How to Solve the Covariant Operator Formalism of Gauge Theories and Quantum Gravity in the Heisenberg Picture. IV: Cauchy Problem Involving Noncommutative Quantities

Abstract

A new mathematical problem-the Cauchy problem involving noncommutative quantities-is encountered in solving higher-order operator equations of nonabelian gauge theories and quantum gravity. In order to make this problem analyzable, a new mathematical technique is introduced, and its effectiveness is demonstrated by showing that the Heaviside-Mikusinski operational calculus is extended to the case involving noncommutative quantities. In a simple nonlinear field-theoretical model, it is explicitly shown how systematically its operator solution is obtained by this technique.