Canonical quantisation and local measure of R ² gravity

This work gives a generalisation of the Ostrogradsky method of bringing the higher-derivative theory to Hamiltonian form suitable for use in gauge theories. The Hamiltonian formalism is established for gravity with Lagrangian L=-g( Lambda --2R+aRmu upsilon Rmu rho +bR2). The structure of the constraints of such a theory is investigated. Depending upon parameters Lambda , a and b, five essentially different cases of the theory are shown to be possible. The canonical quantisation is performed in each of five cases and the corresponding local measures in the path integral are derived, which all differ from the measure in Einstein gravity. The general expression for the local measure is obtained in the arbitrary bosonic theory coupling to gravity.