  We define and study a set of operators that compute statistical properties of
the Brownian Loop Soup, a conformally invariant gas of random Brownian loops
(Brownian paths constrained to begin and end at the same point) in two
dimensions. We prove that the correlation functions of these operators have
many of the properties of conformal primaries in a conformal field theory, and
compute their conformal dimension. The dimensions are real and positive, but
have the novel feature that they vary continuously as a periodic function of a
real parameter. We comment on the relation of the Brownian Loop Soup to the
free field, and use this relation to establish that the central charge of the
Loop Soup is twice its intensity.
