Comment on ''Field-Enhanced Diamagnetism in the Pseudogap State of the Cuprate Bi2Sr2CaCu2O8+\delta Superconductor in an Intense Magnetic Field''

In the above mentioned letter by Wang et al. [Phys. Rev. Lett, 95, 247002 (2005)], magnetization measurements on two Bi_2Sr_2caCu_2O_8+delta samples are reported. They claim that these experimental results support the vortex scenario for the loss of phase coherence at Tc. On the contrary, we show in this comment that they can be explained by means of the Ginzburg Landau theory (under a total-enery cutoff) for the superconducting fluctuations above Tc.Comment: Final versio

On Scale Invariance and Anomalies in Quantum Mechanics

We re-consider the quantum mechanics of scale invariant potentials in two dimensions. The breaking of scale invariance by quantum effects is analyzed by the explicit evaluation of the phase shift and the self-adjoint extension method. We argue that the breaking of scale invariance reported in the literature for the $\delta$(r) potential, is an example of explicit and not an anomaly or quantum mechanical symmetry breaking.Comment: 15 pp in latex, no figure

Faddeev-Jackiw approach to gauge theories and ineffective constraints

The general conditions for the applicability of the Faddeev-Jackiw approach to gauge theories are studied. When the constraints are effective a new proof in the Lagrangian framework of the equivalence between this method and the Dirac approach is given. We find, however, that the two methods may give different descriptions for the reduced phase space when ineffective constraints are present. In some cases the Faddeev-Jackiw approach may lose some constraints or some equations of motion. We believe that this inequivalence can be related to the failure of the Dirac conjecture (that says that the Dirac Hamiltonian can be enlarged to an Extended Hamiltonian including all first class constraints, without changes in the dynamics) and we suggest that when the Dirac conjecture fails the Faddeev-Jackiw approach fails to give the correct dynamics. Finally we present some examples that illustrate this inequivalence.Comment: 21 pages, Latex. To be published in Int. J. Mod. Phys.

Weyl-Wigner-Moyal formulation of a Dirac quantized constrained system

An extension of the Weyl-Wigner-Moyal formulation of quantum mechanics suitable for a Dirac quantized constrained system is proposed. In this formulation, quantum observables are described by equivalent classes of Weyl symbols. The Weyl product of these equivalent classes is defined. The new Moyal bracket is shown to be compatible with the Dirac bracket for constrained systems

Classical Noncommutative Electrodynamics with External Source

In a $U(1)_{\star}$-noncommutative (NC) gauge field theory we extend the Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external current and formulate - to the first order in the NC parameter - gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size $a$, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, $1/r$ included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size $a$. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form-factors. We also analyze the ambiguity in the SW map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation

Colonias comerciales de Lavanda: Estudio cromatográfico cuantitativo mediante cromatografía en capa fina

Se han estudiado mediante C.C.F. 24. marcas de colonias de Lavanda distintas, procedentes del comercio español. Para ello nos centramos en el principal componente del aceite esencial de lavanda, que como es sabido es el linalol, obteniendo unos tantos por ciento absolutos de dicho componente en las diversas colonias estudiadas. Analizando los resultados obtenidos, podemos concluir, que existen dos colonias que se separan claramente del resto, por su porcentaje en linalol, siendo estos porcentajes de 2,77 por 100 correspondiente a la colonia LAVANDA INGLESA ATKINSONS y 2,72 por 100 que corresponde a la colonia LAVANDA INGLESA YARDLEY.It has been studied twenty four different makes of perfumes of Lavan dae from the Spanish market by T.L.C. We have paid attention to the main component of essential oils of Lavan dae, the Linalol. The percentas-es obtained of this component have been absolute in all studied perfumes. There were two perfumes clearly separated from the rest because their percentage in Linalol: Lavanda inglesa Atkinsons 2,77 por 100 and Lavanda Inglesa Yardley 2,72 por 100