Noncommutative gauge fields coupled to noncommutative gravity

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted star product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and twisted gauge transformations. The Seiberg-Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor D_{IJK}.Comment: 18 pages, LaTe

Physical Wigner functions

In spite of their potential usefulness, the characterizations of Wigner functions for Bose and Fermi statistics given by O'Connell and Wigner himself almost thirty years ago has drawn little attention. With an eye towards applications in quantum chemistry, we revisit and reformulate them in a more convenient way.Comment: Latex, 10 page

Star product formula of theta functions

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.Comment: 12 page

Feynman Path Integral on the Noncommutative Plane

We formulate Feynman path integral on a non commutative plane using coherent states. The propagator for a free particle exhibits UV cut-off induced by the parameter of non commutativity.Comment: 7pages, latex 2e, no figures. Accepted for publication on J.Phys.

Coherent States and N Dimensional Coordinate Noncommutativity

Considering coordinates as operators whose measured values are expectations between generalized coherent states based on the group SO(N,1) leads to coordinate noncommutativity together with full $N$ dimensional rotation invariance. Through the introduction of a gauge potential this theory can additionally be made invariant under $N$ dimensional translations. Fluctuations in coordinate measurements are determined by two scales. For small distances these fluctuations are fixed at the noncommutativity parameter while for larger distances they are proportional to the distance itself divided by a {\em very} large number. Limits on this number will lbe available from LIGO measurements.Comment: 16 pqges. LaTeX with JHEP.cl

The Statistics of Supersonic Isothermal Turbulence

We present results of large-scale three-dimensional simulations of supersonic Euler turbulence with the piecewise parabolic method and multiple grid resolutions up to 2048^3 points. Our numerical experiments describe non-magnetized driven turbulent flows with an isothermal equation of state and an rms Mach number of 6. We discuss numerical resolution issues and demonstrate convergence, in a statistical sense, of the inertial range dynamics in simulations on grids larger than 512^3 points. The simulations allowed us to measure the absolute velocity scaling exponents for the first time. The inertial range velocity scaling in this strongly compressible regime deviates substantially from the incompressible Kolmogorov laws. The slope of the velocity power spectrum, for instance, is -1.95 compared to -5/3 in the incompressible case. The exponent of the third-order velocity structure function is 1.28, while in incompressible turbulence it is known to be unity. We propose a natural extension of Kolmogorov's phenomenology that takes into account compressibility by mixing the velocity and density statistics and preserves the Kolmogorov scaling of the power spectrum and structure functions of the density-weighted velocity v=\rho^{1/3}u. The low-order statistics of v appear to be invariant with respect to changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of v is -1.69, and the exponent of the third-order structure function of v is unity. We also directly measure the mass dimension of the "fractal" density distribution in the inertial subrange, D_m = 2.4, which is similar to the observed fractal dimension of molecular clouds and agrees well with the cascade phenomenology.Comment: 15 pages, 19 figures, ApJ v665, n2, 200

On the B\"acklund Transformation for the Moyal Korteweg-de Vries Hierarchy

We study the B\"acklund symmetry for the Moyal Korteweg-de Vries (KdV) hierarchy based on the Kuperschmidt-Wilson Theorem associated with second Gelfand-Dickey structure with respect to the Moyal bracket, which generalizes the result of Adler for the ordinary KdV.Comment: 9 pages, Revte

Quantum deformation of the Dirac bracket

The quantum deformation of the Poisson bracket is the Moyal bracket. We construct quantum deformation of the Dirac bracket for systems which admit global symplectic basis for constraint functions. Equivalently, it can be considered as an extension of the Moyal bracket to second-class constraints systems and to gauge-invariant systems which become second class when gauge-fixing conditions are imposed.Comment: 18 pages, REVTe

Remarks on the Formulation of Quantum Mechanics on Noncommutative Phase Spaces

We consider the probabilistic description of nonrelativistic, spinless one-particle classical mechanics, and immerse the particle in a deformed noncommutative phase space in which position coordinates do not commute among themselves and also with canonically conjugate momenta. With a postulated normalized distribution function in the quantum domain, the square of the Dirac delta density distribution in the classical case is properly realised in noncommutative phase space and it serves as the quantum condition. With only these inputs, we pull out the entire formalisms of noncommutative quantum mechanics in phase space and in Hilbert space, and elegantly establish the link between classical and quantum formalisms and between Hilbert space and phase space formalisms of noncommutative quantum mechanics. Also, we show that the distribution function in this case possesses 'twisted' Galilean symmetry.Comment: 25 pages, JHEP3 style; minor changes; Published in JHE

The noncommutative degenerate electron gas

The quantum dynamics of nonrelativistic single particle systems involving noncommutative coordinates, usually referred to as noncommutative quantum mechanics, has lately been the object of several investigations. In this note we pursue these studies for the case of multi-particle systems. We use as a prototype the degenerate electron gas whose dynamics is well known in the commutative limit. Our central aim here is to understand qualitatively, rather than quantitatively, the main modifications induced by the presence of noncommutative coordinates. We shall first see that the noncommutativity modifies the exchange correlation energy while preserving the electric neutrality of the model. By employing time-independent perturbation theory together with the Seiberg-Witten map we show, afterwards, that the ionization potential is modified by the noncommutativity. It also turns out that the noncommutative parameter acts as a reference temperature. Hence, the noncommutativity lifts the degeneracy of the zero temperature electron gas.Comment: 11 pages, to appear in J. Phys. A: Math. Ge