Noncommutative quantum mechanics: uniqueness of the functional description

The generalized Weyl transform of index $\alpha$ is used to implement the time-slice definition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter $\alpha$. We succeed in proving that the $\alpha$-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian involves products of noncommuting operators originating from the non-commutativity. The antisymmetry of the matrix parameterizing the non-commutativity plays a key role in the cancelation mechanism of the $\alpha$-dependent terms.Comment: 13 page

Einstein gravity as a 3D conformally invariant theory

We give an alternative description of the physical content of general relativity that does not require a Lorentz invariant spacetime. Instead, we find that gravity admits a dual description in terms of a theory where local size is irrelevant. The dual theory is invariant under foliation preserving 3-diffeomorphisms and 3D conformal transformations that preserve the 3-volume (for the spatially compact case). Locally, this symmetry is identical to that of Horava-Lifshitz gravity in the high energy limit but our theory is equivalent to Einstein gravity. Specifically, we find that the solutions of general relativity, in a gauge where the spatial hypersurfaces have constant mean extrinsic curvature, can be mapped to solutions of a particular gauge fixing of the dual theory. Moreover, this duality is not accidental. We provide a general geometric picture for our procedure that allows us to trade foliation invariance for conformal invariance. The dual theory provides a new proposal for the theory space of quantum gravity.Comment: 27 pages. Published version (minor changes and corrections

Dynamical Compactification and Inflation in Einstein-Yang-Mills Theory with Higher Derivative Coupling

We study cosmology of the Einstein-Yang-Mills theory in ten dimensions with a quartic term in the Yang-Mills field strength. We obtain analytically a class of cosmological solutions in which the extra dimensions are static and the scale factor of the four-dimensional Friedmann-Lemaitre-Robertson-Walker metric is an exponential function of time. This means that the model can explain inflation. Then we look for solutions that describe dynamical compactification of the extra dimensions. The effective cosmological constant $\lambda_1$ in the four-dimensional universe is determined from the gravitational coupling, ten-dimensional cosmological constant, gauge coupling and higher derivative coupling. By numerical integration, the solution with $\lambda_1=0$ is found to behave as a matter-dominated universe which asymptotically approaches flat space-time, while the solution with a non-vanishing $\lambda_1$ approaches de Sitter space-time in the asymptotic future.Comment: 30 pages, 7 figure

Exact General Relativistic Thick Disks

A method to construct exact general relativistic thick disks that is a simple generalization of the ``displace, cut and reflect'' method commonly used in Newtonian, as well as, in Einstein theory of gravitation is presented. This generalization consists in the addition of a new step in the above mentioned method. The new method can be pictured as a ``displace, cut, {\it fill} and reflect'' method. In the Newtonian case, the method is illustrated in some detail with the Kuzmin-Toomre disk. We obtain a thick disk with acceptable physical properties. In the relativistic case two solutions of the Weyl equations, the Weyl gamma metric (also known as Zipoy-Voorhees metric) and the Chazy-Curzon metric are used to construct thick disks. Also the Schwarzschild metric in isotropic coordinates is employed to construct another family of thick disks. In all the considered cases we have non trivial ranges of the involved parameter that yield thick disks in which all the energy conditions are satisfied.Comment: 11 pages, RevTex, 9 eps figs. Accepted for publication in PR

Quantum Mechanics as an Approximation to Classical Mechanics in Hilbert Space

Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Classical mechanics can now be viewed as a deformation of quantum mechanics. The forms of semiquantum approximations to classical mechanics are indicated.Comment: 10 pages, Latex2e file, references added, minor clarifications mad

Compact Lattice QED and the Coulomb Potential

The potential energy of a static charge distribution on a lattice is rigorously computed in the standard compact quantum electrodynamic model. The method used follows closely that of Weyl for ordinary quantum electrodynamics in continuous space-time. The potential energy of the static charge distribution is independent of temperature and can be calculated from the lattice version of Poisson's equation. It is the usual Coulomb potential.Comment: 6 pages, includes one figure in Topdrawer, NUB 3054/9

Periastron shift in Weyl class spacetimes

The periastron position advance for geodesic motion in axially symmetric solutions of the Einstein field equations belonging to the Weyl class of vacuum solutions is investigated. Explicit examples corresponding to either static solutions (single Chazy-Curzon, Schwarzschild and a pair of them), or stationary solution (single rotating Chazy-Curzon and Kerr black hole) are discussed. The results are then applied to the case of S2-SgrA$^*$ binary system of which the periastron position advance will be soon measured with a great accuracy.Comment: To appear on General Relativity and Gravitation, vol. 37, 200

When Black Holes Meet Kaluza-Klein Bubbles

We explore the physical consequences of a recently discovered class of exact solutions to five dimensional Kaluza-Klein theory. We find a number of surprising features including: (1) In the presence of a Kaluza-Klein bubble, there are arbitrarily large black holes with topology S^3. (2) In the presence of a black hole or a black string, there are expanding bubbles (with de Sitter geometry) which never reach null infinity. (3) A bubble can hold two black holes of arbitrary size in static equilibrium. In particular, two large black holes can be close together without merging to form a single black hole.Comment: 23 pages, 5 figures, v2: few comments on stability modifie

Wormhole geometries supported by a nonminimal curvature-matter coupling

Wormhole geometries in curvature-matter coupled modified gravity are explored, by considering an explicit nonminimal coupling between an arbitrary function of the scalar curvature, R, and the Lagrangian density of matter. It is the effective stress-energy tensor containing the coupling between matter and the higher order curvature derivatives that is responsible for the null energy condition violation, and consequently for supporting the respective wormhole geometries. The general restrictions imposed by the null energy condition violation are presented in the presence of a nonminimal R-matter coupling. Furthermore, obtaining exact solutions to the gravitational field equations is extremely difficult due to the nonlinearity of the equations, although the problem is mathematically well-defined. Thus, we outline several approaches for finding wormhole solutions, and deduce an exact solution by considering a linear R nonmiminal curvature-matter coupling and by considering an explicit monotonically decreasing function for the energy density. Although it is difficult to find exact solutions of matter threading the wormhole satisfying the energy conditions at the throat, an exact solution is found where the nonminimal coupling does indeed minimize the violation of the null energy condition of normal matter at the throat.Comment: 8 pages, 3 figures. V2: 9 pages, error and typos corrected; discussion and references added; to appear in PR

Applications of quantum integrable systems

We present two applications of quantum integrable systems. First, we predict that it is possible to generate high harmonics from solid state devices by demostrating that the emission spectrum for a minimally coupled laser field of frequency $\omega$ to an impurity system of a quantum wire, contains multiples of the incoming frequency. Second, evaluating expressions for the conductance in the high temperature regime we show that the caracteristic filling fractions of the Jain sequence, which occur in the fractional quantum Hall effect, can be obtained from quantum wires which are described by minimal affine Toda field theories.Comment: 25 pages of LaTex, 4 figures, based on talk at the 6-th international workshop on conformal field theories and integrable models, (Chernogolovka, September 2002