Reviews in Mathematical Physics

The existence of states enjoying a weak form of the Reeh–Schlieder property has been recently established on curved backgrounds and in the framework of locally covariant quantum field theory. Since only the example of a real scalar field has been discussed, we extend the analysis to the case of massive and massless free fields either of spin-½ or of spin-1. In the process, it is also shown that both the vector potential and the Proca field can be described as a locally covariant quantum field theory.

We consider (M, d) a connected and compact manifold and we denote by the Bernoulli space M^ℤ. The analogous problem on the half-line ℕ is also considered. Let be an observable. Given a temperature T, we analyze the main properties of the Gibbs state .

In order to do our analysis, we consider the Ruelle operator associated to , and we get in this procedure the main eigenfunction . Later, we analyze selection problems when the temperature goes to zero: (a) existence, or not, of the limit , a question about selection of subactions, and, (b) existence, or not, of the limit , a question about selection of measures.

The existence of subactions and other properties of Ergodic Optimization are also considered.

The case where the potential depends just on the coordinates (x₀, x₁) is carefully analyzed. We show, in this case, and under suitable hypotheses, a Large Deviation Principle, when T → 0, graph properties, etc. Finally, we will present in detail a result due to van Enter and Ruszel, where the authors show, for a particular example of potential A, that the selection of measure in this case, does not happen.

Short distance scaling limits of a class of integrable models on two-dimensional Minkowski space are considered in the algebraic framework of quantum field theory. Making use of the wedge-local quantum fields generating these models, it is shown that massless scaling limit theories exist, and decompose into (twisted) tensor products of chiral, translation-dilation covariant field theories. On the subspace which is generated from the vacuum by the observables localized in finite light ray intervals, this symmetry can be extended to the Möbius group. The structure of the interval-localized algebras in the chiral models is discussed in two explicit examples.