A Type Of Gravitational Schrodinger Equation And The Minimum Distance Vs The Phase Space

[Dubbelosix]

Minimum Distance in Relativity and the Non-Commutation of the Phase Space

 

 

First... a blow by blow account of what led up to the proposal of the curve equation: Anandan proposes an equation

 

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

 

and I offer also true

 

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

 

That was after correcting the constant of proportionality, and after I derived a following inequality  - The Mandelstam-Tamm inequality for instance

 

I have shown can be written in the following way, with [math]c = 8 \pi G = 1[/math] (as usual), we can construct the relationship: (changing notation only slightly)

 

[math]|<\psi(0)|\psi(t)>|^2 \geq \cos^2(\frac{[<\Gamma^2> - <\psi|\Gamma^2|\psi> ]\Delta t}{\hbar}) = \cos^2(\frac{[<H> - <\psi|H|\psi> ]\Delta t}{\hbar}) = \cos^2(\frac{\Delta H \Delta t}{\hbar})[/math]

 

This linking of geometry to the energy of the system can be understood through a curve equation I derived using the same principles by making use of the Wigner function

 

[math]\frac{ds}{dt} \equiv \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]

 

By understanding that the curve equation when squared provides solutions to the time-dependent Schrodinger equation (in the following way) ~

 

[math]\frac{1}{ i \hbar}H|\psi>\ = |\dot{\psi}>[/math]

 

Then you can construct a more serious equation that may be seen as a gravitational analogue to the Schrodinger equation when the covariant derivative acts on the tensor components and I calculate it as:

 

[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>\ d\mathbf{q}d\mathbf{p} \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]

 

(remember to renomralize the density in the stress energy tensor)

 

Also keep in mind, when you take the bra solution [math]<\dot{\psi}| \nabla[/math] and it;s product with its conjugate above, you could understand the product as producing an object like [math]<\dot{\psi}|[\nabla,\nabla]|\dot{\psi}>[/math] which shows the non-commutation between covariant derivatives - the wave functions, in its most simplest form can be understood as

 

[math]|\psi>\ =  e^{iHt}|q>[/math]

 

[math]<\psi| =\ <q| e^{-iHt}[/math]

 

When we take the product of the bra and ket solutions, we get an analogous identity found in General relativity - we simply take the tangent vector [math]\frac{dx^{\mu}}{d\tau}[/math] and allow the covariant derivative to act on this (and further set it to zero) and defines the minimum curve, or better yet, as a geodesic for a minimum distance,

 

[math]\nabla_n\frac{dx^{\mu}}{d\tau} \equiv\ min\ \sqrt{<\dot{\psi}|[\nabla,\nabla]|\dot{\psi}>}[/math]

 

We know how the covariant derivative acts on the curve from the following equation

 

[math]T_{nm}(y) = \nabla_n V_m = \frac{\partial V_m}{\partial y^{n}} + \Gamma_{nm} V_{r}(x)[/math]

 

The interesting thing about setting this to zero and using this as the definition of the minimum distance is that non-commutation between the covariant derivatives in a phase space is generally not zero [math][\nabla_i \nabla_j] \ne 0[/math]! The Von Neumann algebra insists that deviation from the classical vacuum relies on the non-commutative  properties of the quantum phase space. 

 

The covariant derivative also acts on rank 2 tensors in the following way

 

[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]

 

It also follows then the stress energy tensor responds in much the same way

 

[math]\nabla_nT^{ij} = \frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i}[/math]

 

which were the primary tools we used to construct our form of the non-linear Schrodinger equation.

 

 

 

 

(a reference to my own work, which contains all relevant references to others not by me)

 

http://www.physicsgre.com/viewtopic.php?f=10&t=127412&p=198855#p198855

Edited December 4, 2017 by Dubbelosix