Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon$. Based on the error bounds, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the FDTD methods share the same $\varepsilon$-scalability on time step: $\tau=O(\varepsilon^3)$. Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability on time step is improved to $\tau=O(\varepsilon^2)$ when $0<\varepsilon\ll 1$. Extensive numerical results are reported to support our error estimates.Comment: 34 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1511.0119

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.0288

Theta dependence of SU(N) gauge theories in the presence of a topological term

We review results concerning the theta dependence of 4D SU(N) gauge theories and QCD, where theta is the coefficient of the CP-violating topological term in the Lagrangian. In particular, we discuss theta dependence in the large-N limit. Most results have been obtained within the lattice formulation of the theory via numerical simulations, which allow to investigate the theta dependence of the ground-state energy and the spectrum around theta=0 by determining the moments of the topological charge distribution, and their correlations with other observables. We discuss the various methods which have been employed to determine the topological susceptibility, and higher-order terms of the theta expansion. We review results at zero and finite temperature. We show that the results support the scenario obtained by general large-N scaling arguments, and in particular the Witten-Veneziano mechanism to explain the U(1)_A problem. We also compare with results obtained by other approaches, especially in the large-N limit, where the issue has been also addressed using, for example, the AdS/CFT correspondence. We discuss issues related to theta dependence in full QCD: the neutron electric dipole moment, the dependence of the topological susceptibility on the quark masses, the U(1)_A symmetry breaking at finite temperature. We also consider the 2D CP(N) model, which is an interesting theoretical laboratory to study issues related to topology. We review analytical results in the large-N limit, and numerical results within its lattice formulation. Finally, we discuss the main features of the two-point correlation function of the topological charge density.Comment: A typo in Eq. (3.9) has been corrected. An additional subsection (5.2) has been inserted to demonstrate the nonrenormalizability of the relevant theta parameter in the presence of massive fermions, which implies that the continuum (a -> 0) limit must be taken keeping theta fixe

MATRIX-MFO Tandem Workshop/Small Collaboration: Rough Wave Equations (hybrid meeting)

The consideration of wave propagation in inhomogeneous media or the modelling of nonlinear waves often requires the study of wave equations with low regularity data and/or coefficients. Several Australian-European collaborations have recently led to deeper analytical understanding of rough wave equations. This tandem workshop provided a platform for such collaborations and brought together early career researchers and leading experts in harmonic analysis, microlocal analysis and spectral theory. The workshop focused on collaboration and technical knowledge exchange on topics such as local smoothing, spectral multipliers, restriction estimates, Hardy spaces for Fourier integral operators, and nonlinear partial differential equations

Physical evolution in Loop Quantum Cosmology: The example of vacuum Bianchi I

We use the vacuum Bianchi I model as an example to investigate the concept of physical evolution in Loop Quantum Cosmology (LQC) in the absence of the massless scalar field which has been used so far in the literature as an internal time. In order to retrieve the system dynamics when no such a suitable clock field is present, we explore different constructions of families of unitarily related partial observables. These observables are parameterized, respectively, by: (i) one of the components of the densitized triad, and (ii) its conjugate momentum; each of them playing the role of an evolution parameter. Exploiting the properties of the considered example, we investigate in detail the domains of applicability of each construction. In both cases the observables possess a neat physical interpretation only in an approximate sense. However, whereas in case (i) such interpretation is reasonably accurate only for a portion of the evolution of the universe, in case (ii) it remains so during all the evolution (at least in the physically interesting cases). The constructed families of observables are next used to describe the evolution of the Bianchi I universe. The performed analysis confirms the robustness of the bounces, also in absence of matter fields, as well as the preservation of the semiclassicality through them. The concept of evolution studied here and the presented construction of observables are applicable to a wide class of models in LQC, including quantizations of the Bianchi I model obtained with other prescriptions for the improved dynamics.Comment: RevTex4, 22 pages, 4 figure