eigenvalue S[q] = ∫ p dq − H dt: the tropical polynomial whose argmax selects the classical path; caustic locus where ∂²S/∂q² = 0

ISA programme 

ACTION:   LABEL[S[q] = int(p dq - H dt)]            -- compute action along each path
SADDLE:   ORBIT[delta_S = 0 | Hamilton's eqs]       -- find stationary-phase path
CAUSTIC?: LABEL[det(d^2S/dq^2) = 0?]               -- caustic = stationary phase failure
MASLOV:   TWIST[mu += 1 | per caustic crossing]     -- Maslov index increment
CORRECT:  TWIST[psi *= exp(i mu pi/2)]              -- WKB phase correction
OUTPUT:   ORBIT[psi_WKB = A(q) exp(iS(q)/hbar + i mu pi/2)]  -- corrected amplitude

Computable output

  • Classical trajectory: the path q(t) solving Hamilton’s equations. This is the β → ∞ (tropical) output of the MGE: the exact ground state orbit, selected by argmin of the action.
  • WKB amplitude: ψ_WKB(q) = A(q) e^{iS(q)/ℏ + iμπ/2}, where A(q) = |det(∂²S/∂q∂q₀)|^{−1/2} is the Van Vleck determinant (the H⁰ density of trajectories) and μ is the Maslov index (the H¹ TWIST correction).
  • Maslov index μ ∈ ℤ: the integer counting the number of caustic crossings along the classical path. μ is the topological invariant of the path in Symp: it counts how many times the Lagrangian submanifold L = {(q, ∂_q S)} has become vertical (∂q/∂p₀ = 0). The Maslov index is the output of the TWIST opcode: it is the H¹ winding number of the Lagrangian path in the Lagrangian Grassmannian Λ(n) = U(n)/O(n).

  • Semiclassical spectrum: the Bohr-Sommerfeld quantisation condition, corrected by the Maslov index, gives the energy levels:

    ∮ p dq = 2πℏ(n + μ/4), n ∈ ℤ

    This is the tropical (H⁰) action quantised by the H¹ TWIST correction μ/4.

The β-plane interpretation (Paper 543)

Maslov dequantisation is the Origami ISA limit of the Meld ISA:

β-plane position ISA Physical meaning Mathematical object
β = it, t large Meld Full quantum Path integral over all paths
β = it → ∞ Meld → Origami boundary WKB regime Stationary phase approximation
β → ∞, real Origami Classical Single classical trajectory
Caustic TWIST event Maslov index H¹ correction at H⁰ breakdown

The Wick rotation and Maslov: rotating β from the imaginary axis (Meld) to the real axis (Origami) is Maslov dequantisation. The β = it → β = T (real) passage corresponds to analytic continuation of the path integral from Minkowski (oscillatory, quantum) to Euclidean (decaying, classical). The Euclidean saddle is the instanton (G01); the Minkowski saddle is the classical trajectory. The soliton (D06) is the β → ∞ (Origami, real) limit; the instanton (G01) is the β → 0 (Meld, imaginary) limit. The Maslov index is the interpolation.

Viscosity and ν = 1/β: the viscous Burgers equation (D07) implements Maslov dequantisation in fluid dynamics: ν plays the role of ℏ. As ν → 0, the smooth viscous flow dequantises to the discontinuous shock (tropical). The Rankine- Hugoniot condition is the fluid-dynamic Maslov index — the H¹ TWIST correction at the caustic of characteristics.

Why this entry belongs in the zoo

Maslov dequantisation is not a physical system — it is the universal meta- theorem connecting every Meld-ISA entry to its Origami-ISA limit. But it deserves a zoo entry because:

  1. It is a concrete, computable object. The Maslov index is an integer computable from the trajectory, not an abstraction. The WKB formula with μ correction gives quantitatively accurate energy levels.
  2. It pins down what “β → ∞” means in physics. The claim that Origami ISA is the classical limit of Meld ISA is not a vague analogy; it is the WKB theorem with the Maslov correction, proved by Maslov (1965) and made rigorous by Hörmander (1971).
  3. It is the source of the TWIST opcode. The Maslov index is the original TWIST: the first known case where a pure H⁰ computation (ORBIT + LABEL) fails at a caustic and needs an H¹ correction (TWIST). Every other TWIST in the zoo is a descendant of this one.

Validation

  • WKB approximation: Wentzel (1926), Kramers (1926), Brillouin (1926). Gives correct energy levels to O(ℏ²) for smooth potentials.
  • Maslov index correction: Maslov (1965). Removes the WKB divergence at caustics; gives correct half-integer quantum numbers for harmonic oscillator (μ = 2 per period → n + ½ levels).
  • Lagrangian intersection Floer homology: the Maslov index is the grading in Floer theory — the mathematical descendant of the original physics, rigorous in infinite dimensions. Floer (1988).
  • Hormander’s microlocal analysis: the full rigorous version of the WKB theorem in any dimension, including the Maslov index as a class in H¹(Λ(n), ℤ).

Part of the ISA Zoo. Categorical foundations: Paper 591.