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# The Curvilinear Relativistic Formula

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This thread will be used to describe the previous model we invented to create a semi classical wave equation, without directly inviting any monopole of the gravitational field. As I said on multiple occasions, gravity isn't a real force but it can be melded into the language of wave mechanics and can follow quantum mechanical rules. While the wave spreads out in space, it does not not necessarily follow from the previous model that we can talk about curvilinear geometric properties. The summary of the previous work was;

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t)

= - ħ/2m[a^2, b^2]^ψ (X,t)

≥ N/N_0  ψ (X,t)

(iff) [a^2, b^2] =0 then (ab +ba = 0) which is the Clifford algebra with

a^2 = 1

b^2 = 1

as unitary matrices. Using Schrodingers wave equation we find the final full equation as

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t) - h^2/(4πm)[a^2, b^2]^2/ ∂x^2 ψ (X,t) + V ≥  N/N_0  ψ (X,t)

where V is the Schrodinger potential.

It's a matter of convention that the Planck constant be the reduced version

[a^2,b^2] ≥ h/(2π

remember.... In order to correct the dimensions, the Planck constant has been absorbed with an energy term as B

i[a^2, b^2] ∂h/ ∂t ψ ≥ N/N_0  ψ

since the operators have to Hermitian quadratic observables, it follows from the Schrodinger Roberton Walker inequalities that

a^2, b^2  ≥ |1/2<{a,b}> - <a><b>|^2

|1/2<{a,b}> |^2

Where this <{a,b}> is the anticommutator respectively written as

<{a,b}> = ab + ba

by doing this, the quantum gravity wave equation is now completed with the unavoidable uncertainty laws. It is semi classical  in that respect and Is modelled on a four dimensional space with respect of the gravitational wave requiring it being a gunctiom of time. Without a Curvilinear interpretation/model, we can say, as unified as it appears, it is still a special case. This is whether equations now needs to be modified for particles in curved backgrounds. I've already written a mathematical model for this. Took me some time. In time, I will write this general theory up, and it's name willbe the Schrodinger-Einstein-Wigner-Dirac equation. Id never name a theory or model directly by my own name unless it was my middle name, Lee, a spin on the Lie algebra for a jokey name. Gareth is too much of a sledgehammer wit when you have the beautiful names above. Anyway, please be patient and hopefully this can all be written in a week or two

✌

Edited by Dubbelosix
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Even though I said I had already calculated the formulas from new representations of these equations in curved spacetime, after brushing them up just now has taken me about seven papers. So without further adue, let's try and make a start on this. You get some idea of how I arrived at some of these results, it would certainly benefit you to read my Christoffel symbol thread. Again. This will be arduous cut and paste so please be patient and maybe we cam finish this I'm one night 🌃

Edited by Dubbelosix
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Whem I spoke about the relativitistic correction,

= (∂ Γ)

It is also important to know how ot arises in general relativity for the Ricci curvature. It appears like

R  = ∂ Γ ΓΓ

because it has those essential space derivatives associated to the gradient with how geometry, more specifically curvature spreads through space with dimensions of inverse length squared . But this identity of the curvature, lacks sone essential ingredients which are important. While doing my essays for a boivector theory for gravity again, to get the full relationship, you can simply expand

(∂ Γ)(∂ Γ)

with appropriate indices, and from it you find the parallel transport from ordinary concepts of curvature and the geodesics that matter couples to. From it you find the antisymmetric part involving torsion. Again. In bivector theory, this part arises even more naturally than what you might expect I'm GR. I think ipersonally, bivector theory is more intuitive in this matter as it avoids unnecessary debates as to whether torsion should vanish as a symmetry of ordinary general relativity. When I imposed the gamma Pauli spin to it, we see it preserves with it the generally accepted laws of Poincare spacetime symmetries. In other words, anything preserving the Poincare symmetries, we should in principle expect it to be a real facet of nature. Expanding it we get a set of commutators which describes an uncertainty relationship between space and tend (this relationship is even supported by experiments)

<∇(1),∇(2)>

(∂ (1)Γ)(2)(∂ (2)Γ(1))

=-[∂ (1), Γ (2)]

+[∂ (2), Γ (1)]

+-[ Γ(1),  Γ (2)]

And cutting a longer set of calculations down, the result is three terms, the first two representing curvature and the final commutator being the non vanishing torsion part. Since commutators belong in the phase space, the torsion may very well be important for quantum interactions. Commutators are not special alone in quantum theory, tjey also appear on classical theory, it's only that commutators in the quantum theory appear to be rife in the Hilbert space. Now, I was a bit ambitious hoping to get all this done tonight, but it's been very taxing and I am getting tired

Hopefully, tomorrow now that I've explained curvature and torsion, we can talk in the next lesson concerning the spin space and Hoe the Pauli matrices are attached to the torsion amd the verbein connection. We'll be taking an interesting different definition to the spin connection as

ω = - Ω/2 =GL/2c^2R^3

where the spin connection is defined exactly by the torsion as Ω weighted by a factor of 2. Further defined by the gravitational.l constant G amd the angular momentum

by doing this we will plug it into the Covariant derivative D with some corrective factors. Ie. i/4 and a spin commutator σ_(ab) which will replace the notion of i[a^2,b^2], and it will soon enough describe our last equation in curved spacetime. So we have some fun threads ahead of us.

Edited by Dubbelosix
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This thread will be used to describe the previous model we invented to create a semi classical wave equation, without directly inviting any monopole of the gravitational field. As I said on multiple occasions, gravity isn't a real force but it can be melded into the language of wave mechanics and can follow quantum mechanical rules. While the wave spreads out in space, it does not not necessarily follow from the previous model that we can talk about curvilinear geometric properties. The summary of the previous work was;

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t)

= - ħ/2m[a^2, b^2]^ψ (X,t)

≥ N/N_0  ψ (X,t)

(iff) [a^2, b^2] =0 then (ab +ba = 0) which is the Clifford algebra with

a^2 = 1

b^2 = 1

as unitary matrices. Using Schrodingers wave equation we find the final full equation as

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t) - h^2/(4πm)[a^2, b^2]^2/ ∂x^2 ψ (X,t) + V ≥  N/N_0  ψ (X,t)

where V is the Schrodinger potential.

It's a matter of convention that the Planck constant be the reduced version

[a^2,b^2] ≥ h/(2π

remember.... In order to correct the dimensions, the Planck constant has been absorbed with an energy term as B

i[a^2, b^2] ∂h/ ∂t ψ ≥ N/N_0  ψ

since the operators have to Hermitian quadratic observables, it follows from the Schrodinger Roberton Walker inequalities that

a^2, b^2  ≥ |1/2<{a,b}> - <a><b>|^2

|1/2<{a,b}> |^2

Where this <{a,b}> is the anticummutator respectively written as

<{a,b}> = ab + ba

by doing this, the quantum gravity wave equation is now completed with the unavoidable uncertainty laws. It is semi classical  in that respect and Is modelled on a four dimensional space with respect of the gravitational wave requiring a full cation of time. Without a Curvilinear interpretation/model, we can say, as unified as it appears, it is still a special case. This is whether equations now needs to be modified for particles in curved backgrounds. I've already written a mathematical model for this. Took me some time. In time, I will write this general theory up, and it's name willbe the Schrodinger-Einstein-Wigner-Dirac equation. Id never name a theory or model directly by my own name unless it was my middle name, Lee, a spin on the Lie algebra for a jokey name. Gareth is too much of a sledgehammer wit when you have the beautiful names above

When I spoke about the relativitistic correction,

= (∂ Γ)

It is also important to know how ot arises in general relativity for the Ricci curvature. It appears like

R  = ∂ Γ ΓΓ

because it has those essential space derivatives associated to the gradient with how geometry, more specifically curvature spreads through space with dimensions of inverse length squared . But this identity of the curvature, lacks sone essential ingredients which are important. While doing my essays for a boivector theory for gravity again, to get the full relationship, you can simply expand

(∂ Γ)(∂ Γ)

with appropriate indices, and from it you find the parallel transport from ordinary concepts of curvature and the geodesics that matter couples to. From it you find the antisymmetric part involving torsion. Again. In bivector theory, this part arises even more naturally than what you might expect I'm GR. I think ipersonally, bivector theory is more intuitive in this matter as it avoids unnecessary debates as to whether torsion should vanish as a symmetry of ordinary general relativity. When I imposed the gamma Pauli spin to it, we see it preserves with it the generally accepted laws of Poincare spacetime symmetries. In other words, anything preserving the Poincare symmetries, we should in principle expect it to be a real facet of nature. Expanding it we get a set of commutators which describes an uncertainty relationship between space and tend (this relationship is even supported by experiments)

<∇(1),∇(2)>

(∂ (1)Γ)(2)(∂ (2)Γ(1))

=-[∂ (1), Γ (2)]

+[∂ (2), Γ (1)]

+-[ Γ(1),  Γ (2)]

And cutting a longer set of calculations down, the result is three terms, the first two representing curvature and the final commutator being the non vanishing torsion part. Since commutators belong in the phase space, the torsion may very well be important for quantum interactions. Commutators are not special alone in quantum theory, tjey also appear on classical theory, it's only that commutators in the quantum theory appear to be rife in the Hilbert space. Now, I was a bit ambitious hoping to get all this done tonight, but it's been very taxing and I am getting tired

Hopefully, tomorrow now that I've explained curvature and torsion, we can talk in the next lesson concerning the spin space and Hoe the Pauli matrices are attached to the torsion amd the verbein connection. We'll be taking an interesting different definition to the spin connection as

ω = - Ω/2 =GL/2c^2R^3

where the spin connection is defined exactly by the torsion as Ω weighted by a factor of 2. Further defined by the gravitational.l constant G amd the angular momentum

by doing this we will plug it into the Covariant derivative D with some corrective factors. Ie. i/4 and a spin commutator σ_(ab) which will replace the notion of i[a^2,b^2], and it will soon enough describe our last equation in curved spacetime

The spin matrice commutator will be written in normal convention of

σ(a,b)=i/2[λ(a)λ(b)]

and is related under close inspection of the localised equation

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t) - h^2/(4πm)[a^2, b^2]^2/ ∂x^2 ψ (X,t) + V ≥  N/N_0  ψ (X,t)

im particular. This part

i/(2π)[a^2, b^2]

and only differs by a factor of pi I'm the denominator as a strict convention to make Plancks constant reduced. Tp make the equation work for curvrd space, we must introduce the Covariant derivative,

D = ∂ - i/4  ω σ(a,b)

where the spin connection is

ω = - Ω/2 =GL/2c^2R^3

.This type of connection isn't new, it's found when applying it to the Dirac relativistic wave equation when applying it to curved geometry. What is new however is how we have defined it, to no less, a torsion and the equating of it to the gravitational constant G and the angular momentum L.

Edited by Dubbelosix
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The following model I will construct will be used to describe the previous model we invented to create a semi classical wave equation, without directly inviting any monopole of the gravitational field (graviton) just as there is no monopole of the magnetic force, so it's interesting to note that gravitomagnetic is a theory about linear gravity couplings at weak and larger perturbation pseudoforce unification with spin. As I said on multiple occasions, gravity isn't a real force but it can be melded into the language of wave mechanics and can follow quantum mechanical rules. While the wave spreads out in space, it does not not necessarily follow from the previous model that we can talk about curvilinear geometric properties.

When I spoke about the relativitistic correction to transition the Dirac spinor verbein connection, there are already noteworthy corrections to speak about, one as a good example is how Einstein added a correction term to spacetime Euclidean geometry,

= (∂ Γ)

It is also important to know how ot arises in general relativity for the Ricci curvature. It appears like

R  = ∂ Γ ΓΓ

because it has those essential space derivatives associated to the gradient with how geometry, more specifically curvature spreads through space with dimensions of inverse length squared . But this identity of the curvature, lacks sone essential ingredients which are important. While doing my essays for a boivector theory for gravity again, to get the full relationship, you can simply expand

(∂ Γ)(∂ Γ)

with appropriate indices, and from it you find the parallel transport from ordinary concepts of curvature and the geodesics that matter couples to. From it you find the antisymmetric part involving torsion. Again. In bivector theory, this part arises even more naturally than what you might expect I'm GR. I think ipersonally, bivector theory is more intuitive in this matter as it avoids unnecessary debates as to whether torsion should vanish as a symmetry of ordinary general relativity. When I imposed the gamma Pauli spin to it, we see it preserves with it the generally accepted laws of Poincare spacetime symmetries. In other words, anything preserving the Poincare symmetries, we should in principle expect it to be a real facet of nature. Expanding it we get a set of commutators which describes an uncertainty relationship between space and tend (this relationship is even supported by experiments)

<∇(1),∇(2)>

(∂ (1)Γ)(2)(∂ (2)Γ(1))

=-[∂ (1), Γ (2)]

+[∂ (2), Γ (1)]

+-[ Γ(1),  Γ (2)]

And cutting a longer set of calculations down, the result is three terms, the first two representing curvature and the final commutator being the non vanishing torsion part. Since commutators belong in the phase space, the torsion may very well be important for quantum interactions. Commutators are not special alone in quantum theory, tjey also appear on classical theory, it's only that commutators in the quantum theory appear to be rife in the Hilbert space. Now, I was a bit ambitious hoping to get all this done tonight but we'll see. In bivector theory, when I constructed a gravitational analogue I found that the torsion is naturally nonvanishing as it is fundamental to the bivector mathematical basis of the theory.

We can talk about the spin space and how the Pauli matrices are attached to the torsion amd the verbein connection. We'll be taking an interesting different definition to the spin connection as

ω = - Ω/2 =GL/2c^2R^3

As we explained before, this where the spin connection is defined exactly by the torsion as Ω weighted by a factor of 2 by convention of Heaviside equations. Further defined by the gravitational constant G and the angular momentum.

By doing this we will plug it into the Covariant derivative D with some corrective factors. Ie. i/4 and a spin commutator σ_(ab) which will replace the notion of i[a^2,b^2], and it will soon enough describe our last equation in curved spacetime. So we have some fun threads ahead of us. We already covered one way to do this, there is another we must speak about.

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t)

= - ħ/2m[a^2, b^2]^ψ (X,t)

≥ N/N_0  ψ (X,t)

(iff) [a^2, b^2] =0 then (ab +ba = 0) which is the Clifford algebra with

a^2 = 1

b^2 = 1

as unitary matrices. Using Schrodingers wave equation we find the final full equation as

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t) - h^2/(4πm)[a^2, b^2]^2/ ∂x^2 ψ (X,t) + V ≥  N/N_0  ψ (X,t)

where V is the Schrodinger potential.

It's a matter of convention that the Planck constant be the reduced version

[a^2,b^2] ≥ h/(2π

remember.... In order to correct the dimensions, the Planck constant has been absorbed with an energy term as B

i[a^2, b^2] ∂h/ ∂t ψ ≥ N/N_0  ψ

since the operators have to Hermitian quadratic observables, it follows from the Schrodinger Roberton Walker inequalities that

a^2, b^2  ≥ |1/2<{a,b}> - <a><b>|^2

|1/2<{a,b}> |^2

Where this <{a,b}> is the anticommutator respectively written as

<{a,b}> = ab + ba

by doing this, the quantum gravity wave equation is now completed with the unavoidable uncertainty laws. It is semi classical  in that respect and Is modelled on a four dimensional space with respect of the gravitational wave requiring it being a gunctiom of time. Without a Curvilinear interpretation/model, we can say, as unified as it appears, it is still a special case. This is whether equations now needs to be modified for particles in curved backgrounds. I've already written a mathematical model for this. Took me some time. In time, I will write this general theory up, and it's name willbe the Schrodinger-Einstein-Wigner-Dirac equation.

following model I will construct will be used to describe the previous model we invented to create a semi classical wave equation, without directly inviting any monopole of the gravitational field (graviton) just as there is no monopole of the magnetic force, so it's interesting to note that gravitomagnetic is a theory about linear gravity couplings at weak and larger perturbation pseudoforce unification with spin. As I said on multiple occasions, gravity isn't a real force but it can be melded into the language of wave mechanics and can follow quantum mechanical rules. While the wave spreads out in space, it does not not necessarily follow from the previous model that we can talk about curvilinear geometric properties.

When I spoke about the relativitistic correction to transition the Dirac spinor verbein connection, there are already noteworthy corrections to speak about, one as a good example is how Einstein added a correction term to spacetime Euclidean geometry,

= (∂ Γ)

It is also important to know how ot arises in general relativity for the Ricci curvature. It appears like

R  = ∂ Γ ΓΓ

because it has those essential space derivatives associated to the gradient with how geometry, more specifically curvature spreads through space with dimensions of inverse length squared . But this identity of the curvature, lacks sone essential ingredients which are important. While doing my essays for a boivector theory for gravity again, to get the full relationship, you can simply expand

(∂ Γ)(∂ Γ)

with appropriate indices, and from it you find the parallel transport from ordinary concepts of curvature and the geodesics that matter couples to. From it you find the antisymmetric part involving torsion. Again. In bivector theory, this part arises even more naturally than what you might expect I'm GR. I think ipersonally, bivector theory is more intuitive in this matter as it avoids unnecessary debates as to whether torsion should vanish as a symmetry of ordinary general relativity. When I imposed the gamma Pauli spin to it, we see it preserves with it the generally accepted laws of Poincare spacetime symmetries. In other words, anything preserving the Poincare symmetries, we should in principle expect it to be a real facet of nature. Expanding it we get a set of commutators which describes an uncertainty relationship between space and tend (this relationship is even supported by experiments)

<∇(1),∇(2)>

(∂ (1)Γ)(2)(∂ (2)Γ(1))

=-[∂ (1), Γ (2)]

+[∂ (2), Γ (1)]

+-[ Γ(1),  Γ (2)]

And cutting a longer set of calculations down, the result is three terms, the first two representing curvature and the final commutator being the non vanishing torsion part. Since commutators belong in the phase space, the torsion may very well be important for quantum interactions. Commutators are not special alone in quantum theory, tjey also appear on classical theory, it's only that commutators in the quantum theory appear to be rife in the Hilbert space. Now, I was a bit ambitious hoping to get all this done tonight but we'll see. In bivector theory, when I constructed a gravitational analogue I found that the torsion is naturally nonvanishing as it is fundamental to the bivector mathematical basis of the theory.

We can talk about the spin space and how the Pauli matrices are attached to the torsion amd the verbein connection. We'll be taking an interesting different definition to the spin connection as

ω = - Ω/2 =GL/2c^2R^3

As we explained before, this where the spin connection is defined exactly by the torsion as Ω weighted by a factor of 2 by convention of Heaviside equations. Further defined by the gravitational constant G and the angular momentum.

By doing this we will plug it into the Covariant derivative D with some corrective factors. Ie. i/4 and a spin commutator σ_(ab) which will replace the notion of i[a^2,b^2], and it will soon enough describe our last equation in curved spacetime. So we have some fun threads ahead of us. We already covered one way to do this, there is another we must speak about.

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t)

= - ħ/2m[a^2, b^2]^ψ (X,t)

≥ N/N_0  ψ (X,t)

(iff) [a^2, b^2] =0 then (ab +ba = 0) which is the Clifford algebra with

a^2 = 1

b^2 = 1

as unitary matrices. Using Schrodingers wave equation we find the final full equation as

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t) - h^2/(4πm)[a^2, b^2]^2/ ∂x^2 ψ (X,t) + V ≥  N/N_0  ψ (X,t)

where V is the Schrodinger potential.

It's a matter of convention that the Planck constant be the reduced version

[a^2,b^2] ≥ h/(2π

remember.... In order to correct the dimensions, the Planck constant has been absorbed with an energy term as B

i[a^2, b^2] ∂h/ ∂t ψ ≥ N/N_0  ψ

since the operators have to Hermitian quadratic observables, it follows from the Schrodinger Roberton Walker inequalities that

a^2, b^2  ≥ |1/2<{a,b}> - <a><b>|^2

|1/2<{a,b}> |^2

Where this <{a,b}> is the anticommutator respectively written as

<{a,b}> = ab + ba

by doing this, the quantum gravity wave equation is now completed with the unavoidable uncertainty laws. It is semi classical  in that respect and Is modelled on a four dimensional space with respect of the gravitational wave requiring it being a gunctiom of time. Without a Curvilinear interpretation/model, we can say, as unified as it appears, it is still a special case. This is whether equations now needs to be modified for particles in curved backgrounds. I've already written a mathematical model for this. Took me some time. In time, I will write this general theory up, and it's name willbe the Schrodinger-Einstein-Wigner-Dirac equation.

Now, we should investigate this sin space under the bivector model I invented for analogy to the one which already exists for the Em field. The bivector theorems I published were as essays sent to the gravity research foundation.

∇D = D +  iσ( D)

Where ∧is the wedge product. Thisrsms it can be rewritten as

∇D = D +  iσ⋅(Γ × D)

Where this time × is the cross product and without any silly agents about whether torsion vanishes in general relatobity, we understand it is a crucial non vanishing component when gravity I'd looked at through the spectacles of bivector. This is because the last term on the previous equations defined a crucial relationship with torsion as

-(Γ × D) =  Ω/∂t

And the energy required to make an object spin I'm curved torsion background requires

K = L × Ω

which is the angular momentum cross product with torsion.

Edited by Dubbelosix
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Now. I'm going to take a fag amd coffee break, and we'll plug the Covariant derivative in to describe curved spacetime and we will get the curved version of the relativistic equation in the style of Wigner Localisation.

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So to finally construct this in curved space we recognise we habe three equations we must use, the first being the special case, or "master equation".... We only call equations a master equation when the rest of physics must follow from it

Equation 1.

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t) - h^2/(4πm)[a^2, b^2]^2/ ∂x^2 ψ (X,t) + V ≥  N/N_0  ψ (X,t)

Equation  2.

The Covariant derivative for curved spacetime

D = ∂ - i/4  ω σ(a,b)

Equation 3.

where the spin connection is

ω = - Ω/2 =GL/2c^2R^3

we first cut equation 1. down to size as

i/(2π)[a^2, b^2] ∂h/ ∂t ψ (X,t) ≥  N/N_0  ψ (X,t)

This part

∂ - i/(2π)[a^2, b^2]

becomes modified to appreciate the terms. A bit like a correction factor from

i∂ - i/4  ω σ(a,b)

Giving

∂ - i/(4π)  ω σ(a,b)

since  ω  has units of inverse time. We see then naturally it is encoded in

∂h/ ∂t

In such a case we must attach h to

∂ - i/(4π)  hω σ(a,b)

where we recognise hω  as an energy.

Of course we have one partial derivative left so the Planck constant should be taken ad a parenthesis.

(∂ - i/(4π) ω σ(a,b))h

Everything inside the parenthesis has units of an inverse length which is now being multiplied by the Planck constant, which is an action.of units of mass x velocity x length. That would mean the object above has units now of momentum mass x velocity

We cannot haphazardly plug in results unless we keep close attention to the units. On the RHS of the master equation,  B  has units of energy times Plancks constant. Tjat is,  mass^2 x velocity^3 x length. Tje ratio has been taken as dimensionless N/N_0. In order toake the dimensions equal, we will habe to do something mathematical physicists call a "smudge factor." obtaining it's units of mass x velocity, the RHS was to be divided by a new quantity of an inverse of mass x velocity squared and a factor of length. In physics. We can interpret these dimensions in a few different ways.

First being  mass x velocity squared is an energy. Energy times a length gives us units of a charge often denoted as either a lower case e ir s higher case Q.

The second will still yield s charge. We would have Plancks constant from the products of mass, length and velocity. With one such unit of velocity left, hv is approximated as s charge. And if it had been hc it would be precisely the Planck charge with units of the Gravitational constant times mass squared. Bearing this on mind, the final (Consistent) form.of the proposed relativistic localisation formula would have to look like l

(∂ - i/(4π) ω σ(a,b)) ψ(X,t)

≥  B/Q N/N_0  ψ(X,t)

≡- iħ N/N_0 (∂ ψ(X,t) /∂ x)

The last term is the equivalent to the momentum operator.

I'll double check in good time to see if my calculations hold up its been very arduous so please be patient.

Edited by Dubbelosix

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