The Physics Travel Guide

Koopman-von-Neumann Mechanics

Prerequisites: Hamiltonian Formalism, Hilbert Space. Full path (14 pages)

Best explanations

Intuitive

Koopman-von-Neumann Mechanics is a reformulation of classical mechanics using the same language that we usually only use in quantum mechanics.

Usually in classical mechanics, our main focus is the trajectories of objects.

In contrast, in quantum mechanics, our main focus are observables, states and measurements. This means, instead of describing the trajectories of objects, we now ask instead for example: If we measure the momentum of this ball, what's the probability that the result will be 1010 kg m/s?

Concrete

Take a system with one degree of freedom and Hamiltonian H(q,p)H(q,p); the generalization to nn degrees of freedom replaces q,pq,p by qa,paq^a,p_a and sums over aa. In Koopman-von Neumann (KvN) mechanics the state is a complex function ψ(q,p,t)\psi(q,p,t) on phase space, square integrable with respect to dq dpdq\,dp and normalized, ∫dq dp ∣ψ(q,p,t)∣2=1.\int dq\,dp\, |\psi(q,p,t)|^2 = 1 . The probability density on phase space is ρ=∣ψ∣2\rho = |\psi|^2. The state evolves with a Schrödinger-type equation i∂ψ∂t=L^ψ,L^=−i(∂H∂p∂∂q−∂H∂q∂∂p).i\frac{\partial \psi}{\partial t} = \hat L \psi, \qquad \hat L = -i\left( \frac{\partial H}{\partial p}\frac{\partial}{\partial q} - \frac{\partial H}{\partial q}\frac{\partial}{\partial p} \right). L^\hat L is called the Liouvillian. No ℏ\hbar appears, because L^\hat L has units of inverse time.

Written out, the equation is ∂tψ=−{ψ,H}\partial_t \psi = -\{\psi, H\} with {f,g}=∂qf ∂pg−∂pf ∂qg\{f,g\} = \partial_q f\,\partial_p g - \partial_p f\,\partial_q g. The operator on the right is a first-order differential operator with real coefficients, so it obeys the product rule and ψ∗\psi^* satisfies the same equation. Therefore ∂ρ∂t=−{ρ,H},\frac{\partial \rho}{\partial t} = -\{\rho, H\}, which is the classical Liouville equation. So KvN reproduces ordinary classical statistical mechanics for ρ\rho. The phase of ψ\psi evolves separately and does not affect ρ\rho.

L^\hat L is Hermitian on L2(dq dp)L^2(dq\,dp). To see it, write L^=−i XH\hat L = -i\,X_H with the vector field XH=∂pH ∂q−∂qH ∂pX_H = \partial_p H\,\partial_q - \partial_q H\,\partial_p. This vector field has zero divergence, ∂q(∂pH)−∂p(∂qH)=0\partial_q(\partial_p H) - \partial_p(\partial_q H) = 0, so integrating by parts moves XHX_H onto the other function with a minus sign, and the factor −i-i turns that into Hermiticity. The evolution ψ(t)=e−iL^tψ(0)\psi(t) = e^{-i\hat L t}\psi(0) is therefore unitary.

Observables are functions f(q,p)f(q,p) acting as multiplication operators, with expectation values ⟨f⟩=∫dq dp f ∣ψ∣2\langle f\rangle = \int dq\,dp\, f\,|\psi|^2. In particular q^\hat q and p^\hat p are both multiplication operators and commute, [q^,p^]=0[\hat q,\hat p] = 0, so there is no uncertainty relation between position and momentum. One also introduces λ^q=−i∂q\hat\lambda_q = -i\partial_q and λ^p=−i∂p\hat\lambda_p = -i\partial_p, with [q^,λ^q]=[p^,λ^p]=i[\hat q,\hat\lambda_q] = [\hat p,\hat\lambda_p] = i, and writes L^=∂pH(q^,p^) λ^q−∂qH(q^,p^) λ^p\hat L = \partial_p H(\hat q,\hat p)\,\hat\lambda_q - \partial_q H(\hat q,\hat p)\,\hat\lambda_p. The λ^\hat\lambda are auxiliary: they generate the dynamics but are not the physical position or momentum.

Example: the free particle. For H=p2/2mH = p^2/2m, L^=−ipm∂q\hat L = -i\frac{p}{m}\partial_q, and the solution is ψ(q,p,t)=ψ(q−pt/m, p, 0).\psi(q,p,t) = \psi(q - pt/m,\, p,\, 0). Each momentum slice of the initial wave function moves rigidly with velocity p/mp/m. The density does the same, which is exactly a cloud of free classical particles. A wave packet in qq at fixed pp never spreads; the spread of ρ\rho in qq comes only from the spread in pp.

The equation of motion in Koopman-von-Neumann mechanics is a complexified reformulation of the classical Liouville equation (which handles probability distributions over the phase space).

Abstract

Let (P,ω)(P,\omega) be a symplectic manifold of dimension 2n2n with Liouville measure μ=ωn/n!\mu = \omega^n/n!, and let Φt\Phi_t be the flow of the Hamiltonian vector field XHX_H (assume it is complete). By Liouville's theorem Φt\Phi_t preserves μ\mu. Koopman's observation (1931) is that the composition operators (Utf)(x)=f(Φt(x))(U_t f)(x) = f(\Phi_t(x)) are then unitary on the Hilbert space L2(P,μ)L^2(P,\mu) and form a one-parameter group, Ut+s=UtUsU_{t+s} = U_t U_s. The group is strongly continuous, so by Stone's theorem Ut=eitL^U_t = e^{it\hat L} for a unique self-adjoint operator L^\hat L, which on smooth compactly supported functions acts as L^=−iXH\hat L = -iX_H. The KvN equation i∂tψ=L^ψi\partial_t\psi = \hat L\psi of the Concrete layer corresponds to ψt=U−tψ0=ψ0∘Φ−t\psi_t = U_{-t}\psi_0 = \psi_0\circ\Phi_{-t}. Since XHX_H is real, complex conjugation CC satisfies CL^C=−L^C\hat L C = -\hat L, so the spectrum of L^\hat L is symmetric about zero and L^\hat L is not bounded from below.

In this language a nonlinear finite-dimensional dynamical system becomes a linear, infinite-dimensional one. The observables of KvN mechanics, the functions f∈L∞(P,μ)f \in L^\infty(P,\mu) acting by multiplication, form a commutative von Neumann algebra, which is maximal abelian in B(L2(P,μ))B(L^2(P,\mu)). That is the algebraic difference to quantum mechanics: in the Schrödinger representation the position and momentum operators on L2(Q)L^2(Q) do not commute.

Dynamical properties of the flow become spectral properties of L^\hat L. Restrict to an energy surface ΣE=H−1(E)\Sigma_E = H^{-1}(E) with the induced invariant measure, normalized to a probability measure (assume EE is a regular value of HH and ΣE\Sigma_E is compact). Then: - The flow is ergodic if and only if the eigenvalue 00 of L^\hat L on L2(ΣE)L^2(\Sigma_E) is simple, i.e. the only invariant functions are constants. - The flow is weakly mixing if and only if L^\hat L has no eigenvectors other than constants, i.e. its spectrum on the orthogonal complement of the constants is purely continuous.

Von Neumann's mean ergodic theorem, proved in this setting, states that time averages 1T∫0TUtf dt\frac{1}{T}\int_0^T U_t f\,dt converge in L2L^2 to the projection of ff onto the kernel of L^\hat L. On the full phase space every function of HH is invariant, which is why one restricts to energy surfaces.

The same construction works for any measure-preserving flow, and the Koopman operator of a general (dissipative) dynamical system on spaces such as L2L^2 or C(X)C(X) is used in applied dynamical systems theory, where it is no longer unitary.

Why is it interesting?

The Koopman-von-Neumann reformulation of classical mechanics is extremely helpful to understand the differences between classical and quantum mechanics. Usually these theories are described in completely different languages which makes it hard to compare them. Using the Koopman-von-Neumann reformulation we describe classical mechanics in exactly the same language that we use in quantum mechanics and thus can perfectly analyze what is different.

Ordinary mechanics must also be statistically formulated: the determinism of classical physics turns out to be an illusion, it is an idol, not an ideal in scientific research. Max Born, 1954 Nobel Prize Lecture.

Path to this page

  1. Calculus
  2. Linear Algebra
  3. Functional
  4. Hilbert Space
  5. Legendre Transformation
  6. Newton's Second Law
  7. Vector Calculus
  8. Newtonian Mechanics
  9. Variational Calculus
  10. Configuration Space
  11. Euler-Lagrange Equations
  12. Lagrangian Formalism
  13. Phase Space
  14. Hamiltonian Formalism