Hamilton–Jacobi–Einstein equation

For a free particle in curved "empty space" or "free space", i.e. in the absence of matter other than the particle itself, the equation can be written:^[6][7][8]

where g is the

the rate of change of action with respect to the field coordinates g_ij(r). The g and π here are analogous to q and p = ∂S/∂q, respectively, in classical Hamiltonian mechanics. See canonical coordinates for more background.

The equation describes how

The quantum mechanical concept, that action is the phase of the wavefunction, can be interpreted from this equation as follows. The phase has to satisfy the principle of least action; it must be stationary for a small change in the configuration of the system, in other words for a slight change in the position of the particle, which corresponds to a slight change in the metric components;

${\displaystyle g_{ij}\rightarrow g_{ij}+\delta g_{ij}\,,}$

the slight change in phase is zero:

${\displaystyle \delta S=\int {\frac {\delta S}{\delta g_{ij}(\mathbf {r} )}}\delta g_{ij}(\mathbf {r} )\mathrm {d} ^{3}\mathbf {r} =0\,,}$

(where d³r is the volume element of the volume integral). So the constructive interference of the matter waves is a maximum. This can be expressed by the superposition principle; applied to many non-localized wavefunctions spread throughout the curved space to form a localized wavefunction:

${\displaystyle \Psi =\sum _{n}c_{n}\psi _{n}\,,}$

for some coefficients c_n, and additionally the action (phase) S_n for each ψ_n must satisfy:

${\displaystyle \delta S=S_{n+1}-S_{n}=0\,,}$

for all n, or equivalently,

${\displaystyle S_{1}=S_{2}=\cdots =S_{n}=\cdots \,.}$

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. So in the EHJE above, each wavefront of constant action is where the particle could be found.

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.

Applications

The equation takes various complicated forms in:

• Quantum gravity
• Quantum cosmology

See also

• Foliation
• Quantum geometry
• Quantum spacetime
• Calculus of variations
• The equation is also related to the Wheeler–DeWitt equation.
• Peres metric

References

Notes