Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories

Abstract

Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel's strict deformations of C * -dynamical systems with automorphic actions of R n , whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita-Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations.

Deformations of Quantum Field Theories and Integrable Models

“…This is the approach taken in the deformation theory of algebras in the mathematics literature [Ger64], which has already led to deformations of quantum field theories in certain examples [GL08,BS08,BLS10].…”

“…But the locality condition (2.21) is only valid if ω annihilates L , the translations U ω (x, 1) satisfy a spectrum condition, and the parameters defining ρ are suitably chosen [GL08,BLS10]. Hence the validity of (2.21) is not a property of ρ alone, but also involves properties of ω going beyond compatibility and vanishing on L .…”

“…To motivate the following steps, it is instructive to recall the known results about the special case R(a) = e ia first. In this context, (3.32) is known to hold for a translationally invariant state ω annihilating L if ω satisfies also the spectrum condition and Q is admissible in the sense that QV + ⊂ W 0 [BLS10]. This interplay of locality and spectral properties can be understood as follows.…”

“…In particular, P R (W ′ 0 ) is generated by all φ R,−Q (f ), f ∈ S (W ′ 0 ). Thus the orbit Q := {ΛQΛ −1 : Λ ∈ L + } provides a coordinatization for the different directions of the wedges [GL07,BLS10], whereas the deformation function R labels the kind of deformation used.…”

“…After its first appearance in the context of deformed free field theories on noncommutative Minkowski space [GL07], it was generalized to an operator-algebraic setting in [BS08], where it is known as warped convolution. In the framework of Wightman field theories, this deformation manifests itself as a deformation of the tensor product of the testfunction algebra [GL08], and later on, the connection to Rieffel's strict deformation quantization [Rie92] was explored [BLS10]. By now, the warped convolution technique has also successfully been applied to the deformation of conformal field theories [DT10] and quantum field theories on curved spacetimes [DLM11].…”