# The algebra of Wick polynomials of a scalar field on a Riemannian manifold

March 04, 2019

On a connected, oriented, smooth Riemannian manifold without boundary we
consider a real scalar field whose dynamics is ruled by $E$, a second order
elliptic partial differential operator of metric type. Using the functional
formalism and working within the framework of algebraic quantum field theory
and of the principle of general local covariance, first we construct the
algebra of locally covariant observables in terms of equivariant sections of a
bundle of smooth, regular polynomial functionals over the affine space of the
parametrices associated to $E$. Subsequently, adapting to the case in hand a
strategy first introduced by Hollands and Wald in a Lorentzian setting, we
prove the existence of Wick powers of the underlying field, extending the
procedure to smooth, local and polynomial functionals and discussing in the
process the regularization ambiguities of such procedure. Subsequently we endow
the space of Wick powers with an algebra structure, dubbed E-product, which
plays in a Riemannian setting the same r\^ole of the time ordered product for
field theories on globally hyperbolic spacetimes. In particular we prove the
existence of the E-product and we discuss both its properties and the
renormalization ambiguities in the underlying procedure. As last step we extend
the whole analysis to observables admitting derivatives of the field
configurations and we discuss the quantum M\o ller operator which is used to
investigate interacting models at a perturbative level.

Keywords:

wick polynomials, Euclidean quantum field theories