Space Inversion Research Articles

Space inversion of spinors revisited: A possible explanation of chiral behavior in weak interactions

We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space M1,3, we have four independent 4-component spinors, each living in a different minimal left ideal of Cl(1,3). We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by considering an action for a generalized spinor field ψαi that has not only a spinor index α but also an extra index i running over four ideals. The covariant derivative of ψαi contains the generalized spin connection, the extra components of which are interpreted as the SU(2) gauge fields of weak interactions and their generalization. We thus arrive at a system that is left–right symmetric due to the presence of a “parallel sector”, postulated a long time ago, that contains mirror particles coupled to mirror SU(2) gauge fields.

Understanding High‐Resolution Spectra of Nonrigid Molecules Using Group Theory

Permutation-inversion group theory has developed to become an important tool in the high-resolution spectroscopy of nonrigid molecules. This large class of molecules is very intriguing to study. Small molecules such as ammonia or Na(3) are known to be nonrigid. With increasing size, however, several large-amplitude motions are possible in a molecule, and can even interact with each other. The high-resolution spectra of nonrigid molecules are known to be quite complicated and very rich in information. Details about the molecule and its internal dynamics can be extracted, such as the molecular structure, the character of the chemical bonds, and the barrier heights to internal rotation and their dependence on the chemical bonds. However, due to the nonrigidity of the molecule and the complexity of such spectra, their analysis is usually quite challenging. Theoretical methods are needed for their prediction and analysis. This Review concentrates on permutation-inversion group theory and its usefulness for the analysis of high-resolution spectra of nonrigid molecules, which is examined in more detail using different examples. In a separate section, a special aspect of molecular symmetry is discussed: the breakdown of symmetry principles. Special emphasis is placed on the breakdown of space inversion symmetry (parity violation) in chiral molecules and its possible implications in high-resolution spectroscopy.

Inequivalent solutions for Dirac spin-12 particles under conservation of parity I

Inequivalent invariance constraints upon solutions for Dirac spin-12 particles under conservation of space inversion, time reversal, and charge conjugation have been established, respectively, from standard representation. For space inversion we explicitly show that the wave functions for zero mass neutrinos satisfy only one constraint. For free particles we show that the conventional plane wave solutions as well as Foldy–Wouthuysen representation conditionally comply with the inversion constraints. As a result, only two positive energy solutions and two negative energy solutions can be established. Instead, by law of conservation of parity one shall obtain four linearly independent plane wave solutions which hold not only for positive energies but also for negative energies. We explicitly point out why conventional approach fails to obtain such result. In contrast with free particles one expects intuitively that in a Coulomb field, one can establish twice as many as conventional solutions. Indeed, one shall find easily from inversion constraints that additional linearly independent degenerate bound states have to be established. We present the explicit result.

Giant transient Nernst–Ettingshausen effect in nearly gapless materials

We have measured a pulsed laser induced transient Nernst-Ettingshausen (TNE) effect for semimetal Bi and for nearly gapless materials Bi-Sb and Bi-Te alloy single crystals at 4.2 K and under magnetic fields up to 4 T. We have found the giant longitudinal-TNE (l-TNE) and transverse-TNE (t-TNE) signals for all materials. The l-TNE signals show a quite unusual antisymmetric magnetic field dependence, which suggests a mechanism of bipolar state caused by breakdown of space inversion symmetry.

Introducing Viewpoints of Mechanics into Basic Growth Analysis-(XI) : Negative Weight Problem in Basic Growth Functions and its Hypothetic Avoidance by Sign Reversal of Relative Growth Rate, Space Inversion and Time Reversal-

This study was conducted to avoid negative weight problem hypothetically by applying sign reversal of relative growth rate, space inversion and time reversal to basic growth functions with negative weight. The negative weight occurred when ± signs are given to each of three terms including weight when the square root was taken to solve a differential equation for basic growth mechanics. The positive weight was the ani- mal body weight and the negative weight was interpreted as the feed whose absolute value was expressed in terms of the animal body weight. The results obtained were as follows. Space inversion suggested that the feed was produced at the external space of the animal body. Time reversal suggested that the feed was produced before fed to the animal. The product of sign reversal of RGR and space inversion or time reversal gave positive weight to the feed. These mathematical operations were summarized by the transformation of coordinates from (r, x, y, z, t, -W0) to (-r, -x, -y, -z, -t, W0). Thus, the animal ate the feed with positive weight. It was suggested from the present study that the negative weight problem in basic growth functions was hypothetically avoided by applying sign reversal, space inversion and time reversal.

Introducing Viewpoints of Mechanics into Basic Growth Analysis-(XII) : Applying Sign Reversal of Phase, Space Inversion and Time Reversal to Wave Functions with Negative Amplitude-

Introducing Viewpoints of Mechanics into Basic Growth Analysis-(XII) : Applying Sign Reversal of Phase, Space Inversion and Time Reversal to Wave Functions with Negative Amplitude-

Parity- and time-odd atomic multipoles in magnetoelectricGaFeO3as seen via soft x-ray Bragg diffraction

We present resonant soft x-ray Bragg diffraction data on the space-group forbidden (010) reflection at the $\text{Fe}\text{ }{L}_{2,3}$ edges of magnetoelectric ${\text{GaFeO}}_{3}$. Observed intensities give direct evidence for strange atomic multipoles that violate both time and space inversion, in particular a magnetoelectric quadrupole. The discovery of magnetoelectric multipoles in the open $3d$ shell is a direct measure of coupling between magnetic and electric properties and a basis to a microscopic understanding of magnetically induced multiferroics.

Integral Geometry on Discrete Grassmannians in Z n

We study the Radon transform R on the discrete Grassmannian of rank-d affine sublattices of Zn for 0 < d < n, extending and building on previous work of the first- and third-named authors in codimension 1. By analogy with the integral geometry on Grassmannians in Rn, various natural questions are treated, such as definition and properties of R and its dual transform R*, function space setting, support theorems and inversion formulas.

Vortices in chiral, spin-triplet superconductors and superfluids

Superconductors exhibit unconventional electronic and magnetic properties if the Cooper pair wave function breaks additional symmetries of the normal phase. Rotational symmetries in spin and orbital spaces, as well as discrete symmetries such as space and time inversion, may be spontaneously broken. When this occurs in conjunction with broken global U(1) gauge symmetry, new physical phenomena are exhibited below the superconducting transition that are characteristic of the broken symmetries of the pair condensate. This is particularly true of vortices and related defects. Superconductors with a multi-component order parameter exhibit a variety of different vortex structures and closely related defects that are not possible in condensates belonging to a one-dimensional (1D) representation. In this paper, we discuss the structure of vortices in fermionic superfluids and superconductors which break chiral symmetry, i.e. combined broken time inversion and 2D parity. In particular, we consider the structure of vortices and defects that might be realized in thin films of 3He-A and the layered superconductor Sr2RuO4, and identify some of the characteristic signatures of broken chiral symmetry that should be revealed by these defects.

Odd-frequency pairings in a non-centrosymmetric system

We study possible pairing symmetries of non-centrosymmetric superconductors in the Hubbard model with the Rashba-type spin–orbit interaction (RSOI). Because of the breakdown of space inversion symmetry due to RSOI, a mixture of pairing states with different symmetries can emerge. We find that the RSOI mixes not only the spin-singlet even-parity pairing and spin-triplet odd-parity pairings with even-frequency symmetry, but it also mixes the spin-singlet odd-parity pairing and spin-triplet even-parity pairings with odd-frequency symmetry.

Rise of the appendage

Rise of the appendage

Study of Surface Rashba Effect by Spin- and Angle-Resolved Photoelectron Spectroscopy

Spin-split energy band structure appears in the surface state due to a broken space inversion symmetry at the surface boundary as well as a strong spin-orbit coupling. Spin- and angle-resolved photoemission spectroscopy (spin-resolved ARPES) has been applied to the study of the spin-split energy band structures of non-magnetic solid surfaces. First of all, some explanation of the spin-splitting mechanism for free electrons, so-called Rashba effect, will be given. Next, our spin-resolved ARPES system equipped with the compact Mott-type spin polarimeter will be introduced. We will show how the surface Rashba effect is observed from the spin-resolved ARPES with the recent result for the surface states of Bi(111). Here, we have observed strong and anti-symmetric spin polarizations with respect to the center of surface Brillouin zone in the surface states of Bi(111).

Direct evidence of spin-polarized band structure of Sb(111) surface

We have directly characterized the spin-polarized energy-band structures of Sb(111) near the Fermi level by spin- and angle-resolved photoemission spectroscopy. The results clearly indicate that the energy bands are spin-split due to the spin-orbit coupling and the broken space inversion symmetry.

Deformed coordinate-space Hartree-Fock-Bogoliubov approach to weakly bound nuclei and large deformations

The coordinate-space formulation of the Hartree-Fock-Bogoliubov (HFB) method enables the self-consistent treatment of mean field and pairing in weakly bound systems whose properties are affected by the particle continuum space. Of particular interest are neutron-rich, deformed drip-line nuclei, which can exhibit novel properties associated with neutron skin. To describe such systems theoretically, we developed an accurate two-dimensional lattice Skyrme-HFB solver HFB-AX based on basis (or B)-splines. Compared to previous implementations, ours incorporated a number of improvements aimed at boosting the solver's performance. These include the explicit imposition of axiality and space inversion, use of the modified Broyden method to solve self-consistent equations, and a partial parallelization of the code. HFB-AX has been compared with other HFB codes, both spherical and deformed, and the accuracy of the B-spline expansion was tested by employing the multiresolution wavelet method. Illustrative calculations are carried out for stable and weakly bound nuclei at spherical and very deformed shapes, including constrained fission pathways. In addition to providing new physics insights, HFB-AX can serve as a useful tool to assess the reliability and applicability of coordinate-space and configuration-space HFB frameworks, both existing and in development.

Magneto-electric optics in non-centrosymmetric ferromagnets

Non-centrosymmetric ferromagnetic materials can show optical and x-ray responsesunique to magneto-electrics. For example, the refractive index and absorptioncoefficient may be dependent on the directions of light (x-ray) propagationk andmagnetization M, and these dependences can be termed directional birefringence and directional dichroism,respectively. Such a kind of magneto-electric (ME) optics has recently been observed inGaFeO3 and CuB2O4. The ME optics can be explained from the microscopic point of view by the interference ofan electric dipole (E1) process with a magnetic dipole (M1) or electric quadrupole (E2) processat non-centrosymmetric magnetic sites. Even in magnetic crystals with global inversionsymmetry, the E1–E2 interference can occur at magnetic sites with local inversion breaking,causing unique x-ray diffraction. Such ME-effect-related x-ray diffraction has been found inGaFeO3 and Fe3O4. Artificial breaking of space inversion by means of currently used techniques suchas lithography can be applied to the exploration of multiferroics, as a way ofinvestigating ME optics. Relatively large ME optical signals at room temperaturehave been successfully observed in diffraction from such tailor-made multiferroics.

A new generation of journals is born

The 21(st) century has provided new technical and educational opportunities as well as challenges which have not been approached and exploited sufficiently by traditional scientific journals. Furthermore, case reports are usually "neglected" by traditional journals for a variety of reasons, including limited space in the hardcopy version of the journal issue and priority over original research and review articles. The Journal of Radiology Case Reports is a new generation of journals, which is dedicated to provide open-access, high-quality, peer-reviewed and interactive Radiology case reports.

Demonstration of the Spin-Statistics Connection in Elementary Quantum Mechanics

A simple demonstration of the spin-statistics connection is presented. The effect of exchange and space inversion operators on two-particle states is reviewed. The connection follows directly from successive application of these operations to the two-particle wave function for identical particles in an s-state, evaluated at spatial coordinates $\pm x$, but at equal time, \emph{i.e.}, at spacelike interval.

Gigantic Optical Magnetoelectric Effect in CuB2O4

Although it has been well known that materials in which both space inversion and time reversal symmetries are broken can host optical magneto-electric effect, i.e., change in optical constants with the reversal of propagating direction of light, the largest change in absorption ever reported on this effect was 0.2%. Here we show that optical absorption in noncentrosymmetric weak ferromagnetic material CuB 2 O 4 changes by more than 100% with reversal of a low magnetic field of 300 Oe. The gigantic optical magneto-electric effect is ascribed to the canted antiferromagnetic spin ordering of square-coordinated Cu 2+ sites, where the local inversion is slightly broken.

The fuzzy symmetries for linear tri-atomic $${\rm B} \cdots {\rm A} \cdots {\rm C}$$ dynamic systems

Based on our previous study on the elementary characterization of the fuzzy symmetry, we inquired the static state fuzzy symmetry of some molecules and their molecular orbitals (MO). Now we will analyze the fuzzy symmetry of some simple linear tri-atomic dynamic systems in connection with the reaction. Three related transformations will mainly be studied in detail. These three transformations are (1) the space inversion transformation about the mid-atom as the center, (2) the reaction reversal transformation in relation to the reaction B + AC→BA + C and (3) the joint transformation of the above two. We examined the variation for the internal configuration of these systems owing to the operation of above three transformations, and then establish methods to calculate the fuzzy symmetry characterization, such as the membership functions for the MOs of such linear tri-atomic dynamic systems in relation to these transformations. We examined the variation regularity in relation to the fuzzy symmetry characterization for the MOs of these systems along the intrinsic reaction coordinate (IRC) and dividing line. The variation regularity and the distribution for the fuzzy symmetry characterization in related internal configuration coordinate space are also analyzed. An IRC-scale is suggested for internal configuration coordinate space in this paper.

Optical magnetoelectric effect induced by a moving medium

In this Brief Report, we report a theoretical investigation of the possibility of realizing the optical magnetoelectric effect with a moving medium. This unique approach is quite distinct from other methods using conventional multiferroic materials or artificial materials with broken space inversion and time reversal symmetries simultaneously. Its advantage lies in the controllability of the optical properties, which change with the change of velocity and intrinsic refractive index of the moving medium. This work in the present frame offers a route to tune this phenomenon.