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 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 ≥ 4 the condition of geometric modular action (CGMA) of Buchholz, Dreyer, Florig and Summers [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.

On the PCT-theorem in the theory of local observables

“…In fact, it is easy to check that wedge duality certainly holds, if M (p) −1 exists for all for all p ∈ H + m and has polynomial matrix elements. A sufficient, and by Lemma V.2 in [25] also necessary, condition for this is that det M (p) is constant on the mass shell. PCT symmetry, on the other hand, requires that M (p) can be written as L(p) * M ′ (p)L(p) with M ′ (p) satisfying (4.10) and L(p) polynomial in p. Although an explicit example satisfying wedge duality without the latter property is not known to us there is hardly a doubt that such examples exist.…”

“…But the theorem does not imply that double cones are mapped onto double cones. In fact, as known from the examples of Yngvason [47] (see also [25]) the action of the modular group of a wedge does not need to be local in the perpendicular direction.…”

“…Locality, spectrum condition and positivity imply (by the Jost-Lehmann Dyson representation [7]) that its Fourier transform can be written According to the results of [25] wedge duality is always violated unless M is constant on the mass shell, i.e., unless Φ is the usual Poincaré covariant, scalar free field. In particular, neither the condition of geometric modular action nor the Bisognano Wichmann property hold for fields with non-constant M .…”

Quantum Field Theory as Dynamical System

“…4.3 to higher dimensions has not been achieved. There is a result due to Gaier and Yngvason [GY00] showing that such generalization might be possible. For generalized free fields they got the following result:…”

The Bisognano–Wichmann Property on Nets of Standard Subspaces, Some Sufficient Conditions

“…Known counterexamples to modular covariance seem very artificial as they imply a breakdown of Poincaré covariance, see [34,19]. In Sect.…”