# Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime

March 20, 2001

In order to have well defined rules for the perturbative calculation of
quantities of interest in an interacting quantum field theory in curved
spacetime, it is necessary to construct Wick polynomials and their time ordered
products for the noninteracting theory. A construction of these quantities has
recently been given by Brunetti, Fredenhagen, and Kohler, and by Brunetti and
Fredenhagen, but they did not impose any ``locality'' or ``covariance''
condition in their constructions. As a consequence, their construction of time
ordered products contained ambiguities involving arbitrary functions of
spacetime point rather than arbitrary parameters. In this paper, we construct
an ``extended Wick polynomial algebra''-large enough to contain the Wick
polynomials and their time ordered products. We then define the notion of a
{\it local, covariant quantum field}, and seek a definition of {\it local} Wick
polynomials and their time ordered products as local, covariant quantum fields.
We impose scaling requirements on our local Wick polynomials and their time
ordered products as well as certain additional requirements-such as commutation
relations with the free field and appropriate continuity properties under
variations of the spacetime metric. For a given polynomial order in powers of
the field, we prove that these conditions uniquely determine the local Wick
polynomials and their time ordered products up to a finite number of
parameters. (These parameters correspond to the usual renormalization
ambiguities occurring in Minkowski spacetime together with additional
parameters corresponding to the coupling of the field to curvature.) We also
prove existence of local Wick polynomials. However, the issue of existence of
local time ordered products is deferred to a future investigation

Keywords:

*none*