Optimal phase measurements with pure Gaussian states

We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisenberg limit for large samples. This proves that this limit can be attained with local projective Von Neuman measurements.Comment: 9 pages. Revised version: two new sections added, revised conclusions. Corrected prose. Corrected reference

Black Hole Horizons and Complementarity

We investigate the effect of gravitational back-reaction on the black hole evaporation process. The standard derivation of Hawking radiation is re-examined and extended by including gravitational interactions between the infalling matter and the outgoing radiation. We find that these interactions lead to substantial effects. In particular, as seen by an outside observer, they lead to a fast growing uncertainty in the position of the infalling matter as it approaches the horizon. We argue that this result supports the idea of black hole complementarity, which states that, in the description of the black hole system appropriate to outside observers, the region behind the horizon does not establish itself as a classical region of space-time. We also give a new formulation of this complementarity principle, which does not make any specific reference to the location of the black hole horizon.Comment: Some minor modifications in text and the title chang

Quantization of Space and Time in 3 and in 4 Space-time Dimensions

The fact that in Minkowski space, space and time are both quantized does not have to be introduced as a new postulate in physics, but can actually be derived by combining certain features of General Relativity and Quantum Mechanics. This is demonstrated first in a model where particles behave as point defects in 2 space dimensions and 1 time, and then in the real world having 3+1 dimensions. The mechanisms in these two cases are quite different, but the outcomes are similar: space and time form a (non-cummutative) lattice. These notes are short since most of the material discussed in these lectures is based on two earlier papers by the same author (gr-qc/9601014 and gr-qc/9607022), but the exposition given in the end is new.Comment: Lectures held at the NATO Advanced Study Institute on ``Quantum Fields and Quantum Space Time", Carg\`ese, July 22 -- August 3, 1996. 16 pages Plain TeX, 6 Figure

Scheme to measure squeezing and phase properties of a harmonic oscillator

We propose a simple scheme to measure squeezing and phase properties of a harmonic oscillator. We treat in particular the case of a the field, but the scheme may be easily realized in ion traps. It is based on integral transforms of measured atomic properties as atoms exit a cavity. We show that by measuring atomic polarizations it is possible, after a given integration, to measure several properties of the field.Comment: Presented at XI Central European Workshop on Quantum Optics, Trieste, Italy, 18-20 July, 200

Complementarity Endures: No Firewall for an Infalling Observer

We argue that the complementarity picture, as interpreted as a reference frame change represented in quantum gravitational Hilbert space, does not suffer from the "firewall paradox" recently discussed by Almheiri, Marolf, Polchinski, and Sully. A quantum state described by a distant observer evolves unitarily, with the evolution law well approximated by semi-classical field equations in the region away from the (stretched) horizon. And yet, a classical infalling observer does not see a violation of the equivalence principle, and thus a firewall, at the horizon. The resolution of the paradox lies in careful considerations on how a (semi-)classical world arises in unitary quantum mechanics describing the whole universe/multiverse.Comment: 11 pages, 1 figure; clarifications and minor revisions; v3: a small calculation added for clarification; v4: some corrections, conclusion unchange

Staggered Fermion, its Symmetry and Ichimatsu-Patterned Lattice

We investigate exact symmetries of a staggered fermion in D dimensions. The Dirac operator is reformulated by SO(2D) Clifford algebra. The chiral symmetry, rotational invariance and parity symmetries are clarified in any dimension. Local scalar and pseudo-scalar modes are definitely determined, in which we find non-standard modes. The relation to Ichimatsu-patterned lattice approach is discussed.Comment: 3 pages, 1 figure, "Talk at Lattice2004(theory), Fermilab, June 21-26, 2004

Shock Waves and Cosmological Matrix Models

We find the shock wave solutions in a class of cosmological backgrounds with a null singularity, each of these backgrounds admits a matrix description. A shock wave solution breaks all supersymmetry meanwhile indicates that the interaction between two static D0-branes cancel, thus provides basic evidence for the matrix description. The probe action of a D0-brane in the background of another suggests that the usual perturbative expansion of matrix model breaks down.Comment: 10 pages, harvmav, v2: some comments on instability added, v3: version to appear in JHE

Black hole entropy: inside or out?

A trialogue. Ted, Don, and Carlo consider the nature of black hole entropy. Ted and Carlo support the idea that this entropy measures in some sense ``the number of black hole microstates that can communicate with the outside world.'' Don is critical of this approach, and discussion ensues, focusing on the question of whether the first law of black hole thermodynamics can be understood from a statistical mechanics point of view.Comment: 42 pages, contribution to proceedings of Peyresq

The Gauge Hierarchy Problem and Higher Dimensional Gauge Theories

We report on an attempt to solve the gauge hierarchy problem in the framework of higher dimensional gauge theories. Both classical Higgs mass and quadratically divergent quantum correction to the mass are argued to vanish. Hence the hierarchy problem in its original sense is solved. The remaining finite mass correction is shown to depend crucially on the choice of boundary condition for matter fields, and a way to fix it dynamically is presented. We also point out that on the simply-connected space $S^2$ even the finite mass correction vanishes.Comment: LaTeX2e. 12 pages, 3 Postscript figures; Added references, some comment