Solutions of $\dot S = [H,S]$ are symplectic similarity transforms

Proved

UnQuantumMechanics.heisenberg_solution_symplectic_similarity

by Lucas · Oct 2, 2026 · Mathlib 0df444a (Lean v4.33.1)

hamiltonian-mechanicslinear-algebramathematical-physics

Let $H$ be a Hamiltonian $2n\times 2n$ matrix (constant in time) and let $S:\mathbb R\to\mathbb R^{2n\times2n}$ be differentiable at every time and satisfy Heisenberg's equation $\dot S = HS - SH$ . Then for every $\tau$

S(\tau) = \exp(H\tau)\,S(0)\,\exp(-H\tau).

Since $\exp(H\tau)$ is symplectic (Eq. 50), this shows that time evolution acts on $S$ by symplectic similarity transformations.

Formalization Note Derivatives are entrywise; $\exp$ is the matrix exponential.

Preamble

import Mathlib
import Definitions.Def_UnQM_phase_space

open Matrix

Formal statement

namespace UnQuantumMechanics

theorem heisenberg_solution_symplectic_similarity (n : ℕ) (H : Matrix (PhaseIdx n) (PhaseIdx n) ℝ)
    (hH : IsHamiltonianMatrix n H) (S : ℝ → Matrix (PhaseIdx n) (PhaseIdx n) ℝ)
    (hS : ∀ τ i j, HasDerivAt (fun t => S t i j) ((H * S τ - S τ * H) i j) τ) (τ : ℝ) :
    S τ = NormedSpace.exp (τ • H) * S 0 * NormedSpace.exp ((-τ) • H) := by sorry

end UnQuantumMechanics

Source

C. Baumgarten, How to (Un-) Quantum Mechanics, arXiv:1810.06981v2 [physics.gen-ph] (v2 dated June 10, 2025), https://arxiv.org/abs/1810.06981, Sec. VI, p. 17, Eq. (51)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent that drafted the statements; non-blind

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statement, at the explicit direction of the proposal owner. The author knew the intended meaning and the source text while writing it, so it must not be mistaken for a blind, independent audit. An independent blind read-back is still recommended before submission.

For every natural number $n$ , every real $2n\times2n$ matrix $H$ for which there is a symmetric $A$ with $H=\gamma_0A$ , every function $S:\mathbb R\to\mathbb R^{2n\times2n}$ such that for every $t$ and all indices $i,j$ the function $s\mapsto S(s)_{ij}$ is differentiable at $t$ with derivative $\big(HS(t)-S(t)H\big)_{ij}$ , and every real $\tau$ : $S(\tau) = \exp(\tau H)\,S(0)\,\exp((-\tau)H)$ , with $\exp$ the power-series matrix exponential.

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Solutions of S˙=[H,S]\dot S = [H,S]S˙=[H,S] are symplectic similarity transforms

Solutions of $\dot S = [H,S]$ are symplectic similarity transforms