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We investigate the state on the C*-algebra of Pauli spins on a onedimensional lattice (infinitely extended in both directions) which gives rise to the thermodynamic limit of the Gibbs ensemble in the two-dimensional Ising model (with nearest neighbour interaction). It is shown that the representation of the Pauli spin algebra associated with the state is factorial above and at the known critical temperature, while it has a two-dimensional center below the critical temperature. As a technical tool, we derive a general criterion for a state of the Pauli spin algebra corresponding to a Fock state of the Fermion algebra to be primary. We also show that restrictions of two quasifree states of the Fermion algebra to its even part are equivalent if and only if the projection operators E ί and E 2 (on the direct sum of two copies of the basic Hubert space) satisfy the following two conditions: (1) E l -E 2 is in the Hubert-Schmidt class, (2) E± Λ (1 -E 2 ) has an even dimension, where the even-oddness of dim £ 1 Λ (1 -E 2

The generating functional of the cyclic representation of the canonical commutation relations for the equilibrium state of the free Boson gas is calculated, using a method due to Kac, as the thermodynamic limit of the grand canonical generating functional. The relation to the work of Araki and Woods is discussed.

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An earlier exact result (Phys. Rev. Lerf. 55 (1985). 2273) for the free energy of an oscillatordipole interacting with the radiation field is obtained using the fluctuation-dissipation theorem. A key feature of the earlier calculation, a remarkable formula for the free energy of the oscillator, is obtained in the form of a corresponding formula for the oscillator energy. This confirms, by a longer, more conventional proof, the earlier result. An advantage of this present method is that separate contributions to the energy can be isolated and discussed. Explicit, closed-form expressions are given and the high-temperature limit is discussed.

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