C. Hinsley

15 November 2025

I’d like to try to use the mathematical apparatus of quantum mechanics to help getting insights about the Koopman operator formalism in dynamical systems, so I set out to find first what the correspondence between Koopman operators for classical systems and quantum mechanics is. This article is a brief exposition of how to set up this correspondence.

Koopman flows in Hamiltonian systems

Let $H$ be a Hamiltonian on a phase space $M$ with vector field $X_H = \{\cdot, H\}$. Let $\mathcal{H}$ denote the Hilbert space of complex-valued $L^2$-integrable smooth functions on $M$, with inner product $\langle f, g \rangle = \int f^\dagger g\ dq\ dp$. Let $\hat{\mathcal{L}} = iX_h$ denote the Liouville operator (the “Liouvillian”) on $\mathcal{H}$. For $f \in \mathcal{H}$, we have by the chain rule

$$ \dot{f} = X_Hf; $$

it follows that

$$ f(t) = \exp(tX_H)f(0). $$

Hence the Koopman flow $U_t: \mathcal{H} \to \mathcal{H}$ is given by $U_t = \exp(tX_h) = \exp(-it\hat{\mathcal{L}})$, the solution of the evolution equation

$$ i\dot{f} = \hat{\mathcal{L}}f. $$

In particular, because the Hamiltonian vector field (which is the infinitesimal generator the Koopman flow) is skew-Hermitian (i.e., the Liouvillian $\hat{\mathcal{L}} = iX_H$ is Hermitian), the Koopman flow is unitary.

Proof that the Hamiltonian vector field is skew-Hermitian

Recall that Hamiltonian flows preserve phase-space measure (i.e., $X_H$ has zero divergence). Let $f, g \in \mathcal{H}$ be compactly supported. Then

$$ \langle f, X_Hg \rangle = \int f^\dagger X_Hg\ dq\ dp = \int (f^\dagger X_Hg + gX_Hf^\dagger)\ dq\ dp - \int gX_Hf^\dagger\ dq\ dp $$

$$ = \int X_H(f^\dagger g)\ dq\ dp - \int (X_Hf)^\dagger g\ dq\ dp $$

$$ = \int (\mathrm{div}(f^\dagger gX_H) - f^\dagger g\ \cancel{\mathrm{div}(X_H))}\ dq\ dp - \langle X_Hf, g\rangle $$

$$ = -\langle X_Hf, g \rangle $$

because $\int \mathrm{div}(f^\dagger gX_H)\ dq\ dp = 0$ by $f, g$ being compactly supported (divergence theorem). Hence $X_H^\dagger = -X_H$. $\square$