Abstract
This paper considers the issue of Bose–Einstein condensation in a weakly interacting Bose gas with a fixed total number of particles. We use an old current algebra formulation of non-relativistic many body systems due to Dashen and Sharp to show that, at sufficiently low temperatures, a gas of weakly interacting Bosons displays Off-diagonal Long Range Order in the sense introduced by Penrose and Onsager. Even though this formulation is somewhat cumbersome it may demystify many of the standard results in the field for those uncomfortable with the conventional broken symmetry based approaches. All the physics presented here is well understood but as far as we know this perspective, although dating from the 60's and 70's, has not appeared in the literature. We have attempted to make the presentation as self-contained as possible in the hope that it will be accessible to the many students interested in the field.
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We note that, by contrast, symmetry breaking also implies ODLRO in the correlation function (1).
Here we will not consider fragmented condensates.
The operator \(\hat P = \Psi \left( {{\text{r}}} \right)\left[ {{1 \mathord{\left/ {\vphantom {1 {\hat n}}} \right. \kern-\nulldelimiterspace} {\hat n}}\left( {{\text{r}}} \right)} \right]\Psi ^\dag \left( {{\text{r}}} \right)\) is a projection operator \(\left( {\hat P^2 = \hat P} \right)\) with eigenvalues λ=0, 1. Since for any finite interaction no bosonic state can be annihilated by \(\hat P,\hat P \equiv {\hat l}\) on all states in the Hilbert space.
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Ruckenstein, A.E. Bose Condensation Without Broken Symmetries. Foundations of Physics 30, 2113–2124 (2000). https://doi.org/10.1023/A:1003745608929
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DOI: https://doi.org/10.1023/A:1003745608929