Abstract:
We review a calculation of the quantum corrections to electrical transport in graphene using the trajectory-based semiclassical method. Compared to conventional metals, for graphene the semiclassical propagator contains an additional pseudospin structure, which influences the results for weak localization, and interaction-induced effects, such as the Altshuler-Aronov correction and dephasing. Our results apply to a sample of graphene that is doped away from the Dirac point and subject to a smooth disorder potential, such that electrons follow classical trajectories. In such system, the Ehrenfest time enters as an additional timescale.

Abstract:
Exact procedures that follow Dirac's constraint quantization of gauge theories are usually technically involved and often difficult to implement in practice. We overview an "effective" scheme for obtaining the leading order semiclassical corrections to the dynamics of constrained quantum systems developed elsewhere. Motivated by the geometrical view of quantum mechanics, our method mimics the classical Dirac-Bergmann algorithm and avoids direct reference to a particular representation of the physical Hilbert space. We illustrate the procedure through the example of a relativistic particle in Minkowski spacetime.

Abstract:
Dirac sea corrections for bulk properties of finite nuclei are computed within a self-consistent scheme in the $\sigma$-$\omega$ model. The valence part is treated in the Hartree approximation whereas the sea contribution is evaluated semiclassically up to fourth order in $\hbar$. Numerically, we find a quick convergence of the semiclassical expansion; the fourth order contributing much less than one percent to the binding energy per nucleon.

Abstract:
We identify three semiclassical parameters in the QCD Dirac operator. Mutual coupling of the different types of degrees of freedom (translational, colour and spin) depends on how the semiclassical limit is taken. We discuss various semiclassical limits and their potential to describe spectrum and spectral statistics of the QCD Dirac operator close to zero virtuality.

Abstract:
We study the coupled system of Maxwell and Dirac equations from a semiclassical point of view. A rigorous nonlinear WKB-analysis, locally in time, for solutions of (critical) order $O(\sqrt{\epsilon})$ is performed, where the small semiclassical parameter $\epsilon $ denotes the microscopic/macroscopic scale ratio.

Abstract:
We study the semiclassical behaviour of a two--dimensional nonintegrable system. In particular we analyze the question of quantum corrections to the semiclassical quantization obtaining up to the second order of perturbation theory an explicit analytical formula for the energy levels, which is the usual semiclassical one plus quantum corrections. We compare the "exact" levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections.

Abstract:
It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order $\hbar $ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.

Abstract:
Quantum effects in statistical mechanics are important when the thermal wavelength is of the order of, or greater than, the mean interatomic spacing. This is examined at depth taking the example of a hard-sphere Boltzmann gas. Using the virial expansion for the equation of state, it is shown that the interaction energy of a classical hard-sphere gas is exactly zero. When the (second) virial coefficient of such a gas is obtained quantum mechanically, however, the quantum contribution to the interaction energy is shown to be substantial. The importance of the semiclassical corrections to the interaction energy shows up dramatically in such a system.

Abstract:
In a semiclassical context we investigate the Zitterbewegung of relativistic particles with spin 1/2 moving in external fields. It is shown that the analogue of Zitterbewegung for general observables can be removed to arbitrary order in \hbar by projecting to dynamically almost invariant subspaces of the quantum mechanical Hilbert space which are associated with particles and anti-particles. This not only allows to identify observables with a semiclassical meaning, but also to recover combined classical dynamics for the translational and spin degrees of freedom. Finally, we discuss properties of eigenspinors of a Dirac-Hamiltonian when these are projected to the almost invariant subspaces, including the phenomenon of quantum ergodicity.

Abstract:
A semiclassical theory of spin dynamics and transport is formulated using the Dirac electron model. This is done by constructing a wavepacket from the positive-energy electron band, and studying its structure and center of mass motion. The wavepacket has a minimal size equal to the Compton wavelength, and has self-rotation about the average spin angular momentum, which gives rise to the spin magnetic moment. Geometric gauge structure in the center of mass motion provides a natural explanation of the spin-orbit coupling and various Yafet terms. Applications of the spin-Hall and spin-Nernst effects are discussed.