Inversion Invariance Research Articles

Intermediate-value property for Assouad dimension by its inversion invariance

We prove that if X is a metric space of Assouad dimension s∈(0,∞), then for every α∈[0,s] there is a countable subset of X of Assouad dimension α. We reduce this to the special case of bounded X due to Wang and Wen in 2016 by using the inversion invariance of Assouad dimension.

Complete Nuclear Permutation Inversion Invariant Artificial Neural Network (CNPI-ANN) Diabatization for the Accurate Treatment of Vibronic Coupling Problems.

A recently developed scheme to produce accurate high-dimensional coupled diabatic potential energy surfaces (PESs) based on artificial neural networks (ANNs) [ J. Chem. Phys. 2018, 149, 204106 and J. Chem. Phys. 2019, 151, 164118] is modified to account for the proper complete nuclear permutation inversion (CNPI) invariance. This new approach cures the problem intrinsic to the highly flexible ANN representation of diabatic PESs to account for the proper molecular symmetry accurately. It turns out that the use of CNPI invariants as coordinates for the input layer of the ANN leads to a much more compact and thus more efficient representation of the diabatic PES model without any loss of accuracy. In connection with a properly symmetrized vibronic coupling reference model, which is modified by the output neurons of the CNPI-ANN, the resulting adiabatic PESs show perfect symmetry and high accuracy. In the present paper, the new approach will be described and thoroughly tested. The test case is the representation and corresponding vibrational/vibronic nuclear dynamics of the low-lying electronic states of planar NO3 for which a large number of ab initio data is available. Thus, the present results can be compared directly with the previous studies.

A characteristic space–time conservation element and solution element method for conservation laws

In this paper, an upwind space–time conservation element and solution element (CE/SE) method is developed to solve conservation laws. In the present method, the mesh quantity and spatial derivatives are the independent marching variables, which is consistent with the original CE/SE method proposed by Chang (1995) [5]. The staggered time marching strategy and the definition of conservation element (CE) also follow Chang's propositions. Nevertheless, the definition of solution element (SE) is modified from that of Chang. The numerical flux through the interface of two different conservation elements is not directly derived by a Taylor expansion in the reversed time direction as proposed by Chang, but determined by an upwind procedure. This modification does not change the local and global conservative features of the original method. Although, the time marching scheme of mesh variables is the same with the original method, the upwind fluxes are involved in the calculation of spatial derivatives, yielding a totally different approach from that of Chang's method. The upwind procedure breaks the space–time inversion invariance of the original scheme, so that the new scheme can be directly applied to capture discontinuities without spurious oscillations. In addition, the present method maintains low dissipation in a wide range of CFL number (from 10−6 to 1). Furthermore, we extend the upwind CE/SE method to solve the Euler equations by adopting three different approximate Riemann solvers including Harten, Lax and van Leer (HLL) Riemann solver, contact discontinuity restoring HLLC Riemann solver and mathematically rigorous Roe Riemann solver. Extensive numerical examples are carried out to demonstrate the robustness of the present method. The numerical results show that the new CE/SE solvers perform improved resolutions.

The Quaternion Structure of Space-Time and Arrow of Time

In fundamental theories of physics, the dynamical equations all have time inversion invariance. Except for the evolution of some simple system which has realistic inverse processes, but for a slightly more complicated system, the evolution processes are irreversible. This is the problem of arrow of time, which is always warmly debated. In different point of view, we find there may have some conceptual misunderstanding in the controversy: 1) The realization of an inverse process does not mean the time of the system goes backward. 2) The principles of relativity and covariance are the constraints to physical laws, but not constraints to specific solutions. The equations must be covariant, but the solutions are not definitely symmetric. 3) Time is a global property of the universe, which is a measurement of the evolution process of the universe. The internal time of a matter system reflecting its internal evolution speed also takes this cosmic time as a unified background and standard of measurement. 4) The universe has a unified cosmic time T and a cosmic space related to this cosmic time. They are objective and absolute. 5) The eigensolution of a spinor is a critical state losing time concept, which responses the interaction of environment with some uncertainty, then the evolution process of the world is not uniquely determined. 6) The non-uniqueness of the evolution process means that the inverse process is absent. So for a world including spinors, the evolution is essentially irreversible. In this paper, according to the widely accepted principles and direct calculations of transformation, we reveal the misunderstandings in the usual controversy, and then give more natural and reasonable explanations for structure of space-time and arrow of time.

Inclusion filters: a class of self-dual connected operators

In this paper, we define a connected operator that either fills or retains the holes of the connected sets depending on application-specific criteria that are increasing in the set theoretic sense. We refer to this class of connected operators as inclusion filters, which are shown to be increasing, idempotent, and self dual (gray-level inversion invariance). We demonstrate self duality for 8-adjacency on a discrete Cartesian grid. Inclusion filters are defined first for binary-valued images, and then the definition is extended to grayscale imagery. It is also shown that inclusion filters are levelings, a larger class of connected operators. Several important applications of inclusion filters are demonstrated-automatic segmentation of the lung cavities from magnetic resonance imagery, user interactive shape delineation in content-based image retrieval, registration of intravital microscopic video sequences, and detection and tracking of cells from these sequences. The numerical performance measures on 100-cell tracking experiments show that the use of inclusion filter improves the total number of frames successfully tracked by five times and provides a threefold reduction in the overall position error.

From Time Inversion to Nonlinear QED

In Minkowski flat space-time, it is perceived that time inversion is unitary rather than antiunitary, with energy being a time vector changing sign under time inversion. The Dirac equation, in the case of electromagnetic interaction, is not invariant under unitary time inversion, giving rise to a “Klein paradox.” To render unitary time inversion invariance, a nonlinear wave equation is constructed, in which the “Klein paradox” disappears. In the case of Coulomb interaction, the revised nonlinear equation can be linearized to give energy solutions for Hydrogen-like ions without singularity when nuclear number Z>137, showing a reversed energy order pending for experimental tests such as Zeeman effects. In non-relativistic limit, this nonlinear equation reduces to nonlinear Schrodinger equation with soliton-like solutions. Moreover, particle conjugation and electron-proton scattering with a nonsingular current-potential interaction are discussed. Finally the explicit form of gauge function is found, the uniqueness of Lorentz gauge is proven and the Lagrangian density of quantum electrodynamics (QED) is revised as well. The implementation of unitary time inversion leads to the ultimate derivation of nonlinear QED.

Correspondence of classical and quantum irreversibilities

The reversibility of classical mechanics is broken by the Carnot principle; in quantum mechanics, the time inversion invariance is broken by the principle of state decoherence. In both cases, a greater complexity produces a faster irreversibility. These principles are applied to problems related to absorption and diffusion of light. It appears that Raman scattering in extremely low-pressure gases shifts the frequency of the incident light, is space coherent and thus may be confused with the Doppler effect: the Raman scattering produces at least a part of the galactic redshift; the expansion of the universe is reduced or possibly cancelled.

Molecular potential energy surfaces by interpolation in Cartesian coordinates

We present a new method for expressing a molecular potential energy surface (PES) as an interpolation of local Taylor expansions. By using only Cartesian coordinates for the atomic positions, this method avoids redundancy problems associated with the use of internal coordinates. The correct translation, rotation, inversion, and permutation invariance are incorporated in the PES via the interpolation method itself. The method is most readily employed for bound molecules or clusters and is demonstrated by application to the vibrational motion of acetylene.

SYMMETRY IN THE PROBLEM OF VIBRATION OF A POLAR-ORTHOTROPIC NON-HOMOGENEOUS PLATE ON AN ELASTIC FOUNDATION

The problem of bending vibration of a polar-orthotropic non-homogeneous elastic plate on an elastic foundation is considered, and the invariance of the problem-equation under an inversion transformation with respect to a circle is proved. As a corollary, the optimization problem (the “best” position of the point mass or point support, which optimizes the plate fundamental frequency) is considered and certain geometrical inequalities, which reduce the optimization domain, are proved. Computational time economy generated by the inequalities for the optimization problem is illustrated with numerical examples: (a) the “best” position of the point mass on segment and ring-sector plates; (b) the “best” radius of the ring support for annular plates. Some other corollaries from the problem-equation inversion invariance are given.

Unification of the forces

Unification of the forces is achieved via an elementary application of the symmetry of parity-time-charge (PTC) inversion invariance to classical physics. The strong color force between quarks is found to be a PTC-inversion of the Coulomb force and to have a strength of about 3.75×105 times the Coulomb force between electrons at a separation distance equal to the electron Coulomb radius, r0/2 ≈ 0.7045×10−13 cm [note that what we call the “electron Coulomb radius” (a convenient unit) differs by a factor of 4 from the usual classical electron radius]. It displays “asymptotic freedom” as predicted and, consistent with the predicted spatial variation, it is found to be constant, or independent of the particle separation distance. The strong color force and other PTC-inverse EM forces are found to exist only between quarks, whereas the EM forces are found to exist only between leptons and composite particles. The weak force is the residual of long range EM and PTC-inverse EM forces (a similar situation is indicated of the short-range nucleon-nucleon force). These residual forces are of mixed parity. The net potentials between particles are polynomials in the separation variable r, which have minima and maxima at the zeros of the first derivative. The minima are the stable orbits, which provides a classical basis for quantum mechanics. Finally, the Coulomb and the gravitational forces are found to be non-reciprocal P-inversions of each other and become indistinguishable at mx = ±√(2α) (qx/e)Mp, where Mp is the Planck mass.

Relativistic quantum mechanics of spin-zero and spin-half particles

We consider the quantum mechanics of directly interacting relativistic particles of spin-zero and spin-half. We introduce a scalar product in the vector space of physical states which is finite, positive definite and relativistically invariant and keeps orthogonal eigenstates of total four momentum belonging to different eigenvalues. This allows us to show that the vector space of physical states is, in fact, a Hilbert space. The case of two particles is explicitly considered and the Cauchy problem of physical wave function illustrated. The problem of a spin-1/2 particle interacting with a spin-zero particle is considered and a new equation is proposed for two spin-1/2 particles interacting via the most general form of interaction possible. The restrictions due to Hermiticity, space inversion and time reversal invariance are also considered.

An isotropic description of the new-found gradient-related magneto-optical effect (invited)

The recent observation of a magnetization-gradient-related birefringence in ferromagnetic materials is reviewed. A new phenomenological description using an isotropic dielectric material law in which the dielectric tensor is a linear function of the components of the propagation vector of the light wave and of the symmetric part of the magnetization gradient tensor is proposed. The symmetry of this interaction is derived in terms of space and time inversion invariance requirements, adding information from perpendicular incidence experiments. The dielectric law contains only one material parameter and displays additional unexpected symmetries. It is compatible with all observations for the perpendicular incidence, in-plane magnetization case. Its predictions for the case of oblique incidence and a gradient in the polar component of magnetization are found not to be in contradiction with experiments when precautions are taken to avoid asymmetric diffraction due to the polar Kerr effect.

Mach's principle and the reciprocal symmetry of nature

This article develops the postulate that spacetime-charge inversion invariance reflects a fundamental reciprocal symmetry in nature between the two long range forces, from which the derivation of Mach's principle (i.e., the principle that the fundamental parameters of the electromagnetically elementary charged particle are related to those describing the electromagnetically observable universe) follows quite easily. Interpreting this result, it is argued that relativity and quantum mechanics can be made conceptually compatible and mathematically consistent by this reciprocal symmetry if one realizes that relativity isboth a macroscopic, semiclassical theory (i.e., the “global half of relativity,” described by Eq. (1.1), including special and general relativity) and a microscopic theory (i.e., the “local half of relativity,” described by Eq. (2.1), including relativistic quantum mechanics and field theory). The reciprocal symmetry of nature, then, promises unique (differential and/or integral) relationships between the coordinate variables of the “observers” of these tworeciprocally related theories, which implies unique, consistent numerical values for the scalar curvatureR, the massM, and the critical density for “closure,”ρ c, of the observable universe [derived from the elementary particle parameters (i.e., the electron mass and Coulomb radius)]. With this symmetry we also postulate a plausible mechanism for spontaneous generation of matter from the ubiquitous (zero-mass ether) “nothingness” of the “Dirac sea” of “filled negative energy states,” and can consistently interpret both the positive and negative-energy state solutions of Dirac's equation for massive, spin-1/2 (i.e., fermion) particles and both the advanced and retarded potential solutions of electromagnetic field equations. It is pointed out that, with this interpretation of the “advanced potential solutions” from electromagnetic field theory, one can actuallyderive causality from electromagnetic theory.

Vanishing of the energy-momentum tensor in euclidean Yang-Mills theories

Certain global properties of the energy-momentum tensor in euclidean field theories are established. These properties combined with the inversion invariance are used to show that the energy-momentum tensor corresponding to finite-action solutions of euclidean YM theories must vanish everywhere.

Field induced charge-transfer exciton transitions

This paper reports the observation of a second zero-phonon line, induced by a static electric field, lying 10 cm−1 above the first singlet charge-transfer exciton state of crystalline anthracene-PMDA. The splitting of these charge-transfer exciton states is a quadratic function of the electric field strength for low fields and becomes linear for fields of order 104 V/cm. The origin of these features is discussed on the basis of a one-dimensional model of the charge-transfer exciton in the linear donor–acceptor stack. The effect of charge-transfer admixture to the neutral donor–acceptor ground state and the effect of creating a local charge-transfer (ionic) state by the absorption of light are considered by use of a second-quantized Hamiltonian for the charge-transfer exciton. The splitting of 10 cm−1 observed by use of the static electric field, which removes the inversion invariance of the one-dimensional donor–acceptor chain along the stack axis arises from the mixing of symmetric and antisymmetric exciton eigenstates by the field. Numerical estimates of the splitting, based on experimental data, are in order of magnitude agreement with observation. The charge-transfer exciton band structure is derived by considering both electron and hole hopping terms and tight binding (nearest unit cell) excitation transfer.

External inversion, internal inversion, and reflection invariance

Having in mind that physical systems have different levels of structure we develop the concept of external, internal and total improper Lorentz transformation (space inversion and time reversal). A particle obtained from the ordinary one by the application of internal space inversion or time reversal is generally a different particle. From this point of view the intrinsic parity of a nuclear particle (`elementary particle') is in fact the external intrinsic parity, if we take into account the internal structure of a particle. We show that non-conservation of the external parity does not necessarily imply non-invariance of nature under space inversion. The conventional theory of beta-decay can be corrected by including the internal degrees of freedom to become invariant under total space inversion, though not under the external one.

A nonlocal variational formulation of the equations of radiative transport

A coordinate invariant formulation and a nonlocal variational principle are obtained for the radiative transport equation. Problems with normal scattering kernels are shown to lead to both space and time inversion invariance relations between the transport function and its formal adjoint. The relations for non-normal kernels are also obtained.

On a relation between unitary transformations in the impact parameter approximation which describes the behaviour of the electrons before and after the trajectory crosses the turning point

A relation is derived between unitary transformations in the impact parameter approximation or in the classical path method. The unitary transformations describe the behaviour of the electrons before and after the trajectory, which is followed by heavy particles, crosses the turning point. The properties used to establish the relation are the assumption that the trajectory of heavy particles is determined by a spherically symmetric potential as well as the invariance of the Hamiltonian under rotation and the time inversion invariance. To facilitate the use of this relation an additional relation is derived using the fact that a trajectory determined by a spherically symmetric potential is constrained to a plane.

On the possibility of « kinematic » (unitarity) origin ofCP-violation in the K L → 2π decay

The problem of adding ray corepresentations of pari ty (P) and t ime reversal (T) to the physically relevant uni tary irreducible representations of the connected Poincar~ group ~?+ (UIRP) was discussed by WIGNV.R (1). I t is noteworthy tha t P is interpreted in ref. (1) not as space inversion, but as the product of space inversion and charge conjugation. A reason for this interpretat ion may be posed as follows. Space inversion alone yields neutrinos of both negative and positive helicities. Since a positive-helicity (electronic or muonie) neutrino does not exist in nature, the Hilbert space ~f~---[(v]] spanned by the states of the neutrino does not (( aeeomodate , space inversion. This implies tha t any process involving one neutrino must violate space inversion invariance. If one insists on having par i ty as a symmetry operator, one has to , correct ~) the s i tuat ion; and this one can achieve by the aforementioned interpretat ion: ~f~ contains both the states of the negative-helicity neutrino and the positiveheliei ty antineutrino. In view of the discovery of processes violat ing par i ty in this new interpretation, the above mot ivat ion seems to lose its grounds. One is tempted therefore to resume the original interpretat ion of P as the space inversion operation, implying thereby the violation of par i ty in all processes involving one neutrino or antineutrino. Moreover, the relevant uni tar i ty sum rule induces/n pr inc ip le -v io la t ion of par i ty for nonleptonic processes as well. As an example, consider the process A°-->~-p. Wri t ing S ~-1 "4iB , uni tar i ty implies i(R* R) ~ BR*, or

Invariance and Conservation Laws in Classical Mechanics

It is commonly supposed that, in classical mechanics, invariance of a physical system to coordinate translation implies conservation of linear momentum. ``Invariance'' may be defined in a number of ways. If it is defined to mean invariance of the equation of motion, it is shown that invariance of this equation with respect to coordinate translation does not imply conservation of linear momentum. The effects of scale transformation and coordinate inversion invariance are also investigated. Both the Lagrangian approach and Newton's (second) law approach are considered. It is shown that each of the above invariances implies a condition on the equation of motion, while a combination of these and time-inversion invariance is needed to obtain ordinary momentum conservation.