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A canonical quantization scheme applied to a classical supersymmetric systemwith quadratic in momentum supercharges gives rise to a quantum anomaly problemdescribed by a specific term to be quadratic in Planck constant. We reveal aclose relationship between the anomaly and the Schwarzian derivative, andspecify a quantization prescription which generates the anomaly-freesupersymmetric quantum system with second order supercharges. We also discussthe phenomenon of a coupling-constant metarmorphosis that associates quantumsystems with first-order supersymmetry to the systems with the second-ordersupercharges.

Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis