Two-dimensional models as testing ground for principles and concepts of local quantum physics

Abstract

In the past two-dimensional models of QFT have served as theoretical laboratories for testing new concepts under mathematically controllable condition. In more recent times low-dimensional models (e.g., chiral models, factorizing models) often have been treated by special recipes in a way which sometimes led to a loss of unity of QFT. In the present work, I try to counteract this apartheid tendency by reviewing past results within the setting of the general principles of QFT. To this I add two new ideas: (1) a modular interpretation of the chiral model Diff(S)-covariance with a close connection to the recently formulated local covariance principle for QFT in curved spacetime and (2) a derivation of the chiral model temperature duality from a suitable operator formulation of the angular Wick rotation (in analogy to the Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational chiral theories. The SL (2, Z) modular Verlinde relation is a special case of this thermal duality and (within the family of rational models) the matrix S appearing in the thermal duality relation becomes identified with the statistics character matrix S. The relevant angular ''Euclideanization'' is done in the setting of the Tomita-Takesaki modular formalism of operator algebras. I find it appropriate to dedicate this work to the memory of J.A. Swieca with whom I shared the interest in two-dimensional models as a testing ground for QFT for more than one decade. This is a significantly extended version of an ''Encyclopedia of Mathematical Physics'' contribution hep-th/0502125.

Addendum to ‘Area density of localization entropy: I. The case of wedge localization’

“…Whereas the commutant in the heat-bath thermal situation remains an abstract thermal shadow world, the commutant in the localized vacuum situation is geometric and in typical cases equal to the algebra of the causal disjoint (Haag duality). The only case in which the modular group acts geometrically (independent of the particular model of QFT) is the wedge situation (A(W ), Ω res ); there are however many "partially geometric" situations in which the reference state is different from the vacuum and the corresponding modular group acts as a diffeomorphism if restricted to the subalgebra [23].…”

“…For the multicomponent abelian current models, for which the n-point functions functions can be expressed in terms of the Jakobi Θ-functions, these modular identities follow from known properties of Θ-functions [35] [23]. The general (structural) derivation is conceptually quite demanding; it can be found in [23] [34].…”

“…Rewriting everything in terms of the ε-parametrization one obtains the result of the introduction (4). The best way to understand the temperature duality relation in an operator setting is to view the partition function as the zero-point correlation function in a unnormalized thermal state and to use modular theory in order to perform an angular Euclideanization [23]. The crucial formula is the identity…”

Pascual Jordan’s legacy and the ongoing research in quantum field theory⋆

“…Hence although the Jordan model, different from the intentions of its protagonist, has no bearing on a neutrino theory of light (for whose validity there is not the slightest hint within the Standard Model, which is our presently best particle theory), it is believed to serve as a useful analogy for the great unsolved problem of quantum chromodynamics which is the problem of quark confinement. A recent description of the Jordan model and its appearance in the massless limit Schwinger model can be found in [39].…”

Nuclearity and Thermal States in Conformal Field Theory

“…We now prove a formula pointed out by Schroer and Wiesbrock [26]. We give here below a proof in the case U is a representation of SL(2, R), a case that covers most needs in this paper.…”