Of the three possible action equations, quantum, classical, and hyperbolic, the action equation of quantum charge is the Schrödinger equation as
The classic gravity Hamilton-Jacobi equation in units of time delay and matter change is
and says that an source follows a determinate path, S, unless acted on by another source by the action dm/dt that changes the source orbital period, tp. Even though gravity exists in the same quantum root reality as charge, the gravity of a GR observer does not act on itself. This means that the geodesics of general relativity are not subject to the uncertainty of quantum futures.
In a quantum reality, even gravity matter has phase coherence and shows interference effects and uncertainty since it is light that is the quantum glue that holds both charge and gravity matter together. The symmetry of the gravity biphoton simply means that quantum phase coherence exists for gravity as well as charge. However, the exchange of two gravity biphotons always results in complementary phases and so the resonances between gravity bodies always exchange complementary phase.
A classical photon only transfers intensity from a classical source to a classical observer and does not transfer quantum phase coherence. A quantum photon represents a resonance between an observer and an excited source that transfers both amplitude and phase coherence. A gravity resonance between an observer and a source also represents both amplitude and phase coherence, but a gravity biphoton resonance involves excited states of both observer and source.
The classical Hamilton-Jacobi equation is the beginning of the geodesics of general relativity and it is the quantum Hamilton-Jacobi equation that shows the time derivative of relativity's action geodesic as a matter wave, Sae, as
The matter-scaled Schrödinger equation Eq. 1 with mR as the Rydberg mass equivalent energy of the hydrogen atom bond provides the matter wave psiae. The strange i = eπ/2 Euler phase factor simply represents a phase shift of pi/2 or 90° between a matter wave and its time derivative, which is the observer and a source. It is just this phase coherence that is what makes the quantum matter waves of Eq. 3 much different from classical matter waves of Eq. 2.
It is ironic that time and space both emerge from the Schrödinger equation and the actual primitives are that of discrete aether, psiae and discrete action, Sdotae. That is, time and space actually emerge as the discrete dimensionless notions of tau/taup or q/qp from the action derivative of the Hamilton-Jacobi-Schrödinger equation .
The classical gravity waves of Eq. 2 also have phase coherence, but classical waves have classical coherence with determinate futures and follow the geodesics of relativity. The quantum path derivative is negative, which points both the arrow of quantum time as well as the phase shift between matter and its derivative of action. The norms or products of complementary quantum matter waves of Eq. 3 result in the classical waves of Eq. 2, but lack quantum phase coherence and uncertainty.
Biphoton exchange applies the same quantum glue of coherent photon phase to gravity. Bonding an electron and proton is due to the exchange of a photon particle of the Rydberg mass, mR, which is the hydrogen bond. That binding photon today has a complementary and entangled photon emitted at the CMB that together form a biphoton quadrupole. Instead of a single photon, gravity is this irreducible coupling of bond and emitted photons as a biphoton quadrupole. Biphotons are the phase coherent quantum glue that bonds neutral particles to the universe with the quadrupole biphoton force scaled from a photon as tB / Tu x e.
The Schrödinger equation shows that a differential change in an object is orthogonal to itself for both charge and gravity. A differential change in a gravity wave biphoton will also be proportional to itself, but since the biphoton has dipoles with entangled phase, the resulting product wavefunction now commutes and satisfies both quantum Eq. 1 and classical Eqn. 2.
The classical action integral of general relativity, S, has a matter-scaled time derivative related to the Lagrangian that is simply equal to the kinetic minus the Hamiltonian interaction energy. Typical objects have very large numbers of such quantum gravity states along with many fewer quantum charge states. Quantum gravity states tend to be incoherent sums of matter wave norms that represent classical gravity and relativity. Unlike the relatively high energy of atomic bonds, quantum gravity bonds are very much weaker and so involve very much lower frequency biphotons. Any phase coherence of a quantum gravity is typically dominated by the phase coherence of quantum charge and so gravity mass exists largely as matter wave norms without coherent phase.
The hyperbolic wave equation is simply the dSae/dt action wave with a change in sign. The hyperbolic equation describes antimatter with a simple change in sign and antimatter is inherently unstable in the matter universe since antimatter's time arrow is opposit and yet antimatter is stable in the antiverse precursor to the matter universe.