Canonical quantization of $D$-dimensional $R^2$ gravity

The canonical formalism of $R^2$ gravity in spacetime of arbitrary dimension $D>4$ is constructed. It is shown that there exist five qualitatively different forms of this theory, in each of which the constraints in the phase space are found. Canonical quantization in a unitary gauge is performed, and the generating functional of the Green's functions is represented as a functional integral over the original fields with local measure. It is shown that in four forms of the theory the local measure has a nonstandard form; moreover, the structure of the local measure in one of these forms has not hitherto been encountered in quantum field theory models.