Trace identities and their semiclassical implications

The compatibility of the semiclassical quantization of area-preserving maps with some exact identities which follow from the unitarity of the quantum evolution operator is discussed. The quantum identities involve relations b etween traces of powers of the evolution operator. For classically integrab le maps, the semiclassical approximation is shown to be compatible with the trace identities. This is done by the identification of stationary phase m anifolds which give the main contributions to the result. The compatibility of the semiclassical quantization with the trace identities demonstrates t he crucial importance of non-diagonal contributions. The same technique is not applicable for chaotic maps, and the compatibility of the semiclassical theory in this case remains unsettled. However, the trace identities are a pplied to maps which appear naturally in the theory of quantum graphs, reve aling some features of the periodic orbit theory for these paradigms of qua ntum chaos.