Junction conditions in general relativity

We examine the three sets of junction conditions commonly used in general relativity: those due to Darmois, to O'Brien and Synge, and to Lichnerowicz. We show that those due to Darmois and Lichnerowicz are equivalent. The O'Brien and Synge set is stronger than the other two and is unsatisfactory in that it may rule out physically plausible junctions. We conclude that the Darmois set is the most convenient and reliable.     

Admissible coordinate systems for static,spherically symmetric stellar models

For static, spherically symmetric stellar models it is shown that imposing the condition that the determinant of the metrical coefficients takes on its flat space-time value everywhere is sufficient to ensure that the coordinates are admissible in the sense of Lichnerowicz. The general method of solution is illustrated by integrating the equations for a star of constant, uniform density.     

An algebraic framework for the supersymmetry theory

In the absence of a rigorous mathematical definition (or proof of existence and consistency) of the “g-dimensional space” whose points are labelled by the four (commutative) Minkowski coordinates x_n and four anticommuting coordinates θ_n, a clearly consistent algebraic scheme is presented in the frame of which the concepts used in the theory of the supersymmetry have a well established mathematical meaning.     

An approximate global solution of Einstein’s equation for a rotating compact source with linear equation of state

We use analytic perturbation theory to present a new approximate metric for a rigidly rotating perfect fluid source with equation of state (EOS) $\epsilon +(1-n)p=\epsilon _0$ . This EOS includes the interesting cases of strange matter, constant density and the fluid of the Wahlquist metric. It is fully matched to its approximate asymptotically flat exterior using Lichnerowicz junction conditions and it is shown to be a totally general matching using Darmois–Israel conditions and properties of the harmonic coordinates. Then we analyse the Petrov type of the interior metric and show first that, in accordance with previous results, in the case corresponding to Wahlquist’s metric it can not be matched to the asymptotically flat exterior. Next, that this kind of interior can only be of Petrov types I, D or (in the static case) O and also that the non-static constant density case can only be of type I. Finally, we check that it can not be a source of Kerr’s metric.     

Gauge equivalence and the inverse spectral problem for the magnetic Schrödinger operator on the torus

We study the inverse spectral problem for the Schrödinger operator H on the two-dimensional torus with even magnetic field B(x) and even electric potential V(x). Guillemin [11] proved that the spectrum of H determines B(x) and V(x). A simple proof of Guillemin’s results was given by the authors in [3]. In the present paper, we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of H determine the magnetic field and the electric potential. We also show that, generically, the spectrum and the magnetic field determine the “extended” gauge equivalence class of the magnetic potential. The proof is a modification of that in [3] with some corrections and clarifications.     

对于“包含在连续谱量子体系中的决定论性”一文的讨论

对于具有连续能谱的单粒子量子体系,“包含在连续谱量子体系中的决定论性”一文用所谓“双波函数”来描述处于能量本征态的粒子系综中各粒子的量子行为,并且在所谓的“等价定理”中称:双波函数描述在经典极限下将化为经典力学描述.然而,此描述所给出的系综力学量观测值统计分布的预言与通常量子力学不相容;并且,该文对其“等价定理”的证明是不正确的,这个“定理”实际上不成立 关键词： 连续能谱量子体系 双波函数 经典极限     

Space-time supersymmetry of the covariant superstring

We explicitly derive the ghost oscillator contribution to the gauge covariant fermion emission vertex. This vertex is used to construct the space-time supersymmetry transformation laws which are shown to be an invariance of the free gauge covariant action of the superstring. We develop methods to deal with the quadratic exponentials which appear in the fermion emission vertex, in order to study the closure of the supersymmetry algebra. As a by-product, we complete the proof of the equivalence between the “old” and “new” formulations of the superstring.     

Generalized Boundary Conditions in an Existence and Uniqueness Proof for the Solution of the Non-Stationary Electron Boltzmann Equation by Means of Operator-Semigroups

Recently, based on the semigroup approach a new proof was presented of the existence of a unique solution of the non-stationary Boltzmann equation for the electron component of a collision dominated plasma. The proof underlies some restrictions which should be overcome to extend the validity range to other problems of physical interest. One of the restrictions is the boundary condition applied. The choice of the boundary condition is essential for the proof because it determines the range of definition of the infinitesimal generator and thus the operator semigroup itself. The paper proves the existence of a unique solution for generalized boundary conditions, this solution takes non-negative values, which is necessary for a distribution function from the physical point of view.     

Nordström’s scalar theory of gravity and the equivalence principle

General Relativity obeys the three equivalence principles, the “weak” one (all test bodies fall the same way in a given gravitational field), the “Einstein” one (gravity is locally effaced in a freely falling reference frame) and the “strong” one (the gravitational mass of a system equals its inertial mass to which all forms of energy, including gravitational energy, contribute). The first principle holds because matter is minimally coupled to the metric of a curved spacetime so that test bodies follow geodesics. The second holds because Minkowskian coordinates can be used in the vicinity of any event. The fact that the latter, strong, principle holds is ultimately due to the existence of superpotentials which allow to define the inertial mass of a gravitating system by means of its asymptotic gravitational field, that is, in terms of its gravitational mass. Nordström’s theory of gravity, which describes gravity by a scalar field in flat spacetime, is observationally ruled out. It is however the only theory of gravity with General Relativity to obey the strong equivalence principle. I show in this paper that this remarkable property is true beyond post-newtonian level and can be related to the existence of a “Nordström-Katz” superpotential.     

Theory of filters

We consider the following statistical problem: suppose we have a light beam and a collection of semi-transparent windows which can be placed in the way of the beam. Assume that we are colour blind and we do not possess any colour sensitive detector. The question is, whether by only measurements of the decrease in the beam intensity in various sequences of windows we can recognize which among our windows are light beam filters absorbing photons according to certain definite rules? To answer this question a definition of physical systems is formulated independent of “quantum logic” and lattice theory, and a new idea of quantization is proposed. An operational definition of filters is given: in the framework of this definition certain nonorthodox classes of filters are admissible with a geometry incompatible to that assumed in orthodox quantum mechanics. This leads to an extension of the existing quantum mechanical structure generalizing the schemes proposed by Ludwig [10] and the present author [13]. In the resulting theory, the quantum world of orthodox quantum mechanics is not the only possible but is a special member of a vast family of “quantum worlds” mathematically admissible. An approximate classification of these worlds is given, and their possible relation to the quantization of non-linear fields is discussed. It turns out to be obvious that the convex set theory has a similar significance for quantum physics as the Riemannian geometry for space-time physics.     

On physical proofs of mathematical theorems

We argue that the alleged “proofs” of mathematical results from physical models logically follow from the mathematics, rather than from the specific physical laws used in working out the behaviour of the model. To support this view, we present an optical derivation of Brouwer's fixed point theorem and illustrate how the physical model can be discarded to obtain a strict mathematical proof. We also analyse an elementary version of Landsberg's model which exhibits more clearly the mathematical steps involved in the physical proof.     

Simulation of periodic mono-sized hard sphere systems under different vibration conditions and resulting compaction

This paper presents a simulation study of confined periodic mono-sized hard sphere systems under different vibration conditions and their influence on the final compaction. An initial random loose packing is submitted to a series of vibration cycles allowing to transform it into a “suspension” in which a given proportion of particles have the possibility to vibrate. This “suspension” is then let to settle down and brought to a new denser packing. Different random local rules are used for the simulation of the displacements of particles during vibration.Firstly, a symmetric vibration is applied in which particles attempt to perform vertical upwards and downwards displacements of equal length. Shocks between particles are simply simulated by random upwards or downwards displacements. It has already been shown that in these conditions, and when the whole system is vibrating, the final packing density can be related to the initial density and to the vibration amplitude (“suspension model”). We show here that for a periodic packing, this model can be extended to partial “suspensions” in which only a proportion of particles is vibrating. An excellent agreement is found between this model and the simulation results as long as the packing is disordered. For large vibration amplitudes, an order appears amongst the system allowing to reach high densities (up to 0.66).Then, the symmetric vibration is replaced by a random vibration in which particles attempt to perform alternatively random upwards and downwards displacements whose inclination with the vertical axis follows a normal distribution. It is shown that in these conditions, the vibration still allows transforming the initial packing into a partial “suspension” but the proportion of vibrating particles is lower than for the symmetric vibration previously used. However, the “suspension model” can be extended with a reasonable agreement to this kind of vibration.     

Luttinger theorem and imbalanced Fermi systems

The proof of the Luttinger theorem, which was originally given for a normal Fermi liquid with equal spin populations formally described by the exact many-body theory at zero temperature, is here extended to an approximate theory given in terms of a “conserving” approximation also with spin imbalanced populations. The need for this extended proof, whose underlying assumptions are here spelled out in detail, stems from the recent interest in superfluid trapped Fermi atoms with attractive inter-particle interaction, for which the difference between two spin populations can be made large enough that superfluidity is destroyed and the system remains normal even at zero temperature. In this context, we will demonstrate the validity of the Luttinger theorem separately for the two spin populations for any “Φ-derivable” approximation, and illustrate it in particular for the self-consistent t-matrix approximation.     

Singular filaments organize chemical waves in three dimensions: III. Knotted waves

This is the third of a series of papers on the anatomy of three-dimensional organizing centers in excitable media. We here ask whether all self-consistent waves in excitable media are topologically equivalent to the experimentally-verified scroll ring, whose axis lies in a plane. As a test case we examine a scroll ring whose axis contains a knot. It proves to be incompatible with the requirements of physical chemistry unless simultaneously twisted by an amount equal to the “writhing number” of its axis (which is zero for planar closed curves). Appropriate initial conditions are suggested for experimentally creating a wave whose source is a scroll ring knotted and twisted in this way.     

Local Acausality

A fair amount of recent scholarship has been concerned with correcting a supposedly wrong, but wide-spread, assessment of the consequences of the empirical falsification of Bell-type inequalities. In particular, it has been claimed that Bell-type inequalities follow from “locality tout court” without additional assumptions such as “realism” or “hidden variables”. However, this line of reasoning conflates restrictions on the spatio-temporal relation between causes and their effects (“locality”) and the assumption of a cause for every event (“causality”). It thus fails to recognize a substantial restriction of the class of theories that is falsified through Bell-type inequalities.     

Entropy uncertainty relations and stability of phase-temporal quantum cryptography with finite-length transmitted strings

Any key-generation session contains a finite number of quantum-state messages, and it is there-fore important to understand the fundamental restrictions imposed on the minimal length of a string required to obtain a secret key with a specified length. The entropy uncertainty relations for smooth min and max entropies considerably simplify and shorten the proof of security. A proof of security of quantum key distribution with phase-temporal encryption is presented. This protocol provides the maximum critical error compared to other protocols up to which secure key distribution is guaranteed. In addition, unlike other basic protocols (of the BB84 type), which are vulnerable with respect to an attack by “blinding” of avalanche photodetectors, this protocol is stable with respect to such an attack and guarantees key security.     

Localization for Linear Stochastic Evolutions

We consider a discrete-time stochastic growth model on the d-dimensional lattice with non-negative real numbers as possible values per site. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of the binary contact path process. We show the equivalence between the slow population growth and a localization property in terms of “replica overlap”. The main novelty of this paper is that we obtain this equivalence even for models with positive probability of extinction at finite time. In the course of the proof, we characterize, in a general setting, the event on which an exponential martingale vanishes in the limit.     

Arnold deformation of Petrov types

The Petrov classification of the Weyl tensor, though very useful in general relativity, is unstable under small perturbations. In analogy to Arnold's canonical forms for families of matrices, canonical forms are here found for families of Weyl tensors. These “stabilised” canonical forms for versal deformations of Petrov types have interesting applications, and may be useful in the problem of classifying space-times up to “almost equivalence.”