The problem of a self-gravitating scalar field

In this paper we begin the study of the global initial value problem for Einstein's equations in the spherically symmetric case with a massless scalar field as the material model. We reduce the problem to a single nonlinear evolution equation. Taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, the local, in retarded time, existence and global uniqueness of classical solutions. We also prove that if the initial data is sufficiently small there exists a global classical solution which disperses in the infinite future.Research supported in part by National Science Foundation grants MCS-8201599 to the Courant Institute and PHY-8318350 to Syracuse University     

Golden Oldie

A form of initial value problem is considered in which the initial hypersurface is not spacelike but null. This approach has the striking advantage over the more usual Cauchy problem that all constraints (initial data equations) are eliminated from the theory, for a wide class of interacting fields in special relativity and also for general relativity. The theory is most naturally described in terms of the two-component spinor calculus, for which an elementary introduction is given here. A general scheme for interacting fields, which holds both in special and general relativity, is presented which describes all fields in terms of sets of irreducible spinors. The concept of an exact set of such spinors is introduced and it is shown that this concept is the appropriate one for an initial value problem on a null cone without constraints. The initial data can be expressed in the form of a complex number, called a null datum, defined at each point of the null cone, one corresponding to each spinor. There is the curious feature of these null data that apparently it is sufficient here, to have onehalf as much information per point as in the corresponding Cauchy problem. The classical Maxwell-Dirac theory and the Einstein-Maxwell theory are two examples that can be put into the form of exact sets. The Einstein empty-space equations are also of particular note, and in this case the null datum describes essentially the intrinsic geometry of the null cone. The argument given here as applied to a general exact set is incomplete in two important respects. Firstly it depends on the null data being analytic, and secondly the initial hypersurface must be a cone. However, both these restrictions are removed in the case of certain elementary fields called basic free fields, examples of which are the Weyl neutrino field, the free Maxwell field, and the linearized gravitational field. For these cases a simple explicit formula is introduced which expresses the field at any point in terms of the null datum, as an integral taken over the intersection of the initial null hypersurface with the null cone of the point.This article originally appeared in 1963 in Aerospace Research Laboratories 63-56 (P.G. Bergmann). It is an important and oft-cited work, but as it has never been published in a widely distributed journal, it is generally inaccessable to the relativity community. This regrettable situation is hereby rectified-Ed.This work was done while the author was at Princeton, Syracuse, and Cornell Universities, visiting under a NATO Fellowship administered by the Department of Scientific and Industrial Research in London. The work at Syracuse was supported by the Aeronautical Research Laboratory and at Cornell by the National Science Foundation.     

Local Well-Posedness for Membranes in the Light Cone Gauge

In this paper we consider the classical initial value problem for the bosonic membrane in light cone gauge. A Hamiltonian reduction gives a system with one constraint, the area preserving constraint. The Hamiltonian evolution equations corresponding to this system, however, fail to be hyperbolic. Making use of the area preserving constraint, an equivalent system of evolution equations is found, which is hyperbolic and has a well-posed initial value problem. We are thus able to solve the initial value problem for the Hamiltonian evolution equations by means of this equivalent system. We furthermore obtain a blowup criterion for the membrane evolution equations, and show, making use of the constraint, that one may achieve improved regularity estimates.     

D-branes on non-abelian threefold quotient singularities

We investigate the classical moduli space of D-branes on a non-abelian Calabi-Yau threefold singularity and find that it admits topology-changing transitions. We construct a general formalism of world-volume field theories in the language of quivers and give a procedure for computing the enlarged Kähler cone of the moduli space. The topology changing transitions achieved by varying the Fayet-Iliopoulos parameters correspond to changes of linearization of a geometric invariant theory quotient and can be studied by methods of algebraic geometry. Quite surprisingly, the structure of the enlarged Kahler cone can be computed by toric methods. By using this approach, we give a detailed discussion of two low-rank examples.     

Semi-classical approximation and the problem of boundary conditions in the theory of relativistic particle radiation

Abstract

The process of relativistic particle radiation in an external field has been studied in the semi-classical approximation rather extensively. The main problem arising in the studies is to express the formula of the quantum theory of radiation in terms of classical quantities, for example of the classical trajectories. However, it still remains unclear how the particle trajectory is assigned, that is which particular initial or boundary conditions determine the trajectory in semi-classical approximation quantum theory of radiation.

We shall try to solve this problem. Its importance comes from the fact that in some cases one and the same boundary conditions may give rise to two or more trajectories. We demonstrate that this fact must necessarily be taken into account on deriving the classical limit for the formulae of the quantum theory of radiation, since it leads to a specific interference effect in radiation.

The method we used to deal with the problem is similar to the method employed by Fock to analyze the problem of a canonical transformation in classical and quantum mechanics.     

Quantum-classical correspondence for motion on a plane with deficit angle

A particle constrained to move on a cone and bound to its tip by harmonic oscillator and Coulomb-Kepler potentials is considered both in the classical as well as in the quantum formulations. The SU(2) coherent states are formally derived for the former model and used for showing some relations between closed classical orbits and quantum probability densities. Similar relations are shown for the Coulomb-Kepler problem. In both cases a perfect localization of the densities on the classical solutions is obtained even for low values of quantum numbers.     

Relativity theory: What is reality?

In classical Newtonian physics there was a clear understanding of “what reality is.? Indeed in this classical view, reality at a certain time is the collection of all what is actual at this time, and this is contained in “the present.? Often it is stated that three-dimensional space and one-dimensional time hare been substituted by four-dimensional space-time in relativity theory, and as a consequence the classical concept of reality, as that which is “present,? cannot be retained. Is reality then the four-dimensional manifold of relativity theory? And if so, what is then the meaning of “change in time?? This problem confronts a geometric view (as the Einsteinian interpretation of relativity theory) with a process view (where reality changes constantly in time). In this paper we investigate this problem, taking into account our insight into the nature of reality as it came by analyzing the problems of quantum mechanics. We show that with an Einsteinian interpretation of relativity theory, reality is indeed four-dimensional, but there is no contradiction with the process view, where this reality changes in time.     

Geometrical pinning of magnetic vortices induced by a deficit angle on a surface: Anisotropic spins on a conic space background

We study magnetic vortex-like excitations lying on a conic space background. Two types of them are obtained. Their energies appear to be linearly dependent on the conical aperture parameter, besides of being logarithmically divergent with the sample size. In addition, we realize a geometrical-like pinning of the vortex, say, it is energetically favorable for it to nucleate around the conical apex. We also study the problem of two vortices on the cone and obtain an interesting effect on such a geometry: excitations of the same charge, then repealing each other, may nucleate around the apex for suitable cone apertures. We also pay attention to the problem of the vortex pair and how its dissociation temperature depends upon conical geometry.     

Asymptotics of trajectories for cone potentials

We study the asymptotics of trajectories of the classical Hamiltonian dynamics. For Hamiltonians with cone potentials we have shown earlier that all trajectories are asymptotically free [5], i.e. the asymptotic velocities exist. Here we show that the generic trajectories are asymptotically uniform, i.e. the asymptotic phases exist.     

The Schwarzschild solution: Some conceptual difficulties

It is shown that inconsistencies arise when we look upon the Schwarzschild solution as the space-time arising from a localized point singularity. The notion of black holes is critically examined, and it is argued that, since black hole formation never takes place within the past light cone of a typical external observer, the discussion of physical behavior of black holes, classical or quantum, is only of academic interest. It is suggested that problems related to the source could be avoided if the event horizon did not form and that the universe only contained quasi-black holes.On leave of absence from the Tata Institute of Fundamental Research, Bombay.     

Localization of light on a cone: theoretical evidence and experimental demonstration for an optical fiber

The classical rays propagating along a conical surface are bounded on the narrower side of the cone and unbounded on its wider side. In contrast, it is shown here that a dielectric cone with a small half-angle γ can perform as a high Q-factor optical microresonator which completely confines light. The theory of the discovered localized conical states is confirmed by the experimental demonstration, providing a unique approach for accurate local characterization of optical fibers (which usually have γ ~ 10(-5) or less) and a new paradigm in the field of high Q-factor resonators.     

Extracting dynamical equations from experimental data is NP hard

The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this Letter, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP hard), both for classical and quantum mechanical systems. As a by-product of this work, we give complexity-theoretic answers to both the quantum and classical embedding problems, two long-standing open problems in mathematics (the classical problem, in particular, dating back over 70?years).     

Classical limit of scattering in quantum mechanics—A general approach

The classical and quantum physics seem to divide nature into two domains macroscopic and microscopic. It is also certain that they accurately predict experimental results in their respective regions. However, the reduction theory, namely, the general derivation of classical results from the quantum mechanics is still a far cry. The outcome of some recent investigations suggests that there possibly does not exist any universal method for obtaining classical results from quantum mechanics. In the present work we intend to investigate the problem phenomenonwise and address specifically the phenomenon of scattering. We suggest a general approach to obtain the classical limit formula from the phase shiftδ _l, in the limiting value of a suitable parameter on whichδ _l depends. The classical result has been derived for three different potential fields in which the phase shifts are exactly known. Unlike the current wisdom that the classical limit can be reached only in the high energy regime it is found that the classical limit parameter in addition to other factors depends on the details of the potential fields. In the last section we have discussed the implications of the results obtained.     

Orthomodular Lattices and a Quantum Algebra

We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular element commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although whether or not it holds in all orthomodular lattices remains an open problem, as it does not fail in any of over 50 million Greechie diagrams we tested.     

The far field asymptotics in the problem of diffraction of an acoustic plane wave by an impedance cone

The work deals with the far field asymptotics of the classical solution for the problem of diffraction by an impedance cone. The incident acoustic plane wave completely illuminates the semi-infinite conical surface. The scattered field contains different components in the asymptotics, namely, the spherical wave from the vertex of the cone, the reflected waves, and, under some conditions, also the surface waves of Rayleigh type. We give integral representations for the scattering diagram of the spherical wave. The uniform (with respect to the observation direction) asymptotic expression for the wave field is also addressed and described by the parabolic cylinder ansatz. Dedicated to the memory of Vladimir Borovikov     

A note on classical and quantum unimodular gravity

We discuss unimodular gravity at a classical level, and in terms of its extension into the UV through an appropriate path integral representation. Classically, unimodular gravity is locally a gauge fixed version of general relativity (GR), and as such it yields identical dynamics and physical predictions. We clarify this and explain why there is no sense in which it can “bring a new perspective” to the cosmological constant problem. The quantum equivalence between unimodular gravity and GR is more of a subtle question, but we present an argument that suggests one can always maintain the equivalence up to arbitrarily high momenta. As a corollary to this, we argue, whenever inequivalence is seen at the quantum level, that just means we have defined two different quantum theories that happen to share a classical limit. We also present a number of alternative formulations for a covariant unimodular action, some of which have not appeared, to our knowledge, in the literature before.     

Using Variational Quantum Algorithm to Solve the LWE Problem

The variational quantum algorithm (VQA) is a hybrid classical–quantum algorithm. It can actually run in an intermediate-scale quantum device where the number of available qubits is too limited to perform quantum error correction, so it is one of the most promising quantum algorithms in the noisy intermediate-scale quantum era. In this paper, two ideas for solving the learning with errors problem (LWE) using VQA are proposed. First, after reducing the LWE problem into the bounded distance decoding problem, the quantum approximation optimization algorithm (QAOA) is introduced to improve classical methods. Second, after the LWE problem is reduced into the unique shortest vector problem, the variational quantum eigensolver (VQE) is used to solve it, and the number of qubits required is calculated in detail. Small-scale experiments are carried out for the two LWE variational quantum algorithms, and the experiments show that VQA improves the quality of the classical solutions.     

No-boundary measure of the universe

We consider the no-boundary proposal for homogeneous isotropic closed universes with a cosmological constant and a scalar field with a quadratic potential. In the semiclassical limit, it predicts classical behavior at late times if the scalar field is large enough. The classical histories may be singular in the past or bounce at a finite radius. This probability measure selects inflationary histories but is biased towards small numbers of e-foldings N. However, to obtain the probability of our observations in our past light cone these probabilities should be multiplied by exp(3N). This volume weighting is similar to that in eternal inflation. In a landscape potential, it would predict that the Universe underwent a large amount of inflation and could have always been semiclassical.     

Random Matrices and Lyapunov Coefficients Regularity

Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a “cone property” are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.     

On the relation between classical and quantum-mechanical entropy

We discuss the problem how to define the classical entropy and prove an inequality that establishes its relation to the quantum-mechanical entropy. Furthermore we show that it is monotonic.