### Abstract

We derive a classical integral representation for the partition function, Z^{Q}, of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentum J → ∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality is Z^{C}(J)≦Z^{Q}(J)≦Z^{C}(J+1).

Original language | English (US) |
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Pages (from-to) | 327-340 |

Number of pages | 14 |

Journal | Communications in Mathematical Physics |

Volume | 31 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1973 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

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## Cite this

Lieb, E. (1973). The classical limit of quantum spin systems.

*Communications in Mathematical Physics*,*31*(4), 327-340. https://doi.org/10.1007/BF01646493