Quantum of action in entangled relativity

In this article, we demonstrate that the novel general theory of relativity named `Entangled Relativity' is more economical than General Relativity in terms of universal dimensionful constants and units when both theories are considered through a path integral formulation. The sole parameter of Entangled Relativity is a quantum of energy squared. However, in order to recover standard Quantum Field Theory when gravity is neglected in the path integral, we show that this quantum of energy corresponds to the reduced Planck energy. But this result also implies that Planck's quantum of action $\hbar$ and Newton's constant $G$ are not fixed constants in this framework but vary proportionally to a gravitational scalar degree-of-freedom, akin to typical scalar-tensor and $f(R)$ theories. In particular, it is derived that $\hbar$ is proportional to $G$ in this framework. This establishes an explicit connection between the quantum and gravitational realms. Given the absence of any free theoretical parameter in the theory, we evaluate the level of variation of $\hbar$ and $G$ in the solar system and for neutron stars. We argue that this type of quantitative predictions might be probed observationally in the future, although their amplitudes are extremely small.