Optimal uncertainty quantification for legacy data observations of Lipschitz functions

We consider the problem of providing optimal uncertainty quantification (UQ) --- and hence rigorous certification --- for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.Comment: 38 page

Pilot-wave theory and quantum fields

Pilot-wave theories provide possible solutions to the measurement problem. In such theories, quantum systems are not only described by the state vector, but also by some additional variables. These additional variables, also called beables, can be particle positions, field configurations, strings, etc. In this paper we focus our attention on pilot-wave theories in which the additional variables are field configurations. The first such theory was proposed by Bohm for the free electromagnetic field. Since Bohm, similar pilot-wave theories have been proposed for other quantum fields. The purpose of this paper is to present an overview and further development of these proposals. We discuss various bosonic quantum field theories such as the Schroedinger field, the free electromagnetic field, scalar quantum electrodynamics and the Abelian Higgs model. In particular, we compare the pilot-wave theories proposed by Bohm and by Valentini for the electromagnetic field, finding that they are equivalent. We further discuss the proposals for fermionic fields by Holland and Valentini. In the case of Holland's model we indicate that further work is required in order to show that the model is capable of reproducing the standard quantum predictions. We also consider a similar model, which does not seem to reproduce the standard quantum predictions. In the case of Valentini's model we point out a problem that seems hard to overcome.Comment: 65 pages, no figures, LaTex; v2 minor changes, some extensions; v3 minor improvements; v4 some typos correcte

Infinitely Many Strings in De Sitter Spacetime: Expanding and Oscillating Elliptic Function Solutions

The exact general evolution of circular strings in $2+1$ dimensional de Sitter spacetime is described closely and completely in terms of elliptic functions. The evolution depends on a constant parameter $b$, related to the string energy, and falls into three classes depending on whether $b<1/4$ (oscillatory motion), $b=1/4$ (degenerated, hyperbolic motion) or $b>1/4$ (unbounded motion). The novel feature here is that one single world-sheet generically describes {\it infinitely many} (different and independent) strings. The world-sheet time $\tau$ is an infinite-valued function of the string physical time, each branch yields a different string. This has no analogue in flat spacetime. We compute the string energy $E$ as a function of the string proper size $S$, and analyze it for the expanding and oscillating strings. For expanding strings $(\dot{S}>0)$: $E\neq 0$ even at $S=0$, $E$ decreases for small $S$ and increases $\propto\hspace*{-1mm}S$ for large $S$. For an oscillating string $(0\leq S\leq S_{max})$, the average energy $$ over one oscillation period is expressed as a function of $S_{max}$ as a complete elliptic integral of the third kind.Comment: 32 pages, Latex file, figures available from the authors under request. LPTHE-PAR 93-5

Quantum correlations in the temporal CHSH scenario

We consider a temporal version of the CHSH scenario using projective measurements on a single quantum system. It is known that quantum correlations in this scenario are fundamentally more general than correlations obtainable with the assumptions of macroscopic realism and non-invasive measurements. In this work, we also educe some fundamental limitations of these quantum correlations. One result is that a set of correlators can appear in the temporal CHSH scenario if and only if it can appear in the usual spatial CHSH scenario. In particular, we derive the validity of the Tsirelson bound and the impossibility of PR-box behavior. The strength of possible signaling also turns out to be surprisingly limited, giving a maximal communication capacity of approximately 0.32 bits. We also find a temporal version of Hardy's nonlocality paradox with a maximal quantum value of 1/4.Comment: corrected versio

Vlasov-Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory

We study analytically the collapse of an initially smooth, cold, self-gravitating collisionless system in one dimension. The system is described as a central "S" shape in phase-space surrounded by a nearly stationary halo acting locally like a harmonic background on the S. To resolve the dynamics of the S under its self-gravity and under the influence of the halo, we introduce a novel approach using post-collapse Lagrangian perturbation theory. This approach allows us to follow the evolution of the system between successive crossing times and to describe in an iterative way the interplay between the central S and the halo. Our theoretical predictions are checked against measurements in entropy conserving numerical simulations based on the waterbag method. While our post-collapse Lagrangian approach does not allow us to compute rigorously the long term behavior of the system, i.e. after many crossing times, it explains the close to power-law behavior of the projected density observed in numerical simulations. Pushing the model at late time suggests that the system could build at some point a very small flat core, but this is very speculative. This analysis shows that understanding the dynamics of initially cold systems requires a fine grained approach for a correct description of their very central part. The analyses performed here can certainly be extended to spherical symmetry.Comment: 20 pages, 9 figures, accepted for publication in MNRA

Brane classical and quantum cosmology from an effective action

Motivated by the Randall-Sundrum brane-world scenario, we discuss the classical and quantum dynamics of a (d+1)-dimensional boundary wall between a pair of (d+2)-dimensional topological Schwarzschild-AdS black holes. We assume there are quite general -- but not completely arbitrary -- matter fields living on the boundary ``brane universe'' and its geometry is that of an Friedmann-Lemaitre-Robertson-Walker (FLRW) model. The effective action governing the model in the mini-superspace approximation is derived. We find that the presence of black hole horizons in the bulk gives rise to a complex action for certain classically allowed brane configurations, but that the imaginary contribution plays no role in the equations of motion. Classical and instanton brane trajectories are examined in general and for special cases, and we find a subset of configuration space that is not allowed at the classical or semi-classical level; these correspond to spacelike branes carrying tachyonic matter. The Hamiltonization and Dirac quantization of the model is then performed for the general case; the latter involves the manipulation of the Hamiltonian constraint before it is transformed into an operator that annihilates physical state vectors. The ensuing covariant Wheeler-DeWitt equation is examined at the semi-classical level, and we consider the possible localization of the brane universe's wavefunction away from the cosmological singularity. This is easier to achieve for branes with low density and/or spherical spatial sections.Comment: Shortened to match version accepted by Phys. Rev. D (unabridged text found in version 2), 42 pages, 9 figures, Rextex