Lorentz Invariance Relations Research Articles

Equation-of-motion and Lorentz-invariance relations for tensor-polarized parton distribution functions of spin-1 hadrons

Structure functions of polarized spin-1 hadrons will be measured at various accelerator facilities in the near future. Recently, transverse-momentum-dependent and collinear parton distribution functions were theoretically proposed at twist 3 and twist 4 in addition to the twist-2 ones, so that full investigations became possible for structure functions of spin-1 hadrons in the same level with those of the spin-1/2 nucleons. Furthermore, twist-3 tensor-polarized multiparton distribution functions were also recently found for spin-1 hadrons. In this work, we show relations among the collinear parton- and multiparton-distribution functions for spin-1 hadrons by using equation of motion for quarks. These relations are valuable in constraining the distribution functions and learning about multiparton correlations in spin-1 hadrons.

New approach to the Sivers effect in the collinear twist-3 formalism

The single-transverse spin asymmetry(SSA) for hadron production in the transversely polarized proton scattering receives major contribution from Sivers effect, which can be systematically described within the collinear twist-3 factorization framework in various processes. Conventional method in the evaluation of the Sivers effect known as pole calculation is technically quite different from non-pole calculation which is another method used in evaluating the final state twist-3 effect. In this paper, we extend the non-pole technique to the Sivers effect, and show the consistency with the conventional method through an explicit calculation of $\mathcal{O}(\alpha_s)$ correction in semi-inclusive deep inelastic scattering. As a result, we clarify that the conventional pole calculation is implicitly using the equation of motion and the Lorentz invariant relations whose importance became widely known in the non-pole calculation. We also clarify some technical advantages in using the new non-pole method.

A 4 × 4 diagonal matrix Hermitian Schrödinger equation from relativistic total energy with a 2 × 2 Lorentz invariant relation between amplitude and phase

In this paper an algebraic method is presented to derive a 4 × 4 Hermitian Schrödinger equation from E=V+cm2c2+(p−ecA)2 with E→iℏ∂∂t and p→−iℏ∇. The latter operator replacement is a common procedure in a quantum description of the total energy. In the derivation we don’t make use of Dirac’s method of four vectors. Moreover, the root operator isn’t squared either. Instead, use is made of the algebra of operators to derive a Hermitian matrix Schrödinger equation. We believe that new physics can be obtained from an alternative quantization of the relativistic total energy. Note e.g. the pion physics behind the Klein–Gordon equation and the antimatter behind the Dirac quantization of the total relativistic energy. In this paper, for the sake of clarity, a time-only dependence of the electromagnetic potential vector is assumed. It is also demonstrated that a “latent” Lorentz invariance exists related to a derived expression for amplitude and phase.

Phenomenological constraints on AN in p↑p → π X from Lorentz invariance relations

We present a new analysis of AN in p↑p→πX within the collinear twist-3 factorization formalism. We incorporate recently derived Lorentz invariance relations into our calculation and focus on input from the kinematical twist-3 functions, which are weighted integrals of transverse momentum dependent (TMD) functions. In particular, we use the latest extractions of the Sivers and Collins functions with TMD evolution to compute certain terms in AN. Consequently, we are able to constrain the remaining contributions from the lesser known dynamical twist-3 correlators.

Operator constraints for twist-3 functions and Lorentz invariance properties of twist-3 observables

We investigate the behavior under Lorentz tranformations of perturbative coefficient functions in a collinear twist-3 formalism relevant for high-energy observables including transverse polarization of hadrons. We argue that those perturbative coefficient functions can, {\it a priori}, acquire quite different yet Lorentz-invariant forms in various frames. This somewhat surprising difference can be traced back to a general dependence of the perturbative coefficient functions on lightcone vectors which are introduced by the twist-3 factorization formulae and which are frame-dependent. One can remove this spurious frame dependence by invoking so-called Lorentz invariance relations (LIRs) between twist-3 parton correlation functions. Some of those relations for twist-3 distribution functions were discussed in the literature before. In this paper we derive the corresponding LIRs for twist-3 fragmentation functions. We explicitly demonstrate that these LIRs remove the lightcone vector dependence by considering transverse spin observables in the single-inclusive production of hadrons in lepton-nucleon collisions, $\ell N\to hX$. With the LIRs in hand, we also show that twist-3 observables in general can be written solely in terms of three-parton correlation functions.

Canonical center and relative coordinates for compact binary systems through second post-Newtonian order

Based on a recent paper by Rothe and Schäfer on compact binary systems (Rothe and Schäfer 2010 J. Math. Phys. 51 082501), explicit expressions for canonical center and relative coordinates in terms of standard canonical coordinates are derived for spinless objects up to second post-Newtonian (PN) approximation of Einstein’s theory of gravity (the third post-Newtonian order expressions are available in the form of supplementary data). The inverse relations, i.e. the dependence of the standard canonical coordinates on the canonical center and relative coordinates, are also given up to the second PN approximation. The famous Pythagorean-theorem-type Lorentz-invariant relation between the system’s total energy or Hamiltonian squared, the rest energy or mass squared—solely depending on relative coordinates—and the total linear momentum squared, are explicitly shown through second PN approximation.

Derivation of Mass Independent Quantum Treatment of Phenomenon

The derivation and applications is presented of a spatial variable or spatial radius which is related to the inertia or mass-energy of any quantum body by a Lorentz invariant relation. Mass independent DeBroglie and Schroedinger equations are derived and applied to the resolution of the linguistic incompatibility between quantum theory and the geometrical weak equivalence principle. The equivalence principle is restated in terms of the spatial radius. The gravitational attraction between bodies and the relativistic energy are both presented in terms of the spatial radius follows. The ratio of the gravitational force to the Coulomb force at the Planck scale and is found to equal unity. The annihilation of particle and antiparticle is presented as an interference effect using the spatial radius.

LORENTZ INVARIANCE RELATIONS AND WANDZURA–WILCZEK APPROXIMATION

A complete list of the so-called Lorentz invariance relations between parton distribution functions is given and some of their consequences are discussed, such as the Burkhardt-Cottingham sum rule. The violation of these relations is considered in a model independent way. It is shown that several Lorentz invariance relations are not violated in a generalized Wandzura–Wilczek approximation, indicating that numerically their violation may be small.

Asymmetrically warped brane models, bulk photons and Lorentz invariance

We present a brief review of our recent work [1] on asymmetrically warped brane models, where the background metric is characterized by different time and space warp factors. In particular we examine the case of bulk photons and we show that the standard Lorentz invariant dispersion relation is corrected by nonlinear terms which lead to an Energy-dependent speed of light. Stringent constraints on the parameters of our models can be set by comparing the results with recent data from high-energy Gamma-Ray Astrophysics, for instance the MAGIC Telescope.

Lorentz invariance relations between parton distributions and the Wandzura–Wilczek approximation

The violation of the so-called Lorentz invariance relations between parton distribution functions is considered in a model independent way. It is shown that these relations are not violated in a generalized Wandzura–Wilczek approximation, indicating that numerically their violation may be small.

Bulk photons in asymmetrically warped spacetimes and non-trivial vacuum refractive index

We consider asymmetrically warped brane models, or equivalently brane models where the background metric is characterized by different time and space warp factors. The main feature of these models is that 4D Lorentz symmetry is violated for fields which propagate in the bulk, such as gravitons. In this paper we examine the case of bulk photons in asymmetrically warped brane models. Although our results are general, we examine here two specific but characteristic solutions: 1) AdS-Schwarzschild 5D Black Hole solution and 2) AdS-Reissner Nordstrom 5D Black Hole solution. We show that the standard Lorentz invariant dispersion relation for 4D photons is corrected by nonlinear terms which lead to an Energy-dependent speed of light. Specifically, we obtain a sub-luminous Energy-dependent refractive index of the form neff(omega) = 1+cG omega2, where omega is the energy of the photon, and the factor cG is always positive and depends on the free parameters of the model. Finally, comparing the results with recent data from the MAGIC Telescope, claiming a delayed arrival of photons from the Active Galactic Nucleus of Mk501, we impose concrete restrictions to the two sets of models examined in this work. We shall also discuss briefly other possible astrophysical constraints on our models.

Predictions for double spin asymmetryALTin semiinclusive DIS

In the leading order of QCD parton model of semiinclusive deep inelastic scattering (SIDIS) the double spin asymmetry ${A}_{LT}$ arises due to the longitudinal polarization of quarks in the transversely polarized nucleon. The corresponding ${k}_{T}^{2}$ weighted distribution function ${g}_{1T}^{(1)}$ can be related to ordinary helicity distribution ${g}_{1}(x)$ measured in DIS. Using recent parametrizations for (un)polarized distribution and fragmentation functions we calculated ${A}_{LT}$ asymmetry on transversely polarized proton and deuteron targets for different types of hadron production. The predictions are given for COMPASS, HERMES, and JLab energies. The role of Lorentz invariance relations and positivity constraints are discussed.

Lorentz invariance relations among parton distributions revisited

We revisit the derivation of the so-called Lorentz invariance relations between parton distributions. In the most important cases these relations involve twist-3 and transverse momentum dependent parton distributions. It is shown that these relations are violated if the path-ordered exponential is taken into account in the quark correlator.

On the Lorentz-invariant time-energy uncertainty relation for a relativistic photon

The time-energy uncertainty relation is discussed for a relativistic massless particle. The Lorentz-invariant uncertainty relation is obtained between the root-mean-square energy deviation and the scatter of registration time. The interconnection between this uncertainty relation and its classical analogue is established.

Pulse interaction in nonlinear vacuum electrodynamics

The energy-momentum conservation law is used to investigate the interaction of pulses in the framework of nonlinear electrodynamics with Lorentz-invariant constitutive relations. It is shown that for the pulses of the arbitrary shape, the interaction results in phase shift only.

Light travel times around a closed Universe

The travel times for light rays to circumnavigate a closed, spherical Universe, as measured in a frame attached to a moving clock, are determined for Einstein’s static Universe and two cases of the pulsating model. It is shown that the ratio of the difference of the travel times in the forward and rearward directions to the sum of the travel times yields the clock’s velocity relative to space. The implication of these results for Einstein's various views on the ether and motion are discussed, as well as their bearing on the synchronization problem. Lorentzinvariant relations are shown to hold between the respective sum and difference travel times for two clocks traveling on the same path with different velocities. The mathematical similarity of the expressions for the sum and difference travel times to expressions for relativistic energy and momentum is pointed out, and the possible bearing on Mach’s principle is briefly considered. There are two appendices: in the first, synchronization of separated clocks with (hypothetical) faster-than-light signals is critically discussed (including some of the author’s earlier work). In the second, synchronization based on using slowly-moved clocks is analyzed anew.

Electromagnetic form factors of spin-⅓ baryons

Relationships are established among current matrix elements, covariant vertex functions, and multipole form-factor decompositions for the case of electromagnetic interactions of spin-3/2 systems. The electromagnetic current matrix element for spin-3/2 baryons is defined in terms of the minimum required four independent covariant vertex functions. Explicit Lorentz-invariant relations for the multipole form factors are derived in terms of the previously defined covariant vertex function coefficients. The derivation does not involve any nonrelativistic approximations. Finally, the multipole form factors are isolated and expressed in terms of the electromagnetic current matrix elements. These results are particularly useful in lattice QCD calculations.

Uncertainty relations for light waves and the concept of photons.

A Lorentz-covariant localization for light waves is presented. The unitary representation for the electromagnetic four-potential is constructed for a monochromatic light wave. A model for covariant superposition is constructed for light waves with different frequencies. It is therefore possible to construct a wave function for light waves carrying a covariant probability interpretation. It is shown that the time-energy uncertainty relation (\ensuremath{\Delta}t)(\ensuremath{\Delta}\ensuremath{\omega})\ensuremath{\simeq}1 for light waves is a Lorentz-invariant relation. The connection between photons and localized light waves is examined critically.