Igor wrote:
On Fri, 15 Aug 2003 12:33:30 +0000, ueb
wrote:

At the try to submit a (less important) contribution to s.p.res.,
I got following decision:

| In electrodynamics with magnetic monopoles, the magnetic
| field is not completely described by a vector potential -- it
| can't be, since B = curl A implies div B = 0, i.e., no monopoles.

If I understand the moderator properly, this would mean, one
introduces additional terms to curl A in order to get the
(non-existing !) magnetic monopoles.
As old-fashioned I am, I'd think that a theory, which forbids
non-existing things, be a proper theory, and a theory, which
predicts non-existing things, should be given up. But as I must
see, it appears conversely. With this background, one may understand
why people hate results gathered according to the old-fashioned
method, in which only observable quantities count. If these
old-fashioned methods even let see known particle numbers,
as in http://home.t-online.de/home/Ulrich.Bruchholz/ ,
nobody may experience such unwanted results.

Ulrich Bruchholz


If you introduce magnetic monopoles, then there will be an additional
scalar potential, the gradient of which contributes to the static
magnetic field, and an additional vector potential, the curl of which
contributes to the electric field. All in all, there will complete
symmetry between E and H. Note that without monopoles, there are
eight Maxwell equations, but only four independent potential
components (one scalar and three vector components). With monopoles,
we would have eight equations in eight potential components. Whether
monopoles exist or not is still an open issue.

May I ask a few questions ?
1.) What do you need complete symmetry between E and H for ?
E & H are quite different quantities, as you can well see in the four-
dimensional tensor equations. If nature says us that there is no
symmetry between E & H, why do people want to force such symmetry ?
2.) I have practically calculated with the Maxwell equations as well
as with the combined Einstein-Maxwell equations (see above reference).
In that, I took any degrees of freedom as given, with success.
In the pure Maxwell equations, four independent equations remain
for the four potential components. I learnt it so on the school,
and it was confirmed with my calculations. -
What do you need eight potential components for ?
- I read stuff in context with the quantization of charge, but that
is crap. In order to see the quantization, I can only name above
reference again. Note also my last reply to Robert J. Kolker.

Ulrich Bruchholz

ueb:
Igor wrote:
On Fri, 15 Aug 2003 12:33:30 +0000, ueb
wrote:

At the try to submit a (less important) contribution to s.p.res.,
I got following decision:

| In electrodynamics with magnetic monopoles, the magnetic
| field is not completely described by a vector potential -- it
| can't be, since B = curl A implies div B = 0, i.e., no monopoles.

If I understand the moderator properly, this would mean, one
introduces additional terms to curl A in order to get the
(non-existing !) magnetic monopoles.
As old-fashioned I am, I'd think that a theory, which forbids
non-existing things, be a proper theory, and a theory, which
predicts non-existing things, should be given up. But as I must
see, it appears conversely. With this background, one may understand
why people hate results gathered according to the old-fashioned
method, in which only observable quantities count. If these
old-fashioned methods even let see known particle numbers,
as in http://home.t-online.de/home/Ulrich.Bruchholz/ ,
nobody may experience such unwanted results.

Ulrich Bruchholz


If you introduce magnetic monopoles, then there will be an additional
scalar potential, the gradient of which contributes to the static
magnetic field, and an additional vector potential, the curl of which
contributes to the electric field. All in all, there will complete
symmetry between E and H. Note that without monopoles, there are
eight Maxwell equations, but only four independent potential
components (one scalar and three vector components). With monopoles,
we would have eight equations in eight potential components. Whether
monopoles exist or not is still an open issue.

May I ask a few questions ?
1.) What do you need complete symmetry between E and H for ?
E & H are quite different quantities, as you can well see in the four-
dimensional tensor equations. If nature says us that there is no
symmetry between E & H, why do people want to force such symmetry ?

The symmetry already exists in maxwell's equations. The apparent
asymmetry is one of convention in defining electric and magnetic
charges. If one assumes that matter contains both electric and
magnetic charge, then if the _ratio_ of electric to magnetic charge
is exactly the same in all matter, it's possible to _redefine_ the
electric and magnetic charges in terms of either electriv or magnetic
charges only as follows:

Assume maxwell's equations are symetric, i.e., there esist both a
magnetic charge and magnetic current so that:

div E = \rho_e
div B = \rho_m
curl E = J_m - dB/dt
curl E = J_e + dE/dt

Now, define new charge densities, \rho_e', \rho_m':

\rho_e' = \rho_e cos(A) + \rho_m sin(A)

\rho_m' = \rho_m cos(A) - \rho_e sin(A)

Require that \rho_m' = 0, so that,

\rho_m = \rho_e tan(A)

Thus, if the ratio of magnetic charge to electric charge is a constant,
(i.e., tan(A)), then the electric charge, \rho_e' is:

\rho_e' = \rho_e cos(A) + \rho_e tan(A) sin(A)

= \rho_e [ cos^2(A) + sin^2(A)]/cos(A) = \rho_e/cos(A)

Which then gives the "asymmetric" maxwell equations that are familiar.
It's only possible to distinguish electric from magnetic charge if
there exists some matter for which the ratio of electric to magnetic
charge is different than other matter, e.g., a magnetic monopole.

On Tue, 19 Aug 2003 12:08:39 +0000, ueb
wrote:

Igor wrote:
On Fri, 15 Aug 2003 12:33:30 +0000, ueb
wrote:

At the try to submit a (less important) contribution to s.p.res.,
I got following decision:

| In electrodynamics with magnetic monopoles, the magnetic
| field is not completely described by a vector potential -- it
| can't be, since B = curl A implies div B = 0, i.e., no monopoles.

If I understand the moderator properly, this would mean, one
introduces additional terms to curl A in order to get the
(non-existing !) magnetic monopoles.
As old-fashioned I am, I'd think that a theory, which forbids
non-existing things, be a proper theory, and a theory, which
predicts non-existing things, should be given up. But as I must
see, it appears conversely. With this background, one may understand
why people hate results gathered according to the old-fashioned
method, in which only observable quantities count. If these
old-fashioned methods even let see known particle numbers,
as in http://home.t-online.de/home/Ulrich.Bruchholz/ ,
nobody may experience such unwanted results.

Ulrich Bruchholz


If you introduce magnetic monopoles, then there will be an additional
scalar potential, the gradient of which contributes to the static
magnetic field, and an additional vector potential, the curl of which
contributes to the electric field. All in all, there will complete
symmetry between E and H. Note that without monopoles, there are
eight Maxwell equations, but only four independent potential
components (one scalar and three vector components). With monopoles,
we would have eight equations in eight potential components. Whether
monopoles exist or not is still an open issue.

May I ask a few questions ?
1.) What do you need complete symmetry between E and H for ?
E & H are quite different quantities, as you can well see in the four-
dimensional tensor equations. If nature says us that there is no
symmetry between E & H, why do people want to force such symmetry ?

Mostly for aesthetic reasons. As far back as Maxwell, people noticed
that there appeared to be missing source terms and so they began
filling them in. You would think that if magnetic monopoles existed,
we probably would have seen them by now. One rather intriguing
possibility that has been proposed is that they may really exist but
the gauge symmetry in Maxwell's equations is so severely broken that
the electric monopoles (electrons, quarks, etc...) have reasonable
masses, and are easier to see, but the magnetic monopoles end up so
massive, we haven't generated sufficient energies in accelerators yet
to see them.



2.) I have practically calculated with the Maxwell equations as well
as with the combined Einstein-Maxwell equations (see above reference).
In that, I took any degrees of freedom as given, with success.
In the pure Maxwell equations, four independent equations remain
for the four potential components. I learnt it so on the school,
and it was confirmed with my calculations. -
What do you need eight potential components for ?

Not that you really need them, but that they are already there in
Maxwell's equations to begin with. Without magnetic monopoles, four
of them just revert to arbitrary constants. The other four become the
active set of four potentials. My whole point was to show that, in
general, there are eight Maxwell equations in a possible eight
potential components. We don't usually look at it that way, but if
you allow for magnetic monopoles, then all eight potential components
will be non-trivial.

- I read stuff in context with the quantization of charge, but that
is crap. In order to see the quantization, I can only name above
reference again. Note also my last reply to Robert J. Kolker.

Ulrich Bruchholz

That was Dirac who derived that. Are you calling it crap? The source
of quantization of charge is still a great mystery. Frankly, I think
Dirac's derivation is, at this point, as good an explanation as any.
If you have one, I'd be glad to hear it.

Igor wrote:
On Tue, 19 Aug 2003 12:08:39 +0000, ueb
wrote:

Igor wrote:
On Fri, 15 Aug 2003 12:33:30 +0000, ueb
wrote:

At the try to submit a (less important) contribution to s.p.res.,
I got following decision:

| In electrodynamics with magnetic monopoles, the magnetic
| field is not completely described by a vector potential -- it
| can't be, since B = curl A implies div B = 0, i.e., no monopoles.

If I understand the moderator properly, this would mean, one
introduces additional terms to curl A in order to get the
(non-existing !) magnetic monopoles.
As old-fashioned I am, I'd think that a theory, which forbids
non-existing things, be a proper theory, and a theory, which
predicts non-existing things, should be given up. But as I must
see, it appears conversely. With this background, one may understand
why people hate results gathered according to the old-fashioned
method, in which only observable quantities count. If these
old-fashioned methods even let see known particle numbers,
as in http://home.t-online.de/home/Ulrich.Bruchholz/ ,
nobody may experience such unwanted results.

Ulrich Bruchholz


If you introduce magnetic monopoles, then there will be an additional
scalar potential, the gradient of which contributes to the static
magnetic field, and an additional vector potential, the curl of which
contributes to the electric field. All in all, there will complete
symmetry between E and H. Note that without monopoles, there are
eight Maxwell equations, but only four independent potential
components (one scalar and three vector components). With monopoles,
we would have eight equations in eight potential components. Whether
monopoles exist or not is still an open issue.

May I ask a few questions ?
1.) What do you need complete symmetry between E and H for ?
E & H are quite different quantities, as you can well see in the four-
dimensional tensor equations. If nature says us that there is no
symmetry between E & H, why do people want to force such symmetry ?

Mostly for aesthetic reasons.

I believe too.

As far back as Maxwell, people noticed
that there appeared to be missing source terms and so they began
filling them in.

Ah. Why do people miss source terms ? If there are no source terms,
is it not good so ? I know only the old method, and this consists
in it to take notice of the facts. I succeeded just in using the
old method.

You would think that if magnetic monopoles existed,
we probably would have seen them by now.

I think indeed so, and the gathered results from numerical simulations
agree with me.

One rather intriguing
possibility that has been proposed is that they may really exist but
the gauge symmetry in Maxwell's equations is so severely broken that
the electric monopoles (electrons, quarks, etc...) have reasonable
masses, and are easier to see,

Has ever one seen quarks ?

but the magnetic monopoles end up so
massive, we haven't generated sufficient energies in accelerators yet
to see them.

A nice hope. Why are you not satisfied with the reality ?

2.) I have practically calculated with the Maxwell equations as well
as with the combined Einstein-Maxwell equations (see above reference).
In that, I took any degrees of freedom as given, with success.
In the pure Maxwell equations, four independent equations remain
for the four potential components. I learnt it so on the school,
and it was confirmed with my calculations. -
What do you need eight potential components for ?

Not that you really need them, but that they are already there in
Maxwell's equations to begin with. Without magnetic monopoles, four
of them just revert to arbitrary constants. The other four become the
active set of four potentials. My whole point was to show that, in
general, there are eight Maxwell equations in a possible eight
potential components. We don't usually look at it that way, but if
you allow for magnetic monopoles, then all eight potential components
will be non-trivial.

The whole mess comes probably from the kind of notation. If you use
the tensor notation with F_{ik} = A_{i,k} - A_{k,i} , which means
nothing else than E = grad \phi and H = curl A together, you
will immediately see what crap it is to make these equations
symmetrical. E & H perform _quite_ different components in the
field tensor. What do you make with the four-dimensional vector
potential ?

- I read stuff in context with the quantization of charge, but that
is crap. In order to see the quantization, I can only name above
reference again. Note also my last reply to Robert J. Kolker.

Ulrich Bruchholz

That was Dirac who derived that. Are you calling it crap?

That may sound frivolously. But crap keeps crap also if a celebrity
did it. (I remember very intelligent remarks from Dirac, so I'd
rather suppose that he saw this proposal not as dogged. It was
probably rather a suggestion.

The source
of quantization of charge is still a great mystery.

Not more. :-)

Frankly, I think
Dirac's derivation is, at this point, as good an explanation as any.

Is that really a derivation ? Because it would immediately fail,
as I briefly demonstrated above.

If you have one, I'd be glad to hear it.

http://home.t-online.de/home/Ulrich.Bruchholz/
It is no closed formula, but you can see the quantization in the results
from numerical simulations.

Ulrich Bruchholz

Bilge wrote:
ueb:
...
May I ask a few questions ?
1.) What do you need complete symmetry between E and H for ?
E & H are quite different quantities, as you can well see in the four-
dimensional tensor equations. If nature says us that there is no
symmetry between E & H, why do people want to force such symmetry ?

The symmetry already exists in maxwell's equations. The apparent
asymmetry is one of convention in defining electric and magnetic
charges. If one assumes that matter contains both electric and
magnetic charge, then if the _ratio_ of electric to magnetic charge
is exactly the same in all matter, it's possible to _redefine_ the
electric and magnetic charges in terms of either electriv or magnetic
charges only as follows:

Assume maxwell's equations are symetric, i.e., there esist both a
magnetic charge and magnetic current so that:

div E = \rho_e
div B = \rho_m
curl E = J_m - dB/dt
curl E = J_e + dE/dt

Now, define new charge densities, \rho_e', \rho_m':

\rho_e' = \rho_e cos(A) + \rho_m sin(A)

\rho_m' = \rho_m cos(A) - \rho_e sin(A)

Require that \rho_m' = 0, so that,

\rho_m = \rho_e tan(A)

Thus, if the ratio of magnetic charge to electric charge is a constant,
(i.e., tan(A)), then the electric charge, \rho_e' is:

\rho_e' = \rho_e cos(A) + \rho_e tan(A) sin(A)

= \rho_e [ cos^2(A) + sin^2(A)]/cos(A) = \rho_e/cos(A)

Which then gives the "asymmetric" maxwell equations that are familiar.
It's only possible to distinguish electric from magnetic charge if
there exists some matter for which the ratio of electric to magnetic
charge is different than other matter, e.g., a magnetic monopole.

These formulae are a great work of art, but have unfortunately
nothing to do with nature. How can the symmetric "Maxwell equations"
fit an equivalent tensor notation ? (See also my remarks to Igor.)

Ulrich Bruchholz

Igor wrote

If you have one [derivation of charge quantization],
I'd be glad to hear it.

ueb replied

http://home.t-online.de/home/Ulrich.Bruchholz/
It is no closed formula, but you can see the quantization in the results
from numerical simulations.

[to Igor, and all persons who should understand what I did]
Are you glad now ?
If yes, please say it aloud.
If not, please say why.
- I should supplement that, together with the value of the charge,
the values of spin and magnetical momentum appear in the most cases.

Ulrich Bruchholz