Bose-Einstein Condensation in a Harmonic Potential

We examine several features of Bose-Einstein condensation (BEC) in an external harmonic potential well. In the thermodynamic limit, there is a phase transition to a spatial Bose-Einstein condensed state for dimension D greater than or equal to 2. The thermodynamic limit requires maintaining constant average density by weakening the potential while increasing the particle number N to infinity, while of course in real experiments the potential is fixed and N stays finite. For such finite ideal harmonic systems we show that a BEC still occurs, although without a true phase transition, below a certain ``pseudo-critical'' temperature, even for D=1. We study the momentum-space condensate fraction and find that it vanishes as 1/N^(1/2) in any number of dimensions in the thermodynamic limit. In D less than or equal to 2 the lack of a momentum condensation is in accord with the Hohenberg theorem, but must be reconciled with the existence of a spatial BEC in D=2. For finite systems we derive the N-dependence of the spatial and momentum condensate fractions and the transition temperatures, features that may be experimentally testable. We show that the N-dependence of the 2D ideal-gas transition temperature for a finite system cannot persist in the interacting case because it violates a theorem due to Chester, Penrose, and Onsager.Comment: 34 pages, LaTeX, 6 Postscript figures, Submitted to Jour. Low Temp. Phy

Boundary Conditions and Quasilocal Energy in the Canonical Formulation of All 1+1 Models of Gravity

Within a first-order framework, we comprehensively examine the role played by boundary conditions in the canonical formulation of a completely general two-dimensional gravity model. Our analysis particularly elucidates the perennial themes of mass and energy. The gravity models for which our arguments are valid include theories with dynamical torsion and so-called generalized dilaton theories (GDTs). Our analysis of the canonical action principle (i) provides a rigorous correspondence between the most general first-order two-dimensional Einstein-Cartan model (ECM) and GDT and (ii) allows us to extract in a virtually simultaneous manner the ``true degrees of freedom'' for both ECMs and GDTs. For all such models, the existence of an absolutely conserved (in vacuo) quantity C is a generic feature, with (minus) C corresponding to the black-hole mass parameter in the important special cases of spherically symmetric four-dimensional general relativity and standard two-dimensional dilaton gravity. The mass C also includes (minimally coupled) matter into a ``universal mass function.'' We place particular emphasis on the (quite general) class of models within GDT possessing a Minkowski-like groundstate solution (allowing comparison between $C$ and the Arnowitt-Deser-Misner mass for such models).Comment: REVTeX, 41 pages, 2 Postscript figures, 10 macro

Science, performance and transformation: performance for a ‘scientific’ age?

This is an accepted manuscript of an article published by Taylor & Francis in International Journal of Performance Arts and Digital Media on 30/09/2014, available online: https://www.tandfonline.com/doi/full/10.1080/14794713.2014.946282 The accepted version of the publication may differ from the final published version.The ‘two cultures’ of science and the arts/humanities are often considered at odds, but digital technology, and the broader implications of digital culture, provides a model for more productive forms of exchange and hybridity. This article applies theories of intercultural theatre practice to performance that works across this cultural divide to explore the types of interaction that take place. Following a historical discussion of the science/art divide, a three-fold model is proposed and explored through case studies including Djerassi and Laszlo's 2003 NO, Eduardo Kac's 1999 Genesis, Reckless Sleepers' 1996/2006 Schrödinger's Box, and John Barrow's 2002 Infinities. It is argued that science operates through the creation of mathematical models of aspects of the physical world, whilst art similarly constructs different kinds of models for understanding the social/cultural and occasionally physical world. Digital technology expands the modelling possibilities both directly, through simulation, virtual reality and integration into ‘live’ activities of augmented and intermedia performance, and through the transformative nature of digital culture

One-Loop Divergences in Simple Supergravity: Boundary Effects

This paper studies the semiclassical approximation of simple supergravity in Riemannian four-manifolds with boundary, within the framework of $\zeta$-function regularization. The massless nature of gravitinos, jointly with the presence of a boundary and a local description in terms of potentials for spin ${3\over 2}$, force the background to be totally flat. First, nonlocal boundary conditions of the spectral type are imposed on spin-${3\over 2}$ potentials, jointly with boundary conditions on metric perturbations which are completely invariant under infinitesimal diffeomorphisms. The axial gauge-averaging functional is used, which is then sufficient to ensure self-adjointness. One thus finds that the contributions of ghost and gauge modes vanish separately. Hence the contributions to the one-loop wave function of the universe reduce to those $\zeta(0)$ values resulting from physical modes only. Another set of mixed boundary conditions, motivated instead by local supersymmetry and first proposed by Luckock, Moss and Poletti, is also analyzed. In this case the contributions of gauge and ghost modes do not cancel each other. Both sets of boundary conditions lead to a nonvanishing $\zeta(0)$ value, and spectral boundary conditions are also studied when two concentric three-sphere boundaries occur. These results seem to point out that simple supergravity is not even one-loop finite in the presence of boundaries.Comment: 37 pages, Revtex. Equations (5.2), (5.3), (5.5), (5.7), (5.8) and (5.13) have been amended, jointly with a few misprint

Theory of Bose-Einstein condensation in trapped gases

The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.Comment: revtex, 69 pages, 38 eps figures, new version with more references, new figures, various changes and corrections, for publ. in Rev. Mod. Phys., available also at http://www-phys.science.unitn.it/bec/BEC.htm

Dilaton Gravity in Two Dimensions

The study of general two dimensional models of gravity allows to tackle basic questions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important examples of spherically symmetric Black Holes, together with string inspired models, belong to this class, valuable knowledge can also be gained for these systems in the quantum case. In the last decade new insights regarding the exact quantization of the geometric part of such theories have been obtained. They allow a systematic quantum field theoretical treatment, also in interactions with matter, without explicit introduction of a specific classical background geometry. The present review tries to assemble these results in a coherent manner, putting them at the same time into the perspective of the quite large literature on this subject.Comment: 144 pages, 16 figures; v2,v3: added refs. and corrected typos, v4: added 2 refs. and corrected typos (published version), v5: added note with some relevant refs., v6: diligent students found still a couple of typos, added 1 ref., v7: last update from Vienna (a couple of typos), v8: Leipzig edition (a dozen typos), v9: MIT edition (4 typos, 1 ref.