Mean-Field- and Classical Limit of Many-Body Schr\"odinger Dynamics for Bosons

We present a new proof of the convergence of the N-particle Schroedinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in the Planck constant , up to an exponentially small remainder. For h=0, the classical dynamics in the mean-field limit is given by the Vlasov equation.Comment: Latex 2e, 18 page

Wigner Functions and Separability for Finite Systems

A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension p^n where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p^2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p^(2n) points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.Comment: Some misprints have been corrected and a proof of the separability of the A matrices has been adde

Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes

We construct a class of solutions to the Cauchy problem of the Klein-Gordon equation on any standard static spacetime. Specifically, we have constructed solutions to the Cauchy problem based on any self-adjoint extension (satisfying a technical condition: "acceptability") of (some variant of) the Laplace-Beltrami operator defined on test functions in an $L^2$-space of the static hypersurface. The proof of the existence of this construction completes and extends work originally done by Wald. Further results include the uniqueness of these solutions, their support properties, the construction of the space of solutions and the energy and symplectic form on this space, an analysis of certain symmetries on the space of solutions and of various examples of this method, including the construction of a non-bounded below acceptable self-adjoint extension generating the dynamics

Property (RD) for Hecke pairs

As the first step towards developing noncommutative geometry over Hecke C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989 to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C*-algebra. Hence they have the same K_0-groups.Comment: A short note added explaining other methods to prove that the subalgebra of rapidly decreasing functions is smooth. This is the final version as published. The published version is available at: springer.co

The Reliability of Global and Hemispheric Surface Temperature Records

The purpose of this review article is to discuss the development and associated estimation of uncertainties in the global and hemispheric surface temperature records. The review begins by detailing the groups that produce surface temperature datasets. After discussing the reasons for similarities and differences between the various products, the main issues that must be addressed when deriving accurate estimates, particularly for hemispheric and global averages, are then considered. These issues are discussed in the order of their importance for temperature records at these spatial scales: biases in SST data, particularly before the 1940s; the exposure of land-based thermometers before the development of louvred screens in the late 19th century; and urbanization effects in some regions in recent decades. The homogeneity of land-based records is also discussed; however, at these large scales it is relatively unimportant. The article concludes by illustrating hemispheric and global temperature records from the four groups that produce series in near-real time

Mixing Quantum and Classical Mechanics

Using a group theoretical approach we derive an equation of motion for a mixed quantum-classical system. The quantum-classical bracket entering the equation preserves the Lie algebra structure of quantum and classical mechanics: The bracket is antisymmetric and satisfies the Jacobi identity, and, therefore, leads to a natural description of interaction between quantum and classical degrees of freedom. We apply the formalism to coupled quantum and classical oscillators and show how various approximations, such as the mean-field and the multiconfiguration mean-field approaches, can be obtained from the quantum-classical equation of motion.Comment: 31 pages, LaTeX2

The quantum state vector in phase space and Gabor's windowed Fourier transform

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.Comment: 36 pages, 6 figures. Revised in light of referees' comments, and further references adde

A Geometric Formulation of Quantum Stress Fields

We present a derivation of the stress field for an interacting quantum system within the framework of local density functional theory. The formulation is geometric in nature and exploits the relationship between the strain tensor field and Riemannian metric tensor field. Within this formulation, we demonstrate that the stress field is unique up to a single ambiguous parameter. The ambiguity is due to the non-unique dependence of the kinetic energy on the metric tensor. To illustrate this formalism, we compute the pressure field for two phases of solid molecular hydrogen. Furthermore, we demonstrate that qualitative results obtained by interpreting the hydrogen pressure field are not influenced by the presence of the kinetic ambiguity.Comment: 22 pages, 2 figures. Submitted to Physical Review B. This paper supersedes cond-mat/000627

Different atmospheric moisture divergence responses to extreme and moderate El Niños

On seasonal and inter-annual time scales, vertically integrated moisture divergence provides a useful measure of the tropical atmospheric hydrological cycle. It reflects the combined dynamical and thermodynamical effects, and is not subject to the limitations that afflict observations of evaporation minus precipitation. An empirical orthogonal function (EOF) analysis of the tropical Pacific moisture divergence fields calculated from the ERA-Interim reanalysis reveals the dominant effects of the El Niño-Southern Oscillation (ENSO) on inter-annual time scales. Two EOFs are necessary to capture the ENSO signature, and regression relationships between their Principal Components and indices of equatorial Pacific sea surface temperature (SST) demonstrate that the transition from strong La Niña through to extreme El Niño events is not a linear one. The largest deviation from linearity is for the strongest El Niños, and we interpret that this arises at least partly because the EOF analysis cannot easily separate different patterns of responses that are not orthogonal to each other. To overcome the orthogonality constraints, a self-organizing map (SOM) analysis of the same moisture divergence fields was performed. The SOM analysis captures the range of responses to ENSO, including the distinction between the moderate and strong El Niños identified by the EOF analysis. The work demonstrates the potential for the application of SOM to large scale climatic analysis, by virtue of its easier interpretation, relaxation of orthogonality constraints and its versatility for serving as an alternative classification method. Both the EOF and SOM analyses suggest a classification of “moderate” and “extreme” El Niños by their differences in the magnitudes of the hydrological cycle responses, spatial patterns and evolutionary paths. Classification from the moisture divergence point of view shows consistency with results based on other physical variables such as SST

The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space

We combine I. background independent Loop Quantum Gravity (LQG) quantization techniques, II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and III. the theory of integrable systems resulting in the invariant Pohlmeyer Charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space. While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge new, non -- trivial solution to the representation problem. This solution exists 1. for any target space dimension, 2. for Minkowski signature of the target space, 3. without tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies (zero central charge), 7. while preserving manifest target space Poincar\'e invariance and 8. without picking up UV divergences. The existence of this stable solution is exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string. Moreover, these new representations could solve some of the major puzzles of string theory such as the cosmological constant problem. The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper we treat the more complicated case of curved target spaces.Comment: 46 p., LaTex2e, no figure