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

TL;DR

This paper performs a Hamiltonian analysis of the $oxed{ ext{Brans-Dicke}}$ theory at $oxed{ ext{omega}=-3/2}$, revealing that the Jordan and Einstein frames are not equivalent under Hamiltonian canonical transformations.

Contribution

It provides a detailed Hamiltonian Dirac's constraint analysis for the $oxed{ ext{Brans-Dicke}}$ theory at $oxed{ ext{omega}=-3/2}$ and shows the non-equivalence of frames at the Hamiltonian level.

Findings

Jordan and Einstein frames have different constraint algebras.

Transformations between frames are not Hamiltonian canonical transformations.

Highlights the non-equivalence of frames in Hamiltonian formalism.

Abstract

We carefully perform a Hamiltonian Dirac's constraint analysis of $\omega=-\frac{3}{2}$ Brans-Dicke theory with Gibbons-Hawking-York (GHY) boundary term. The Poisson brackets are computed via functional derivatives. After a brief summary of the results for $\omega\neq-\frac{3}{2}$ case, we derive all Hamiltonian Dirac's constraints and constraint algebra both in the Jordan and Einstein frames. Confronting and contrasting Dirac's constraint algebra in both frames, it is shown that they are not equivalent. This highlights the transformations from the Jordan to the Einstein frames are not Hamiltonian canonical transformations.