The Hausdorff moments in statistical mechanics

A new method for solving the Hausdorff moment problem is presented which makes use of Pollaczek polynomials. This problem is severely ill posed; a regularized solution is obtained without any use of prior knowledge. When the problem is treated in the L 2 space and the moments are finite in number and affected by noise or round‐off errors, the approximation converges asymptotically in the L 2 norm. The method is applied to various questions of statistical mechanics and in particular to the determination of the density of states. Concerning this latter problem the method is extended to include distribution valued densities. Computing the Laplace transform of the expansion a new series representation of the partition function Z(β) (β=1/k BT ) is obtained which coincides with a Watson resummation of the high‐temperature series for Z(β)

Reciprocal relativity of noninertial frames and the quaplectic group

Newtonian mechanics has the concept of an absolute inertial rest frame. Special relativity eliminates the absolute rest frame but continues to require the absolute inertial frame. General relativity solves this for gravity by requiring particles to have locally inertial frames on a curved position-time manifold. The problem of the absolute inertial frame for other forces remains. We look again at the transformations of frames on an extended phase space with position, time, energy and momentum degrees of freedom. Under nonrelativistic assumptions, there is an invariant symplectic metric and a line element dt^2. Under special relativistic assumptions the symplectic metric continues to be invariant but the line elements are now -dt^2+dq^2/c^2 and dp^2-de^2/c^2. Max Born conjectured that the line element should be generalized to the pseudo- orthogonal metric -dt^2+dq^2/c^2+ (1/b^2)(dp^2-de^2/c^2). The group leaving these two metrics invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is 'reciprocal' in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group. The Heisenberg group itself is the semidirect product of two translation groups. This provides the noncommutative properties of position and momentum and also time and energy that are required for the quantum mechanics that results from considering the unitary representations of the quaplectic group.Comment: Substantial revision, Publicon LaTe

Self-Interacting Electromagnetic Fields and a Classical Discussion on the Stability of the Electric Charge

The present work proposes a discussion on the self-energy of charged particles in the framework of nonlinear electrodynamics. We seek magnet- ically stable solutions generated by purely electric charges whose electric and magnetic fields are computed as solutions to the Born-Infeld equa- tions. The approach yields rich internal structures that can be described in terms of the physical fields with explicit analytic solutions. This suggests that the anomalous field probably originates from a magnetic excitation in the vacuum due to the presence of the very intense electric field. In addition, the magnetic contribution has been found to exert a negative pressure on the charge. This, in turn, balances the electric repulsion, in such a way that the self-interaction of the field appears as a simple and natural classical mechanism that is able to account for the stability of the electron charge.Comment: 8 pages, 1 figur

The Application of Feedback in Measurement

Instrument errors, error reduction, and elements of measurements for measurement systems with feedback instrumentatio

Anharmonic stabilization of the high-pressure simple cubic phase of calcium

The phonon spectrum of the high-pressure simple cubic phase of calcium, in the harmonic approx- imation, shows imaginary branches that make it mechanically unstable. In this letter, the phonon spectrum is recalculated using density-functional theory (DFT) ab initio methods fully including anharmonic effects up to fourth order at 50 GPa. Considering that perturbation theory cannot be employed with imaginary harmonic frequencies, a variational procedure based on the Gibbs- Bogoliubov inequality is used to estimate the renormalized phonon frequencies. The results show that strong quantum anharmonic effects make the imaginary phonons become positive even at zero temperature so that the simple cubic phase becomes mechanically stable, as experiments suggest. Moreover, our calculations find a superconducting Tc in agreement with experiments and predict an anomalous behavior of the specific heat.Comment: 5 pages, 3 figure

Born-Regulated Gravity in Four Dimensions

Previous work involving Born-regulated gravity theories in two dimensions is extended to four dimensions. The action we consider has two dimensionful parameters. Black hole solutions are studied for typical values of these parameters. For masses above a critical value determined in terms of these parameters, the event horizon persists. For masses below this critical value, the event horizon disappears, leaving a ``bare mass'', though of course no singularity.Comment: LaTeX, 15 pages, 2 figure

Nonperturbative calculation of Born-Infeld effects on the Schroedinger spectrum of the hydrogen atom

We present the first nonperturbative numerical calculations of the nonrelativistic hydrogen spectrum as predicted by first-quantized electrodynamics with nonlinear Maxwell-Born-Infeld field equations. We also show rigorous upper and lower bounds on the ground state. When judged against empirical data our results significantly restrict the range of viable values of the new electromagnetic constant which is introduced by the Born-Infeld theory. We assess Born's own proposal for the value of his constant.Comment: 4p., 2 figs, 1 table; submitted for publicatio

Spontaneous Generation of Photons in Transmission of Quantum Fields in PT Symmetric Optical Systems

We develop a rigorous mathematically consistent description of PT symmetric optical systems by using second quantization. We demonstrate the possibility of significant spontaneous generation of photons in PT symmetric systems. Further we show the emergence of Hanbury-Brown Twiss (HBT) correlations in spontaneous generation. We show that the spontaneous generation determines decisively the nonclassical nature of fields in PT symmetric systems. Our work can be applied to other systems like plasmonic structure where losses are compensated by gain mechanisms.Comment: 4 pages, 5 figure

Indeterminacy of Holographic Quantum Geometry

An effective theory based on wave optics is used to describe indeterminacy of position in holographic spacetime with a UV cutoff at the Planck scale. Wavefunctions describing spacetime positions are modeled as complex disturbances of quasi-monochromatic radiation. It is shown that the product of standard deviations of two position wavefunctions in the plane of a holographic light sheet is equal to the product of their normal separation and the Planck length. For macroscopically separated positions the transverse uncertainty is much larger than the Planck length, and is predicted to be observable as a "holographic noise" in relative position with a distinctive shear spatial character, and an absolutely normalized frequency spectrum with no parameters once the fundamental wavelength is fixed from the theory of gravitational thermodynamics. The spectrum of holographic noise is estimated for the GEO600 interferometric gravitational-wave detector, and is shown to approximately account for currently unexplained noise between about 300 and 1400Hz. In a holographic world, this result directly and precisely measures the fundamental minimum interval of time.Comment: 4 pages, LaTeX. Considerably shortened from earlier version. Conclusions are unchanged. Submitted to PR

Black holes in extended gravity theories in Palatini formalism

We consider several physical scenarios where black holes within classical gravity theories including $R^2$ and Ricci-squared corrections and formulated \`a la Palatini can be analytically studied.Comment: 4 pages, contribution to the "Spanish Relativity Meeting in Portugal 2012 (Progress in Mathematical Relativity, Gravitation and Cosmology)", Springer Proceedings in Mathematics (to appear