  We consider the problem of representing in Hilbert space commutation
relations of the form $$ a_ia_j^*=\delta_{ij}{\bold1} +
\sum_{k\ell}T_{ij}^{k\ell} a_\ell^*a_k \quad,$$ where the $T_{ij}^{k\ell}$ are
essentially arbitrary scalar coefficients. Examples comprise the $q$-canonical
commutation relations introduced by Greenberg, Bozejko, and Speicher, and the
twisted canonical (anti-)commutation relations studied by Pusz and Woronowicz,
as well as the quantum group S$_\nu$U$(2)$. Using these relations, any
polynomial in the generators $a_i$ and their adjoints can uniquely be written
in ``Wick ordered form'' in which all starred generators are to the left of all
unstarred ones. In this general framework we define the Fock representation, as
well as coherent representations. We develop criteria for the natural scalar
product in the associated representation spaces to be positive definite, and
for the relations to have representations by bounded operators in a Hilbert
space. We characterize the relations between the generators $a_i$ (not
involving $a_i^*$) which are compatible with the basic relations. The relations
may also be interpreted as defining a non-commutative differential calculus.
For generic coefficients $T_{ij}^{k\ell}$, however, all differential forms of
degree 2 and higher vanish. We exhibit conditions for this not to be the case,
and relate them to the ideal structure of the Wick algebra, and conditions of
positivity. We show that the differential calculus is compatible with the
involution iff the coefficients $T$ define a representation of the braid group.
This condition is also shown to imply improved bounds for the positivity of the
Fock representation. Finally, we study the KMS states of the group of gauge
transformations defined by $a_j\mapsto\exp(it)a_j$.
