The Relation Between KMS-states for Different Temperatures

Given a thermal field theory for some temperature $\beta^{-1}$, we construct the theory at an arbitrary temperature $1 / \beta'$. Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region $\O_\circ \subset \rr^4$, but coincide with the given KMS state in the space-like complement of a slightly larger region $\hat{\O}$. By a weak*-compactness argument there always exists a convergent subnet of states as the size of $\O_\circ$ and $\hat{\O}$ tends towards $\rr^4$. Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between the boundaries of $\O_\circ$ and $\hat{\O}$. We show that this surface energy can be controlled by a generalized cluster condition.