Related papers: Uncertainty and Wigner negativity in Hilbert-space…
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…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
The correspondence principle states that classical mechanics emerges from quantum mechanics in the appropriate limits. However, beyond this heuristic rule, an information-theoretic perspective reveals that classical mechanics is a…
Relativistic invariance in Euclidean formulations of quantum mechanics is discussed. Relativistic treatments of quantum theory are needed to study hadronic systems at sub-hadronic distance scales. Euclidean formulations of relativistic…
Using quantum-classical analogies, we find that dynamical pictures of quantum mechanics have precise counterparts in classical mechanics. In particular, the Eulerian and Lagrangian descriptions of fluid dynamics in classical mechanics are…
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics,…
Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes…
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert…
Following an article by John von Neumann on infinite tensor products, we develop the idea that the usual formalism of quantum mechanics, associated with unitary equivalence of representations, stops working when countable infinities of…
Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum coupling. The proposed model not only…
We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a…
Viewed as approximations to quantum mechanics, classical evolutions can violate the positive-semidefiniteness of the density matrix. The nature of this violation suggests a classification of dynamical systems based on classical-quantum…
Quantum--mechanical operators corresponding to canonical momentum and position of a point--like particle, which follow from the quantum field theory in the general Riemannian space-time, satisfy generally to a deformation of the canonical…
In the Hilbert space formulation of classical mechanics (CM), pioneered by Koopman and von Neumann (KvN), there are potentially more observables that in the standard approach to CM. In this paper we show that actually many of those extra…
Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert…
This study investigates pseudo-Hermitian quantum mechanics, where the Hamiltonian satisfies a modified Hermiticity condition. We extend the uncertainty relation for such systems, demonstrating its equivalence to the standard Hermitian case…
In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum…
A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear…
A central feature in the Hilbert space formulation of classical mechanics is the quantisation of classical Liouville densities, leading to what may be termed term Groenewold operators. We investigate the spectra of the Groenewold operators…
Classical limits of quantum systems are shown to lead to different conceptions of spaces different from the classical one underlying the process of quantization of such systems. The accent is put in situations where traces of…