The Ground State of the Bose Gas

Abstract

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.