It is named for Albert Einstein, Carl Gustav Jacob Jacobi, and William Rowan Hamilton.
The EHJE contains as much information as all ten Einstein field equations (EFEs).
Substitution of this into the quantum general Schrödinger equation (SE): and taking the limit ħ → 0 yields the classical HJE: which is one aspect of the correspondence principle.
In RQM and QFT, position returns to the usual spatial coordinates alongside the time coordinate, although these theories are consistent only with SR in four-dimensional flat Minkowski space, and not curved space nor GR.
It is possible to formulate quantum field theory in curved spacetime, yet even this still cannot incorporate GR because gravity is not renormalizable in QFT.
[4] There is theoretical and experimental evidence from QFT that vacuum does have energy since the motion of electrons in atoms is fluctuated, this is related to the Lamb shift.
[4] In any case, a four-dimensional curved spacetime continuum is a well-defined and central feature of general relativity, but not in quantum mechanics.
The equation describes how wavefronts of constant action propagate in superspace - as the dynamics of matter waves of a free particle unfolds in curved space.
Like the Einstein field equations, it is non-linear in the metric because of the products of the metric components, and like the HJE it is non-linear in the action due to the product of variational derivatives in the action.
The quantum mechanical concept, that action is the phase of the wavefunction, can be interpreted from this equation as follows.
This can be expressed by the superposition principle; applied to many non-localized wavefunctions spread throughout the curved space to form a localized wavefunction: for some coefficients cn, and additionally the action (phase) Sn for each ψn must satisfy: for all n, or equivalently, Regions where Ψ is maximal or minimal occur at points where there is a probability of finding the particle there, and where the action (phase) change is zero.
This equation still does not "unify" quantum mechanics and general relativity, because the semiclassical Eikonal approximation in the context of quantum theory and general relativity has been applied, to provide a transition between these theories.