Isospectral flow and lax pairs

Certain non-linear differential equations may be considered as compatibility conditions for a linear system of equations, that is, as the vanishing-curvature condition for some connection. It is shown how one can obtain this connection from an isospectral-flow condition using a Foldy-Wouthuysen transformation.     

Conformal transformations in metric‐affine gravity and ghosts

Conformal transformations play a widespread role in gravity theories in regard to their cosmological and other implications. In the pure metric theory of gravity, conformal transformations change the frame to a new one wherein one obtains a conformal‐invariant scalar–tensor theory such that the scalar field, deriving from the conformal factor, is a ghost. In this work, conformal transformations and ghosts will be analyzed in the framework of the metric‐affine theory of gravity. Within this framework, metric and connection are independent variables, and, hence, transform independently under conformal transformations. It will be shown that, if affine connection is invariant under conformal transformations, then the scalar field of concern is a non‐ghost, non‐dynamical field. It is an auxiliary field at the classical level, and might develop a kinetic term at the quantum level. Alternatively, if connection transforms additively with a structure similar to yet more general than that of the Levi‐Civita connection, the resulting action describes the gravitational dynamics correctly, and, more importantly, the scalar field becomes a dynamical non‐ghost field. The equations of motion, for generic geometrical and matter‐sector variables, do not reduce connection to the Levi‐Civita connection, and, hence, independence of connection from metric is maintained. Therefore, metric‐affine gravity provides an arena in which ghosts arising from the conformal factor are avoided thanks to the independence of connection from the metric.     

Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms.While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.     

The Quantum Geometric Tensor in a Parameter-Dependent Curved Space

We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.     

Massive self-duality solution associated with invariant one-forms

A massive self-duality solution associated with invariant 1-forms is presented. At the zero mass limit the massive self-dual theory of the SO(3) gauge group on 4 dimensions cannot be reduced to that of massless self-duality.In such a case the self-dual connection turns to the flat connection and one cannot obtain a massless theory in such an approach.     

On Almost Norden Statistical Manifolds

We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian metric, in particular the Norden metric. We investigate almost Norden (statistical) manifolds under the condition that the almost complex structure J is ∇-recurrent. We provide one example of a complex statistical connection.     

Paths of spinning particles in general relativity as geodesics of an Einstein connection

This paper deals with the Papapetrou-Pirani equations of motion for a spinning test particle in general relativity. The motion of the center of mass can be represented by the geodesic equation of an affine connection that is the sum of the Christoffel connection and a tensor that depends on the Riemann-Christoffel curvature tensor, the mass of the particle, its 4-velocity, and its spin tensor. The connection is not unique, and here it is chosen to satisfy one of the basic geometrical principles of Einstein's unified field theory: The symmetric part of the fundamental tensor of the geometry is specified to be the metric tensor of general relativity. The special case of conformally flat space-times is discussed.     

Formal Connections for Families of Star Products

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection, which was introduced by the first author. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled on the formal symplectic vector fields. Moreover, we showthat if the parameter space has trivial first cohomology group, any two flat formal connections are related by an automorphism of the family of star products.     

Parallelogram space

Two pairs of parallel geodesic intervals can be generated from two such intervals with a common endpoint by suitable application of parallel transfer. When the connection coefficients are symmetric, a parallelogram-like figure with a closed perimeter can be constructed with the two pairs of sides. When the connection coefficients are asymmetric, combining the two pairs of parallel intervals yields a figure with a coordinate gap. To avoid an open perimeter, a fiber that contains many coordinate-like labels is constructed for each point. Then the gap can occur in the fiber of a single point. Thus a closed figure is obtained even when the connection is asymmetric. A process to obtain the coordinate-like quantities in the fiber of one point with respect to a nearby point is described. A discussion of transformations of the coordinate-like quantities follows and the fundamental curvature-like quantity is obtained.     

Energy-Work Connection Integration Scheme for Nonholonomic Hamiltonian Systems

This paper focuses on studying a new energy-work relationship numerical integration scheme of nonholonomic Hamiltonian systems. The signal-stage numerical, multi-stage and parallel composition numerical integration schemes are presented. The high-order energy-work relation scheme of the system is constructed by a parallel connection of n multi-stage schemes of order 2, its order of accuracy is 2n. The connection, which is discrete analogue of usual case, between the change of energy and work of nonholonomic constraint forces is obtained for nonholonomic Hamiltonian systems. This paper also gives that there is smaller error of the scheme when taking a large number of stages than a less one. Finally, an applied example is discussed to illustrate these results.     

On a certain solution of the Einstein-Maxwell equations

A connection is established between a certain exterior solution of the Einstein-Maxwell equations and an interior cylindrically symmetric stationary solution found earlier by one of the authors.     

Quasiparticle kinetic equation in geometry of local base vectors

Kinetic equations for quasiparticle excitations in ideal crystals, known from solid state physics, are generalized to the case of material bodies the crystal structure of which is distorted by the existence of continuously distributed defects. Distribution of defects is described by a field of local base vectors of a primitive crystal lattice. The form of conservation laws implied by such kinetic equations is discussed using the example of energy balance in a phonon system. It is shown that energy balance can be written either with respect to lattice connection or with respect to the Euclidean connection, having a vanishing source term in both cases. Transition from one version to another involves a redefinition of the heat flux vector.     

Determining and uniformly estimating the gauge potential corresponding to a given gauge field on M 4

In an earlier paper on the field copy problem, we proved that there exists a generic set of connections (gauge potentials) on a principle bundle with a semi-simple structure group over a four-dimensional base manifold for which the connection is uniquely determined by its curvature (gauge field). We conjectured that there exists a smaller, but still generic, set of connections for which the curvature map sending a connection to its curvature admits a continuous inverse with respect to the appropriate function space topologies. The conjecture says, in other words, that restricting to certain generic curvature 2-forms, one can determine and uniformly estimate the connection and its derivatives from the curvature and uniform estimates of its derivatives. In this Letter we give an affirmative answer to the conjecture and show, moreover, that the set of such connections contains an open dense set in the Whitney C topology.     

The significance of curvature in general relativity

In General Relativity, one has several traditional ways of interpreting the curvature of spacetime, expressed either through the curvature tensor or the sectional curvature function. This essay asks what happens if curvature is treated on a more primitive level, that is, if the curvature is prescribed, what information does one have about the metric and associated connection of space-time? It turns out that a surprising amount of information is available, not only about the metric and connection, but also, through Einstein's equations, about the algebraic structure of the energy-momentum tensor.     

Yang–Mills theory à la string

A surface of codimension higher than one embedded in an ambient space possesses a connection associated with the rotational freedom of its normal vector fields. We examine the Yang–Mills functional associated with this connection. The theory it defines differs from Yang–Mills theory in that it is a theory of surfaces. We focus, in particular, on the Euler-Lagrange equations describing this surface, introducing a framework which throws light on their relationship to the Yang–Mills equations. Dedicated to Octavio Obregón, on the occasion of his 60th birthday.     

Gauge-invariant localization of infinitely many gravitational energies from all possible auxiliary structures

The problem of finding a covariant expression for the distribution and conservation of gravitational energy–momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann’s realization that there are infinitely many conserved gravitational energy–momenta. Initially use is made of a flat background metric (or rather, all of them) or connection, because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of pseudotensors of a given type, such as the Einstein pseudotensor, in every coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression (Landau–Lifshitz, Goldberg, Papapetrou or the like) or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether’s theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many.     

Phänomenologische Betrachtung zur Photon-Elektron-Wechselwirkung in einem Plasma

The question at stake is, whether a simple physical connection may be found between Richardson equation for thermionic emission on the one hand, and Richardson equation for photoelectric emission on the other hand. The proposition of such a connection is based on the following supposition: that electrons are not only elements of a (Fermi-Dirac-) statistical ensemble and, as such, cause thermionic phenomena; but that they can also interact with a radiation field, thereby causing an additional emission current, according to Richardson (photoelectric) equation. — It is shown in detail that the current emitted from a metal of 2000 °K is determined by the complete radiation of this metal only to a very slight degree. It is then estimated however, that within temperatures of some million degrees Kelvin the excitation of electrons caused by complete radiation will be much greater than that caused by interaction.     

Connection Strength of Bolt End Plate of Steel Structures Under Earthquake Action

Russian Physics Journal - Bolt end plate connection is one of the most widely used connection methods in steel structures. It is mainly used for beam-column connection and beam-to-beam splicing in...     

Auditory paradox based on fractal waveform

The synthesis of certain paradoxical acoustic waveforms is described that will sound lower when reproduced at higher speeds (and vice versa). Specifically, one such waveform, when recorded on magnetic tape, will sound a semitone lower (rather than an octave higher) when played back at twice the tape speed. The connection with fractal objects in geometry is illustrated.