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Unifying Temperature Into Hilbert Space Through Geometrizing The Model


Dubbelosix

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Part One: A bit more on this gravity stuff?

 

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;. 

 
[math]\mathbf{D}V^n = dV^n + \Gamma^n_{mr}V^rdx^m[/math]
 
For a smooth manifold, the tangent bundle of [math]M[/math] is the affine connection, itself distinguishes a class of curves called affine geodesics, (Kobayashi and Nomizu). The curve is given as:
 
[math]\Gamma(\gamma)\dot{\gamma}(s) = \dot{\gamma}(t)[/math]
 
and the derivative yields the ordinary notation, which featured in my previous work on a non-linear Schrodinger equation
 
[math]\nabla_{\gamma(t)} \dot{\gamma} = 0[/math]
 
Where [math]\Gamma[/math] is the usual gravitational field (connection) and [math]\nabla[/math] is the Covariant derivative. 
 
[math]\nabla_{\gamma(t)} \dot{\gamma}(t) \equiv\ min\ g^{ij}\sqrt{<\dot{\psi}|[\nabla_i,\nabla_j]|\dot{\psi}>}[/math]
 
The previous equation is a curve-distance equation, defining the minimum of the geodesic. The product of commutators [math][\nabla_i,\nabla_j][/math] not only has intrinsic uncertainty attached to the spacetime, but as is well-known, they also form the Riemann tensor [math]R_{ij}[/math]. It simply takes form as
 
[math]R_{ij} = [\nabla_i,\nabla_j] = [\nabla_i \nabla_j - \nabla_j \nabla_i] \geq g_{ij}^{-}\ (\frac{1}{\ell^2})[/math]
 
With [[math]\ell[/math]] a notation for the Planck length. The commutation relationships are calculated the following (usual) way, equivalent to the Riemann curvature tensor:
 
[math][\nabla_i, \nabla_j] = (\partial_i + \Gamma_i)(\partial_j + \Gamma_j) - (\partial_j + \Gamma_j)(\partial_i + \Gamma_i)[/math]
 
[math]= (\partial_i \partial_j + \Gamma_i \partial_j + \partial_i \Gamma_j + \Gamma_i\Gamma_j) - (\partial_j \partial_i + \partial_j\Gamma_i + \Gamma_j \partial_i + \Gamma_j \Gamma_i)[/math]
 
[math]= -(\partial_i, \Gamma_j) + (\partial_j, \Gamma_i) + (\Gamma_i, \Gamma_j)[/math]
 
Also with, I found a non-trivial inequality bound identical in form to the quantum bound:
 
[math]<\psi|[\nabla_i, \nabla_j] |\psi>\ =\ <\psi| R_{ij}| \psi>\ \leq 2 \sqrt{|<\nabla^2_i><\nabla^2_j>|}[/math]
 
Concerning the curve equation [math]\nabla_{\gamma(t)}\dot{\gamma} = 0[/math] the product of the wave functions which have lengths of velocity in the Hilbert space is given by:
 
[math]K_BT = \frac{1}{2}(\frac{dx^{\mu}}{d\tau} \cdot \frac{dx^{\mu}}{d\tau}) \equiv\ \frac{1}{2}<\dot{\psi}|\dot{\psi}>[/math]
 
... 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, 
 
please no one reply until I gave finished all parts. 
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Part Two

 

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):

 

[math]\nabla_n|\dot{\psi}>\ = \frac{c^4}{8 \pi G} \int \int\ |W(q,p)^2|\ (\frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i})|\psi>\ dqdp \geq \int\ \frac{1}{\hbar}(\frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i})dV|\psi>[/math]

 

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 [math]|W(q,p)^2|[/math] appears, we intend the quantum uncertainty in [math]~[\pi (\hbar)]^{-1}[/math]

 

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:

 

[math]H|\psi> = i \hbar| \dot{\psi}>[/math]

 

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

 


[math]\nabla_n \dot{\gamma}(t) \equiv\ min\ g^{ij}\sqrt{<\dot{\psi}|[\nabla_i,\nabla_j]|\dot{\psi}>} = 0[/math]

 


 

REFERENCES:

 


 


 

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Part Three: Let's see how I got there!

 

 little bit on information theory. Entropy is defined in the usual way:

 

[math]S_A= -\sum_i |c_i|^2 \log |c_i|^2 = S_B[/math]

 

[math]S_A = -\sum_i |c_i|^2 \log |c_i|^2 = \sum_i |c_i|^2 \log(\frac{1}{|c_i|^2})[/math]

 

and can be understood also as

 

[math]S = -\sum \frac{1}{N} \ln(\frac{1}{N})[/math]

 

[math]= - N \frac{1}{N} \ln(\frac{1}{N})= \ln(N)[/math]

 

In terms of information theory, the entropy can be interpreted as an thermodynamic property. If [math]\ln(2)[/math] is the conversion factor from the base of 2 in Shannon entropy to the natural base of e, then the energy in terms of information theory is

 

[math]E = N k_B T \ln(2)[/math]

 

where N is the amount of information in bits. The relative entropy is a distance measure between probability distributions and is

 

[math]D(p|q) = \sum_i p_i(\log_2 p_i - \log_2 q_i)[/math]

 

Another similar formula by I. Białynicki-Birula, J. Mycielski explains the difference of statistical properties of two systems as

 

[math]- \int |\psi|^2 ln[|\psi(q)|^2]\ dq - \int |\hat{\psi}|^2 \ln[|\hat{\psi}(p)|^2]\ dp \geq 1 + \ln \pi[/math]

 

Anandan proposes a difference of geometry equation:

 

[math]<\Delta \Gamma^2> = \sum <\psi| (\Gamma - <\psi |\Gamma| \psi>)^2|\psi >[/math]

 

and suspects an energy equation can be derived from it

 

[math]E = \frac{k}{G} \Delta \Gamma^2[/math]

 

which when you correct the constant of proportionality I get

 

[math]\Delta E = \frac{c^4}{8 \pi G} \int < \Delta \Gamma^2>\ dV =\frac{c^4}{8 \pi G} \int <\psi|(\Gamma - <\psi |\Gamma| \psi>)^2|\psi>\ dV[/math]

 

[math] = \frac{c^4}{8 \pi G} \int \frac{1}{R^2} \frac{d\phi}{dR} (R^2 \frac{d\phi}{dR})\ dV[/math]

 

Of course, difference of geometries leads to survival probabilities linked to decoherence so has application similar to Penroses own model of a gravitational collapse. Could it be possible that the zeno effect (the act of collapsing the wave function under the ill-defined ''observation'' upon it), may have something still instrinsically related to the stability of two superpositioned geometries. 

 

 

 

Not Finished Yet

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Part Four: Anti-commutation and Covariant Deriatives and their Connection to the Gravitational weak-field Limit.

 

 
The Parallel transport in cuviliear space for a vector is given s:
 
[math]\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu \lambda} \frac{dx^{\nu}}{d \tau} \frac{dx^{\lambda}}{d\tau} = 0[/math]
 
was in fact, satisfied in a very concise way using the bra-ket notation, 
 
[math]\nabla_n \dot{\gamma}(t) = \nabla_n\frac{dx^{\mu}}{d\tau} \equiv\ min\ \sqrt{<\dot{\psi}|[\nabla_j,\nabla_j]|\dot{\psi}>} = 0[/math]
 
There is classical commutation going on, on the RHS 
 
[math][\nabla_j, \nabla_j] = 0[/math]                             ]
 
which just means in the usual sense, that the order of the operators does not matter, for obvious reasons. This is why it should be noted, that space time non-commutivity is subtly hinted at when you consider connections with derivatives in both space and time. It appears that spacetime uncettainty has been at least supportd from particle scattering experiments. Let's describe curve equations more accurately now:
 

The difference between two points is given as:

 

[math]|\psi - \psi'|[/math]

 

This still doesn't yet describe curvilinear space - this is actually a Euclidean measure of the distance between two states. The length of a curve however can be given as:

 

[math]\frac{ds}{d\tau} = \sqrt{<\dot{\psi}|\dot{\psi}>}[/math]

 

And so our metric is,

 

[math]ds = (\tau_1 - \tau_2) \sqrt{<\dot{\psi}|\dot{\psi}>}[/math]

 

This involves the tangent vector \dot{\psi} has lengths that is the velocity which travels in the Hilbert space. I can construct a an equation related the evolution of the Schrodinger equation from this. With [math]c = 8 \pi G = 1[/math]

 

[math]\frac{ds}{dt} = \frac{1}{\hbar} \sqrt{<\psi|R^2|\psi>} = \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math]

 

and the metric of the solution is[

 

[math]ds = \frac{1}{\hbar}(t_1 - t_2)\sqrt{<\psi|H^2|\psi>}[/math]

 

Anandan, a physicist whome I used an equation that described the difference of geometries (but made it within the context of the curvature tensor) has proposed that the Euclidean length is an intrinsic parameter of the Hilbert space.

 

Assuming I have done all this right so far, I did find an interesting continuation which uses the Wigner function. The curve of the length can be related to an inequality by pulling out the terms to write a Wigner function. Again, in natural units,

 

[math]\frac{ds}{dt} = |W(q,p)| \sqrt{<\psi|R^2|\psi>}\ \geq \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math]

 

or as

 

[math]\sqrt{<\dot{\psi}|\dot{\psi}>} = |W(q,p)| \sqrt{<\psi|R^2|\psi>}\ \geq \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math]

 

due to the inequality relationship:

 

[math]|W(q,p)|^2\ dpdq \geq \frac{1}{\pi \hbar}[/math]

 

Anyway, moving on, from the second equation we can construct two solutions for the time-dependent Schrodinger equation which satisfies 

 
[math]\frac{1}{ i \hbar}H|\psi>\ = |\dot{\psi}>[/math]
 
When you consider the geometry related to the Hamiltonian in such a way:
 
[math]\sqrt{<\dot{\psi}|\dot{\psi}>} = \int \int\ |W(q,p)^2| \sqrt{<\psi|\Gamma^2|\psi>}\ dqdp \geq \frac{1}{\hbar}\sqrt{<\psi|H^2|\psi>}[/math]
 
Which was derived by myself using the Wigner function, then it becomes more apparent that it was possible to construct a wave equation encoding the rank 2 tensor dynamics of the connection and the stress energy tensor in the following way
 
[math]\nabla_n|\dot{\psi}>\ =  \frac{c^4}{8 \pi G} \int \int\ |W(q,p)^2|\ (\frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i})|\psi>\ dqdp \geq \int\  \frac{1}{\hbar}(\frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i})dV|\psi>[/math]
 
This is in fact, totally analogous to an acceleration/curve in the general theory of relativity, except this time, it satisfies an equality bound and it also satisfies the wave dynamics of a Schrodinger equation, albeit, a non-linear one. It is not immediately obvious there are terms in this last equation which can satisfy an inequality bound, but in my early investigation, the bounds where found as:
 
[math]\sqrt{|<\nabla^2_i><\nabla^2_j>|} \geq \frac{1}{2}<\psi|[\nabla_i, \nabla_j] |\psi>\ = \frac{1}{2} <\psi|R^2_{ij}| \psi>[/math]
 
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Part Four; Concise Result for Curved Space

 

Which is basically a mean deviation that can reach twice the classical upper bound. This bound holds importance for our equation which satisfied the geodesic equation

 
[math]\nabla_n \dot{\gamma}(t) = \nabla_n\frac{dx^{\mu}}{d\tau} \equiv\ min\ \sqrt{<\dot{\psi}|[\nabla_j,\nabla_j]|\dot{\psi}>} = 0[/math]
 
The product of [math]\ <\dot{\phi}|\dot{\psi}>[/math] has a length in the Hilbert space... with velocity comes acceleration:
 
[math]\frac{\partial v}{\partial \tau}  \cdot \frac{\partial v}{\partial \tau} = \frac{\hbar^2}{2m^2}[\nabla \Gamma] =\ <\dot{\psi}|\dot{\psi}>[/math]
 
with 
 
[math]v^2 \equiv |U|^2 = U^{\mu}U_{\mu}[/math]
 
It's surprising that replacing canonical operators for their quantum parameters, with geometric aspects captured on both sides. 
 
Most surprisingly also it could unify the concept of phases of particles;
 
Which can have a Berry curve definition ~
 
[math]\gamma = i \oint <n(\mathbf{R})\nabla_{\mathbf{R}}|n(\mathbf{R})>\ d\mathbf{R}[/math]
 
Obtained from a second quantization on momentum operator, while replacing the d,Alambertian for its gravitational connections and involves phase changes of particles. 
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Geometry Link With Temperature

 

 
This equation, was a classical momentum equation, which we second quantized and replaced spatial derivatives with Cristoffel symbols. 
 
[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] = \mathbf{U}^{\mu}\mathbf{U}_{\mu}[/math]
 
where we find a familiar form from previous equations:
 
[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] =\ <\dot{\psi}|\dot{\psi}>[/math]
 
Wither way, they will describe gravitational physics. The second rank tensor of the Covariant derivative acting on the Christoffel symbol is :
 
[math] \nabla_n\Gamma^{ij} = \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}[/math]
 
In the case of 
 
[math]v^2 =  \frac{\hbar^2}{2m^2}\ [ \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}][/math]
 
The left hand side will be re-written as the four velocity. 
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The Nature Behind The Spatial Derivatives

 

You may have realized for case which involve derivatives [math]\frac{\partial}{\partial x} \rightarrow\ \Gamma\frac{and}{or} \nabla [/math]. We see why this is so, and we should uncover some new physics along the way. The velocity equation squared as featured befoe may be written in the following form, they do define the acceleration/geodesic in non-inertial frames in the wave function but it's source relies in the covariant derivative of the connection. The equation I arrived at was:

 
[math]v^2 =  \frac{\hbar^2}{2m^2}\ [\nabla^{\mu}, \nabla_{\mu}][/math]
 
 
SIDELINE INVESTIGATION FOR LATER: This appears like it could be entirely deterministic if the velocity squared term can be interpreted under guidance waves. 
 
[math]\nabla_{\gamma(t)}\ \dot{\gamma}(t) \equiv\ |min| g^{\mu \nu} \sqrt{<\dot{\psi}|\dot{\psi}>}[/math]
 
From kinetic quantizaion of momentum operator with two connections of the gravitational force.
 
[math]<E_k>\ = \frac{\hbar^2}{2m^2}\Gamma^2 =  \frac{\hbar^2}{2m^2}[\Gamma, \Gamma][/math]
 
It's an operator in form because connections of the gravtational field do not generally commute. From the Virial theorem and equipartiction we find the relationsip
 
[math]\phi \approx \frac{3}{5}\ g_{00} \approx  \frac{3}{2} k_BT  = \frac{1}{2} mv^2[/math]
 
scalar field defined through ideal gas law and virial theorm:
 
[math]\phi = \frac{3}{5} \frac{Gm}{r}[/math] 
 
[math]=  \frac{3}{2}   m(\frac{\partial x^{\mu}}{\partial \tau} \cdot \frac{\partial x_{\mu}}{\partial \tau})[/math]
 
[math]= \frac{\hbar^2}{2m^2}\Gamma^2 [/math]
 
These equations are so generic in set-up for linear gravitational fields, it would also be later difficult not to see this in a modified approach using the Sciama definition of the gravielectromagnetic field. 
 
Some notes...
 
 A linear wave  equation is akin to a classical formulation of wave mechanics in special relativity, it is simply not general while it may not possess curvature. The addition of Einstein's stress energy showed (may not so easily) that you cannot have a surface geometry without a notion of curvature, gravity and an extra temporal degree of freedom. 
 
The first equation could also be theorized as:
 
[math]<E_k>\ = \frac{\hbar^2}{2m^2}[\nabla_{\mu}, \Gamma^{\mu}][/math]
 
or even as
 
[math]<KE_{\mu \nu}> = \frac{\hbar^2}{2m^2}[\nabla_{\mu}, \nabla_{\nu}] = \frac{\hbar^2}{2m^2}\ <\psi|R_{\mu \nu}|\psi>[/math]
 
 
[math]\gamma gamma from nabla nabla fits directive on velocity equation. Now introduce wavefunctions 
 
 
[math]<\psi|KE_{\mu \nu}|\psi> =  \frac{\hbar^2}{2m^2}\ <\psi|R_{\mu \nu}|\psi>[/math]
 
 
Just to note, my ultimate intention is to find a [[full]] relativistic of my Schrodinger wave equation. Doing so would unify gravity to U(1) gauge parameter. But most of all, there are some nice aspects possible for a forward to unification for the crucial generalized curve equation:
 
[math]\nabla_{\mu \nu}\frac{ds}{dt}  \equiv |min| g^{\mu \nu} \sqrt{<\dot{\psi}|\dot{\psi}>}[/math]
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So let’s get into the subject of Pilot waves and what it has to do with what we have talked about. Let's begin talking about the guiding equation for deterministic quantum mechanics. That equation takes the form:

 

[math]\frac{dQ}{dt} = \frac{\hbar}{m}\ \mathbf{Im} \frac{\psi* \partial_k \psi}{\psi^{*} \psi}(Q_1... Q_N)[/math]

 

Where Q(t) is the first order evolution for the position of the particles. The distribution can be understood through the Born law, [math]\int |\psi|^2 = \psi^{*}\psi [/math]  in which case the ratio can be understood in terms of their respective probability density distributions in which case, you can form a time dependent Schrodinger equation taking into respect a guiding wave function:

 

[math]i \hbar \frac{\partial \psi}{\partial t} = (-\frac{h^2}{2m}\nabla^2 + V + \frac{\hbar^2}{2m}\frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}})\psi[/math]

 

Notice, the statistical dependence is removed (or normalized) from the guidance equation

 
Under Bohmian Pilot waves, the equaion of interest is:
 
[math]\frac{\partial s}{\partial t} = \frac{s^2}{2m}\nabla^2[/math]
 
the metric is: [math]s = ct[/math]
 
Since the geodesice of the main quantification;
 
[math]\nabla^{\mu \nu}\nabla_{\sigma \lambda}\ (\frac{ds^{\sigma}}{dt} \cdot \frac{ds^{\lambda}}{dt})  \equiv |min| g^{\mu \nu} <\dot{\psi}|\dot{\psi}>[/math]
 
After squaring solves for two solutions those for conjugate ket's and bra (ie. [math]|\dot{\psi}>[/math]\ and [math]<\dot{\psi}|[/math] )solutions. 
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More On Bohmian Mechanics

 

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

 

[math]v(x,t) = \frac{\nabla(x,t)}{m}[/math]

 

In our case, we would like to seek solutions of 

 

[math]|U|^2 = \mathbf{U}^{\mu}\mathbf{U}_{\mu} [/math]

 

Which is the magnitude of velocity. 

 

The next term within Bohmian model, we have the equation

 

[math]Q = - \frac{\hbar}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}}[/math]

 

Which is a potential. The statistical part has been normalized but there exists the coefficient [math]\frac{\hbar}{2m}[/math] which is close to a coefficient we ave been using [math]\frac{\hbar}{2m}[/math].

 

These things I will look into

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Gravitational Guiding Waves

 

 

The guiding wave equation is:

 

[math]v = \frac{\partial s}{\partial t} = \frac{S^2}{2m}\nabla^2[/math]
 
Two equations that catch the eye with the format of the previous equation also introduces a simple solve for velocity:
 
[math]\frac{\partial v}{\partial \tau} \cdot \frac{\partial v}{\partial \tau} = \frac{\hbar^2}{2m^2}\ [ \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}][/math]

 

The velocity in the first equation when squared is simply

 

[math]v^2 = (\frac{\partial S}{\partial t})^2 = \frac{S^4}{4m^2}\nabla^4[/math]

 

The operator at the end to the forth power can more interestingly given as

 

[math]v^2 = (\frac{\partial S}{\partial t})^2 = \frac{S^4}{4m^2}[\nabla\, \Gamma][\nabla\, \Gamma][/math]

 

This equation appears to be mixed and separability should be possible, such that

 

[math]v = \frac{\partial S}{\partial t} = \frac{S^2}{2m}[\nabla, \Gamma][/math]

 

 

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BRIEF LOOK AT WHAT HAS BEEN WRITTEN SO FAR

 

Out of all these interesting equations, the first being the geometric Schrodinger equation defining the curve:

 
[math]\nabla_n|\dot{\psi}>\ =  \frac{c^4}{8 \pi G} \int \int\ |W(q,p)^2|\ (\frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i})|\psi>\ dqdp \geq \int\  \frac{1}{\hbar}(\frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i})dV|\psi>[/math]
 
Under another derivation 
 
[math]\nabla_n v^2 =  \frac{\hbar^2}{2m^2}\ [ \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}][/math]
 
And:

 

[math]v = \frac{\partial S}{\partial t} = \frac{S^2}{2m}[\nabla, \Gamma][/math]

 
Why do they interest me? Well, the velocity guiding equation is now described geometrically so we should be able to see the left hand side in context of it?
 
[math]\frac{S^2}{2m}[\Gamma, \Gamma] =  \frac{\partial S}{\partial t} =  \sqrt{\frac{\hbar^2}{2m^2}\ [ \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}]}[/math]
 
The indices remain on the right until we clarify the left hand side. 
 
[math]\frac{S^4}{4m^2}[\Gamma, \Gamma][\Gamma, \Gamma] =  (\frac{\partial S}{\partial t})^2 = \frac{\hbar^2}{2m^2}\ [ \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}][/math]
 
But before we get ahead of ourselves, let us first analyse how we obtained this geometric second quantization on the momenum operator;
 
This equation, was a classical momentum equation, which we second quantized and replaced spatial derivatives with Cristoffel symbols. 
 
[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] = \mathbf{U}^{\mu}\mathbf{U}_{\mu}[/math]
 
The right hand side established clearly we are dealing with the four velocity. We find a familiar form from previous equations:
 
[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] =\ <\dot{\psi}|\dot{\psi}>[/math]
 
[math]\frac{\hbar^2}{2m^2}[\nabla\ \Gamma_{\mu}, \nabla\ \Gamma_{\mu}] =\ <\dot{\psi}|R_{\mu \nu}|\dot{\psi}>[/math]
 
Both sides yields geometric properties, the right defines the velocity of the wave function traversing through the curved Hilbert space, while the left defines either a commuting or anticommuting set of Christoffel symbols. It will make an interesting investigation. When in context of temperature, its hard not to see the obvious similarities for curved trajectories:
 

[math]K_BT = \frac{1}{2}(\frac{dx^{\mu}}{d\tau} \cdot \frac{dx^{\mu}}{d\tau}) \equiv\ \frac{1}{2}<\dot{\psi}|\dot{\psi}>[/math]

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Please bear with me, but I think I know for sure how the dynamics in the temperature relationship  geometry:

 

[math]K_BT = \frac{\hbar^2}{2m^2}[\Gamma, \Gamma] =\ \frac{1}{2} <\dot{\psi}|\dot{\psi}>[/math]

 

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

 

1) Though it the Bohmian wave is indeed determinstic, this written in spacetime 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] ):
 

[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] =\ \frac{1}{2} <\dot{\psi}|\dot{\psi}>[/math]

 

decomposing into bra-ket notation we get:

 

[math]|\dot{\psi}_{\mu}>\ = \frac{\hbar}{m}\Gamma_{\mu}[/math]

 

and it's conjugate

 

[math]<\dot{\psi}^{\mu}|\ = \frac{\hbar}{m}\Gamma^{\mu}[/math]

 

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/2017/09/30/wavefunctions-as-gravitational-waves/

 

 

The velocity equation as well

 

 

[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] =\ \frac{1}{2} <\dot{\psi}|\dot{\psi}>[/math]

 

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

 

[math]\nabla_n \dot{\gamma}(t) = \nabla_n \frac{dx}{d\tau} \equiv\ min\ \sqrt{<\dot{\psi}|[\nabla_j,\nabla_j]|\dot{\psi}>} = 0[/math]

Edited by Dubbelosix
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And finally, the velocity equation

 

[math]\nabla_n v^2 =  \frac{\hbar^2}{2m^2}\ [ \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}][/math]

 

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

 

[math]U^{i}U^{j} =  \frac{\hbar^2}{2m^2}\ [ \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}][/math]

Edited by Dubbelosix
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As it turned out, I had a whole free hour at a friends house, pot smoker type who lies to watch quiz TV shows ... anyway, I was capable of scribbling down the basis of the theory: I realized the ''master'' equation was:

 

[math]k_BT = \frac{\hbar^2}{2m^2}[\Gamma, \Gamma] = \frac{1}{2}<\dot{\psi}|\dot{\psi}>[/math]

 

I considered Covariant derivatives inside of those commutators, but I realized it became incredibly complex to calculate, so the idea was to simply it from the physics we established. One such being

 

[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] =\ <\dot{\psi}|\dot{\psi}>[/math]

 

The left hand side, encodes the geometry, arisen of course through the presence of the gravitational field connections which also so happens to hold possibilities for space and time uncertainties. To break the equation up, we find the presence of the symmetry again 

 

[math]|\dot{\psi}_{\mu}>\ = \frac{\hbar}{m} \Gamma_{\mu} = \sqrt{k_BT_{\mu}}[/math]

 

We obtain the square root from knowing there are two such parts in the wave function. It was a bit of a surprise when I realised that when the Covariant derivative acts on this equation, we get back a curve equation, we know this because we have used such notation before, in the non-linear wave equation satisfying Wigner bounds:

 

[math]\nabla_n|\dot{\psi}>\ =  \frac{c^4}{8 \pi G} \int \int\ |W(q,p)^2|\ (\frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i})|\psi>\ dqdp \geq \int\  \frac{1}{\hbar}(\frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i})dV|\psi>[/math]
 
Recall the [gravitational] guidance wave equation we defined was:
 
[math]\vec{v} = \frac{S(r,t)}{m}\Gamma[/math]
 
The [math]S(r,t)[/math] is a solution from Hamiltons equation of motion, but notice we have created it in a gravitational form - make aware also, there could be a question about complexifying the gravitational field, a feature not present in the Wheeler de Witt equation and has caused wonder with physicists ever since. A quantization just requires an imaginary coefficient:
 
[math]|\dot{\psi}_{\mu}>\  = \frac{i\ \hbar}{m} \Gamma_{\mu}[/math]
 
If gravity is a true quantum field, it should be complexified to allow the existence of gravitons. I don't personally think that way and may come back to a post explaining why. 
 
The squared solution of the wave equation is
 
[math]\mathbf{U}_{\mu}\mathbf{U}^{\mu} = (\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}][/math]
 
The four velocity in such a case, can be entirely termed through some geometric properties arising from this commutation. It's basically a theory of curvature in the phase space. This last equation is interesting even further we have covariant derivatives acting on both velocity terms, but with indices in time:
 
[math]\nabla_0\ \mathbf{U}_{\mu}\nabla^0\ \mathbf{U}^{\mu} = (\frac{S(r,t)}{m})^2[\nabla_0\ \Gamma_{\mu}, \nabla^0 \Gamma^{\mu}][/math]
 
Albeit, a more complicated object to work around with; but why a covariant derivative of time: First of all, a covariant derivative in time is no more less than [math]\frac{\partial \mathbf{U}^{\mu}}{\partial t} +\ correcting\ terms[/math] and this defines a velocity curve (squared), being an acceleration squared. :
 
[math]\frac{dv}{dt} \cdot \frac{dv}{dt} = a_{\mu}a^{\mu}[/math]
 
This isn't quite the four velocity since that is defined as:
 
[math]\frac{\mathbf{D}\mathbf{U}^{\lambda}}{\partial \tau} = \frac{\partial \mathbf{U}^{\lambda}}{\partial \tau} + \Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}\mathbf{U}^{\nu}[/math]
 
Simple enough right? It is after all equivalent to the curve found from the Hilbert space as a modified quadratic solution:
 
[math]\frac{1}{4}<\dot{\psi}|\dot{\psi}><\dot{\psi}|\dot{\psi}>[/math]
 
If all the wave functions are governed by ordinary rules, the identity operator falls out of this,
 
[math]\mathbf{I} = |\dot{\psi}><\dot{\psi}|[/math]
 
This is known as a Dyad. The equation that took me the most was
 
[math]|\dot{\psi}_{\mu}>\ = \frac{\hbar}{m} \Gamma_{\mu} = \sqrt{k_BT_{\mu}}[/math]
 
Because I am considering its form when there is a covariant derivative acting on the wave vector... same as before, doing so defines a curve in spacetime. We had done this with the modified Schrodinger equation for Wigner functions, so why not this? I'll leave this task to a later date, but a hint is that it follows tensors of rank 2, which follow the transformation laws:
 
[math] \nabla_n\Gamma^{ij} = \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}[/math]
Edited by Dubbelosix
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Did anyone do it? 

 

You will recall that we obtained the velocity squared formula. It might interest you to know that this formula is a covariant derivative away from  curve equation: It not only guides the particle using gravitational waves but it seems to contribute to the four velocity or more accurately, a four acceleration 

 
[math] (\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]
 
And that four-acceleration is in fact implemented like: 
 
[math]\frac{\mathbf{D}\mathbf{U}^{\lambda}}{\partial \tau} = \frac{\partial \mathbf{U}^{\lambda}}{\partial \tau} + \Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}\mathbf{U}^{\nu}[/math]
 
The idea follows as:
 
[math]\mathbf{D}^{\lambda} (\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] =  \frac{\partial \mathbf{U}^{\lambda}}{\partial \tau} + \Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}\mathbf{U}^{\nu}[/math]
 
Pretty attractive equation, with guiding gravitational wave mechanics playing the role of a four-velocity on the particle. Now a careful study of the Christoffel commutations, to see whether they suit the commutation or anticommutation rules. It will be fun. There was however, an alternative interesting derivation. It was feasible theoretically to consider the derivatives under the form for the guidance equation:
 
[math]v^2 = \frac{\partial x}{\partial t} \cdot \frac{\partial x}{\partial t} = (\frac{\partial s}{\partial t})^2 = \frac{S^2}{2m}[\frac{\partial \Gamma}{\partial x}, \frac{\partial \Gamma}{\partial x}] [/math]
 
This term is still acting under commutation rules as the gravitational field is made from the derivatives of the metric:
 
[math][\frac{\partial \Gamma}{\partial x}, \frac{\partial \Gamma}{\partial x}] [/math]
 
It can also be noted again that the product equals the Hilbert gravitational wavelength:
 
[math]\frac{\partial x}{\partial t} \cdot \frac{\partial x}{\partial t} =\ <\dot{\psi}|\dot{\psi}>[/math]
 
This finally suggests that there could also be possible physics under combination of terms:
 
[math] <\dot{\psi}|\dot{\psi}>\ =  \frac{S^2}{2m}[\frac{\partial \Gamma}{\partial x}, \frac{\partial \Gamma}{\partial x}] [/math]
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