Spherically symmetric Jordan-Brans-Dicke quantum gravity with de Broglie Bohm pilot wave perspective

We obtain two dimensional analogue of the Jordan-Brans-Dicke (JBD) gravity action described in four dimensional spherically symmetric curved space time metric. There will be two scalar fields, namely, the Brans Dicke (BD) $\phi$ and scale factor of 2-sphere part of the space time $\psi.$ We obtained suitable duality transformations between $(\psi,\phi)$ and $(\rho,S)$ where $\rho$ and $S$ is respectively amplitude and phase part of the corresponding de Broglie pilot wave function $\Psi(\rho,S)=\sqrt{\rho}e^{iS}.$ Covariant conservation of mass-energy current density of particles ensemble $J_a=\rho\partial_aS,$ is established by applying a particular dynamical conformal frame described by $(\rho,S).$