Reconciling Quantum Mechanics and General Relativity through Non-Linear Time-Energy Relationships

A class of unified theories due to Apsel, Ryff, and others treat quantum phase frequency as a particle's local clock. However, this frequency scales linearly with energy in QM, but exponentially in gravitational time dilation, so they only match to first order. We reconcile this discrepancy by proposing a non-linear modification to the Schrödinger equation with an exponential dependence of time on energy. Perhaps surprisingly, this maintains the eigenfunctions of the original Hamiltonian while altering only the time evolution, thereby preserving essential quantum properties such as superposition and linearity in the wavefunction. This approach extends naturally to the Dirac equation, suggesting a broader applicability. This offers a potential pathway to unify Quantum Mechanics with General Relativity, and invites further exploration into the foundational aspects of quantum gravity.