International Journal of Modern Physics A

This review paper is devoted to analyzing the main properties of the cosmological singularity associated with the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the cosmological issue, we review in detail the main results obtained over the last forty years on the Mixmaster topic. We first assess the classical picture of the homogeneous chaotic cosmologies and, after a presentation of the canonical method for the quantization, we develop the quantum Mixmaster behavior. Finally, we extend both the classical and the quantum features to the fully inhomogeneous case. Our survey analyzes the fundamental framework of the Mixmaster picture and completes it by accounting for recent and peculiar outstanding results.

We have studied the quasinormal modes and the late-time tail behaviors of scalar, electromagnetic and gravitational perturbations in the Schwarzschild black hole pierced by a cosmic string. Although the metric is locally identical to that of the Schwarzschild black hole so that the presence of the string will not imprint in the motion of test particles, we found that quasinormal modes and the late-time tails can reflect physical signatures of the cosmic string. Compared with the scalar and electromagnetic fields, the gravitational perturbation decays slower, which would be more interesting to disclose the string effect in this background.

We obtain explicit time-dependent brane solutions in M-theory as well as in string theory by solving the reduced equations of motion (which follow, as in Int. J. Mod. Phys. A17, 4647 (2002), from 11-dimensional supergravity) for a class of brane solutions in curved backgrounds. The behavior of our solutions in both asymptotic and near-horizon limits are studied. It is shown that our time-dependent solutions serve as explicit examples of branes in singular, cosmological backgrounds. In some special cases the asymptotic and the boundary AdS solutions can be identified as Milne × Rn space–time.

We describe a theory of gravitation on canonical noncommutative space–times. The construction is based on θ-twisted general coordinate transformations and local Lorentz invariance.

It is shown that a massless scalar probe reveals a universal near-horizon conformal structure for a wide class of black holes, including the BTZ. The central charge of the corresponding Virasoro algebra contains information about the black hole. With a suitable quantization condition on the central charge, the CFT associated with the black hole in our approach is consistent with the recent observation of Witten, where the dual theory for the BTZ in the AdS/CFT framework has been identified with the construction of Frenkel, Lepowsky and Meurman. This CFT admits the Fischer–Griess monster group as its symmetry. The logarithm of the dimension of a specific representation of the monster group has been identified by Witten as the entropy of the BTZ black hole. Our algebraic approach shows that a wide class of black holes share the same near-horizon conformal structure as that for the BTZ. With a suitable quantization condition, the CFT's for all these black holes in our formalism can be identified with the FLM model, although not through the AdS/CFT correspondence. The corresponding entropy for the BTZ provides a lower bound for the entropy of this entire class of black holes.

New quantum modes of the free scalar field are derived in a special time-evolution picture that may be introduced in moving charts of de Sitter backgrounds. The wave functions of these new modes are solutions of the Klein–Gordon equation and energy eigenfunctions, defining the energy basis. This completes the scalar quantum mechanics where the momentum basis is well known for long time. In this enlarged framework the quantization of the scalar field can be done in canonical way obtaining the principal conserved one-particle operators and the Green functions.

The existence of an exactly scale invariant sector possessing a nontrivial infrared fixed point at a higher energy scale and its possible communication with the Standard Model particles through a heavy messenger sector has been shown to lead to curious unparticle effects. We demonstrate that top physics at the Tevatron can already constrain such theories. We also consider possible improvements at the LHC and delineate some striking signatures.

We clearly show that the symplectic structures deformations lead, upon quantization, to quantum theories of noncommutative fields. Two variants of deformations are considered. The quantization is performed and the modes expansions of the quantum fields are derived. The Hamiltonians are given and the degeneracies lifting induced by the deformation is also discussed. As an illustration, we consider the noncommutative chiral boson fields in the context of fractional quantum Hall effect. A generalized fractional filling factor is derived and shown to reproduce the Jain Hall states. We also show that the coupling of left and right edge excitations of a quantum Hall sample, gives rise a noncommutative chiral boson theory. The coupling or the noncommutativity induces a shift of the chiral components velocities. A nonlinear dispersion relation is obtained, corroborating some recent analytical and numerical analysis.

We assume that the noncommutativity starts to be visible continuously from a scale ΛNC. According to this assumption, a two-loop effective action is derived for noncommutative ϕ⁴ and ϕ³ theories from a Wilsonian point of view. We show that these effective theories are free of UV/IR mixing phenomena. We also investigate the positivity constraint on coefficients of higher dimension operators present in the effective theory. This constraint makes the low energy theory to be UV completion of a full theory. Finally, we discuss noncommutativity and extra dimensions. In our effective theories formulated on noncommutative extra dimensions, if the campactification scale Λc is less than the scale ΛNC, the theory will not suffer from UV/IR mixing.

We derive a trace formula for Green's functions of position-dependent (effective) mass Schrödinger equations that are defined on a real, finite interval and connected by a Darboux transformation of arbitrary order. Our findings generalize former results (J. Phys. A37, 10287 (2004)) on constant mass Schrödinger equations to the effective mass case.

In this paper, we first we present a general spherical symmetry solution, independent of any coordinate condition, to the Einstein equations. By using this solution, we obtain different cases in different coordinate conditions, such as Schwarzschild solution, Fock solution, which have different forms of f(r). Then, we obtain a modified formula with respect to f(r) of the Newtonian gravity. This formula may describe the gravity at small length scale.