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Geometric Modular Action, Wedge Duality and Lorentz Covariance are Equivalent for Generalized Free Fields

Johanna Gaier·Jakob Yngvason
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Abstract

The Tomita-Takesaki modular groups and conjugations for the observable algebras of space-like 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 $d\geq 4$ the condition of geometric modular action (CGMA) of Buchholz, Dreyer, Florig and Summers \cite{BDFS}, 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.

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Johanna Gaier, Jakob Yngvason. 1999-10-20. Geometric Modular Action, Wedge Duality and Lorentz Covariance are Equivalent for Generalized Free Fields. https://doi.org/10.1063/1.1286174

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