POSITIVE REPRESENTATIONS OF GENERAL COMMUTATION RELATIONS ALLOWING WICK ORDERING

We consider the problem of representing in Hilbert space commutation r elations of the form [GRAPHICS] where the T-ij(Kl) are essentially arb itrary scalar coefficients. Examples comprise the q-canonical commutat ion relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti-)commutation relations studied by Pusz and Wor onowicz, as well as the quantum group SnuU(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 ar e to the left of all unstarred ones. In this general framework we defi ne the Fock representation, as well as coherent representations. We de velop criteria for the natural scalar product in the associated repres entation spaces to be positive definite and for the relations to have representations by bounded operators in a Hilbert space. We characteri ze the relations between the generators a(i) (not involving a(i)) whi ch are compatible with the basic relations. The relations may also be interpreted as defining a non-commutative differential calculus. For g eneric coefficients T-ij(kl), however, all differential forms of degre e 2 and higher vanish. We exhibit conditions for this not to be the ca se and relate them to the ideal structure of the Wick algebra and cond itions of positivity. We show that the differential calculus is compat ible with the involution iff the coefficients T define a representatio n of the braid group. This condition is also shown to imply improved b ounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by a(j) --> exp(it)a(j). (C) 1995 Academic Press, Inc