Operator ordering and symbol representations of enveloping algebras

Phase space methods in quantum theory make use of symbol representations of the Weyl algebra. The method can be generalized to enveloping algebras U(L) of Lie algebras L different from the Heisenberg Lie algebra. The method starts from a one-to-one correspondence between the linear space of the enveloping algebra U(L) and that of the symmetric algebra S(L) over L. The algebra S(L) can be equipped with a so-called twisted product so that it becomes isomorphic to U(L). Different ordering rules correspond to different bases in U(L). The elements of U(L) are represented by different symbols in dependence on the ordering rule chosen. We generalize the notion of an ordering defining function φ. This helps to determine a transformation between different types of symbols and allows to use the same fast multiplication algorithm for different orderings. 14 refs