Two finite difference schemes for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian are presented. Both utilize a single-cone staggered space-time grid which ensures a single-cone energy dispersion to formulate a numerical treatment of the mixed-state dynamics within the von Neumann equation. The first scheme executes the time-derivative according to the product rule for "bra" and "ket" indices of the density operator. It therefore directly inherits all the favorable properties of the difference scheme for the pure-state Dirac equation and conserves positivity. The second scheme proposed here performs the time-derivative in one sweep. This direct scheme is investigated regarding stability and convergence. Both schemes are tested numerically for elementary simulations using parameters which pertain to topological insulator surface states. Application of the schemes to a Dirac Lindblad equation and real-space-time Green's function formulations are discussed.Two finite difference schemes for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian are presented. Both utilize a single-cone staggered space-time grid which ensures a single-cone energy dispersion to formulate a numerical treatment of the mixed-state dynamics within the von Neumann equation. The first scheme executes the time-derivative according to the product rule for "bra" and "ket" indices of the density operator. It therefore directly inherits all the favorable properties of the difference scheme for the pure-state Dirac equation and conserves positivity. The second scheme proposed here performs the time-derivative in one sweep. This direct scheme is investigated regarding stability and convergence. Both schemes are tested numerically for elementary simulations using parameters which pertain to topological insulator surface states. Application of the schemes to a Dirac Lindblad equation and real-space-time Green's function formulations are discussed.

arXiv:1503.02685Mar 6, 20153 citations

On the relation between Schrödinger and von Neumann equations

Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan +1 more

Journal of Russian Laser ResearchSep 1, 199922 citations

Symplectic structure of the von Neumann equation

Dariusz Chruściński

Reports on Mathematical PhysicsFeb 1, 19912 citations

Quantum mechanics, non-locality, and the space discreteness hypothesis

Wilson A. Zúñiga-Galindo

The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space $\mathcal{X}$ as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})^{3}$ as a model of a space-time. For simplicity, we work out our models using $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})$ as the configuration space. Quantum mechanics (QM), in the sense of Dirac-von Neumann, on the Hilbert space $L^{2}(\mathbb{R}\times\mathcal{X})$ is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region $\mathbb{R}\times\mathbb{R}$ . We apply the developed formalism to the measurement problem. We propose a new mechanism for the collapse of the wavefunction. The mechanism resembles the one proposed by Ghirardi, Ramini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment, where bright and dark states of light (proposed recently) naturally occur.The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space $\mathcal{X}$ as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})^{3}$ as a model of a space-time. For simplicity, we work out our models using $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})$ as the configuration space. Quantum mechanics (QM), in the sense of Dirac-von Neumann, on the Hilbert space $L^{2}(\mathbb{R}\times\mathcal{X})$ is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region $\mathbb{R}\times\mathbb{R}$. We apply the developed formalism to the measurement problem. We propose a new mechanism for the collapse of the wavefunction. The mechanism resembles the one proposed by Ghirardi, Ramini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment, where bright and dark states of light (proposed recently) naturally occur.

arXiv:2508.14836Aug 20, 20250 citations

Quantum simulation of the Dirac equation

R. Gerritsma, Gerhard Kirchmair, F. Zähringer +3 more

arXiv:0909.0674NatureJan 1, 2010725 citations

\mathcal{X}

as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose

\mathbb{R}\times(\mathbb{R}\times\mathcal{X})^{3}

as a model of a space-time. For simplicity, we work out our models using

\mathbb{R}\times(\mathbb{R}\times\mathcal{X})

as the configuration space. Quantum mechanics (QM), in the sense of Dirac-von Neumann, on the Hilbert space

L^{2}(\mathbb{R}\times\mathcal{X})

is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region

\mathbb{R}\times\mathbb{R}

. We apply the developed formalism to the measurement problem. We propose a new mechanism for the collapse of the wavefunction. The mechanism resembles the one proposed by Ghirardi, Ramini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment, where bright and dark states of light (proposed recently) naturally occur.