MG14 - Talk detail

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A quantum field theory in its algebraic description (AQFT) typically admits many irregular states. To select the physically reasonable states one may seek a criterion in the so-called nuclearity condition of Buchholz and Wichmann, which express the idea that the phase space volume should not grow too rapidly with increasing energy. For free fields, however, one usually selects the class of Hadamard states, which are characterised by having a smooth expectation value for the renormalised stress tensor. In this talk we consider free scalar fields and we discuss wether the Hadamard condition is necessary and/or sufficient for a certain nuclearity condition. Our desire to pose this question for general Hadamard states in globally hyperbolic spacetimes prevents us from using a Hamilton operator to measure the energy. Instead we use a modular operator, associated to the state and the algebra of a bounded diamond region. Such modular nuclearity conditions have recently played an important role in constructive two-dimensional algebraic QFT’s, and they are known to be satisfied by the Minkowski vacuum on wedge shaped regions. We present some precise results at the one-particle level, which show that all Hadamard states satisfy the modular nuclearity condition. The converse, however, is false.

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