Abstract
A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g., Buchholz-Wichmann nuclearity). We propose instead to use a modular ℓp -condition, which is an extension of a strengthened modular nuclearity condition to generally covariant theories. The modular nuclearity condition was previously introduced in Minkowski space, where it played an important role in constructive two dimensional algebraic QFT’s. We show that our generally covariant extension of this condition makes sense for a vast range of theories, and that it behaves well under causal propagation and taking mixtures. In addition we show that our modular ℓp -condition holds for every quasi-free Hadamard state of a free scalar quantum field (regardless of mass or scalar curvature coupling). However, our condition is not equivalent to the Hadamard condition.
Highlights
The observables and the states of a system are the two basic ingredients in any physical theory.In quantum field theory, the observables can conveniently be described as elements of a ∗ -algebra, and encode fundamental features such as causality into their algebraic relations.The states, essential to make contact with empirical results, are taken to be expectation value functionals on this algebra of observables.To ensure a consistent probabilistic interpretation, states are required to satisfy the basic algebraic requirements of linearity, positivity, and normalization
We have shown that all quasi-free Hadamard states of a free scalar field satisfy the modular and the same is true for convex combinations of such states, by Proposition 2.10
The main conclusion that one may draw from our investigation is that the modularp -condition, which we introduced as an extension of the modular nuclearity condition known from Minkowski space, is an interesting additional tool in the study of generally covariant quantum field theories
Summary
The observables and the states of a system are the two basic ingredients in any physical theory. There exists a local version of it, which instead of the Hamiltonian rather uses modular operators arising from applying Tomita-Takesaki theory [18] to the local observable algebras This “modular nuclearity condition” [15] can be formulated as follows: Consider an inclusion Õ ⊂ O ⊂ M of bounded open regions in Minkowski spacetime M, and a state ω on a quantum field theory on M. The behavior under spacetime deformations has the nice effect that to verify the modular p -condition for a suitable class of states in a generally covariant theory, it suffices to consider simple spacetimes such as ultra-static ones For such spacetimes, a strong energy nuclearity condition for the theory of a free massive Klein-Gordon field in the GNS representation of its canonical vacuum state was already proven by Verch [27]. A discussion of our results in Section 6 concludes the article
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