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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

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

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

According to Dirac’s ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator D0. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane (J. Phys. B 22, 3791–3814 (1989)), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is the solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.

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Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation

The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.

The bcs functional for general pair interactions

We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in . One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdős, Schlein, and Yau. © 2015 Wiley Periodicals, Inc.

Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti

For the BCS equation with local two-body interaction λV(x), we give a rigorous analysis of the asymptotic behavior of the critical temperature as γ»0. We derive necessary and sufficient conditions on V(x) for the existence of a nontrivial solution for all values of λ>0.

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The Critical Temperature for the BCS Equation at Weak Coupling

We derive upper and lower bounds on the critical temperature ${T}_{c}$ and the energy gap $\ensuremath{\Xi}$ (at zero temperature) for the BCS gap equation, describing spin-$\frac{1}{2}$ fermions interacting via a local two-body interaction potential $\ensuremath{\lambda}V(x)$. At weak coupling $\ensuremath{\lambda}⪡1$ and under appropriate assumptions on $V(x)$, our bounds show that ${T}_{c}\ensuremath{\sim}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ and $\ensuremath{\Xi}\ensuremath{\sim}C\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ for some explicit coefficients $A$, $B$, and $C$ depending on the interaction $V(x)$ and the chemical potential $\ensuremath{\mu}$. The ratio $A∕C$ turns out to be a universal constant, independent of both $V(x)$ and $\ensuremath{\mu}$. Our analysis is valid for any $\ensuremath{\mu}$; for small $\ensuremath{\mu}$, or low density, our formulas reduce to well-known expressions involving the scattering length of $V(x)$.

Christian Hainzl

Papers

Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation

The bcs functional for general pair interactions

Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti

The Critical Temperature for the BCS Equation at Weak Coupling

Critical temperature and energy gap for the BCS equation