A Path-Integral Analysis of Interacting Bose Gases and Loop Gases

Abstract

We review some recent results on interacting Bose gases in thermal equilibrium. In particular, we study the convergence of the grand-canonical equilibrium states of such gases to their mean-field limits, which are given by the Gibbs measures of classical field theories with quartic Hartree-type self-interaction, and to the Gibbs states of classical gases of point particles. We discuss various open problems and conjectures concerning, e.g., Bose–Einstein condensation, polymers and \(\vert \varvec{\phi } \vert ^{4}\)-theory.

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Notes

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    For mathematical results concerning Bose gases at zero temperature and the convergence of the quantum dynamics to the mean-field dynamics the reader is referred to the literature quoted in [1, 2, 5, 8].

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    A broad introduction to equilibrium statistical mechanics, including a mathematical discussion of the equivalence of the three standard ensembles—micro-canonical, canonical and grand-canonical—can be found in [36].

  3. 3.

    Readers concerned with mathematical rigor may want to introduce a lattice regularization of the expressions considered below and let the lattice spacing tend to 0 at the end of the calculations; see [2].

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Acknowledgements

We thank David Brydges, Alessandro Pizzo and Daniel Ueltschi for very useful discussions and some correspondence on problems related to the ones studied in this paper. We are grateful to Mathieu Lewin, Phan Thành Nam and Nicolas Rougerie for informing us about their beautiful results [16] prior to publication.

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Fröhlich, J., Knowles, A., Schlein, B. et al. A Path-Integral Analysis of Interacting Bose Gases and Loop Gases. J Stat Phys 180, 810–831 (2020). https://doi.org/10.1007/s10955-020-02543-x

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Keywords

  • Interacting Bose gases
  • Functional integral representation of equilibrium states
  • Loop gases
  • Ginibre and Symanzik representations
  • Mean-field limit
  • Gibbs measures