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.

https://arxiv.org/pdf/cond-mat/9806038.pdf

Theory of Bose-Einstein condensation in trapped gases

Modified Gravity and Cosmology

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

Quantum field theory in curved spacetime has been remarkably fruitful. It can be used to explain how the large-scale structure of the universe and the anisotropies of the cosmic background radiation that we observe today first arose. Similarly, it provides a deep connection between general relativity, thermodynamics, and quantum field theory. This book develops quantum field theory in curved spacetime in a pedagogical style, suitable for graduate students. The authors present detailed, physically motivated, derivations of cosmological and black hole processes in which curved spacetime plays a key role. They explain how such processes in the rapidly expanding early universe leave observable consequences today, and how in the context of evaporating black holes, these processes uncover deep connections between gravitation and elementary particles. The authors also lucidly describe many other aspects of free and interacting quantized fields in curved spacetime.

/pdf/quantum-field-theory-in-curved-spacetime-quantized-fields-32xdrurrwq.pdf

Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity

We consider a class of grand unified theories (GUT's) based on the Georgi-Glashow model in curved spacetime. We are particularly concerned with the coupling constants involving the curvature. These include the cosmological and gravitational constants, as well as coupling constants appearing in terms quadratic in the curvature and in terms which link the Higgs bosons to the scalar curvature. For asymptotically free theories, we use the renormalization group to obtain expressions for these effective coupling constants at high curvature (between the GUT and Planck scales). We discuss the role of the effective coupling constants in the gravitational field equations. These results may be of importance for cosmology.

Renormalization Group Analysis of Grand Unified Theories in Curved Space-time

We present an analysis of Bose-Einstein condensation for a system of noninteracting spin-0 particles in a harmonic-oscillator confining potential trap. We discuss why a confined system of particles differs both qualitatively and quantitatively from an identical system which is not confined. One crucial difference is that a confined system is not characterized by a critical temperature in the same way as an unconfined system such as the free boson gas. We present the results of both a numerical and analytic analysis of the problem of Bose-Einstein condensation in a general anisotropic harmonic-oscillator confining potential. \textcopyright{} 1996 The American Physical Society.

Bose-Einstein condensation of atomic gases in a general harmonic-oscillator confining potential trap

We study the question of the gauge dependence of the quantum gravity contribution to the running gauge coupling constant for electromagnetism. The calculations are performed using dimensional regularization in a manifestly gauge-invariant and gauge-condition-independent formulation of the effective action. It is shown that there is no quantum gravity contribution to the running charge, and hence there is no alteration to asymptotic freedom at high energies as predicted by Robinson and Wilczek.

David J. Toms

Papers

Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity

Renormalization Group Analysis of Grand Unified Theories in Curved Space-time

Bose-Einstein condensation of atomic gases in a general harmonic-oscillator confining potential trap

Quantum gravity and charge renormalization

Quantized bulk scalar fields in the Randall–Sundrum brane model