An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise

We study a linear quadratic problem for a system governed by the heat equation on a halfline with Dirichlet boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem.Comment: 16 pages. Many misprints have been correcte

An Hilbert space approach for a class of arbitrage free implied volatilities models

We present an Hilbert space formulation for a set of implied volatility models introduced in \cite{BraceGoldys01} in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price $T$ an $K$, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtained fixing a strike $K$ and varying $T$. In order to give conditions to prove an existence-and-uniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given.Comment: 21 page

Static quantum corrections to the Schwarzschild spacetime

We study static quantum corrections of the Schwarzschild metric in the Boulware vacuum state. Due to the absence of a complete analytic expression for the full semiclassical Einstein equations we approach the problem by considering the s-wave approximation and solve numerically the associated backreaction equations. The solution, including quantum effects due to pure vacuum polarization, is similar to the classical Schwarzschild solution up to the vicinity of the classical horizon. However, the radial function has a minimum at a time-like surface close to the location of the classical event horizon. There the g_{00} component of the metric reaches a very small but non-zero value. The analysis unravels how a curvature singularity emerges beyond this bouncing point. We briefly discuss the physical consequences of these results by extrapolating them to a dynamical collapsing scenario.Comment: 10 pages; Talk given at QG05, Cala Gonone (Italy), September 200

The synchrotron foreground and CMB temperature-polarization cross correlation power spectrum from the first year WMAP data

We analyse the temperature-polarization cross-correlation in the Galactic synchrotron template that we have recently developed, and between the template and CMB temperature maps derived from WMAP data. Since the polarized synchrotron template itself uses WMAP data, we can estimate residual synchrotron contamination in the CMB $C_\ell^{TE}$ angular spectrum. While $C_2^{TE}$ appears to be contamined by synchrotron, no evidence for contamination is found in the multipole range which is most relevant for the fit of the cosmological optical depth.Comment: Accepted for pubblication on MNRAS Lette

Semiclassical zero-temperature corrections to Schwarzschild spacetime and holography

Motivated by the quest for black holes in AdS braneworlds, and in particular by the holographic conjecture relating 5D classical bulk solutions with 4D quantum corrected ones, we numerically solve the semiclassical Einstein equations (backreaction equations) with matter fields in the (zero temperature) Boulware vacuum state. In the absence of an exact analytical expression for in four dimensions we work within the s-wave approximation. Our results show that the quantum corrected solution is very similar to Schwarzschild till very close to the horizon, but then a bouncing surface for the radial function appears which prevents the formation of an event horizon. We also analyze the behavior of the geometry beyond the bounce, where a curvature singularity arises. In the dual theory, this indicates that the corresponding 5D static classical braneworld solution is not a black hole but rather a naked singularity.Comment: 26 pages, 4 figures; revised version (title changed, conclusions shortened), published as Phys. Rev. D73, 104023 (2006

Entanglement generation in relativistic quantum fields

We present a general, analytic recipe to compute the entanglement that is generated between arbitrary, discrete modes of bosonic quantum fields by Bogoliubov transformations. Our setup allows the complete characterization of the quantum correlations in all Gaussian field states. Additionally, it holds for all Bogoliubov transformations. These are commonly applied in quantum optics for the description of squeezing operations, relate the mode decompositions of observers in different regions of curved spacetimes, and describe observers moving along non-stationary trajectories. We focus on a quantum optical example in a cavity quantum electrodynamics setting: an uncharged scalar field within a cavity provides a model for an optical resonator, in which entanglement is created by non-uniform acceleration. We show that the amount of generated entanglement can be magnified by initial single-mode squeezing, for which we provide an explicit formula. Applications to quantum fields in curved spacetimes, such as an expanding universe, are discussed.Comment: 8 pages, 2 figures, Ivette Fuentes previously published as Ivette Fuentes-Guridi and Ivette Fuentes-Schuller; v2: published version (online), to appear in the J. Mod. Opt. Special Issue on the Physics of Quantum Electronic

Entanglement of Dirac fields in an expanding spacetime

We study the entanglement generated between Dirac modes in a 2-dimensional conformally flat Robertson-Walker universe. We find radical qualitative differences between the bosonic and fermionic entanglement generated by the expansion. The particular way in which fermionic fields get entangled encodes more information about the underlying space-time than the bosonic case, thereby allowing us to reconstruct the parameters of the history of the expansion. This highlights the importance of bosonic/fermionic statistics to account for relativistic effects on the entanglement of quantum fields.Comment: revtex4, 7 figures, I.F. previously published as Fuentes-Guridi and Fuentes-Schuller. Journal reference update