Global Dynamics for Newton and Planck

We discuss recently introduced scale-free Einstein equations, where the information from their trace part is lost. These equations are classically equivalent to General Relativity, yet the Newton constant becomes a constant of integration or a global dynamical degree of freedom. Thus, from the point of view of standard quantization, this effective Newton constant is susceptible to quantum fluctuations. This is similar to what happens to the cosmological constant in the unimodular gravity where the trace part of the Einstein equations is lost in a different way. Using analogy with the Henneaux-Teitelboim covariant action for the unimodular gravity, we consider different general-covariant actions resulting in these dynamics. This setup allows one to formulate the Heisenberg uncertainty relations for the Newton constant and canonically conjugated quantities. Unexpectedly, one of such theories also promotes the Planck's quantum constant to a global degree of freedom, which is subject to quantum fluctuations. Following analogy with the unimodular gravity, we discuss non-covariant "unimatter" and "unicurvature" gravities describing the scale-free Einstein equations. Finally, we show that in some limit of the Yang-Mills gauge theory a "frozen" axion-like field can emulate the gravitational Newton constant or even of the quantum Planck constant.