Spin Coordinates Research Articles

Twistorial versus spacetime formulations: Unification of various string models

We introduce the $D=4$ twistorial tensionfull bosonic string by considering the canonical twistorial 2-form in two-twistor space. We demonstrate its equivalence to two bosonic string models: due to Siegel (with covariant world-sheet vectorial string momenta ${P}_{\ensuremath{\mu}}^{m}(\ensuremath{\tau},\ensuremath{\sigma})$) and the one with tensorial string momenta ${P}_{[\ensuremath{\mu}\ensuremath{\nu}]}(\ensuremath{\tau},\ensuremath{\sigma})$. We show how to obtain in mixed spacetime-twistor formulation the Soroka-Sorokin-Tkach-Volkov (SSTV) string model and subsequently by harmonic gauge fixing the Bandos-Zheltukhin (BZ) model, with constrained spinorial coordinates.

Master higher-spin particle

We propose a ‘master’ higher-spin (HS) particle system. The particle model relevant to the unfolded formulation of HS theory, as well as the HS particle model with a bosonic counterpart of supersymmetry, follows from the master model as its two different gauges. Quantization of the master system gives rise to a new form of the massless HS equations in an extended space involving, besides extra spinorial coordinates, a complex scalar one. As solutions to these equations we recover the massless HS multiplet with fields of all integer and half-integer helicities, and obtain new multiplets with a non-zero minimal helicity. The HS multiplets are described by complex wavefunctions which are holomorphic in the scalar coordinate and carry an extra U(1) charge q. The latter fully characterizes the given multiplet by fixing the minimal helicity as q/2. We construct a twistorial formulation of the master system and present the general solution of the associate HS equations through an unconstrained twistor ‘prepotential’.

Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation, Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple Jordan algebras of degree three correspond to extensions of Minkowskian spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra (2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal triple systems defined over these Jordan algebras describe conformally covariant phase spaces. Following hep-th/0008063, we give a unified geometric realization of the quasiconformal groups that act on their conformal phase spaces extended by an extra "cocycle" coordinate. For the generic Jordan family the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are given. The minimal unitary representations of the quasiconformal groups F_4(4), E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our earlier work hep-th/0409272.

Addendum to “Sufficient conditions for three-particle entanglement and their tests in recent experiments”

A recent paper [] presented a bound for the three-qubit Mermin inequality such that the violation of this bound indicates genuine three-qubit entanglement. We show that this bound can be improved for a specific choice of observables. In particular, if spin observables corresponding to orthogonal directions are measured at the qubits (e.g., $X$ and $Y$ spin coordinates), then the bound is the same as the bound for states with a local hidden variable model. As a consequence, it can straightforwardly be shown that in the experiment described by ], genuine three-qubit entanglement was detected.

Field theory in noncommutative Minkowski superspace

There is much discussion of scenarios where the space-time coordinates x^\mu are noncommutative. The discussion has been extended to include nontrivial anticommutation relations among spinor coordinates in superspace. A number of authors have studied field theoretical consequences of the deformation of N=1 superspace arising from nonanticommutativity of coordinates \theta, while leaving \bar{theta}'s anticommuting. This is possible in Euclidean superspace only. In this note we present a way to extend the discussion by making both \theta and \bar{theta} coordinates non-anticommuting in Minkowski superspace. We present a consistent algebra for the supercoordinates, find a star-product, and give the Wess-Zumino Lagrangian L_{WZ} within our model. It has two extra terms due to non(anti)commutativity. The Lagrangian in Minkowski superspace is always manifestly Hermitian and for L_{WZ} it preserves Lorentz invariance.

Spin and statistics in classical mechanics

The spin–statistics connection is obtained for classical particles. The connection holds within pseudomechanics, a theory of particle motion that extends classical physics to include anticommuting Grassmann variables, and that exhibits classical analogs of both spin and statistics. Classical realizations of Lie groups are constructed in a canonical formalism generalized to include Grassmann variables. The theory of irreducible canonical realizations of the Poincaré group is developed in this framework, with particular emphasis on the rotation subgroup. The behavior of irreducible realizations under time inversion and charge conjugation is obtained. The requirement that the Lagrangian retain its form under the combined operation of charge conjugation and time reversal leads directly to the spin–statistics connection by an adaptation of Schwinger’s 1951 proof to irreducible canonical realizations of the Poincaré group of spin j: Generalized spin coordinates and momenta satisfy fundamental Poisson bracket relations for 2j even, and fundamental Poisson antibracket relations for 2j odd.

Non(anti)commutative superspace

We investigate the most general non(anti)commutative geometry in N = 1 four-dimensional superspace, invariant under the classical (i.e., undeformed) supertranslation group. We find that a nontrivial non(anti)commutative superspace geometry compatible with supertranslations exists with non(anti)commutation parameters which may depend on the spinorial coordinates. The algebra is in general nonassociative. Imposing associativity introduces additional constraints which, however, allow for nontrivial commutation relations involving fermionic coordinates. We obtain explicitly the first three terms of a series expansion in the deformation parameter for a possible associative ⋆-product. We also consider the case of N = 2 Euclidean superspace where the different conjugation relations among spinorial coordinates allow for a more general supergeometry.

Virtual Reality and Elite Athletes

This paper will provide a discussion on virtual reality in sport. The terms virtual reality (VR), virtual environment (VE), and simulator, will be defined and discussed with regard to their potential as tools for the enhancement of elite athletic performance. Issues related to the need for these tools and the challenges of developing them, plus current examples of VR tools already in use by elite athletes, will be presented. VR refers to “an alternative world filled with 3-D computer generated images that respond to human movement” (Kaufman Broida & Germann, 1999, p. 95); this is similar to a virtual environment that is “made of three-dimensional [3-D] graphic images that are generated by computers for the expressed purpose of cognitive and physical interaction” (Caird, 1996, p. 124). According to Spring (1991), these are not new concepts, they are integral to work on scientific visualization, graphical user interfaces, hypertext, hypermedia, and visual languages. A truly virtual environment is one in which the user is a participant in an abstract space where physical machinery does not exist; only images and illusions are used to create a sense of reality. Helsel and Roth (1991) refer to the impact that resulted from the presentation of one of the first black and white movies, “The Arrival of the Train at the Station” (circa 1895), in which audience members physically jumped out of their seats to avoid being hit by the train. This is the kind of impact that researchers and developers would like to achieve in today's virtual environments, however audiences are now much more sophisticated and require environments that not only visually re-create reality, but also stimulate all the senses activated in the real environment. Simulators tend to be defined as either equipment or software that simulate all or part of the physical or cognitive elements of a specific act or skill. Caird (1996) argues that VE's are not very different from simulators that have essentially the same purpose. The difference is that VE's incorporate 3-D computer graphics with input devices that allow people to interact with the 3-D environment created around them whereas simulators do not. VR is immersive and takes simulators to another level. The types of input devices used to allow interaction are varied. For example, NASA uses devices such as eye phones, data gloves, and full body data suits to allow astronauts to explore various 3-D worlds (Spring, 1991). Paradiso and Hasan (2001) used compact, wireless, low pressure sensor cards inserted into a running shoe to monitor and transmit analog data, pressure measurements, acceleration data, angular and vertical spin coordinates, and orientation data which allow athletes to create interactive performances in simulated environments. As noted before, VE's need to stimulate as many as possible, if not all, of the senses stimulated by reality for the environment to be truly convincing and an effective training tool. This is one of the greatest challenges for virtual reality developers within the sports environments. The examples presented in this paper have been successful in stimulating a number of the senses as they are stimulated in reality. Wall, Bertrand, Gale, and Saunders (1998) asked sailors to sail a VR sailing simulator. The results found that performance in the simulator correlated highly with the real physical performance of the sailors (r = 0.99). Qualitative questionnaires given to the sailors also found that they perceived the correlation between the simulated and real physical performances to be high. However, they noted that the correlation between the results of the simulator performance and reality on the decision-making processes required during sailing was not as high. Another VR simulator, Computrainer created by RacermateTM is used by a number of elite cyclists, including athletes, at the Olympic Oval. The simulator applies pressure to the rear wheel of the rear wheel of the athletes' bike to simulate going uphill, on the flat, or downhill, and eases off when the athlete drafts behind another bike. The bike is also linked to a computer that provides the athlete with a 3-D visual picture of where they are and where their competitor is. This computerized aspect also provides a series of data on the athlete such as heart rate, rotations per minute, and calorie expenditure. A number of studies have found the simulator to be a very reliable and effective tool for simulating cycling (Carey & LaPort, 1999; Prasuhn, 2000). Preliminary work conducted at the Olympic Oval, University of Calgary, has been successful in creating a virtual oval that skaters can train the cognitive and visual components of speed skating at a variety of speeds over various distances.

Mesoscopics, superconductivity, and fundamental properties of space-time

We show that investigations of mesoscopic metal particles can give an evidence of the reality of superspace that is of Grassmann spinor coordinates of free space-time which were introduced in addition to x, y, z, t in quantum field theory to realize supersymmetry. We formulate conditions under which all degrees of freedom of the metal particle which are active in thermodynamics and low frequency dynamics correspond to “motion” of the particle along these additional coordinates. Just these degrees of freedom are responsible for mesoscopic superconductivity under condition that the superconductivity transition temperature is much smaller than the single-electron energy level spacing. Superconductivity is defined as a spontaneous breaking of the gauge invariance and is realized in thermodynamic equilibrium states with a non-integral average number of electrons in the particle. Superconducting order parameter can be characterized by pair or (and) single-electron phases. Possible experiments to check the proposed description are discussed which would at the same time be an experimental check of the concept of superspace.

Spin-charge decoupling and orthofermi quantum statistics

Currently Gutzwiller projection technique and nested Bethe ansatz are two main methods used to handle electronic systems in the $U$ infinity limit. We demonstrate that these two approaches describe two distinct physical systems. In the nested Bethe ansatz solutions, there is a decoupling between the spin and charge degrees of freedom. Such a decoupling is absent in the Gutzwiller projection technique. Whereas in the Gutzwiller approach, the usual antisymmetry of space and spin coordinates is maintained, we show that the Bethe ansatz wave function is compatible with a new form of quantum statistics, viz., orthofermi statistics. In this statistics, the wave function is antisymmetric in spatial coordinates alone. This feature ultimately leads to spin-charge decoupling.

Theoretical study on magnetic quantum tunneling of anisotropic spin systems with magnetic field

We propose a path integral method to investigate the quantum spin dynamics. A centroid with spin is introduced as an average spin coordinate of a closed isomorphic necklace of each spin space, so-called a spin centroid. We calculate the magnetization of a system with total spin 1 and observe magnetic quantum tunneling with magnetic field.

Quantum spin dynamics of antiferromagnetic ring with central excess spin

We propose a path integral method to investigate the quantum spin dynamics. A centroid with spin is introduced as an average spin coordinate of a closed isomorphic necklace of each spin space, so-called a spin centroid. We calculate the magnetization of a system with total spin 1/2 and observe magnetic quantum tunneling amplitude becomes zero.

Quantum Spin Dynamics by Path Integral Centroid Molecular Dynamics Method

Quantum spin dynamics in mesoscopic magnets has received much attention over the recent years, both experimentally and theoretically. Especially, molecular magnets such as the ferric wheel or Mn12 have emerged as promising candidates for the experimental observation of macroscopic quantum phenomena such as the tunneling of the magnetization. We investigated tunneling rates of model manganese systems by means of the nonlinear σ model. 1) The path integral centroid molecular dynamics (CMD) 2),3) method has been shown to be capable of simulating the real time quantum dynamics of complicated many-body systems, even for reasonably long time, albeit approximately. So far, this method has been applied to calculating dynamical properties in systems consisting of both Fermi and Bose particles. 4) In this study, we demonstrate that the centroid-based path integral method is feasible to investigate real time quantum spin dynamics at low temperature. We introduce a centroid corresponding to a spin variable as an average spin coordinate of a closed isomorphic necklace of each spin space, so-called spin centroid. We calculated magnetization of a system with S = 1.

Higher-order effects in polarized proton dynamics

So far, polarized proton beams have never been accelerated to energies higher than 25 GeV. During the acceleration process, the beam polarization is quite undisturbed, when the accelerator is well adjusted, except at first-order depolarizing spin orbit resonances. At some accelerators other effects have been observed but first-order resonances have always been dominant. At these resonances the spin tune plus or minus one of the orbit tunes is an integer. These beams have usually been investigated by theories which correspondingly lead to an undisturbed polarization during acceleration, except at such resonances. Therefore we speak of ``first-order theories.'' The first frequently used first-order theory is the single resonance model, which is usually used for simulating the acceleration process. Here the equation of spin motion is simplified drastically by dropping all but the dominant Fourier component of the driving term of that differential equation. The second frequently used first-order theory, the linearized spin-orbit motion theory, is also quite crude. It is based on a linearization of the spin and orbit equation of motion with respect to the phase space coordinates and two suitably chosen spin coordinates. Because of linearization this method cannot be used close to resonances but at fixed energies it is a useful tool. It will be shown that the validity of these first-order theories is restricted at Hadron Electron Ring Accelerator (HERA) energies of up to 820 GeV. An overview of the available theories which go beyond the first-order resonances is given and we explain which of these approaches are applicable for the analysis of polarization in the HERA proton ring. Since these theories include more than one Fourier harmonic in the driving term of the equation of motion, we refer to them as ``non-first-order'' or ``higher-order'' theories. Finally, the higher-order effects observed while simulating polarized beams in HERA with these advanced methods are illustrated.

TENSORIAL CENTRAL CHARGES AND NEW SUPERPARTICLE MODELS WITH FUNDAMENTAL SPINOR COORDINATES

We consider firstly simple D=4 superalgebra with six real tensorial central charges Zμν, and discuss its possible realizations in massive and massless cases. Massless case is dynamically realized by generalized Ferber–Shirafuji (FS) model with fundamental bosonic spinor coordinates. The Lorentz invariance is not broken due to the realization of central charges generators in terms of bosonic spinors. The model contains four fermionic coordinates and possesses three κ-symmetries thus providing the BPS configuration preserving 3/4 of the target space supersymmetries. We show that the physical degrees of freedom (eight real bosonic and one real Grassmann variable) of our model can be described by OSp (8|1) supertwistor. The relation with recent superparticle model by Rudychev and Sezgin is pointed out. Finally we propose a higher-dimensional generalization of our model with one real fundamental bosonic spinor. D=10 model describes massless superparticle with composite tensorial central charges and in D=11 we obtain zero-superbrane model with nonvanishing mass which is generated dynamically.

SELF-DUALITY OF SUPER D3-BRANE ACTION ON AdS5×S5 BACKGROUND

We show that the super D3-brane action on AdS5×S5 background recently constructed by Metsaev and Tseytlin is exactly invariant under the combination of the electric–magnetic duality transformation of the world-volume gauge field and the SO(2) rotation of N=2 spinor coordinates. The action is shown to satisfy the Gaillard–Zumino duality condition, which is a necessary and sufficient condition for an action to be self-dual. Our proof needs no gauge fixing for the κ-symmetry.

Super D-string action on AdS5× S5

We present a supersymmetric and κ-symmetric D-string action on AdS 5× S 5 in supercoset construction. As in the previous work of the super D-string action in the flat background, the super D-string action on AdS 5× S 5 can be transformed to a form of the IIB Green-Schwarz superstring action with the SL(2, Z) covariant tension on AdS 5× S 5 through a duality transformation. In order to understand a part of the duality transformation as SO(2) rotation of N=2 spinor coordinates, it seems to be necessary to fix the κ-symmetry in a gauge condition which simplifies the classical action. This is the article showing for the first time that there exists S-duality in type IIB superstring theory in a curved background whose validity has been conjectured in the past but not shown so far in an explicit way.

Mesoscopic superconductivity in superspace

In the theory of mesoscopic superconductivity it is necessary to supplement the coordinates x, y, z with some additional physical characteristics (coordinates) of free space. An adequate description of mesoscopic superconductivity is obtained by introducing a nonrelativistic version of Grassmann spinor coordinates, which are studied in supersymmetric field theories and convert ordinary space into superspace. An experimental check of the proposed description of the superconducting and magnetic properties of metallic nanoparticles would provide a direct confirmation of the concept of superspace.

Background harmonic superfields inN=2 supergravity

A modification of the harmonic superfield formalism in $D=4, N=2$ supergravity using a subsidiary condition of covariance under the background supersymmetry with a central charge ($B$-covariance) is considered. Conservation of analyticity together with the $B$-covariance leads to the appearance of linear gravitational superfields. Analytic prepotentials arise in a decomposition of the background linear superfields in terms of spinor coordinates and transform in a nonstandard way under the background supersymmetry. The linear gravitational superfields can be written via spinor derivatives of nonanalytic spinor prepotentials. The perturbative expansion of the extended supergravity action in terms of the $B$-covariant superfields and the corresponding version of the differential-geometric formalism are considered. We discuss the dual harmonic representation of the linearized extended supergravity, which corresponds to the dynamical condition of Grassmann analyticity.

Classification of local generalized symmetries for the vacuum Einstein equations

A local generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all local generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the local generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's “exact set of fields” for the vacuum equations.