Geometric modular action, wedge duality, and Lorentz covariance are equivalent for generalized free fields

Abstract

The Tomita–Takesaki modular groups and conjugations for the observable algebras of spacelike wedges and the vacuum state are computed for translationally covariant, but possibly not Lorentz covariant, generalized free quantum fields in arbitrary space–time dimension d. It is shown that for d⩾4 the condition of geometric modular action (CGMA) of Buchholz, Dreyer, Florig, and Summers [Rev. Math. Phys. (2000)], Lorentz covariance and wedge duality are all equivalent in these models. The same holds for d=3 if there is a mass gap. For massless fields in d=3, and for d=2 and arbitrary mass, CGMA does not imply Lorentz covariance of the field itself, but only of the maximal local net generated by the field.