Wigner Rotation Research Articles

Image rotation, Wigner rotation, and the fractional Fourier transform

In this study the degree p = 1 is assigned to the ordinary Fourier transform. The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row. Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform. Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms. I approach the subject of fractional Fourier transforms in two other ways. First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming. Second, I propose two optical setups that are able to perform a fractional Fourier transform.

The proton spin and the Wigner rotation

It is shown that in both the gluonic and strange sea explanations of the Ellis-Jaffe sum rule violation discovered by the European Muon Collaboration (EMC), the spin of the proton, when viewed in in its rest reference frame, could by fully provided by quarks and antiquarks within a simple quark model picture, taken into account the relativistic effect from the Wigner rotation.

Torsion angle analysis of glycolipid order at membrane surfaces

A method is presented for determining the average glycosidic torsion angles and motion about those angles for a glycolipid headgroup at a model membrane surface. Dipolar and quadrupolar coupling constants were previously collected on the headgroup of beta-dodecyl glucoside embedded in phospholipid/detergent bilayers which orient in a magnetic field (Sanders, C.R., and J.H. Prestegard. 1991. J. Am. Chem. Soc. 113:1987-1996). These observables are expressed as averages of second order spherical harmonics, and Wigner rotation matrices are used here to transform the spherical harmonics from the laboratory frame to a set of frames which allow motional averaging to be described as the result of simple bond rotations. Euler angles corresponding to rotations about glycosidic torsion angles phi and psi are chosen to best reproduce experimental coupling constants, using models which have varying degrees of motional averaging. These models include a rigid headgroup, axially symmetric headgroup motion, and independent motion about each torsion angle in a square well potential. The square well model proves to be significantly better than the rigid model in reproducing experimental observations and it offers a more physically meaningful description of motion than the axially symmetric model. The structures obtained, assuming a square well potential, are compared to potential energy maps for the glycolipid torsional angles to illustrate the need for inclusion of the membrane interface in energetic modeling of glycolipid conformations.

The reflection of light by planar stratified media: the groupoid of amplitudes and a phase 'Thomas precession'

The reflection coefficient of light on stratified planar structures can be obtained by postulating the use of a complex generalization of Einstein's addition theorem for parallel velocities. The algebraic properties of the 'composition law of amplitudes' show that the set of all complex amplitudes of the electromagnetic field in heterostructures forms a weakly associative-commutative groupoid. The first concrete application of this abstract concept was found only in 1988 in special relativity. This work exhibits another example in a quite different field of physics. It also puts into evidence that the 'phase rotation' of light in stratified planar structures is to be considered as a 'Thomas rotation'.

On the relativistic velocity composition paradox and the Thomas rotation

The non-commutativity and the non-associativity of the composition law of the non-colinear velocities lead to an apparent paradox, which in turn is solved by the Thomas rotation. A 3×3 parametric, unimodular and orthogonal matrix elaborated by Ungar is able to determine the Thomas rotation. However, the algebra involved in the derivation of the Thomas rotation matrix is overwhelming. The aim of this paper is to present a direct derivation of the Thomas angle as the angle between the composite vectors of the non-colinear velocities, thus obtaining a simplicity with which the rotation can be expressed. This allows the formulation of an alternative to the statement related to the necessity of the Thomas rotation of the Cartesian axes by the statement implying the necessity of the rotation of the direct (inverse) relativistic composite velocity to coincide with the inverse (direct) relativistic composite velocity.

Measuring the Wigner rotation with electron beams.

The Wigner phase-space formalism has been shown to provide the natural tools to treat charged-beam transport. In this article we show how simple experiments, exploiting electron beams propagating through quadrupole magnets and drift-tube sections, can be designed to measure the Wigner rotation

Effective heavy quark theory

We show how heavy quark effective theory, including 1/ M Q corrections, may be matched onto dynamical quarks models by making a specific choice of K μ , m and ν μ in the p μ = mν μ + K μ expansion. We note that Wigner rotations of heavy quark spins arise at O( p 2/ m 2) in non-relativistic models by at O( Λ QCD/ M Q) orO(velocity-transfer) in HQET and so are necessary for a consistent treatment.

Factorization relations and Wigner's rotation matrices

A factorization formula for the squared Wigner rotation function is derived. General properties of the coefficients and special cases are discussed. The connection with recent factorization relations for vibrational transition probabilities in harmonic and anharmonic oscillators is studied.

Quantum calculation of thermal rate constants for the H+D2 reaction

Thermal rate constants for the H+D2 reaction on the LSTH potential-energy surface are determined quantum mechanically over T=300–1500 K using the quantum flux–flux autocorrelation function of Miller [J. Chem. Phys. 61, 1823 (1974)]. Following earlier works [T. J. Park and J. C. Light, J. Chem. Phys. 91, 974 (1989); T. J. Park and J. C. Light, ibid. 94, 2946 (1991)], we use the adiabatically adjusted principal axis hyperspherical coordinates of Pack [Chem. Phys. Lett. 108, 333 (1984)] and a direct product C2v symmetry-adapted discrete variable representation to evaluate the Hamiltonian and flux. The initial representation of the J=0 Hamiltonian in the ℒ2 basis of ∼14 000 functions is sequentially diagonalized and truncated to yield ∼600 accurate eigenvalues and eigenvectors for each symmetry species block. The J>0 Hamiltonian is evaluated in the direct product basis of truncated J=0 eigenvectors and parity decoupled Wigner rotation functions. Diagonalization of the J>0 Hamiltonian is performed separately for each KJ block by neglecting Coriolis coupling and approximating K coupling by perturbation. Both eigenvalues and eigenvectors are corrected by the perturbation. Thermal rate constants for each J, kJ(T), are then determined by the flux–flux autocorrelation function considering nuclear spins. Due to the eigenvector corrections, both parity calculations are required to determine kJ(T). Overall thermal rate constants k(T) are obtained by summing kJ(T) over J with the weight of 2J+1 up to J=30. The results show good agreement with experiments.

Vector parametrization of the N-body problem in quantum mechanics: Polyspherical coordinates.

The configuration of an N-body system can be entirely represented by N-1 relative position vectors after separation of the center-of-mass motion. Many of the sets of coordinates that are commonly used for describing molecular configurations can be viewed as spherical coordinates, for the various vectors, collected together. The spherical angles are local; i.e., they are defined for frames that change from one vector to another. Each particular set of coordinates of that nature (polyspherical coordinates) consists of three Euler angles for the overall rotation of the body-fixed frame and 3N-6 internal coordinates: the N-1 vector lengths, N-2 planar angles between pairs of vectors, and N-3 dihedral angles between two vectors around a third one. This article aims at developing an example of this type of parametrization, where the body-fixed-frame z axis is parallel to one vector. The quantum-mechanical kinetic-energy operator for the system so described is derived. The operator action on the angular part of the functional basis set is studied (Wigner rotation matrix elements for the Euler angles and spherical harmonics for the internal angles), and the structure of the matrix representing the kinetic-energy operator is described in detail. The advantages and drawbacks of the present vector parametrization and the polyspherical coordinates are discussed. The principal advantage is in numerically calculating the matrix elements of the kinetic-energy operator: The integration over all angles turns out to be analytically achieved, so that the numerical effort is to be concentrated only on the N-1 radial coordinates. Radial basis functions are to be selected according to the physical context (collisonal or vibrational, or any other). Thus the angular basis set proposed constitutes an adequate finite-basis representation for the kinetic-energy operator and, combined with a discrete-variable representation for the potential energy, is likely to provide an efficient collocation framework for the dynamical study of more-than-three particle systems.

Boost, recoil, and Wigner rotation effects on no-pair analyses of proton elastic scattering.

The relativistic ``no-pair'' potential for elastic proton scattering is shown to reduce to a conventional t\ensuremath{\rho} form using optimal factorization of the optical potential and the impulse approximation. The nucleon-nucleon t matrix is needed in the Breit frame and it may be obtained by boosting the NN c.m. frame t matrix. It is shown that the general form of the boost involves a Mo?ller factor, Wigner rotation operators acting on spins, and a Lorentz boost of momentum arguments. The reduction to t\ensuremath{\rho} form clarifies how the no-pair analysis differs from conventional Watson or Kerman, McManus, and Thaler (KMT) analyses which usually omit Wigner rotation effects and use a Galilean boost of momentum arguments of the t matrix instead of a Lorentz boost. Recent calculations in momentum space have found rather minor off-shell effects based on the no-pair analysis but other calculations have reported substantial effects based on the KMT formalism. Off-shell effects owing to the momentum dependence of the NN t matrix are found to be considerably smaller than the difference between published no-pair and KMT calculations. It is shown that technical differences, predominantly caused by use of a Galilean boost and inaccurate Coulomb corrections, are capable of explaining the disparate results at least at 200 MeV. Wigner rotations are found to cause negligible differences in proton scattering observables at 200 and 500 MeV. KMT calculations based on a Lorentz boost and accurate Coulomb corrections in momentum space yield rather minor off-shell effects owing to the momentum dependence of the NN t matrix in essential agreement with the no-pair analysis.

Geometrical interpretation for relativistic composition of velocities and the Thomas precession

The relativistic composition of velocities is interpreted in terms of the rapidity triangle, as a generalization of the non-relativistic parallelogram rule. Some geometrical experiments are presented for grasping essential properties of the rapidity triangle, by preparing a curved sheet of paper on which the composition of rapidities is mapped. The Wigner rotation and accordingly the Thomas precession are interpreted also in the same scheme.

The Relativistic Kinematic and Electromagnetic Problems of Thomas Rotation

The Relativistic Kinematic and Electromagnetic Problems of Thomas Rotation

Successive Lorentz transformations of the electromagnetic field

A velocity-orientation formalism to deal with compositions of successive Lorentz transformations, emphasizing analogies shared by Lorentz and Galilean transformations, has recently been developed. The emphasis in the present article is on the convenience of using the velocity-orientation formalism by resolving a paradox in the study of successive Lorentz transformations of the electromagnetic field that was recently raised by Mocanu. The paradox encountered by Mocanu results from the omission of the Thomas rotation (or, precession) which is involved in the composition of two Lorentz transformations with corresponding noncollinear velocity parameters. By resolving this paradox, we expose (i) the central role that the Thomas rotation plays in special relativity, (ii) the need to consider in special relativity orientations in addition to velocities between inertial frames, and (iii) the power and elegance of the novel velocity-orientation formalism. A similar paradox in STR that Mocanu pointed out, also resulting from the omission of the Thomas rotation, has been resolved in a previous communication.

Multipole theory of composite pulses. II. Time-reversal and offset-alternating sequences with experimental confirmation

In the multipole basis, a pure δ pulse has the effect of a rotation of the multipole polarizations with the rotations described by Wigner rotation matrices. By retaining the form of these Wigner rotation matrices, an exact analytical expression for a composite pulse, described by a product of rotations, is given with its explicit dependence to offset Δω RF amplitude ω 1 and phase φ. By the analysis of the symmetries of the tilt angle β and azimuthal angle α to these parameters, composite sequences in which off-resonance and phase-distortion effects are minimized can be developed. Experimental confirmation for up to six composite pulses is given.

Interaction induced contributions to the far infrared spectrum of a polar fluid: Zeroth and second spectral moments

The zeroth and second spectral moments for the far infrared spectral intensity of a polar liquid are analyzed from first principles, taking into account the first-order dipole–induced dipole contribution. The results, based on evaluating the equilibrium averages of appropriate correlation functions, are simplified by considering that the liquid pair distribution function can be approximated by a truncated expansion in terms of Wigner rotation functions. Averages involving three- and four-molecule distribution functions are not considered in the present calculation. The calculated zeroth moment for acetonitrile, M(0)=17.5 D2, compares favorably with the experimental value of 20.7 D2. The difference probably reflects the neglect of three- and four-molecule terms as well as higher order induction effects.

Modeling some two-dimensional relativistic phenomena using an educational interactive graphics software

This paper describes several relativistic phenomena in two spatial dimensions that can be modeled using the collision program of Spacetime Software. These include the familiar aberration, the Doppler effect, the headlight effect, and the invariance of the speed of light in vacuum, in addition to the rather unfamiliar effects like the dragging of light in a moving medium, reflection at moving mirrors, Wigner rotation of noncommuting boosts, and relativistic rotation of shrinking and expanding rods. All these phenomena are exhibited by tracings of composite computer printouts of the collision movie. It is concluded that an interactive educational graphics software with pleasing visuals can have considerable investigative power packed within it.

Group-Like Structure Underlying the Unit Ball in Real Inner Product Spaces

Abstraction of the relativistic velocity addition law and of the Thomas rotation of the special theory of relativity yields a means of endowing the unit ball in any real inner product space with a group- like structure, in which the standard associative- commutative laws are relaxed by means of the Thomas rotation. The resulting group- like object is called a complete weakly associative- commutative groupoid. Any complete WACG can be extended to a group analogous to the Lorentz group of the special theory of relativity.

SO(n,1) Wigner rotation as an SL(2,R) problem

We show that the composition of not only two SO(3,1) boosts, but also that of two SO(n,1) boosts for anyn ≥ 2, is basically an SO(2,1) problem and hence can be analysed completely using SL(2,R) matrices. By computing the expression for the Thomas/Wigner angle directly using SL(2,R) matrices we show that this approach results in considerable economy of algebra.

The relativistic velocity composition paradox and the Thomas rotation

The relativistic velocity composition paradox of Mocanu and its resolution are presented. The paradox, which rests on the bizarre and counterintuitive non-communtativity of the relativistic velocity composition operation, when applied to noncollinear admissible velocities, led Mocanu to claim that there are “some difficulties within the framework of relativistic electrodynamics.” The paradox is resolved in this article by means of the Thomas rotation, shedding light on the role played by composite velocities in special relativity, as opposed to the role they play in Galilean relativity.