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The compatibility of the semiclassical quantization of area-preserving mapswith some exact identities which follow from the unitarity of the quantumevolution operator is discussed. The quantum identities involve relationsbetween traces of powers of the evolution operator. For classically {\itintegrable} maps, the semiclassical approximation is shown to be compatiblewith the trace identities. This is done by the identification of stationaryphase manifolds which give the main contributions to the result. The sametechnique is not applicable for {\it chaotic} maps, and the compatibility ofthe semiclassical theory in this case remains unsettled. The compatibility ofthe semiclassical quantization with the trace identities demonstrates thecrucial importance of non-diagonal contributions.Comment: LaTeX - IOP styl

Trace identities and their semiclassical implications