Generalized Quantization, its Representations and Relations to Field Theory

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Although quantization is one of the fundamental concepts in physics, its deep meaning is not yet very clear. Especially, it is known that the canonical quantization usually adopted is not applicable to a system constrained on a finite manifold. For this reason, about 50 years ago, modifying the canonical formalism Dirac proposed a new method of quantization for a constrained system. However it explicitly depend on the dynamics under consideration. On the other hand several years ago we argued another approach to this problem from somewhat different point of view. Applying it to a system constrained to move on SィイD1DィエD1 we have found emergence of a remarkable type of gauge potentials in quantizing the system. We have also shown that they can be obtained as a connection on the cosset space SO(D+1)/SO(D)〜SィイD1DィエD1, Using this technique we have examined the gauge structures in quantizing the respective systems on the chiral manifold SUィイD2LィエD2(n)×UィイD2RィエD2(n)/UィイD2VィエD2(n) and the Gras … More smann manifold SU(m+n)/SU(n)×SU(m). As a result it has been shown that gauge potentials in the former case become position-independent under suitable choice of the gauge while in the latter case they are given in integral forms. A review talk about our study on these problems was delivered at the International Workshop held in Varna in 1998, and published, in 1999, in the Proceedings of the Workshop. Through these investigations some limits of applicability of our quantization have also become clear. Extension of our quantization to a system on a manifold without geometric symmetry has been concluded to be impossible. Ii comes frome non-integrable structure of displacements connecting of two point on the manifold. In spite of this when combining our quantization technique to Dirac formalism we can detemine all possible irreducible representations of the Dirac algebra for a system constrained on a deformed manifold diffeomorphic to SィイD1DィエD1 under the Hamiltonian with potential interaction. The result has been reported in the International Conference held at Kiev in 1999. Furthermore generalization of our quantization scheme to filed theory has been found to be impossible without deserting the global structure of the manifold on which the field is constrained. A detail of this argument was delivered at the 6th Winger Symposium held at Istanbul in 1999. Less