Lorentzian 2D CFT from the pAQFT Perspective

Abstract

We provide a detailed construction of the quantum theory of the massless scalar field on two-dimensional, globally hyperbolic (in particular, Lorentzian) manifolds using the framework of perturbative algebraic quantum field theory. From this we obtain subalgebras of observables isomorphic to the Heisenberg and Virasoro algebras on the Einstein cylinder. We also show how the conformal version of general covariance, as first introduced by Pinamonti as an extension of the construction due to Brunetti, Fredenhagen and Verch, may be applied to natural Lagrangians, which allow one to specify a theory consistently across multiple spacetimes, in order to obtain a simple condition for the conformal covariance of classical dynamics, which is then shown to quantise in the case of a quadratic Lagrangian. We then compare the covariance condition for the stress-energy tensor in the classical and quantum theory in order to obtain a transformation law involving the Schwarzian derivative of the new coordinate, in accordance with a well-known result in the Euclidean literature.

The Unruh–DeWitt model and its joint interacting Hilbert space

“…Second, we need to assume that there exists no zero mode [77]. In the context of massless scalar field in a twodimensional Einstein cylinder, it is known that there is a zero mode [105,106] that admits no Fock representation, which leads to the ambiguity of the ground state or KMS state [107].…”