Mathematics Colloquium

In a first (perhaps second) course on quantum mechanics one learns to quantize a classical mechanical system in the operator formalism via "canonical quantization." However, when dealing with classical systems with non-flat configuration spaces, canonical quantization may be ambiguous due to problems with "operator orderings." On the other hand at first blush, Feynman's path integral interpretation of quantum mechanics does not seem to suffer from these ambiguities. However, there is no free lunch and the same ambiguities reappear in the Feynman picture when one actually tries to precisely define these path integral expressions. This talk will describe some attempts to mathematically interpret Feynman's picture for quantum mechanical systems in geometric settings. Choices will have to be made and these choices lead to different quantizations of the same classical system. (All terms in this abstract will be explained at least at the level needed for this talk.)