Realization of the Three-dimensional Quantum Euclidean Space by Differential Operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by $q$-deformation. Simultaneously, angular momentum is deformed to $so_q(3)$, it acts on the $q$-Euclidean space that becomes a $so_q(3)$-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on $C^{\infty}$ functions on $\mathbb{R}^3$. On a factorspace of $C^{\infty}(\mathbb{R}^3)$ a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a $q$-lattice.Comment: 13 pages, late

On semiclassical approximation and spinning string vertex operators in AdS_5 x S^5

Following earlier work by Polyakov and Gubser, Klebanov and Polyakov, we attempt to clarify the structure of vertex operators representing string states which have large (``semiclassical'') values of AdS energy (equal to 4-d dimension \Delta) and angular momentum J in S^5 or spin S in AdS_5. We comment on the meaning of semiclassical limit in the context of \alpha' perturbative expansion for the 2-d anomalous dimensions of the corresponding vertex operators. We consider in detail the leading-order 1-loop renormalization of these operators in AdS_5 x S^5 sigma model (ignoring fermionic contributions). We find new examples of operators for which, as in the case considered in hep-th/0110196, the 1-loop anomalous dimension can be made small by tuning quantum numbers. We also comment on a possibility of deriving the semiclassical relation between \Delta and J or S from the marginality condition for the vertex operators, using a stationary phase approximation in the path integral expression for their 2-point correlator on a complex plane.Comment: 35 pages, harvmac; v2: references adde

Epicyclic orbital oscillations in Newton's and Einstein's dynamics

We apply Feynman's principle, ``The same equations have the same solutions'', to Kepler's problem and show that Newton's dynamics in a properly curved 3-D space is identical with that described by Einstein's theory in the 3-D optical geometry of Schwarzschild's spacetime. For this reason, rather unexpectedly, Newton's formulae for Kepler's problem, in the case of nearly circular motion in a static, spherically spherical gravitational potential accurately describe strong field general relativistic effects, in particular vanishing of the radial epicyclic frequency at the marginally stable orbit.Comment: 8 page

The Parton Orbital Angular Momentum: Status and Prospects

Theoretical progress on the formulation and classification of the quark and gluon orbital angular momenta (OAM) is reviewed. Their relation to parton distributions and open questions and puzzles are discussed. We give a status report on the lattice calculation of the parton kinetic and canonical OAM and point out several strategies to calculate the quark and gluon canonical OAM on the lattice.Comment: 16 pages, contribution to the EPJA speical issue on "3D Structure of the Nucleon

Generalised uncertainty relations for angular momentum and spin in quantum geometry

We derive generalised uncertainty relations (GURs) for angular momentum and spin in the smeared-space model of quantum geometry. The model implements a minimum length and a minimum linear momentum, and recovers both the generalised uncertainty principle (GUP) and the extended uncertainty principle (EUP) within a single formalism. In this paper, we investigate the consequences of these results for particles with extrinsic and intrinsic angular momentum, and obtain generalisations of the canonical ${\rm so(3)}$ and ${\rm su(2)}$ algebras. We find that, although ${\rm SO(3)}$ symmetry is preserved on three-dimensional slices of an enlarged phase space, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, $\hbar \rightarrow \hbar + \beta$. The value of the new parameter, $\beta \simeq \hbar \times 10^{-61}$, is determined by the ratio of the dark energy density to the Planck density. Here, we assume the former to be of the order of the Planck length and the latter to be of the order of the de Sitter momentum $\sim \hbar\sqrt{\Lambda}$, where $\Lambda$ is the cosmological constant. In the smeared-space model, $\hbar$ and $\beta$ are interpreted as the quantisation scales for matter and geometry, respectively, and a quantum state vector is associated with the spatial background. We show that this also gives rise to a rescaled Lie algebra for generalised spin operators, together with associated subalgebras that are analogous to those for orbital angular momentum. Remarkably, consistency of the algebraic structure requires the quantum state associated with a flat background to be fermionic, with spin eigenvalues $\pm \beta/2$. Finally, the modified spin algebra leads to GURs for spin measurements.Comment: 28 pages of main text, plus 12 additional pages split between 4 appendices and 3 pages of references. No figures. Invited contribution to the Universe special issue "Rotation Effects in Relativity", Matteo Ruggiero ed. Published versio

Electric charge in the field of a magnetic event in three-dimensional spacetime

We analyze the motion of an electric charge in the field of a magnetically charged event in three-dimensional spacetime. We start by exhibiting a first integral of the equations of motion in terms of the three conserved components of the spacetime angular momentum, and then proceed numerically. After crossing the light cone of the event, an electric charge initially at rest starts rotating and slowing down. There are two lengths appearing in the problem: (i) the characteristic length $\frac{q g}{2 \pi m}$, where $q$ and $m$ are the electric charge and mass of the particle, and $g$ is the magnetic charge of the event; and (ii) the spacetime impact parameter $r_0$. For $r_0 \gg \frac{q g}{2 \pi m}$, after a time of order $r_0$, the particle makes sharply a quarter of a turn and comes to rest at the same spatial position at which the event happened in the past. This jump is the main signature of the presence of the magnetic event as felt by an electric charge. A derivation of the expression for the angular momentum that uses Noether's theorem in the magnetic representation is given in the Appendix.Comment: Version to appear in Phys. Rev.