Wigner Rotation Research Articles

Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons

In this work, we present the inverted harmonic oscillator (IHO) Hamiltonian as a paradigm to understand the quantum mechanics of scattering and time-decay in a diverse set of physical systems. As one of the generators of area preserving transformations, the IHO Hamiltonian can be studied as a dilatation generator, squeeze generator, a Lorentz boost generator, or a scattering potential. In establishing these different forms, we demonstrate the physics of the IHO that underlies phenomena as disparate as the Hawking–Unruh effect and scattering in the lowest Landau level (LLL) in quantum Hall systems. We derive the emergence of the IHO Hamiltonian in the LLL in a gauge invariant way and show its exact parallels with the Rindler Hamiltonian that describes quantum mechanics near event horizons. This approach of studying distinct physical systems with symmetries described by isomorphic Lie algebras through the emergent IHO Hamiltonian enables us to reinterpret geometric response in the lowest Landau level in terms of relativistic effects such as Wigner rotation. Further, the analytic scattering matrix of the IHO points to the existence of quasinormal modes (QNMs) in the spectrum, which have quantized time-decay rates. We present a way to access these QNMs through wave packet scattering, thus proposing a novel effect in quantum Hall point contact geometries that parallels those found in black holes.

Spinors of spin-one-half fields

This paper reviews how a two-state, spin-one-half system transforms under rotations. It then uses that knowledge to explain how momentum-zero, spin-one-half annihilation and creation operators transform under rotations. The paper then explains how a spin-one-half field transforms under rotations. The momentum-zero spinors are found from the way spin-one-half systems transform under rotations and from the Dirac equation. Once the momentum-zero spinors are known, the Dirac equation immediately yields the spinors at finite momentum. The paper then shows that with these spinors, a Dirac field transforms appropriately under charge conjugation, parity, and time reversal. The paper also describes how a Dirac field may be decomposed either into two 4-component Majorana fields or into a 2-component left-handed field and a 2-component right-handed field. Wigner rotations and Weinberg’s derivation of the properties of spinors are also discussed.

Successive Lorentz transformations for energy and momentum. Application: relativistic elastic scattering of two particles having non-collinear velocities and its dependence on Thomas rotation

Successive Lorentz transformations (SLT) for energy and momentum are derived, and the involved Thomas rotation angle (TRA) is highlighted. In particular, if the axes of the coordinate systems related by SLT are suitably chosen, the SLT for energy and momentum do not depend on the TRA explicitly and take a form analogous to that of the Lorentz transformation for energy and momentum. Then the study of dynamical problems involving SLT becomes as easy as that of dynamical problems involving a Lorentz transformation. Those results are applied to the study of a relativistic elastic scattering of two particles having non-collinear velocities as a dynamical problem involving SLT. The expressions for the energies and momentums of the scattered particles in the laboratory system are obtained in terms of the TRA. If the considered particles are protons with given non-collinear velocities, the TRA, the energies and directions of the scattered protons are determined. Original diagrams for momentum are obtained.

Electrostatics of charged dielectric spheres with application to biological systems. III. Rigorous ionic screening at the Debye-Hückel level.

The unequivocal role of electrostatic forces in biological (and colloidal) systems underscores the importance of attaining accurate and rapid calculations of electrostatic forces if one wishes to faithfully simulate the electrostatic aspect of a biological system. This paper makes significant progress toward this aspect as it rigorously incorporates ionic screening at the Debye-Hückel level for an electrolyte system containing dielectric spheres of finite radii. We investigated earlier this system without mobile ions via a surface charge method. However, the need for computing a large number of Wigner rotation matrix elements per configuration can significantly slow down the numerical calculations. This difficulty was recently overcome by our Wigner-matrix-free formalism. Unfortunately, in that method ions can only be included individually, making it impractical to investigate, for example, ionic screening in a system modeled by charged dielectric spheres immersed in a solution of mobile ions. Here, we overcome this difficulty by extending the surface charge method to treat ions implicitly. Previous treatments of charged dielectric spheres in a solution of mobile ions did not emphasize the energy reciprocity of electrostatics and are largely limited to a few spheres and/or special symmetries. Our new formalism respects reciprocity and accommodates arbitrarily many dielectric spheres of different dielectric constants and sizes while being rigorous at the Debye-Hückel level. The differences, and the relationship, between our new implicit ion treatment and our previous ion-free (or explicit ion) approach are described. A closed form for the electrostatic energy with implicit ions is also provided. This new formalism speeds up the computation of the electrostatic energy in the presence of ions, and accommodates permanent and induced multipoles that are very important when the polarization effect needs to be correctly included. We also mention how the proposed method can be transformed to a numerical method for use with arbitrary nonspherical surfaces.

The relativistic mechanism of the Thomas–Wigner rotation and Thomas precession

We consider the Thomas–Wigner rotation of coordinate systems under successive Lorentz transformations of inertial reference frames, and disclose its physical mechanism on the basis of the relativistic contraction of moving scale, and the relativity of the simultaneity of events for different inertial observers. This result allows us to better understand the physical meaning of the Thomas precession, and to indicate some overlooked aspects of the physical interpretation of this effect, as related to two specific examples: the circular motion of a classical electron around a heavy nucleus, and the motion of a classical electron along an open path, where its initial velocity and acceleration are mutually orthogonal to each other.

Tailoring optical pulling forces with composite microspheres

Optical pulling forces or tractor beams can pull particles against light propagation by redirecting the incident photons forward. This is typically achieved using Bessel beams with very small half-cone angles, which considerably limits its applicability. One can circumvent such issue by using a superposition of plane waves. In order to investigate optical pulling forces exerted by a pair of non-colinear plane waves, we develop a theoretical framework based on Mie theory, Debye potentials and Wigner rotation matrices. We apply this framework to calculate the optical pulling force on metallo-dielectric composite particles, which we put forward as an alternative material platform to optimize and tailor tractor beams. Indeed we demonstrate that by adding a few plasmonic inclusions to low-refractive index dielectric particles of arbitrary sizes, we are able to produce polarization dependent optical pulling forces that cannot occur in the corresponding homogeneous particles. Altogether our findings not only provide innovative theoretical methods to compute optical pulling forces, but also provide new strategies to tailor and optimize them, paving the way to increase their applicability.

Composition of velocities and momentum transformations in a scator-deformed Lorentz metric

The transformations of coordinates, velocities and momenta between inertial frames are described via real scator algebra. This algebra imposes a hyperbolic metric that is identical to the Lorentz metric in $$1+1$$ dimensions but becomes deformed in higher dimensions. Differences in the transformations of transverse quantities arise due to this departure in two and three spatial dimensions. The structure admits velocities arbitrarily close to c in orthogonal directions, but nonetheless, the scator velocity magnitude is always subluminal. The transformation between frames is described by an isometric scator product mapping. The product operation satisfies commutative group properties in a restricted subspace. Due to these properties, a sequence of Lorentz boosts can be synthesized as a single Lorentz boost without involving a rotation. A consequence of this algebraic structure is that the scator composition of velocities does not produce a Thomas rotation. The scenarios where this deformed metric could play a relevant role have not been yet assessed.

Reexamining the Thomas precession

We review the Thomas precession exhibiting the exact form of the Thomas rotation in the axis-angle parameterization. Assuming three inertial frames S, S′, and S″ moving with arbitrary velocities and with S and S″ having their axes parallel to the axes of S′, we focus our attention on the two essential elements of the Thomas precession: (i) there is a rotation between the axes of frames S and S″ and (ii) the combination of two Lorentz transformations from S to S′ and from S′ to S″ fails to produce a pure Lorentz transformation from S to S″. The physical consequence of (i) and (ii) refers to the impossibility of arbitrary frames S, S′, and S″ moving with non-parallel relative velocities having their axes mutually parallel. Then, we reexamine the validity of (i) and (ii) under the conjecture that time depends on the state of motion of the frames and we show that the Thomas precession assumes a different form as formulated in (i) and (ii).

Conformational Entropy from Restricted Bond-Vector Motion in Proteins: The Symmetry of the Local Restrictions and Relation to NMR Relaxation.

Locally mobile bond-vectors contribute to the conformational entropy of the protein, given by Sk ≡ S/k = −∫(Peq ln Peq)dΩ – ln∫dΩ. The quantity Peq = exp(−u)/Z is the orientational probability density, where Z is the partition function and u is the spatially restricting potential exerted by the immediate internal protein surroundings at the site of the motion of the bond-vector. It is appropriate to expand the potential, u, which restricts local rotational reorientation, in the basis set of the real combinations of the Wigner rotation matrix elements, D0KL. For small molecules dissolved in anisotropic media, one typically keeps the lowest even L, L = 2, nonpolar potential in axial or rhombic form. For bond-vectors anchored at the protein, the lowest odd L, L = 1, polar potential is to be used in axial or rhombic form. Here, we investigate the effect of the symmetry and polarity of these potentials onSk. For L = 1 (L = 2), Sk is the same (differs) for parallel and perpendicular ordering. The plots of Sk as a function of the coefficients of the rhombic L = 1 (L = 2) potential exhibit high-symmetry (specific low-symmetry) patterns with parameter-range-dependent sensitivity. Similar statements apply to analogous plots of the potential minima. Sk is also examined as a function of the order parameters defined in terms of u. Graphs displaying these correlations, and applications illustrating their usage, are provided. The features delineated above are generally useful for devising orienting potentials that best suit given physical circumstances. They are particularly useful for bond-vectors acting as NMR relaxation probes in proteins, when their restricted local motion is analyzed with stochastic models featuring Wigner-function-made potentials. The relaxation probes could also be molecules adsorbed at surfaces, inserted into membranes, or interlocked within metal–organic frameworks.

Effect of Thomas Rotation on the Lorentz Transformation of Electromagnetic fields

A relativistic particle undergoing successive boosts which are non collinear will experience a rotation of its coordinate axes with respect to the boosted frame. This rotation of coordinate axes is caused by a relativistic phenomenon called Thomas Rotation. We assess the importance of Thomas rotation in the calculation of physical quantities like electromagnetic fields in the relativistic regime. We calculate the electromagnetic field tensor for general three dimensional successive boosts in the particle’s rest frame as well as the laboratory frame. We then compare the electromagnetic field tensors obtained by a direct boost overrightarrow{beta }+delta overrightarrow{beta } and successive boosts overrightarrow{beta } and Delta overrightarrow{beta } and check their consistency with Thomas rotation. This framework might be important to situations such as the calculation of frequency shifts for relativistic spin-1/2 particles undergoing Larmor precession in electromagnetic fields with small field non-uniformities.

Dalitz-plot decomposition for three-body decays

We present a general formalism to write the decay amplitude for multibody reactions with explicit separation of the rotational degrees of freedom, which are well controlled by the spin of the decay particle, and dynamic functions on the subchannel invariant masses, which require modeling. Using the three-particle kinematics we demonstrate the proposed factorization, named the Dalitz-plot decomposition. The Wigner rotations, that are subtle factors needed by the isobar modeling in the helicity framework, are simplified with the proposed decomposition. Consequently, we are able to provide them in an explicit form suitable for the general case of arbitrary spins. The only unknown model-dependent factors are the isobar lineshapes that describe the subchannel dynamics. The advantages of the new decomposition are shown through three examples relevant for the recent discovery of the exotic charmonium candidate $Z_c(4430)$, the pentaquarks $P_c$, and the intriguing $\Lambda_c^+\to pK^-\pi^+$ decay.

Wigner numbers.

All reduced Wigner rotation matrix elements dM'M J(θ) can be evaluated very efficiently as a linear combination of either cos(Nθ) or sin(Nθ) terms as N runs in unit steps from either 0 or 12 to J. Exact, infinite-precision formulas are derived here for the Fourier coefficients in these dM'M J(θ) expressions by finding remarkable analytic solutions for the normalized eigenvectors of arbitrarily large matrices that represent the ĴY angular momentum operator in the basis of ĴZ eigenstates. The solutions involve collections of numbers Wm,n J for (m, n) = (J-M, J-N) ∈ [0, 2J] that satisfy the recursion relation (m+1)Wm+1,n J-2(J-n)Wm,n J+(2J-m+1)Wm-1,n J=0. These quantities, designated here as Wigner numbers, are proved to be integers that exhibit myriad intriguing mathematical properties, including various closed combinatorial formulas, (M, N) sum rules, three separate M-, N-, and J-recursion relations, and a large-J limiting differential equation whose applicable solutions are products of a polynomial and a Gaussian function in the variable z = -21/2(J + 1)-1/2M. Accordingly, the Wigner numbers constitute a new thread of mathematics extending outside the context of their immediate discovery. In the midst of the Wm,n J proofs, a class of previously unknown combinatorial summation identities is also found from Wigner number orthonormalization conditions.

Helicity rotation induced by Lorentz boosts

In this paper, we calculate the helicity rotation angle induced by Lorentz boosts. This is relevant for the study of Lorentz boost effects on quantum entanglement encoded in pairs of massive fermions, which are described in terms of positive energy solutions of the Dirac equation with definite helicity. A Lorentz boost describing the change to an inertial frame moving at uniform speed will in general rotate the particle’s helicity. We obtain the coefficients of the helicity superposition in the boosted frame and specialize our results for a perpendicular boost geometry. We verify that the helicity rotation angle can be obtained in terms of the Wigner rotation angle for spin [Formula: see text] states, bridging the framework considered in our previous works to the one of the Wigner rotations. Finally, we calculate the boost-induced spin-parity entanglement for a single particle.

Thin film-based Gires–Tournois Resonator (GTR) as quasi-optical analogue of the Thomas rotation angle effect in special relativity

We present an unconventional approach to generate an optical analogue of the Thomas rotation angle found in Special Relativity (SR) by using a thin-film–based Gires–TournoisResonator (GTR). At first glance, the phase response of the GTR seems to fail as an optical analogue for the Thomas rotation angle in the phase domain but we demonstrate that an analogue can be successfully constructed in the intensity domain by combining the GTR with a Michelson interferometer (MI), thus forming an interferometer known as the Michelson–Gires–Tournois Interferometer (MGTI). Thomas rotation angle is a spatial rotation of the reference frame due to the Einstein velocity addition (EVA) law of two successive relativistic velocities travelling in non-collinear directions, whereas, the GTR is a thin-film based asymmetric Fabry–Perot resonator (FPR) with a partially reflecting front mirror, and a 100% rear-reflecting mirror. We investigate in detail, both analytically and numerically, the Thomas rotation angle’s full behaviour under various parameter conditions. This approach offers the following advantages: (i) it leads to a simpler overall configuration, (ii) it measures the Thomas rotation angle directly requiring only a single optical parameter instead of multiple parameters compared with other methods, and (iii) it requires no sophisticated multilayer thin film design.

Local Ordering at the N-H Sites of the Rho GTPase Binding Domain of Plexin-B1: Impact of Dimerization.

We have developed a new molecular dynamics (MD) based method for describing analytically local potentials at mobile N-H sites in proteins. Here we apply it to the monomer and dimer of the Rho GTPase binding domain (RBD) of the transmembrane receptor plexin-B1 to gain insight into dimerization, which can compete with Rho GTPase binding. In our method, the local potential is given by linear combinations, u(DL,K), of the real combinations of the Wigner rotation matrix elements, DL,K, with L = 1-4 and appropriate symmetry. The combination that "fits best" the corresponding MD potential of mean force, u(MD), is the potential we are seeking, u(DL,K-BEST). For practical reasons the fitting process involves probability distributions, Peq ∝ exp(-u), instead of potentials, u. The symmetry of the potential, u(DL,K), may be related to the irreducible representations of the D2h point group. The monomer (dimer) potentials have mostly Ag and B2u (B1u and B2u) symmetry. For the monomer, the associated probability distributions are generally dispersed in space, shallow, and centered at the "reference N-H orientation" (defined in section 3.1. below); for the dimer many are more concentrated, deep and centered away from the "reference N-H orientation". The u(DL,K) functions provide a consistent description of the potential energy landscape at protein N-H sites. The L1-loop of the plexin-B1 RBD is not seen in the crystal structure, and many resonances of the L4 loop are missing in the NMR 15N-1H HSQC spectrum of the dimer; we suggest reasons for these features. An allosteric signal transmission pathway was reported previously for the monomer. We find that it has shallow N-H potentials at its ends, which become deeper as one proceeds toward the middle, complementing structurally the previously derived dynamic picture. Prospects of this study include correlating u(DL,K-BEST) with MD force-fields, and using them without further adjustment in NMR relaxation analysis schemes.

Electrostatics of charged dielectric spheres with application to biological systems. II. A formalism bypassing Wigner rotation matrices.

Given the crucial role of electrostatic forces in biological systems, accurate and rapid calculations of electrostatic forces are imperative in faithfully simulating biological systems. More than a decade ago, we proposed a surface charge method, applied it to a system of an arbitrary number of charged dielectric spheres, and obtained an exact solution for arbitrary configuration of the spheres. The precision depends only on the number of terms kept in a series expansion, and can therefore be controlled at will. However, the numerical implementation can be significantly slowed down by the need to compute a large number of Wigner rotation matrix elements each time the electrostatic energy is calculated. In this paper, we provide the proof of a formula introduced in 1992 and apply it to arrive at a useful closed form for the electrostatic interaction energy without computing any Wigner rotation matrix elements, hence significantly improving the efficiency for numeric implementation of the rigorous surface charge method. This new Wigner-matrix-free formalism may also be used to speed up the computation of the electrostatic energy in the presence of permanent and induced multipoles, which can be very important for atomic modeling with polarization effect included.

General Wigner Rotations in D Dimensions**Supported by the National Natural Science Foundation of China under Grant No. 11475016, and by the Scientific Research Foundation for Returned Scholars, Ministry of Education of China

We construct general Wigner rotations for both massive and massless particles in D-dimensional spacetime. We work out the explicit expressions of these Wigner rotations for arbitrary Lorentz transformations. We study the relation between the electromagnetic gauge invariance and the non-uniqueness of Wigner rotation.

Quantum information processing in the neighborhood of a black hole

Using an imperfectly prepared state, we show that in relativistic settings, the evolution of a massive spin-1/2 particle violates many standard assumptions made in quantum information theory, including complete positivity. Unlike other recent endeavors in relativistic quantum information, we are able to quantify and maximize how much information can be transferred through such a quantum process by calculating its scope. We show that, surprisingly, relativistic noise can increase the amount of information that can be transferred, and in fact, even if the initial state is arbitrarily close to the completely mixed state, information can still be transferred perfectly. Additionally, we explore the relativistic effects of velocity and gravity on quantum information processing, and we briefly discuss how quantum computation is affected by general relativity. In particular, we show that the large Wigner rotation caused by a black hole as described in the Schwarzchild metric can greatly increase the informatic content of a qubit.

Black hole geodesic parallel transport and the Marck reduction procedure

The Wigner rotations arising from the combination of boosts along two different directions are rederived from a relative boost point of view and applied to gyroscope spin precession along timelike geodesics in a Kerr spacetime, clarifying the geometrical properties of Marck's recipe for describing parallel transport along such world lines expressed in terms of the constants of the motion. His final angular velocity isolates the cumulative spin precession angular velocity independent of the spacetime tilting required to keep the spin 4-vector orthogonal to the gyro 4-velocity. As an explicit example the cumulative precession effects are computed for a test gyroscope moving along both bound and unbound equatorial plane geodesic orbits.

Local Ordering at Mobile Sites in Proteins: Combining Perspectives from NMR Relaxation and Molecular Dynamics.

We report on progress toward improving NMR relaxation analysis in proteins in terms of the slowly relaxing local structure (SRLS) approach by developing a method that combines SRLS with molecular dynamics (MD) simulations. 15N-H bonds from the Rho GTPase binding domain of plexin-B1 are used as test case. We focus on the locally restricting/ordering potential of mean force (POMF), u(θ,φ), at the N-H site (θ and φ specify the orientation of the N-H bond in the protein). In SRLS, u(θ,φ) is expanded in the basis set of the real linear combinations of the Wigner rotation matrix elements with M = 0, D L,| K|(θ,φ). Because of limited data sensitivity, only the lowest ( L = 2) terms are preserved; this potential function is denoted by u(SRLS). In MD, the force-field-parametrized POMF is the potential, u(MD), defined in terms of the probability distribution, Peq(MD) ∝ exp(- u(MD)). Peq(MD), and subsequently u(MD), can be derived from the MD trajectory as histograms. One might contemplate utilizing u(MD) instead of u(SRLS); however, histograms cannot be used in SRLS analyses. Here, we approximate u(θ,φ) in terms of linear combinations of the D L,| K| functions with L = 1-4 and appropriate symmetry, denoted by u(DLK), and optimize the latter (via Peq) against u(MD). This yields for every N-H bond an analytical ordering potential, u(DLK-BEST), which exceeds u(SRLS) considerably in accuracy. u(DLK-BEST) can be used fixed in SRLS data fitting, thereby enabling the determination of additional parameters. This yields a substantially improved picture of structural dynamics, which is a significant benefit. The primary achievement of this work is to have employed for the first time MD data to derive a suitable (in terms of composition and symmetry) approximation to the SRLS POMF.