An apologetic discourse on the quantum picture of torsion

[Dubbelosix]

The technical issue i came across, is fixable, which is a good thing, but I need to weed out the exact problem itself, which appeared when I was analysing the dimensions of an equation from further work I was intending for thi lin of investigation. I can't work out whether the spin orbit equation, which is quite standard has its units from (which was extracted from wiki) or whether its something I've done wrong and its continued on through the work. Let's recall I had came to an equation from a quasi static solution for the gravimahnetic field:

∇x Γ = - 1/2c² ⋅∂Ω/∂t

B = 1/2mc² ⋅ 1/r ⋅(m/r² + 1/c² [Φ + φ]Ω x v) ⋅ Jq

I wrote down, in later work that I had not published on line, that if we recognise 

1/2mc² ⋅ 1/r = 1/e² 

Then the equation can be concisely written as

B = K (m/r² + 1/c² [Φ + φ]Ω x v)/e

where, 

K = J/2e 

A Josephson constant analogue. Believing this to be true, I went to the basics, we established that torsion was identified with 

Ω = 1/2mc² ⋅(dU/dt)

giving it units of inverse time, and is an object itself encoded in the original and standard equation which describes the spin orbit interaction formula. I also showed in my notes, that the cross product on B would give

B x v = 1/2emc² ⋅ 1/r ⋅ (dU/dt) Jv

and Jv has the same units as mc²r, or as we showed early, the square of the charge e². But heres the thing, I crushed these down to their dimensional analysis and it is, after a little rearranging to find the Lorentz force,

e(B x v) = 1/2mc² ⋅ 1/r ⋅ (dU/dt) Jv = (units of :  electric charge squared over one unit of time)

Its OK that we are missing the Coulomb constant, because in any reasonable argument we can set that to 1 anyway, what we cannot miss is that the correct dimensions requires one more unit of inverse time, and its also missing a squared inverse unit of celeritas (sped of light) because to make the inverse time a length, it must couple to a velocity term. Its not impossible to fix this of course, but I've been so careful throughout my last work, j am highly curious as to where the missing tim and extra speed of light squared went to, or another way to put it which is equally right, you can argue that we are simply missing an inverse squared factor of length. So how would we fix it? First of all, we might make an argument for the former case, that we are missing an inverse of time, that would mean that the equation for torsion would have to make a squared value and then we'd need to plug in the inverse speed of light squared to properly obtain the inverse square law to obtain the Coulomb equation, so we would have 

Ω² = 1/2mc² ⋅(d²U/dt²)

B = 1/2emc² ⋅ 1/c² (d²U/dt²) Jv

But there is an easier correction to the dimensions, one which I favor more because this object, is called the central potential, and its potential is a function of the radius

d²U(r)/dr²

The central potential deals with systems that are moving around a fixed axis. Without all the corrections, of another inverse time and that with another inverse squared speed of light, we can still argue, that while 

 1/2mc² ⋅(d²U/dr²)

does not have inverse time explicitly shown, we can loosely say it indirectly related to torsion, because we have hindsight in the equation

Ω = 1/2mc² ⋅(dU/dt)

Now the dimensions are correct, 

e(B x v) = 1/2mc² ⋅ 1/r ⋅ (dU/dr) Jv

we don't even need to express the parenthesis as a derivative that is second in nature, theoretically we can drag one of those inverse terms out and imply

e(B x v) = 1/2mc² ⋅ 1/r ⋅ (dU/dr) Jv

To make clear and light work of this, the units are now: charge squared by an inverse law of length squared which does match the units of a Lorentz force.

Edited June 22, 2021 by Dubbelosix

[Dubbelosix]

With these tweaks in mind, I will carefully show you now the full equation, which would feature the Coulomb constant. It simply is

k = 1/4πε

of course, plugging it in I childsplay, since the charge squared ratio with inverse length squared returns on the RHS, we get now 

e(B x v) = 1/8πε ⋅ 1/mc² ⋅ 1/r ⋅ (dU/dr) Jv = k/mc² ⋅ 1/r ⋅ (dU/dr) Jv

The factor of eight appears, because the already existing factor of 2 is multiplied through accordingly.

Edited June 22, 2021 by Dubbelosix

[Dubbelosix]

And now, at least, I've been able to work out why the dimensions where coming out wrong, because the torsion was identified appropriately, but in an admittedly lazy way, because the link to the spin orbit equation clearly specifies the radius term, where I lazily carried on the definition of torsion, without corrected the inverse length, so yes, it was an error, thankfully, one able to be corrected.

https://en.m.wikipedia.org/wiki/Spin–orbit_interaction

[Dubbelosix]

Noticing this small blunder, did have one upside, it shows the wiki article has failed also to plug in the Coulomb constant. This is either an oversight, or a ack of information based on the constant being set to the natural unit of 1. So its a bit educational from that viewpoint.