| Research Abstract |
Quantization problem on a closed manifold have been studied so far on the basis of Dirac's method, which was formulated long time ago by generalizing the usual method of canonical quantization Recently, however, more fundamental approach to this problem was independently achieved by Landsman-Linden and by Ohnuki-Kitakado. As a result of these investigations, it was revealed that the theory derived through this quantization can have a so remarkable property that it is automatically equipped with a certain type of gauge potential. Applying the induced representation technique of group theory to perform this quantization we succeeded in constructing the explicit form of the gauge potential on S^D (D=1,2, -). Based on this result we have made clear those mathematical properties of the gauge potentials, which may be stated as follows :
1. The gauge potential emerging in quantization on S^D (D=1,2, -) satisfies the Yang-Mills equation on this sphere. 2. For D=1 and D=2n (n=1,2, -) the gauge p
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otentials become topologically non-trivial. They are shown to be the same as the Aharonov-Bohm gauge potential for D=1, the magnetic monopole gauge potential for D=2, the instanton solution for D=4, and Fujii's generalized instanton configuration for D=6,8,10, -. Especially, for D=4p (p=1,2, -) we can define a duality (or anti-duality) relation among the gauge fields. 3. On the other hand, for D=2n+1 (n=1,2, -) the gauge potentials are found to be all topologically trivial.
It is interesting to note that the topologically non-trivial solutions to the Yang-Mills equation on S^D, which have been known already, are completely exhausted by our gauge potentials obtained by an algebraic method. The results were presented in several international conferences and people working on basic problems of quantum mechanics seemed to have much interest in our approach. Related to this it would be quite important from physical and mathematical view to examine in what extent the above properties of gauge potentials hold true. Very recently we have derived out all possible forms of the gauge potential induced in quantizing a particle moving on the Grassmann manifold U (n+m) /U (n) *U (m). A study of its mathematical properties is in progress. The details will be published in near future together with related topics. Less
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