The Relation between KMS-States for Different Temperatures

Abstract. Given a thermal field theory for a certain temperature, we present a method to construct the theory at any finite positive temperature provided the number of local degrees of freedom is restricted in a physically sensible manner. Our work is based on a construction invented by Buchholz and Junglas [BJu 89]. Starting from the vacuum representation, they established the existence of thermal equilibrium states (KMS-states) for a large class of quantum field theories. The KMS-states were constructed as limit points of nets of states which represent strictly localized excitations of the vacuum. We adjust their method to the general structure of thermal quantum field theory. In a first step we construct states which can not be distinguished from KMS-states for the new temperature by measurements in a local region O◦⊂IR 4 but coincide with the given KMS-state in the space-like complement of a slightly larger region Ô. By a weak ∗ -compactness argument there always exsits a convergent subsequence of states as we increase the size of O◦ and Ô. Whether or not the limit state of such a subsequence is a global KMS-states for the new temperature, depends on the surface energy contained in the layer in between the boundaries of O (i) ◦ and Ô as i→∞. Introducing an auxiliary structure and applying a generalized cluster theorem, we control the relevant surface effects in all thermal theories which satisfy a certain “nuclearity condition”.