Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Bose condensation with negative chemical potential

  • 38 Accesses


There is widespread prejudice that the existence of Bose-condensed equilibrium states of infinite ideal boson gas requires chemical potential to be strictly zero. This is not true, in general. Using standard techniques of algebraic QFT only, we show that there exists eith invariant extensions T of Schwartz's space D(ℝ3) and Bose-condensed KMS states on the CCR algebra \(\mathfrak{A}\) (T) for every chemical potential μ≤0 (h=−Δ−μ, the one-particle Hamiltonian). The corresponding condensation fields are, in general, of rapid growth at infinity, with suggested physical implications.

This is a preview of subscription content, log in to check access.


  1. 1.

    Doplicher, S., Haag, R., and Roberts, J., Commun. Math. Phys. 23, 199 (1971).

  2. 2.

    Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics, Vol. 2, Springer-Verlag, Berlin, 1981.

  3. 3.

    Manuceau, J., Ann. Inst. Henri Poincaré 8, 139 (1968).

  4. 4.

    Rocca, F., Sirugue, M., and Testard, D., Commun. Math. Phys. 19, 119 (1970).

  5. 5.

    Dubin, D. A., Solvable Models in Algebraic Statistical Mechanics, Oxford Univ. Press, 1974.

  6. 6.

    Hein, S. and Roepstorff, G., Ann. Inst. Henri Poincaré 32, 21 (1980).

  7. 7.

    Bogoliubov, N. N., Lectures on Quantum Statistics, Vol. 2, New York, 1970.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hein, S. Bose condensation with negative chemical potential. Lett Math Phys 10, 263–268 (1985). https://doi.org/10.1007/BF00420565

Download citation


  • Statistical Physic
  • Equilibrium State
  • Rapid Growth
  • Group Theory
  • Standard Technique