Quantized Gravitational Field

Abstract

A gravitational action operator is constructed that is invariant under general coordinate transformations and local Lorentz (gauge) transformations. To interpret the formalism the arbitrariness in description must be restricted by introducing gauge conditions and coordinate conditions. The time gauge is defined by locking the time axes of the local coordinate systems to the general coordinate time axis. The resulting form of the action operator, including the contribution of a spinless matter field, enables canonical pairs of variables to be identified. There are four field variables that lack canonical partners, in virtue of differential constraint equations, which can be interpreted as space-time coordinate displacements. In a physically distinguished class of coordinate system the gravitational field variables are not explicit functions of the coordinate displacement parameters. There remains the freedom of Lorentz transformation. The generators of spatial translations and rotations have the correct commutation properties. The question of Lorentz invariance is left undecided since the energy density operator is only given implicitly.