The exponential of a Hamiltonian matrix is symplectic

Proved

UnQuantumMechanics.exp_hamiltonian_symplectic

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

hamiltonian-mechanicslinear-algebramathematical-physics

Let $H$ be a Hamiltonian $2n\times 2n$ matrix and $\tau\in\mathbb R$ . Then $M(\tau) = \exp(H\tau)$ is symplectic:

M(\tau)\,\gamma_0\,M(\tau)^T = \gamma_0 .

This is the statement that linear Hamiltonian time evolution preserves the symplectic structure.

Formalization Note $\exp$ is the matrix exponential (power series).

Preamble

import Mathlib
import Definitions.Def_UnQM_phase_space

open Matrix

Formal statement

namespace UnQuantumMechanics

theorem exp_hamiltonian_symplectic (n : ℕ) (H : Matrix (PhaseIdx n) (PhaseIdx n) ℝ) (hH : IsHamiltonianMatrix n H) (τ : ℝ) :
    IsSymplecticMatrix n (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, Eqs. (49)–(50); Sec. VI.B, p. 17

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 $2n\times 2n$ matrix $H$ for which there is a symmetric $A$ with $H=\gamma_0A$ , and every real $\tau$ : the matrix $M = \exp(\tau H)$ (power-series matrix exponential) satisfies $M\,\gamma_0\,M^T = \gamma_0$ .

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).

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