Notes on Wick's theorem in many-body theory

In these pedagogical notes I introduce the operator form of Wick's theorem, i.e. a procedure to bring to normal order a product of 1-particle creation and destruction operators, with respect to some reference many-body state. Both the static and the time- ordered cases are presented.Comment: 6 page

Simple conformally recurrent space-times are conformally recurrent PP-waves

We show that in dimension n>3 the class of simple conformally recurrent space-times coincides with the class of conformally recurrent pp-waves.Comment: Dedicated to the memory of professor Witold Rote

Extended Derdzinski-Shen theorem for the Riemann tensor

We extend a classical result by Derdzinski and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms) as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"), typical of some well known differential structures.Comment: 5 page

Twisted Lorentzian manifolds, a characterization with torse-forming time-like unit vectors

Robertson-Walker and Generalized Robertson-Walker spacetimes may be characterized by the existence of a time-like unit torse-forming vector field, with other constrains. We show that Twisted manifolds may still be characterized by the existence of such (unique) vector field, with no other constrain. Twisted manifolds generalize RW and GRW spacetimes by admitting a scale function that depends both on time and space. We obtain the Ricci tensor, corresponding to the stress-energy tensor of an imperfect fluid.Comment: 6 pages, marginal errors corrected, reference update

Non-Hermitian spectra and Anderson localization

The spectrum of exponents of the transfer matrix provides the localization lengths of Anderson's model for a particle in a lattice with disordered potential. I show that a duality identity for determinants and Jensen's identity for subharmonic functions, give a formula for the spectrum in terms of eigenvalues of the Hamiltonian with non-Hermitian boundary conditions. The formula is exact; it involves an average over a Bloch phase, rather than disorder. A preliminary investigation of non-Hermitian spectra of Anderson's model in D=1,2 and on the smallest exponent is presented.Comment: 8 pages, 10 figure

Cosmological perfect-fluids in f(R) gravity

We show that an n-dimensional generalized Robertson-Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress-energy tensor for any f(R) gravity model. Furthermore we prove that a conformally flat GRW space-time is still a perfect fluid in both f(R) and quadratic gravity where other curvature invariants are considered.Comment: 13 pages, final versio

Devil's staircase phase diagram of the fractional quantum Hall effect in the thin-torus limit

After more than three decades the fractional quantum Hall effect still poses challenges to contemporary physics. Recent experiments point toward a fractal scenario for the Hall resistivity as a function of the magnetic field. Here, we consider the so-called thin-torus limit of the Hamiltonian describing interacting electrons in a strong magnetic field, restricted to the lowest Landau level, and we show that it can be mapped onto a one-dimensional lattice gas with repulsive interactions, with the magnetic field playing the role of a chemical potential. The statistical mechanics of such models leads to interpret the sequence of Hall plateaux as a fractal phase diagram, whose landscape shows a qualitative agreement with experiments.Comment: 5 pages main text, 11 pages supplementary, 2 figure