For Weber electrodynamics, we don't need to specify a specific gauge. I understand that in Maxwell's electrodynamics, in order to have Gauss's law mathematically correct when the scalar and vector potential functions are explicitly specified, the Coulomb gauge is used. No need to do this in Weber electrodynamics. Gauge freedom applies to Weber's electrodynamics. No need to specify a gauge.

- Researcher720
**Posts:**15**Joined:**Mon Jun 18, 2018 12:29 pm

After talking to other researcher's, I think I understand their point of view on Maxwell much better. They effectively say:

1) Maxwell's equations are postulated to hold true. So you can not mathematically verify them. They are stated to be true as the starting postulate.

2) Then, writing the fields E and B in terms of scalar and vector potential functions, phi and A, is postulated to be true (without specifying what the scalar and vector potential functions are) and so you can't mathematically verify Maxwell's equations of the fields in terms of the potential functions.

3) But once you explicitly write the scalar and vector potential functions in terms of the charge density and current density functions, then Maxwell's equations are not mathematically valid, in general. But some of Maxwell's equations may be valid in specific gauges. But not all of Maxwell's equations, for a specific fixed gauge, are mathematically valid at the same time. For example, Gauss's law (electric) is valid in the Coulomb gauge, but that doesn't mean any of the other Maxwell's equations are valid in the Coulomb gauge. And Gauss's Law is not necessarily mathematically valid in other gauges.

It seems a bit convoluted and confusing to me.... On the other hand, for Weber's electrodynamics, we don't have to deal with such issues. Weber is mathematically valid right from the start, assuming only that the energy function for the energy between two charged bodies, is valid. Everything else in Weber's electrodynamics follows from that single function. Seems like a much simpler approach.

1) Maxwell's equations are postulated to hold true. So you can not mathematically verify them. They are stated to be true as the starting postulate.

2) Then, writing the fields E and B in terms of scalar and vector potential functions, phi and A, is postulated to be true (without specifying what the scalar and vector potential functions are) and so you can't mathematically verify Maxwell's equations of the fields in terms of the potential functions.

3) But once you explicitly write the scalar and vector potential functions in terms of the charge density and current density functions, then Maxwell's equations are not mathematically valid, in general. But some of Maxwell's equations may be valid in specific gauges. But not all of Maxwell's equations, for a specific fixed gauge, are mathematically valid at the same time. For example, Gauss's law (electric) is valid in the Coulomb gauge, but that doesn't mean any of the other Maxwell's equations are valid in the Coulomb gauge. And Gauss's Law is not necessarily mathematically valid in other gauges.

It seems a bit convoluted and confusing to me.... On the other hand, for Weber's electrodynamics, we don't have to deal with such issues. Weber is mathematically valid right from the start, assuming only that the energy function for the energy between two charged bodies, is valid. Everything else in Weber's electrodynamics follows from that single function. Seems like a much simpler approach.

- Researcher720
**Posts:**15**Joined:**Mon Jun 18, 2018 12:29 pm

Hi Jack, There is an idea in physics that using mathematics can help discover new physical things and help explore new ideas about the physical Universe. And that if the mathematics is contradictory then there is something wrong with the mathematical description of the physics indicating that we don't understand what is going on. The idea is that once the physical can be comprehended by the logical mind, then it can be written down in mathematics so that other people can comprehend it. But if the math leads to a logical/mathematical contradiction, then there is something wrong with the understanding of the physical Universe. But, don't forget, doing math is not necessarily doing physics. Need experimentation, for sure. Experimentation and math work together to make up "physics"....

So, I think it is very important to have a logical, verifiable mathematical description of electrodynamics. Maxwell's mathematical description doesn't appear to be adequate. Weber seems to be better, but is not the last word for electrodynamics. I am hoping others will start looking at Weber electrodynamics

http://www.weberelectrodynamics.com/

and help extend it.

So, I think it is very important to have a logical, verifiable mathematical description of electrodynamics. Maxwell's mathematical description doesn't appear to be adequate. Weber seems to be better, but is not the last word for electrodynamics. I am hoping others will start looking at Weber electrodynamics

http://www.weberelectrodynamics.com/

and help extend it.

- Researcher720
**Posts:**15**Joined:**Mon Jun 18, 2018 12:29 pm

Hi celeste, For (3), consider Gauss's Law (does anyone know how to enter math on this forum?)

div E = (1/epsilon_0)rho

This is one of Maxwell's equations (SI units). This is assumed to be true for electrodynamics, but note it was derived only for electrostatics (for example, in Chapter 1 of Jackson on electrostatics). When E is specified in terms of the scalar and vector potential functions

E = -grad Phi - @ A/@t

(here "@" is the partial derivative symbol) Gauss's law above is still assumed to be true.

But once you write the scalar potential function, Phi, in terms of the charge density function, and the vector potential function, A, in terms of the current density function, then it is easy to show that

div E = (1/epsilon_0) rho - @(div A)/@t

which is not Gauss's law because of the @(div A)/@t part. So, in general, once the potential functions are written in terms of the charge and current density functions, Maxwell's equations are not mathematically valid, in general.

However, if the special condition (Coulomb gauge condition) that div A = 0 is imposed, then Gauss's law remains valid. But 1) this fixes the gauge, removing the "gauge freedom" of Maxwell's electrodynamics, and 2) there is no guarantee that any of the other Maxwell's equations remain valid. Indeed, the Ampere's Law is not valid....

I've written a lot about this on my Weber electrodynamics website at

http://www.weberelectrodynamics.com/

In particular, for this topic

http://www.weberelectrodynamics.com/Max ... xwell.html

Weber electrodynamics doesn't have this problem.

div E = (1/epsilon_0)rho

This is one of Maxwell's equations (SI units). This is assumed to be true for electrodynamics, but note it was derived only for electrostatics (for example, in Chapter 1 of Jackson on electrostatics). When E is specified in terms of the scalar and vector potential functions

E = -grad Phi - @ A/@t

(here "@" is the partial derivative symbol) Gauss's law above is still assumed to be true.

But once you write the scalar potential function, Phi, in terms of the charge density function, and the vector potential function, A, in terms of the current density function, then it is easy to show that

div E = (1/epsilon_0) rho - @(div A)/@t

which is not Gauss's law because of the @(div A)/@t part. So, in general, once the potential functions are written in terms of the charge and current density functions, Maxwell's equations are not mathematically valid, in general.

However, if the special condition (Coulomb gauge condition) that div A = 0 is imposed, then Gauss's law remains valid. But 1) this fixes the gauge, removing the "gauge freedom" of Maxwell's electrodynamics, and 2) there is no guarantee that any of the other Maxwell's equations remain valid. Indeed, the Ampere's Law is not valid....

I've written a lot about this on my Weber electrodynamics website at

http://www.weberelectrodynamics.com/

In particular, for this topic

http://www.weberelectrodynamics.com/Max ... xwell.html

Weber electrodynamics doesn't have this problem.

Last edited by Researcher720 on Sun Jul 08, 2018 4:58 am, edited 1 time in total.

- Researcher720
**Posts:**15**Joined:**Mon Jun 18, 2018 12:29 pm

"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina

- Higgsy
**Posts:**217**Joined:**Wed Mar 08, 2017 3:32 pm

"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina

- Higgsy
**Posts:**217**Joined:**Wed Mar 08, 2017 3:32 pm

"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina

- Higgsy
**Posts:**217**Joined:**Wed Mar 08, 2017 3:32 pm

- Higgsy
**Posts:**217**Joined:**Wed Mar 08, 2017 3:32 pm

Hi Higgsy, Thanks for your comments. You wrote: Weber "... does not lead to the prediction of electromagnetic waves and the correct speed of light." Yes it does. The "old" or original interpretation of Weber did not include waves. It was an instantaneous action at a distance theory. But when I wrote Weber in terms of "fields", it looks like Maxwell's theory (in symbols). And when I introduced time retardation (following J. P. Wesley's method for introducing time retardation) Weber's theory gives waves with speed c. These are new results of my work, so I don't expect others to know about it all.

You wrote: "By definition the magnetic field remain unchanged under transformations of the vector potential by addition of the gradient of a scalar function, and the electric field remains unchanged under transformation of the scalar potential by the addition of the time derivative of a scalar function. So the classical theory of electromagnetism has this gauge freedom and is a gauge theory." Yes. And that, now, also holds for Weber's electrodynamics. So then why must we specify the Coulomb gauge (in Maxwell) for Gauss's law to be valid? If the gauge choice doesn't change the E field, then Gauss's law should remain valid regardless of the gauge specified. But Gauss's law, in Maxwell, is only valid in the Coulomb gauge.

And Ampere's Law doesn't hold mathematically. Take the mathematical definition of the vector potential function, A, and put it in left-hand side of Ampere's law

(\nabla x B = \nabla x (\nabla x A))

and you can't get the right hand side of Ampere's law. Is there some other gauge that makes this work, because it doesn't work in the Coulomb gauge. So, picking a gauge may make some of Maxwell's 4 equations mathematically correct (left-hand side equals right-hand side) but doesn't make all 4 Maxwell equations correct.

My approach, which is very simple, makes all 4 "corrected" Maxwell's equations correct for the current density and charge density functions.

You wrote: "Note that in QFT the magnetic vector potential has physical meaning in that it can be used to predict experimentally verifiable physical effects such as the Aharonov-Bohm effect." Thanks for reminding me. I meant to look at the Aharonov-Bohm effect in terms of Weber. Note that Weber's force law includes the Lorentz force law and other terms. Maybe these other terms give the Aharonov-Bohm effect.... I'll look at it.

You wrote: "As far as I am concerned the problem isn't whether there is gauge freedom in the theory or not, but whether the theory can be shown to be an accurate representation of experimental results, which test Maxwell's theory has passed with flying colours for more than 100 years." Yes, but those same tests also verifies Weber's electrodynamics. Note that Lorentz force law is contained in Weber's force law (as dominate terms). For experiments based physics, note that Ampere's force equation (not Ampere's law of Maxwell equations) was developed **from experimentation** by Ampere. (See Assis's book Ampere's electrodynamics, Graneau's book Ampere-Neumann Electrodynamics of Metals, Graneau's book Newtonian Electrodynamics, Wesley's book Selected Topics In Scientific Physics, and papers in the literature...) Why then isn't Ampere's force law in mainstream physics textbooks? It was developed from experiments... Oh, and Ampere's force law is also in Weber's force law. See http://www.weberelectrodynamics.com/Web ... Force.html

The typical experiment showing Lorentz force law alone is lacking are the so-called "exploding wire" experiments. This requires a longitudinal force along the current carrying wire to rip apart the wire. But the Lorentz force equation doesn't have a longitudinal force. Ampere's force equation (and hence Weber's force equation) does. There are other experiments that indicate the Lorentz force is not enough to describe all electrodynamics effects.

You wrote: "So I ask again, in your attempted derivation of Gauss's law using Maxwell's theory, which gauge are you working in?" I am not working in any particular gauge. I consider Maxwell's equations and potentials and Lorentz force and continuity equation as a single mathematical **system** of equations. Meaning that they **should** all be mathematically consistent with one another. They aren't. Just try (as I know you have) Gauss's law with the E field written in terms of grad Phi and @A/@t . It doesn't hold. Why should it? It was derived only for the electrostatic case (E = -grad Phi) and not for the electrodynamics case. If you want to use the Coulomb gauge, o.k. Then try to get Ampere's law to hold in the same Coulomb gauge. It doesn't work. (It all works with Weber's electrodynamics.)

You wrote: "By definition the magnetic field remain unchanged under transformations of the vector potential by addition of the gradient of a scalar function, and the electric field remains unchanged under transformation of the scalar potential by the addition of the time derivative of a scalar function. So the classical theory of electromagnetism has this gauge freedom and is a gauge theory." Yes. And that, now, also holds for Weber's electrodynamics. So then why must we specify the Coulomb gauge (in Maxwell) for Gauss's law to be valid? If the gauge choice doesn't change the E field, then Gauss's law should remain valid regardless of the gauge specified. But Gauss's law, in Maxwell, is only valid in the Coulomb gauge.

And Ampere's Law doesn't hold mathematically. Take the mathematical definition of the vector potential function, A, and put it in left-hand side of Ampere's law

(\nabla x B = \nabla x (\nabla x A))

and you can't get the right hand side of Ampere's law. Is there some other gauge that makes this work, because it doesn't work in the Coulomb gauge. So, picking a gauge may make some of Maxwell's 4 equations mathematically correct (left-hand side equals right-hand side) but doesn't make all 4 Maxwell equations correct.

My approach, which is very simple, makes all 4 "corrected" Maxwell's equations correct for the current density and charge density functions.

You wrote: "Note that in QFT the magnetic vector potential has physical meaning in that it can be used to predict experimentally verifiable physical effects such as the Aharonov-Bohm effect." Thanks for reminding me. I meant to look at the Aharonov-Bohm effect in terms of Weber. Note that Weber's force law includes the Lorentz force law and other terms. Maybe these other terms give the Aharonov-Bohm effect.... I'll look at it.

You wrote: "As far as I am concerned the problem isn't whether there is gauge freedom in the theory or not, but whether the theory can be shown to be an accurate representation of experimental results, which test Maxwell's theory has passed with flying colours for more than 100 years." Yes, but those same tests also verifies Weber's electrodynamics. Note that Lorentz force law is contained in Weber's force law (as dominate terms). For experiments based physics, note that Ampere's force equation (not Ampere's law of Maxwell equations) was developed **from experimentation** by Ampere. (See Assis's book Ampere's electrodynamics, Graneau's book Ampere-Neumann Electrodynamics of Metals, Graneau's book Newtonian Electrodynamics, Wesley's book Selected Topics In Scientific Physics, and papers in the literature...) Why then isn't Ampere's force law in mainstream physics textbooks? It was developed from experiments... Oh, and Ampere's force law is also in Weber's force law. See http://www.weberelectrodynamics.com/Web ... Force.html

The typical experiment showing Lorentz force law alone is lacking are the so-called "exploding wire" experiments. This requires a longitudinal force along the current carrying wire to rip apart the wire. But the Lorentz force equation doesn't have a longitudinal force. Ampere's force equation (and hence Weber's force equation) does. There are other experiments that indicate the Lorentz force is not enough to describe all electrodynamics effects.

You wrote: "So I ask again, in your attempted derivation of Gauss's law using Maxwell's theory, which gauge are you working in?" I am not working in any particular gauge. I consider Maxwell's equations and potentials and Lorentz force and continuity equation as a single mathematical **system** of equations. Meaning that they **should** all be mathematically consistent with one another. They aren't. Just try (as I know you have) Gauss's law with the E field written in terms of grad Phi and @A/@t . It doesn't hold. Why should it? It was derived only for the electrostatic case (E = -grad Phi) and not for the electrodynamics case. If you want to use the Coulomb gauge, o.k. Then try to get Ampere's law to hold in the same Coulomb gauge. It doesn't work. (It all works with Weber's electrodynamics.)

- Researcher720
**Posts:**15**Joined:**Mon Jun 18, 2018 12:29 pm

- Higgsy
**Posts:**217**Joined:**Wed Mar 08, 2017 3:32 pm

- Higgsy
**Posts:**217**Joined:**Wed Mar 08, 2017 3:32 pm

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