Jordan and Einstein frames from the perspective of $\omega =-3/2$ Hamiltonian Brans-Dicke theory

We carefully perform a Hamiltonian Dirac’s constraint analysis of the $\omega =-\frac{3}{2}$ Brans-Dicke theory with the Gibbons-Hawking-York boundary term. The Poisson brackets are computed via functional derivatives. After a brief summary of the results for the $\omega \ne -\frac{3}{2}$ case [G. Gionti S. J., Canonical analysis of Brans-Dicke theory addresses Hamiltonian inequivalence between the Jordan and Einstein frames, Phys. Rev. D 103, 024022 (2021)] we derive all Hamiltonian Dirac’s constraints and constraint algebra in both the Jordan and the Einstein frames. Confronting and contrasting Dirac’s constraint algebra in both frames, it is shown that they are not equivalent. This highlights that the transformations from the Jordan to the Einstein frames are not Hamiltonian canonical transformations.