Dirac and Lagrangian reductions in the canonical approach to the first-order form of the Einstein–Hilbert action

It is shown that the Lagrangian reduction, in which solutions of equations of motion that do not involve time derivatives are used to eliminate variables, leads to results quite different from the standard Dirac treatment of the first-order form of the Einstein–Hilbert action when the equations of motion correspond to the first class constraints. A form of the first-order formulation of the Einstein–Hilbert action which is more suitable for the Dirac approach to constrained systems is presented. The Dirac and reduced approaches are compared and contrasted. This general discussion is illustrated by a simple model in which all constraints and the gauge transformations which correspond to first class constraints are completely worked out using both methods to demonstrate explicitly their differences. These results show an inconsistency in the previous treatment of the first-order Einstein–Hilbert action which is likely responsible for problems with its canonical quantization.