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

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

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

Black Hole Complementarity vs. Locality

The evaporation of a large mass black hole can be described throughout most of its lifetime by a low-energy effective theory defined on a suitably chosen set of smooth spacelike hypersurfaces. The conventional argument for information loss rests on the assumption that the effective theory is a local quantum field theory. We present evidence that this assumption fails in the context of string theory. The commutator of operators in light-front string theory, corresponding to certain low-energy observers on opposite sides of the event horizon, remains large even when these observers are spacelike separated by a macroscopic distance. This suggests that degrees of freedom inside a black hole should not be viewed as independent from those outside the event horizon. These nonlocal effects are only significant under extreme kinematic circumstances, such as in the high-redshift geometry of a black hole. Commutators of space-like separated operators corresponding to ordinary low-energy observers in Minkowski space are strongly suppressed in string theory.Comment: 32 pages, harvmac, 3 figure

Wilson Loops as Precursors

There is substantial evidence that string theory on AdS_5 x S_5 is a holographic theory in which the number of degrees of freedom scales as the area of the boundary in Planck units. Precisely how the theory can describe bulk physics using only surface degrees of freedom is not well understood. A particularly paradoxical situation involves an event deep in the interior of the bulk space. The event must be recorded in the (Schroedinger Picture) state vector of the boundary theory long before a signal, such as a gravitational wave, can propagate from the event to the boundary. In a previous paper with Polchinski, we argued that the "precursor" operators which carry information stored in the wave during the time when it vanishes in a neighborhood of the boundary are necessarily non-local. In this paper we argue that the precursors cannot be products of local gauge invariant operators such as the energy momentum tensor. In fact gauge theories have a class of intrinsically non-local operators which cannot be built from local gauge invariant objects. These are the Wilson loops. We show that the precursors can be identified with Wilson loops whose spatial size is dictated by the UV-IR connection.Comment: 23 pages, no figure