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

TL;DR

This paper demonstrates that for generalized free quantum fields in higher dimensions, geometric modular action, wedge duality, and Lorentz covariance are mathematically equivalent conditions, with some distinctions in lower dimensions and massless cases.

Contribution

It establishes the equivalence of geometric modular action, wedge duality, and Lorentz covariance for generalized free fields in various space-time dimensions, extending previous results.

Findings

For d≥4, geometric modular action, wedge duality, and Lorentz covariance are equivalent.

In d=3 with a mass gap, the same equivalence holds.

For massless fields in d=3 and all cases in d=2, CGMA implies covariance of the net, not the field itself.

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.