Quantum symmetries from quantum phases, fermions from bosons, a Z2 anomaly and Galilean invariance

In the presence of a topologically nontrivial Wess-Zumino term, the wave functions are not functions on the configuration space Q of the system; instead they are functions on a U(1) bundle Q over this configuration space. The space Q is obtained by associating circles to points of Q in a suitable way. If a group G operates on Q and is a symmetry of the action, then it can happen that G does not operate on Q and on wave functions. Instead, the 'quantum mechanical' symmetry group GQM which does act on Q, and thereby wave functions, is in general a central extension of G; it may not even contain G as a subgroup. In this paper, we give an explicit and elementary construction of GQM. The construction is applied to the charge-monopole system and the Skyrme solitons, and used to explain why the states of these systems may be spinorial in nature even though the action contains only tensorial (integral spin) variables. The well known Z2 anomaly of certain SU(2) gauge theories is also established using a modification of this construction. When a classical system with Galilei invariance is quantized, the quantum theoretic Galilei group GQM is different from the classical Galilei group G. Again the former is a central extension of the latter even though there is no Wess-Zumino term in this case. We present a novel explanation of this fact as well by constructing the lift of G from (galilean) spacetime Q to the appropriate U(1) bundle Q. (orig.)