Snellius meets Schwarzschild - Refraction of brachistochrones and time-like geodesics

The brachistochrone problem can be solved either by variational calculus or by a skillful application of the Snellius' law of refraction. This suggests the question whether also other variational problems can be solved by an analogue of the refraction law. In this paper we investigate the physically interesting case of free fall in General Relativity that can be formulated as a variational problem w. r. t. proper time. We state and discuss the corresponding refraction law for a special class of spacetime metrics including the Schwarzschild metric

Reduction and extremality of finite observables

We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one observables and consider various constructions that reduce observables to simpler ones. However, these constructions do not suffice to generate all finite extremal observables, as we show by means of counter-examples

Generalized Bell inequalities and frustrated spin systems

We find a close correspondence between generalized Bell inequalities of a special kind and certain frustrated spin systems. For example, the Clauser-Horn-Shimony-Holt inequality corresponds to the frustrated square with the signs +++- for the nearest neighbor interaction between the spins. Similarly, the Pearle-Braunstein-Cave inequality corresponds to a frustrated even ring with the corresponding signs +...+-. Upon this correspondence, the violation of such inequalities by the entangled singlet state in quantum mechanics is equivalent to the spin system possessing a classical coplanar ground state, the energy of which is lower than the Ising ground state's energy. We propose a scheme which generates new inequalities and give further examples, the frustrated hexagon with additional diagonal bonds and the frustrated hypercubes in n=3,4,5 dimensions. Surprisingly, the hypercube in n=4 dimensions yields an inequality which is not violated by the singlet state. We extend the correspondence to other entangled states and XXZ-models of spin systems.Comment: 15 pages, 6 figure

The Floquet theory of the two level system revisited

We reconsider the periodically driven two level system and especially the Rabi problem with linear polarization. The Floquet theory of this problem can be reduced to its classical limit, i.e., to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to J.H. Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approximations to physical quantities of the Rabi problem with linear polarization as well as asymptotic formulas for various limit cases.Comment: Minor correction

Theory of ground states for classical Heisenberg spin systems I

We formulate part I of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. The main result is that all ground states can be constructed from the eigenvectors of a real, symmetric matrix with entries comprising the coupling constants of the spin system as well as certain Lagrange parameters. The eigenvectors correspond to the unique maximum of the minimal eigenvalue considered as a function of the Lagrange parameters. However, there are rare cases where all ground states obtained in this way have unphysical dimensions $M>3$ and the theory would have to be extended. Further results concern the degree of additional degeneracy, additional to the trivial degeneracy of ground states due to rotations or reflections. The theory is illustrated by a couple of elementary examples.Comment: The latest version contains an additional theorem on the existence of symmetric ground states (section IV

Why do all the curvature invariants of a gravitational wave vanish ?

We prove the theorem valid for (Pseudo)-Riemannian manifolds $V_n$: "Let $x \in V_n$ be a fixed point of a homothetic motion which is not an isometry then all curvature invariants vanish at $x$." and get the Corollary: "All curvature invariants of the plane wave metric ds \sp 2 \quad = \quad 2 \, du \, dv \, + \, a\sp 2 (u) \, dw \sp 2 \, + \, b\sp 2 (u) \, dz \sp 2 identically vanish." Analysing the proof we see: The fact that for definite signature flatness can be characterized by the vanishing of a curvature invariant, essentially rests on the compactness of the rotation group $SO(n)$. For Lorentz signature, however, one has the non-compact Lorentz group $SO(3,1)$ instead of it. A further and independent proof of the corollary uses the fact, that the Geroch limit does not lead to a Hausdorff topology, so a sequence of gravitational waves can converge to the flat space-time, even if each element of the sequence is the same pp-wave.Comment: 9 page

Floquet theory of the analytical solution of a periodically driven two-level system

We investigate the analytical solution of a two-level system subject to a monochromatical, linearly polarized external field that was published a couple of years ago. In particular, we derive an explicit expression for the quasienergy. Moreover, we calculate the time evolution of a typical two-level system over a full period by evaluating series solutions of the confluent Heun equation. This is possible without invoking the connection problem of this equation since the complete time evolution of the system under consideration can be reduced to that of the first quarter-period

Periodic thermodynamics of the Rabi model with circular polarization for arbitrary spin quantum numbers

We consider a spin $s$ subjected to both a static and an orthogonally applied oscillating, circularly polarized magnetic field while being coupled to a heat bath, and analytically determine the quasi\-stationary distribution of its Floquet-state occupation probabilities for arbitrarily strong driving. This distribution is shown to be Boltzmannian with a quasitemperature which is different from the temperature of the bath, and independent of the spin quantum number. We discover a remarkable formal analogy between the quasithermal magnetism of the nonequilibrium steady state of a driven ideal paramagnetic material, and the usual thermal paramagnetism. Nonetheless, the response of such a material to the combined fields is predicted to show several unexpected features, even allowing one to turn a paramagnet into a diamagnet under strong driving. Thus, we argue that experimental measurements of this response may provide key paradigms for the emerging field of periodic thermodynamics.Comment: 12 figures, 2 additional appendice

A framework for sequential measurements and general Jarzynski equations

We formulate a statistical model of two sequential measurements and prove a so-called J-equation that leads to various diversifications of the well-known Jarzynski equation including the Crooks dissipation theorem. Moreover, the J-equation entails formulations of the Second Law going back to Wolfgang Pauli. We illustrate this by an analytically solvable example of sequential discrete position-momentum measurements accompanied with the increase of Shannon entropy. The standard form of the J-equation extends the domain of applications of the quantum Jarzynski equation in two respects: It includes systems that are initially only in local equilibrium and it extends this equation to the cases where the local equilibrium is described by microcanononical, canonical or grand canonical ensembles. Moreover, the case of a periodically driven quantum system in thermal contact with a heat bath is shown to be covered by the theory presented here. Finally, we shortly consider the generalized Jarzynski equation in classical statistical mechanics

Fourier-Taylor series for the figure eight solution of the three body problem

We provide an analytical approximation of a periodic solution of the three body problem in celestial mechanics, the so-called figure eight solution, discovered 1993 by C. Moore. This approximation has the form of a Fourier series whose components are in turn Taylor series w. r. t. some parameter. The method is first illustrated by application to two other problems, (1) the problem of oscillations of a particle in a cubic potential that has a well-known analytic solution in terms of elliptic functions and (2) periodic solutions and corresponding eigenvalues of a generalized Mathieu equation that cannot be solved analytically. When applied to the three body problem it turns out that the Fourier-Taylor series, evaluated up to 30th order, represents un-physical solutions except for a particular value of the series parameter. For this value the series approximates the numerical solution known from the literature up to a relative error of $1.6\times 10^{-3}$.Comment: 11 figure