De Donder condition and the Poincare group in quantized general relativity

The De Donder coordinate condition for an unquantized metrical field is discussed first. While the coordinate transformations allowed by this condition do not form a group in the ordinary sense, it is possible in an infinite number of ways to make these transformations elements of a group of metric-dependent coordinate transformations. In the unquantized theory, the question what coordinate transformations the De Donder condition will allow remains not generally answered, but it is not of primary importance, because the imposition of the De Donder condition in the unquantized theory is at best a question of convenience. The question is more important in the quantized ''altered theory'' of gravity, in which the De Donder condition serves for selecting the physical part of a theory that without this condition would also describe unphysical states. There are as many ''altered'' quantum theories of gravity, as there are affine orbits in the space of coordinate systems. Each orbit has its own altered theory, and in each separate altered theory with its own De Donder condition the only coordinate transformations that make sense are the affine ones, or, with proper boundary conditions imposed, Poincare transformations. The coordinate transformation between two frames in which the De Donder condition is valid, either weakly or strongly, must be an affine (or Poincare) transformation, if the coordinates are required to be c numbers