Quantization of Systems with Constraints

Abstract

The quantization of systems with constraints is of considerable importance in a variety of applications. Let $\left\{ {pj,{q^j}} \right\},1 \leqslant j \leqslant J$ denote a set of dynamical variables, $\left\{ {{\lambda ^a}} \right\},1 \leqslant a \leqslant A \leqslant 2J$, a set of Lagrange multipliers, and $\left\{ {{\phi _a}\left( {p,q} \right)} \right\}$ a set of constraints. Then the dynamics of a constrained system may be summarized in the form of an action principle by means of the classical action (summation implied)

$$I = \int {\left[ {pj{{\dot q}^j} - H\left( {p,q} \right) - {\lambda ^a}{\phi _a}\left( {p,q} \right)} \right]} dt. $$

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