Koopman wavefunctions and Clebsch variables in Vlasov–Maxwell kinetic theory

Abstract

Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here, we present different approaches to Koopman's Hilbert-space formulation of classical mechanics in the context of Vlasov–Maxwell kinetic theory. The celebrated Koopman–von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is non-canonical and appears to overcome certain issues emerging in the canonical formalism. Furthermore, the canonical structure is restored for a variant of the Koopman–von Neumann construction that carries a different phase dynamics. Going back to van Hove's prequantum theory, the corresponding Koopman–van Hove equation provides an alternative Clebsch representation which is then coupled to the electromagnetic fields. Finally, the role of gauge transformations in the new context is discussed in detail.

Evolution of hybrid quantum-classical wavefunctions

“…However, one may insist that the KvN prescription ρ = |χ| 2 is perfectly sensible so that the question becomes to identify an alternative variational formulation overcoming the issues emerging from to the canonical structure presented in Section 2.1. This alternative construction was presented in [78] and here we will show how this can be obtained as a closure model emerging from KvH theory.…”

“…More importantly, if one initializes a zero phase S = 0 in (2.2), then the Hamiltonian functional (2.4) collapses entirely, thereby indicating that this variational formulation may need special care. This difficulty was recently overcome in [78], where an alternative variational formulation was presented. This variant will be discussed later on.…”

“…For example, as noticed in [11], the symplectic potential A(z) = Jz/2 is particularly useful in the case of quadratic Hamiltonians. At this point, as presented extensively in [78], one can perform the replacement D = |χ| 2 . Taking the square root of (3.3) yields the following Lagrange-to-Euler map for Koopman-von Neumann wavefunctions:…”

“…In this construction, no ambiguity occurs when the Koopman phase is set to zero and the total energy coincides with the KvN prescription ´H|χ| 2 d 2 z. For further details, we address the reader to [78].…”

“…As shown in Section 3.1, the application of this gauge-invariance principle requires techniques from continuum theories, such as the Lagrangian flow paths widely used in hydrodynamics. Then, the gauge-invariance principle leads to an alternative variational formulation of KvN theory, which represents the immediate extension of the well-known variational setting of the classical Liouville equation [78]. This alternative variational formulation corresponds to a noncanonical Hamiltonian structure that is presented explicitly in Section 3.3.…”