  We provide a detailed construction of the quantum theory of the massless
scalar field on 2-dimensional, globally-hyperbolic (in particular, Lorentzian)
manifolds using the framework of perturbative algebraic quantum field theory.
From this we 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
the concept of natural Lagrangians 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 Minkowksi
space, obtaining a transformation law dependent on the Schwarzian derivative of
the transformed coordinate, in accordance with a well-known result in the
Euclidean literature.
