SYMMETRY AND QUANTIZATION - HIGHER-ORDER POLARIZATION AND ANOMALIES

The concept of (full) polarization subalgebra in a Group Approach to Quantization on a Lie group G as a generalization of the analogous concept in geometric or standard quantization is discussed. The lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition. A generalization of the subalgebra of (full) polarization is then provided, made out of higher-order differential operators in the enveloping algebra of G. Higher-order polarizations can also be used to quantize nonanomalous theories in different "representations." Numerous examples are analyzed, including the finite-dimensional dynamics associated with the Schrodinger group, which presents an anomaly, and an infinite-dimensional anomalous system associated with the Virasoro group. In the last example, the operators in the higher-order polarization are in one-to-one correspondence with the null vectors in the Verma module approach.