Quantization of Higher-Order Constrained Lagrangian Systems Using the WKB Approximation

A general theory is given for solving the Hamilton–Jacobi partial differential equations (HJPDEs) for both constrained and unconstrained systems with arbitrarily higher-order Lagrangians. The Hamilton–Jacobi function is obtained for both types of systems by solving the appropriate set of HJPDEs. This is used to determine the solutions of the equations of motion. The quantization of both systems is then achieved using the WKB approximation. In constrained systems, the constraints become conditions on the wave function to be satisfied in the semiclassical limit.