I in fact worked on a topic related to this for my first paper to the gravitational research foundation, in which I derived an equation satisfying the Schrodinger equation for Parallel Transport in a curved spacetime interval (see also third reference):

Which had been derived under fundamental assumptions concerning the uncertainty of the system, due to the presence of the Wigner function ~ giving theoretically, a quantum solution to non-commutative geometries.

When appears, we intend the quantum uncertainty in

This uncertainty is known as the Wigner function, found here to have implication with ''quantum gravity'' using non-commutation rules.

it isn't hard to see why these are solutions similar to the ordinary Schrodinger equation of the form:

Showing also that it was bound by the commutation rules applied (surprisingly and remarkably concisely) as:

I've been studying gravielectromagnetism and since felt it is probably the strongest contestant to unifying gravity ''in some way.'' I feel like getting back into the non-linear Hilbert space I had been investigating for the first essay to the gravitational research foundation;.

For a smooth manifold, the tangent bundle of is the affine connection, itself distinguishes a class of curves called affine geodesics, (Kobayashi and Nomizu). The curve is given as:

and the derivative yields the ordinary notation, which featured in my previous work on a non-linear Schrodinger equation

Where is the usual gravitational field (connection) and is the Covariant derivative.

The previous equation is a curve-distance equation, defining the minimum of the geodesic. The product of commutators not only has intrinsic uncertainty attached to the spacetime, but as is well-known, they also form the Riemann tensor . It simply takes form as

With [] a notation for the Planck length. The commutation relationships are calculated the following (usual) way, equivalent to the Riemann curvature tensor:

Also with, I found a non-trivial inequality bound identical in form to the quantum bound:

Concerning the curve equation the product of the wave functions which have lengths of velocity in the Hilbert space is given by:

... But (maybe) more fundamentally, from the same geometrized Hilbert space, we have found a definition of ''how temperature arises'' within the theory. Certainly motion is included for those wave functions who have a velocity and time derivative in the length of the Hilbert space... and motion of atomic and molecular systems is the reigning explanation to how objects may heat up. Some of these equations will be provided for the next parts,

Geometry Link With Temperature

This equation, was a classical momentum equation, which we second quantized and replaced spatial derivatives with Christoffel symbols formed from a Bohm guiding equation.

where we find a familiar form from previous equations:

Wither way, they will describe gravitational physics. The second rank tensor of the Covariant derivative acting on the Christoffel symbol is :

In the case of

The left hand side will be re-written as the four velocity.

The velocity of the classical wave under the Hamilton-Jacobi form as

Is known as the Bohmian guiding equation, In our case, we would like to seek solutions of

I think I know for sure how the dynamics in the temperature relationship geometry:

This equation in my eyes, established all the dynamics but first some hard premises I came to:

1) Though the Bohmian wave is indeed deterministic, this iswritten in spacetime itself... and requires that the wave functions are in fact very small gravitational perturbance, known as gravitational waves. So in other words, particles create ripples in in the space as gravitational waves - in fact, it is under current research to see if the detector can from subatomic gravitational waves, since the detector and detected are coupled in such ways. I postulated this soley from the equation of form:

The reason why I came to this conclusion, are for a number of things, but the real min reason, is just like the full curved Schrodinger equation I derived it has two solutions of the form (if you consider spacetime itself as an observable, as I do follow [1] ):

decomposing into bra-ket notation we get:

and it's conjugate

Both these curves encoded in the wave functions are linked to a geometric argument concerning spacetime. In other words, it is possible to say that the wave creates the curvature, as much as it is valid to say a particle does. The unifying idea here is that as the mass moves through spacetime, it inexorably causes the waves, and on and on the cycle goes.

We came from a totally logical, not add-hoc but some assumptions thrown in. The second quantization of a momentum operator with derivatives related to the gravitational field, is at least a tantalizing approach.

After my assumption of the wave function being gravitational wave ripples in spacetime, came to a surprising shock to find the following article too see if anyone was mad enough as me to think this way:

https://readingfeynman.org/2017/09/3...ational-waves/

The velocity equation as well

Is really similar in structure to the accelertion/curve equation, save for the fact it features the Riemann curvature tensor.

The Conclusion Equations Are:

I managed to get the generalistic form on the left and takes form