J. David Vergara

The Kepler Problem and Noncommutativity

We investigate the Kepler problem using a symplectic structure consistent with the commutation rules of the noncommutative quantum mechanics. We show that a noncommutative parameter of the order of 10-58m2 gives observable corrections to the movement of the solar system. In this way, modifications in the physics of smaller scales imply modifications at large scales, something similar to the uv/ir mixing.

Newton's second law in a non-commutative space

In this Letter we show that corrections to Newtons second law appear if we assume a symplectic structure consistent with the commutation rules of the non-commutative quantum mechanics. For central field we find that the correction term breaks the rotational symmetry. For the Kepler problem, this term is similar to a Coriolis force.

Reality conditions for Ashtekar variables as Dirac constraints

We show that the reality conditions to be imposed on Ashtekar variables to recover real gravity can be implemented as second-class constraints following Dirac. Thus, counting gravitational degrees of freedom follows accordingly. Some constraints of the real theory turn out to be non-polynomial, regardless of the form, polynomial or non-polynomial, taken for the reality conditions. We comment upon the compatibility of our approach with the recently proposed Wick transform point of view, as well as on some alternatives for dealing with such second-class constraints.

A membrane action for WM2 / OM theory

Abstract Through direct examination of the effect of the OM limit on the M2-brane worldvolume action, we derive a membrane action for OM theory, and more generally, for the eleven-dimensional M-theoretic construct known as Galilean or Wrapped M2-brane (WM2) theory, which contains OM theory as a special class of states. In the static gauge, the action in question implies a discrete spectrum for the closed membrane of WM2 theory, which under double dimensional reduction is shown to reproduce the known NCOS/Wound closed string spectrum. We examine as well open membranes ending on each of the three types of M5-branes in WM2 theory (OM theory arising from the ‘longitudinal’ type), and show that the ‘fully transverse’ fivebrane is tensionless. As a prelude to the membrane, we also study the case of the string, where we likewise obtain a reparametrization-invariant action, and make contact with previous work.

Note about the quantum of area in a noncommutative space

In this note we show that in a two-dimensional noncommutative space the area operator is quantized; this outcome is compared with the result obtained by loop quantum gravity methods.

Weyl invariant p-brane and Dp-brane actions

Abstract Conformal invariant new forms of p -brane and D p -brane actions are proposed. The field content of these actions are: an induced metric, gauge fields, an auxiliary metric and an auxiliary scalar field that implements the Weyl invariance. This scalar field transforms with a conformal weight that depends on the brane dimension. The proposed actions are Weyl invariant in any dimension and the elimination of the auxiliary metric and the scalar field reproduces the Nambu–Goto action for p -branes and the Born–Infeld action for D p -branes. As a consequence of the fact that in the p -brane case, the action is quadratic in ∂ 0 X , we develop a complete construction for the associated canonical formalism, solving the problem in previous formulations of conformal p -brane actions where the Hamiltonian can not be constructed. We obtain the algebra of constraints and identify the generator of the Weyl symmetry. In the construction of the corresponding supersymmetric generalization of this conformal p -brane action we show that the associated κ -symmetry is consistent with the conformal invariance. In the D p -brane case, the actions are quadratic in the gauge fields. These actions can be used to construct new conformal couplings in any dimension p to the auxiliary scalar field now promoted as a dynamical variable.

The -value equation and Wigner distributions in noncommutative Heisenberg algebras

We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature.

Real sector of the nonminimally coupled scalar field to self-dual gravity

A scalar field nonminimally coupled to gravity is studied in the canonical framework, using self-dual variables. The corresponding constraints are first class and polynomial. To identify the real sector of the theory, reality conditions are implemented as second class constraints, leading to three real configurational degrees of freedom per space point. Nevertheless, this realization makes nonpolynomial some of the constraints. The original complex symplectic structure reduces to the expected real one, by using the appropriate Dirac brackets. For the sake of preserving the simplicity of the constraints, an alternative method preventing the use of Dirac brackets, is discussed. It consists of converting all second class constraints into first class by adding extra variables. This strategy is implemented for the pure gravity case.

Canonical quantization, spacetime noncommutativity and deformed symmetries in field theory

Within the spirit of Diracs canonical quantization, noncommutative spacetime field theories are introduced by making use of the reparametrization invariance of the action and of an arbitrary non-canonical symplectic structure. This construction implies that the constraints need to be deformed, resulting in an automatic Drinfeld twisting of the generators of the symmetries associated with the reparametrized theory. We illustrate our procedure for the case of a scalar field in (1+1)-spacetime dimensions, but it can be readily generalized to arbitrary dimensions and arbitrary types of fields.

Dynamical origin of the ⋆θ-noncommutativity in field theory from quantum mechanics

We show that introducing an extended Heisenberg algebra in the context of the Weyl-Wigner-Groenewold-Moyal formalism leads to a deformed product of the classical dynamical variables that is inherited to the level of quantum field theory, and that allows us to relate the operator space noncommutativity in quantum mechanics to the quantum group inspired algebra deformation noncommutativity in field theory.