A Generalization of Quantum Stein’s Lemma

Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein’s Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ. We present a generalization of quantum Stein’s Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein’s Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.     

A Generalization of the Bargmann’s Theory of Ray Representations

The paper contains a complete theory of factors for ray representations acting in a Hilbert bundle, which is a generalization of the known Bargmanns theory. With its help, we have reformulated the standard quantum theory so that the gauge freedom emerges naturally from the very nature of quantum laws. The theory is of primary importance in the investigations of covariance (in contradistinction to symmetry) of a quantum theory which possesses a nontrivial gauge freedom. In that case the group in question is not any symmetry group but a covariance group only – the case not yet investigated in depth. It is shown that the factor of a covariance group representation depends on space and time when the system in question possesses gauge freedom. In nonrelativistic theories, the factor depends on time only. In relativistic theory, the Hilbert bundle is built over spacetime while in the nonrelativistic case-over time. We explain two applications of this generalization: in the theory of a quantum particle in gravitational field in the nonrelativistic limit, and in quantum electrodynamics.     

Niels Bohr’s Generalization of Classical Mechanics

We clarify Bohrs interpretation of quantum mechanics by demonstrating the central role played by his thesis that quantum theory is a rational generalization of classical mechanics. This thesis is essential for an adequate understanding of his insistence on the indispensability of classical concepts, his account of how the quantum formalism gets its meaning, and his belief that hidden variable interpretations are impossible.     

Relativistic generalization of Bell’s inequalities in Wigner’s form

A relativistic generalization of Bell’s inequalities in Wigner’s form was obtained for the decays of a pseudoscalar and a scalar particle to two particles having a nonzero spin (fermions and photons). Both inequalities involving a full anticorrelation of final-particle spins and having a nonrelativistic analog and inequalities involving a full correlation of spins are considered. It is shown that Bohr’s complementarity principle may be tested experimentally in the relativistic region.     

A Mechanical Model for Fourier’s Law of Heat Conduction

Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier’s law for heat conduction from the laws of mechanics remains thus a major unsolved problem. In this note we present a deterministic mechanical model of a heat-conducting chain with nontrivial interactions, where kinetic energy fluctuations at the nodes of the chain are removed. In this model the derivation of Fourier’s law can proceed rigorously.     

Well Behaved Charged Generalization of Buchdahl’s Fluid Spheres

In the present article, we have obtained a class of well behaved charged analogues of Buchdahl (Phys. Rev. 116:1027–1034, 1959) neutral perfect fluid solution, which reduces to its neutral counter part in the absence of charge. The solutions so obtained are utilized to depict the super-dense stars models such as models for neutron stars and strange star. It is observed that the models are well behaved for restricted range of the parameter K (1<K≤1.64). Over all the maximum mass and corresponding radius is 2.4495M _Θ and 16.7289 respectively and moment of inertia . Also the pulsars character of the super-dense stars so obtained and has been analyzed with the help of moment of inertia. The analysis of the models reveals both vela and crab pulsars.     

A Note on Fick’s Law with Phase Transitions

Journal of Statistical Physics - We characterize the non equilibrium stationary states in two classes of systems where phase transitions are present. We prove that the interface in the limit is a...     

Generalization of Luttinger’s theorem for strongly correlated electron systems

By on the analyzing the general structure of the Green function of a strongly correlated electron system, it is shown that, for the regime of strong correlations, Luttinger’s theorem should be generalized in the following way: the volume of the Fermi surface of the system of noninteracting particles is equal to that of the quasiparticles in the strongly correlated system with due regard for the spectral weight of the quasiparticles. An investigation of the t-J model and of the Hubbard model, as applied to the paramagnetic nonsuperconducting phase, shows that the generalized Luttinger theorem is valid for these models.     

Restoration of Wigner’s Sypersymmetry in Heavy and Superheavy Nuclei

Physics of Atomic Nuclei - Various aspects of restoration of Wigner’s supersymmetry [SU(4) symmetry] in heavy and superheavy nuclei are analyzed by comparing the results of calculations with...     

Generalization of Biot’s equations with allowance for shear relaxation of a fluid

On the basis of the generalized variational principle for dissipative continuum mechanics, a system of generalized Biot’s equations is derived to describe the wave propagation in a two-phase porous permeable medium in the presence of shear relaxation in the pore-filling fluid. It was shown that the inclusion of shear viscoelasticity of the fluid leads to the appearance of two transverse modes in addition to two longitudinal modes described by the Biot theory. One of the transverse modes is an acoustic mode, whereas the other is a diffusion mode characterized by the linear frequency dependence of phase velocity and attenuation coefficient in the low-frequency region.     

Consciousness and the Wigner’s Friend Problem

It is generally agreed that decoherence theory is, if not a complete answer, at least a great step forward towards a solution of the quantum measurement problem. It is shown here however that in the cases in which a sentient being is explicitly assumed to take cognizance of the outcome the reasons we have for judging this way are not totally consistent, so that the question has to be considered anew. It is pointed out that the way the Broglie–Bohm model solves the riddle suggests a possible clue, consisting in assuming that even very simple systems may have some sort of a proto-consciousness, but that their “internal states of consciousness” are not predictive. It is, next, easily shown that if we imagine the systems get larger, in virtue of decoherence their internal states of consciousness progressively gain in predictive value. So that, for macro-systems, they may be identified (in practice) with the predictive states of consciousness on which we ground our observational predictions. The possibilities of carrying over this idea to standard quantum mechanics are then investigated. Conditions of conceptual consistency are considered and found rather strict, and, finally, two solutions emerge, differing conceptually very much from one another but in both of which the, possibly non-predictive, generalized internal states of consciousness play a crucial role.     

Entropic Corrections to Coulomb’s Law

Two well-known quantum corrections to the area law have been introduced in the literatures, namely, logarithmic and power-law corrections. Logarithmic corrections, arises from loop quantum gravity due to thermal equilibrium fluctuations and quantum fluctuations, while, power-law correction appears in dealing with the entanglement of quantum fields in and out the horizon. Inspired by Verlinde’s argument on the entropic force, and assuming the quantum corrected relation for the entropy, we propose the entropic origin for the Coulomb’s law in this note. Also we investigate the Uehling potential as a radiative correction to Coulomb potential in 1-loop order and show that for some value of distance the entropic corrections of the Coulomb’s law is compatible with the vacuum-polarization correction in QED. So, we derive modified Coulomb’s law as well as the entropy corrected Poisson’s equation which governing the evolution of the scalar potential ϕ. Our study further supports the unification of gravity and electromagnetic interactions based on the holographic principle.     

Loschmidt, Stefan, and Stigler’s Law of Eponymy

Josef Loschmidt (1821–1895) and Josef Stefan (1835–1893) were eminent scientists in the Institute of Physics at the University of Vienna during the second half of the nineteenth century but are not well known today, as their legacies have been recognized differently by the scientific community. Loschmidt first described the structure of the benzene molecule and determined the size of air molecules, from which the number of molecules per unit volume can easily be determined, yet others received the credit for these achievements. Stefan posited the fourth-power temperature radiation law, but neither he nor his student Ludwig Boltzmann (1844–1906) calculated the proportionality constant now known as the Stefan-Boltzmann constant. These are instances of Stigler’s Law of Eponymy. Besides these achievements, perhaps the greatest unheralded contribution of both Loschmidt and Stefan was the experimental evidence they provided in support of the emerging kinetic theory of gases.     

Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators

We consider a Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice , and subjected to stochastic forcing mimicking heat baths of temperatures T ₁ and T ₂ on the hyperplanes at 0 and N. We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for N large, is approximately a local equilibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile. Partially supported by the Academy of Finland.     

A model of Jupiter’s magnetodisk

An axially symmetric equilibrium model of Jupiter’s magnetodisk is developed in the MHD approximation that takes the plasma corotation and the centrifugal force into account. The model is constructed for two cases: (1) the magnetodisk plasma is assumed to have a uniform temperature; (2) the plasma pressure is assumed to be an adiabatic function of density. Analytical expressions for the magnetic field, current density, and magnetodisk temperature and thickness distributions are obtained as functions of the system parameters, viz., the radial distribution of plasma pressure in the equatorial plane, the transverse magnetic field in the center of the layer, and the angular velocity of the plasma rotation.     

Einstein, Wigner and Feynman: From E=mc 2 to Feynman’s decoherence via Wigner’s little groups

The 20th-century physics starts with Einstein and ends with Feynman. Einstein introduced the Lorentz-covariant world with E=mc ². Feynman observed that fast-moving hadrons consist of partons which interact incoherently with external signals. If quarks and partons are the same entities observed in different Lorentz frames, the question then is why partons are incoherent while quarks are coherent. This is the most puzzling question Feynman left for us to solve. In this report, we discuss Wigner’s role in settling this question. Einstein’s E=mc ², which takes the form $E = \sqrt {m^2 + p^2 } $ , unifies the energy-momentum relations for massive and massless particles, but it does not take into account internal space-time structure of relativistic particles. It is pointed out Wigner’s 1939 paper on the inhomogeneous Lorentz group defines particle spin and gauge degrees of freedom in the Lorentz-covariant world. Within the Wigner framework, it is shown possible to construct the internal space-time structure for hadrons in the quark model. It is then shown that the quark model and the parton model are two different manifestations of the same covariant entity. It is shown therefore that the lack of coherence in Feynman’s parton picture is an effect of the Lorentz covariance.     

Charge-exchange resonances and restoration of Wigner’s supersymmetry in heavy and superheavy nuclei

Various facets of the question of whether Wigner’s supersymmetry [SU(4) symmetry] may be restored in heavy and superheavy nuclei are analyzed on the basis of a comparison of the results of calculations with experimental data. The energy difference between the giant Gamow–Teller resonance and the analog resonance (the difference of E _G and E _A) according to calculations based on the theory of finite Fermi systems is presented for the case of 33 nuclei for which experimental data are available. The calculated difference ΔE _G–A of E _G and E _A tends to zero in heavier nuclei, showing evidence of the restoration of Wigner’s SU(4) symmetry. Also, the isotopic dependence of the Coulomb energy difference between neighboring isobaric nuclei is analyzed within the SU(4) approach for more than 400 nuclei in the mass-number range of A = 5–244. The restoration of Wigner’s SU(4) symmetry in heavy nuclei is confirmed. It is shown that the restoration of SU(4) symmetry is compatible with the possible existence of the stability island in the region of superheavy nuclei.     

A Possible Reinterpretation of Einstein’s Equations

In this paper, we first review Huei’s formulation in which it is shown that the linearized Einstein equations can be written in the same form as the Maxwell equations. We eliminate some imperfections like the scalar potential which is ill linked to the electric-type field, the Lorentz-type force which is obtained with a time independence restriction and the undesired factor 4 which appears in the magnetic-type part. Second, from these results and in the light of a recent work by C.C. Barros, we propose an extension of the equivalence principle and we suggest a new interpretation for Einstein’s equations by showing that the electromagnetic Maxwell equations can be derived from a new version of Einstein’s ones.     

New Analytical Solution of the Equilibrium Ampere’s Law Using the Walker’s Method: a Didactic Example

This work aims to demonstrate the analytical solution of the Grad-Shafranov (GS) equation or generalized Ampere’s law, which is important in the studies of self-consistent 2.5-D solution for current sheet structures. A detailed mathematical development is presented to obtain the generating function as shown by Walker (RSPSA 91, 410, 1915). Therefore, we study the general solution of the GS equation in terms of the Walker’s generating function in details without omitting any step. The Walker’s generating function g(ζ) is written in a new way as the tangent of an unspecified function K(ζ). In this trend, the general solution of the GS equation is expressed as exp(??2Ψ) =?4|K ^′(ζ)|²/cos²[K(ζ) ? K(ζ ^?)]. In order to investigate whether our proposal would simplify the mathematical effort to find new generating functions, we use Harris’s solution as a test, in this case K(ζ) = arctan(exp(i ζ)). In summary, one of the article purposes is to present a review of the Harris’s solution. In an attempt to find a simplified solution, we propose a new way to write the GS solution using g(ζ) = tan(K(ζ)). We also present a new analytical solution to the equilibrium Ampere’s law using g(ζ) = cosh(b ζ), which includes a generalization of the Harris model and presents isolated magnetic islands.     

Analytical Derivation of the Stefan–Boltzmann Law for Integral Radiance from Planck’s Law for Spectral Radiance

The Stefan–Boltzmann law for the integral radiation flux was analytically and mathematically exactly derived based on Planck’s law for the spectral radiation flux of an object.