Wigner Rotation Research Articles

On the difference between Wigner's and Møller's approaches to the description of Thomas precession

An account is given of the Wigner concept of particle spin and velocity rotations and of the variation of the angle between them under Lorentz transformations with noncollinear velocities. It is shown that Møller's description of spin rotation can be reduced to the Wigner rotation, and Møller's formula for the angle of spin rotation in the curvilinear motion of a particle is corrected. The permutation asymmetry of the relativistic velocity addition law distinguishes the Wigner sequence of Lorentzian boosts by its applicability to the description of spin and velocity rotations in curvilinear motion.

Rotation matrices for real spherical harmonics: general rotations of atomic orbitals in space-fixed axes

The angular factors of atomic orbitals are real spherical harmonics. This is independent of the choice of basis function. In the course of molecular electronic structure calculations, numerous rotations of real spherical harmonics are required in a suitably defined space-fixed coordinate system. The origin and axes are space-fixed and rotation matrices defined on the basis of spherical harmonics. Typically, rotations are used to align the local axes for atoms with a unique interatomic direction in a fixed frame for integral evaluations. In this work, a highly compact expression and efficient evaluation of the rotation matrices are given for a real spherical harmonic basis. Relations to Gaunt coefficients are shown explicitly as are recurrence formulae for rotation matrices. This leads to extremely rapid and precise rotation algorithms. The Wigner rotation matrices which are still often used in orbital rotations are shown to be completely surpassed by this approach. The present work is related to a method described by Kautz in the field of image processing but significant improvements have been made, especially in the study of structure and storage of the rotation matrices. After complete testing using computer algebra, a numerical program was written in C. Numerical tests are cited in the closing sections of this work.

Gravitational Spin Entropy Production

Spin entropy production for particles with arbitrary spin moving in a curved spacetime is discussed. There is a Wigner rotation due to both the acceleration an the curvature, which causes an initial pure state to transform into a final mixed state. Depending on the spacetime characteristics, one may find paths on which there is no Wigner rotation and the state remains pure.

Relativity and Lorentz Invariance of Entanglement Distillability

We study entanglement distillability of bipartite mixed spin states under Wigner rotations induced by Lorentz transformations. We define weak and strong criteria for relativistic isoentangled and isodistillable states to characterize relative and invariant behavior of entanglement and distillability. We exemplify these criteria in the context of Werner states, where fully analytical methods can be achieved and all relevant cases presented.

Numerical generation of hyperspherical harmonics for tetra-atomic systems

A numerical generation method of hyperspherical harmonics for tetra-atomic systems, in terms of row-orthonormal hyperspherical coordinates-a hyper-radius and eight angles-is presented. The nine-dimensional coordinate space is split into three three-dimensional spaces, the physical rotation, kinematic rotation, and kinematic invariant spaces. The eight-angle principal-axes-of-inertia hyperspherical harmonics are expanded in Wigner rotation matrices for the physical and kinematic rotation angles. The remaining two-angle harmonics defined in kinematic invariant space are expanded in a basis of trigonometric functions, and the diagonalization of the kinetic energy operator in this basis provides highly accurate harmonics. This trigonometric basis is chosen to provide a mathematically exact and finite expansion for the harmonics. Individually, each basis function does not satisfy appropriate boundary conditions at the poles of the kinetic energy operator; however, the numerically generated linear combination of these functions which constitutes the harmonic does. The size of this basis is minimized using the symmetries of the system, in particular, internal symmetries, involving different sets of coordinates in nine-dimensional space corresponding to the same physical configuration.

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs on the unit sphere $S^2$, in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group $SU(1,1) = Sp(2, R) = SL(2,R)$, the double cover of SO(2,1). The present work develops a theory of turns for $SL(2,C)$, the double and universal cover of SO(3,1) and $SO(3,C)$, rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.

Fast and accurate determination of the Wigner rotation matrices in the fast multipole method

In the rotation based fast multipole method the accurate determination of the Wigner rotation matrices is essential. The combination of two recurrence relations and the control of the error accumulations allow a very precise determination of the Wigner rotation matrices. The recurrence formulas are simple, efficient, and numerically stable. The advantages over other recursions are documented.

Pion Electromagnetic Form Factor in the KT Factorization Formulae

Based on the light-cone (LC) framework and the kT factorization formalism, the transverse momentum effects and the different helicity components' contributions to the pion form factor Fπ(Q2) are recalculated. In particular, the contribution to the pion form factor from the higher helicity components (λ1 + λ2 = ±1), which come from the spin-space Wigner rotation, are analyzed in the soft and hard energy regions respectively. Our results show that the right power behavior of the hard contribution from the higher helicity components can only be obtained by fully keeping the kT dependence in the hard amplitude, and that the kT dependence in LC wavefunction affects the hard and soft contributions substantially. A model for the twist-3 wavefunction ψp(x,k⊥) of the pion has been constructed based on the moment calculation by applying the QCD sum rules, whose distribution amplitude has a better end-point behavior than that of the asymptotic one. With this model wavefunction, the twist-3 contributions including both the usual helicity components (λ1 + λ2 = 0) and the higher helicity components (λ1 + λ2 = ±1) to the pion form factor have been studied within the modified pQCD approach. Our results show that the twist-3 contribution drops fast and it becomes less than the twist-2 contribution at Q2~10GeV2. The higher helicity components in the twist-3 wavefunction will give an extra suppression to the pion form factor. When all the power contributions, which include higher order in αs, higher helicities, higher twists in DA and etc., have been taken into account, it is expected that the hard contributions will fit the present experimental data well at the energy region where pQCD is applicable.

Maple procedures for the coupling of angular momenta. IX. Wigner D-functions and rotation matrices

The Wigner D-functions, Dpqj(α,β,γ), are known for their frequent use in quantum mechanics. Defined as the matrix elements of the rotation operator Rˆ(α,β,γ) in R3 and parametrized in terms of the three Euler angles α, β, and γ, these functions arise not only in the transformation of tensor components under the rotation of the coordinates, but also as the eigenfunctions of the spherical top. In practice, however, the use of the Wigner D-functions is not always that simple, in particular, if expressions in terms of these and other functions from the theory of angular momentum need to be simplified before some computations can be carried out in detail.

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

As is well-known, Lorentz boosts do not form a group: the composite of two boosts is, in general, equivalent to the composite of a rotation — the so-called Thomas (or Wigner) rotation — and a single boost. Using spacetime algebra, the geometric algebra for Minkowski spacetime, we show in this paper how to relate the Thomas rotation to the Doppler shift experienced between two observers in relative motion. Both observers are moving away from a third observer, set in the laboratory, with relative non-parallel velocities.

Rotations associated with Lorentz boosts

It is possible to associate two angles with two successive non-collinear Lorentz boosts. If one boost is applied after the initial boost, the result is the final boost preceded by a rotation called the Wigner rotation. The other rotation is associated with Wigner's O(3)-like little group. These two angles are shown to be different. However, it is shown that the sum of these two rotation angles is equal to the angle between the initial and final boosts. This relation is studied for both low-speed and high-speed limits. Furthermore, it is noted that the two-by-two matrices which are under the responsibility of other branches of physics can be interpreted in terms of the transformations of the Lorentz group, or vice versa. Classical ray optics is mentioned as a case in point.

Relativistic effects in neutron-deuteron elastic scattering

We solved the three-nucleon Faddeev equation including relativistic features such as relativistic kinematics, boost effects and Wigner spin rotations. As dynamical input a relativistic nucleon-nucleon interaction exactly on-shell equivalent to the AV18 potential has been used. The effects of Wigner rotations for elastic scattering observables were found to be small. The boost effects are significant at higher energies.They diminish the transition matrix elements at higher energies and lead in spite of the increased relativistic phase-space factor as compared to the nonrelativistic one to rather small effects in the cross section, which are mostly restricted to the backward angles.

Computing spin networks

We expand a set of notions recently introduced providing the general setting for a universal representation of the quantum structure on which quantum information stands. The dynamical evolution process associated with generic quantum information manipulation is based on the (re)coupling theory of SU (2) angular momenta. Such scheme automatically incorporates all the essential features that make quantum information encoding much more efficient than classical: it is fully discrete; it deals with inherently entangled states, naturally endowed with a tensor product structure; it allows for generic encoding patterns. The model proposed can be thought of as the non-Boolean generalization of the quantum circuit model, with unitary gates expressed in terms of 3 nj coefficients connecting inequivalent binary coupling schemes of n + 1 angular momentum variables, as well as Wigner rotations in the eigenspace of the total angular momentum. A crucial role is played by elementary j-gates (6 j symbols) which satisfy algebraic identities that make the structure of the model similar to “state sum models” employed in discretizing topological quantum field theories and quantum gravity. The spin network simulator can thus be viewed also as a Combinatorial QFT model for computation. The semiclassical limit (large j) is discussed.

Towards a relativistic description of exotic meson decays

This work analyzes hadronic decays of exotic mesons, with a focus on the lightest one, the ${J}^{PC}={1}^{\ensuremath{-}+}$ ${\ensuremath{\pi}}_{1}$, in a fully relativistic formalism, and makes comparisons with nonrelativistic results. We also discuss Coulomb gauge decays of normal mesons that proceed through their hybrid components. The relativistic spin wave functions of mesons and hybrids are constructed based on unitary representations of the Lorentz group. The radial wave functions are obtained from phenomenological considerations of the mass operator. Fully relativistic results (with Wigner rotations) differ significantly from nonrelativistic ones. We also find that the decay channels ${\ensuremath{\pi}}_{1}\ensuremath{\rightarrow}\ensuremath{\pi}{b}_{1},\ensuremath{\pi}{f}_{1},K{K}_{1}$ are favored, in agreement with results obtained using other models.

Thomas Rotation and Thomas Precession

Exact and simple calculation of Thomas rotation and Thomas precessions along a circular world line is presented in an absolute (coordinate-free) formulation of special relativity. A straightforward derivation of the Fermi–Walker equation is also given. Besides the simplicity of calculations the absolute treatment of spacetime allows us to make a clear conceptual distinction between the phenomena of Thomas rotation and Thomas precession.

Greenberger–Horne–Zeilinger correlation and Bell-type inequality seen from a moving frame

The relativistic version of the Greenberger–Horne–Zeilinger experiment with massive particles is proposed. We point out that, in the moving frame, GHZ correlations of spins in original directions transfer to different directions due to the Wigner rotation. Its effect on the degree of violation of Bell-type inequality is also discussed.

Pion form factor in thekTfactorization formalism

Based on the light-cone (LC) framework and the $k_T$ factorization formalism, the transverse momentum effects and the different helicity components' contributions to the pion form factor $F_{\pi}(Q^2)$ are recalculated. In particular, the contribution to the pion form factor from the higher helicity components ($\lambda_1+\lambda_2=\pm 1$), which come from the spin-space Wigner rotation, are analyzed in the soft and hard energy regions respectively. Our results show that the right power behavior of the hard contribution from the higher helicity components can only be obtained by fully keeping the $k_T$ dependence in the hard amplitude, and that the $k_T$ dependence in LC wave function affects the hard and soft contributions substantially. As an example, we employ a model LC wave function to calculate the pion form factor and then compare the numerical predictions with the experimental data. It is shown that the soft contribution is less important at the intermediate energy region.

Relativistic velocity space, Wigner rotation, and Thomas precession

We develop a relativistic velocity space called rapidity space from the single assumption of Lorentz invariance, and use it to visualize and calculate effects resulting from the successive application of non-collinear Lorentz boosts. In particular, we show how rapidity space provides a geometric approach to Wigner rotation and Thomas precession in the same way that space–time provides a geometrical approach to kinematic effects in special relativity.

Sivers function in light-cone quark model and azimuthal spin asymmetries in pion electroproduction

We perform a calculation of Sivers function in a light-cone SU(6) quark–diquark model with both scalar diquark and vector diquark spectators. We derive the transverse momentum dependent light-cone wave function of the proton by taking into account the Melosh–Wigner rotation. By adopting one-gluon exchange, we obtain a non-vanishing Sivers function of down quark from interference of proton spin amplitudes. We analyze the |P h⊥| M weighted Sivers asymmetries in π +, π − and π 0 electroproduction off transverse polarized proton target, averaged and not averaged by the kinematics of HERMES experiment.

WIGNER ROTATIONS, BELL STATES, AND LORENTZ INVARIANCE OF ENTANGLEMENT AND VON NEUMANN ENTROPY

We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein–Podolsky–Rosen–ell entangled states and their behaviour under the Lorentz group are analyzed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transformations can then be described succinctly. Reduced density matrices applicable to systems of identical particles are defined through Yang's prescription. The von Neumann entropy of each of the reduced density matrix is Lorentz invariant; and its relevance as a measure of entanglement is discussed, and illustrated with an explicit example. A regularization of the entropy in terms of generalized zeta functions is also suggested.