quantization of electric charge [Archive]

alistair

Aug18-04, 11:33 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nWhy would the existence of magnetic monopoles mean that electric\ncharge is quantized as Paul Dirac thought?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Why would the existence of magnetic monopoles mean that electric
charge is quantized as Paul Dirac thought?

Urs Schreiber

Aug19-04, 04:33 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"alistair" <alistair@goforit64.fsnet.co.uk> schrieb im Newsbeitrag\nnews:861c1b21.0408180826.71e002b8@pos ting.google.com...\n>\n> Why would the existence of magnetic monopoles mean that electric\n> charge is quantized as Paul Dirac thought?\n\nThis is very easy to see once you know the differential geometric\nformulation of the electromagnetic field.\n\nLet me first give the answer in that language and then comment on some\ndetails.\n\nWhen F denotes the Faraday 2-form, the electric charge e enclosed in some\nsurface S is\n\ne = int_S *F,\n\nthe integral of the Hodge dual *F of F over that surface (this is true only\nin 4 spacetime dimensions but has straightforward generalizations in higher\ndimensions).\n\nSimilarly the magnetic charge g enclosed in S is computed by\n\ng = int_S F\n\nthe integral of F itself over that surface.\n\nNow assume that S is convex and that inside of S a single magnetic monopole\nis sitting and no other sources are present. It follows that\n\ndF = 0\n\neverywhere apart from the location of the monopole.\n\nBy the Poincare lemma a locally closed form is locally exact on a starshaped\nregion. This means that there is a 1-form "vector potential" A defined\neverywhere except on a ray originating at the monopole and going out to\ninfinity (or outside the coordinate patch that we are working in, anyway)\nsuch that\n\nF = dA\n\nwhere A is defined.\n\nThe line on which A is not defined is called the "Dirac string".\n\nThis has nothing to do with strings in the sense of string theory, it could\njust as well be called the "Dirac ray". Note furthermore that the direction\nin which the Dirac ray runs is completely arbitrary. We can find infinitely\nmany A which satisfy F = dA outside _some_ "Dirac ray". This is essentially\nsomething like a choice of coordinate patch and the "Dirac ray" is something\nlike a coordinate singularity. It has no physical meaning whatsoever but is\njust an artifact of our description of reality.\n\nSince we assumed S to be convex the "Dirac ray" pokes through S in a single\npoint, the single point on S on which A is not defined.\n\nNow cut out of S a small disk D being a neighbourhood of that point. By\nmaking D arbitrarily small the integral\n\nint_{S-D} F\n\ncomes arbitrarily close to the true integral int_S F that we are integrested\nin, so we can just as well try to compute int_{S-D} F for a very small\ndisk D cut out of S.\n\nThe advantage is that by the above considerations we can write F on the\nintegration region S-D as F = dA. So the integral we want to compute now\nreads\n\ng = int_{S-D} dA ,\n\ni..e. it is the integral of an exact form over a bounded surface. By the\ncelebrated Stokes\' theorem this is equal to\n\ng = int_{@(S-D)} A,\n\nwhere @(S-D) is the boundary of S-D, namely the small circle around D which\nruns around the "Dirac ray", i.e. this is equal to the integral of the\n1-form A over that circle.\n\nFine. But now we must recall that the Dirac ray must not have any physical\nconsequences, since it is just an artifact of our language. So whenever you\nfind any physically relevant expression which does involve int_{@(S-D)} A\nyou get, by demanding the invisibility of this term, a certain condition.\n\nSuch a situation does occur as soon as we put electrically charged quantum\nparticles in the field of the magnetic monopole. Namely the action of an\nelectrically charged relativistic particle in the field described by a\nvector potential A is\n\nS = int_L (m ds + eA)\n\nwhere the integral runs over the worldline L of the particle.\n\nIn the path integral for the particle we integrate over\n\nexp(iS) .\n\nNow, if the magnetic monopole were absent, paths going around the small\ncircle @(S-D) would contribute a vanishingly small action S, since int_L ds\nis nothing but the circumference of the small circle, which we send to 0.\n\nBut if the magnetic monopole is present the above tells us that even for\ninfinitely small circumference of the circle @(S-D) around the Dirac ray,\nthe action has the value\n\nS = e int_L A = e int_{@(S-A)} A .\n\nBut except for the prefactor e this is just the expression for the charge of\nthe magnetic monopole enclosed in S, as derived above. Hence we find that\nfor these infinitesimally small paths the action is just the product of the\nelectric charge of the particle and the magnetic charge of the magnetic\nmonopole:\n\nS = eg for L = @(S-D)\n\nBut since the Dirac string must be "invisible" we cannot allow this action\nto give a different contribution to the path integral that the action S=0\nwhich we obtain when an infinitely small worldline L does not encircle the\n"Dirac ray". But since\n\nexp(i S )\n\nfor S=0 is just equal to 1, we find that also in the presence of the "Dirac\nray" this has to equal unity, i.e. we find that\n\nexp(i eg) = 1 .\n\nBut this is true if and only if the product of the electric and the magnetic\ncharge satisfies\n\ne g = n 2pi\n\nfor any integer n.\n\nThis is the famous Dirac quantization condition. Solved for e it says that\ngiven any magnetic monopole with magnetic charge g in the world, we\nimmediately know that the electric charge must be quantized as\n\ne = n 2 pi/g\n\nfor some integer n.\n\n\nNow some comments: In case you don\'t know what a 2-form is, or what the\nFaraday 2-form F is, or what exact and closed forms are, or what Stokes\'\ntheorem says, pick up the book\n\nTheodore Frankel,\nThe Geometry of Phyiscs\nCambridge (1997)\n\nand look these things up. In order to understand what I have written above\nit suffices to be familiar with the content of\n\nsection 2.5 The Grassmann or Exterior Algebra\n\nsection 2.6 Exterior Differentiation\n\nsection 3 Integration of Differential Forms\n\nin particular\n\nsection 3.3.b Stokes\' Theorem\n\nand of course\n\nsection 3.5 Maxwell\'s equations.\n\nIf you are interested in explicit expressions for the vector potential A of\na monopole field read section 5.5 of Frankel\'s book.\n\nIn general: Read Frankel\'s book!\n\nIn my above exposition I have mainly followed page 21 of\n\nMaximilian Kreuzer:\nGeometrische Methoden der Theoretischen Physik\nhttp://hep.itp.tuwien.ac.at/~kreuzer/inc/gmtp.pdf\n\n(the text is in English, only the title is German).\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"alistair" <alistair@goforit64.fsnet.co.uk> schrieb im Newsbeitrag
news:861c1b21.0408180826.71e002b8@posting.google.c om...
>
> Why would the existence of magnetic monopoles mean that electric
> charge is quantized as Paul Dirac thought?

This is very easy to see once you know the differential geometric
formulation of the electromagnetic field.

Let me first give the answer in that language and then comment on some
details.

When F denotes the Faraday 2-form, the electric charge e enclosed in some
surface S is

e = \int_S *F,

the integral of the Hodge dual *F of F over that surface (this is true only
in 4 spacetime dimensions but has straightforward generalizations in higher
dimensions).

Similarly the magnetic charge g enclosed in S is computed by

g = \int_S F

the integral of F itself over that surface.

Now assume that S is convex and that inside of S a single magnetic monopole
is sitting and no other sources are present. It follows that

dF =

everywhere apart from the location of the monopole.

By the Poincare lemma a locally closed form is locally exact on a starshaped
region. This means that there is a 1-form "vector potential" A defined
everywhere except on a ray originating at the monopole and going out to
infinity (or outside the coordinate patch that we are working in, anyway)
such that

F = dA

where A is defined.

The line on which A is not defined is called the "Dirac string".

This has nothing to do with strings in the sense of string theory, it could
just as well be called the "Dirac ray". Note furthermore that the direction
in which the Dirac ray runs is completely arbitrary. We can find infinitely
many A which satisfy F = dA outside _some_ "Dirac ray". This is essentially
something like a choice of coordinate patch and the "Dirac ray" is something
like a coordinate singularity. It has no physical meaning whatsoever but is
just an artifact of our description of reality.

Since we assumed S to be convex the "Dirac ray" pokes through S in a single
point, the single point on S on which A is not defined.

Now cut out of S a small disk D being a neighbourhood of that point. By
making D arbitrarily small the integral

\int_{S-D} F

comes arbitrarily close to the true integral \int_S F that we are integrested
in, so we can just as well try to compute \int_{S-D} F for a very small
disk D cut out of S.

The advantage is that by the above considerations we can write F on the
integration region S-D as F = dA. So the integral we want to compute now
reads

g = \int_{S-D} dA ,

i..e. it is the integral of an exact form over a bounded surface. By the
celebrated Stokes' theorem this is equal to

g = \int_{@(S-D)} A,

where @(S-D) is the boundary of S-D, namely the small circle around D which
runs around the "Dirac ray", i.e. this is equal to the integral of the
1-form A over that circle.

Fine. But now we must recall that the Dirac ray must not have any physical
consequences, since it is just an artifact of our language. So whenever you
find any physically relevant expression which does involve \int_{@(S-D)} A
you get, by demanding the invisibility of this term, a certain condition.

Such a situation does occur as soon as we put electrically charged quantum
particles in the field of the magnetic monopole. Namely the action of an
electrically charged relativistic particle in the field described by a
vector potential A is

S = \int_L (m ds + eA)

where the integral runs over the worldline L of the particle.

In the path integral for the particle we integrate over

\exp(iS) .

Now, if the magnetic monopole were absent, paths going around the small
circle @(S-D) would contribute a vanishingly small action S, since \int_L ds
is nothing but the circumference of the small circle, which we send to .

But if the magnetic monopole is present the above tells us that even for
infinitely small circumference of the circle @(S-D) around the Dirac ray,
the action has the value

S = e \int_L A = e \int_{@(S-A)} A .

But except for the prefactor e this is just the expression for the charge of
the magnetic monopole enclosed in S, as derived above. Hence we find that
for these infinitesimally small paths the action is just the product of the
electric charge of the particle and the magnetic charge of the magnetic
monopole:

S = eg[/itex] for [itex]L = @(S-D)

But since the Dirac string must be "invisible" we cannot allow this action
to give a different contribution to the path integral that the action S=0
which we obtain when an infinitely small worldline L does not encircle the
"Dirac ray". But since

\exp(i S )

for S=0 is just equal to 1, we find that also in the presence of the "Dirac
ray" this has to equal unity, i.e. we find that

\exp(i eg) = 1 .

But this is true if and only if the product of the electric and the magnetic
charge satisfies

e g = n 2pi

for any integer n.

This is the famous Dirac quantization condition. Solved for e it says that
given any magnetic monopole with magnetic charge g in the world, we
immediately know that the electric charge must be quantized as

e = n 2 \pi/g

for some integer n.

Now some comments: In case you don't know what a 2-form is, or what the
Faraday 2-form F is, or what exact and closed forms are, or what Stokes'
theorem says, pick up the book

Theodore Frankel,
The Geometry of Phyiscs
Cambridge (1997)

and look these things up. In order to understand what I have written above
it suffices to be familiar with the content of

section 2.5 The Grassmann or Exterior Algebra

section 2.6 Exterior Differentiation

section 3 Integration of Differential Forms

in particular

section 3.3.b Stokes' Theorem

and of course

section 3.5 Maxwell's equations.

If you are interested in explicit expressions for the vector potential A of
a monopole field read section 5.5 of Frankel's book.

In general: Read Frankel's book!

In my above exposition I have mainly followed page 21 of

Maximilian Kreuzer:
Geometrische Methoden der Theoretischen Physik
http://hep.itp.tuwien.ac.at/~kreuzer/inc/gmtp.pdf

(the text is in English, only the title is German).

Danny Ross Lunsford

Aug19-04, 04:51 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0408180826.71e002b8@posting.google. com>...\n> Why would the existence of magnetic monopoles mean that electric\n> charge is quantized as Paul Dirac thought?\n\nFor the best discussion of this topic, see Feynman & Weinberg,\n"Elementary Particles and the Laws of Physics", Dirac Memorial\nLectures. Feynman talks about it and shows how a hypothetical spin-0\npole + spin-0 charge pair implies both Fermi statistics and charge\nquantization. The important point is that a pole-charge system has an\nintrinstic angular momentum not derived from their individual spins,\nwhich must come in half-units of hbar. The value is not dependent on\ntheir spatial separation.\n\n-drl\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0408180826.71e002b8@posting.google.com>...
> Why would the existence of magnetic monopoles mean that electric
> charge is quantized as Paul Dirac thought?

For the best discussion of this topic, see Feynman & Weinberg,
"Elementary Particles and the Laws of Physics", Dirac Memorial
Lectures. Feynman talks about it and shows how a hypothetical spin-0
pole + spin-0 charge pair implies both Fermi statistics and charge
quantization. The important point is that a pole-charge system has an
intrinstic angular momentum not derived from their individual spins,
which must come in half-units of \hbar. The value is not dependent on
their spatial separation.

-drl

Charles J. Quarra

Aug19-04, 04:51 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0408180826.71e002b8@posting.google. com>...\n> Why would the existence of magnetic monopoles mean that electric\n> charge is quantized as Paul Dirac thought?\n\n\n\nif you consider Qe an electric charge object, and Qm an magnetic\ncharge object (aka monopole) then the transformation\n\nQe\' = cos(a)Qe + sin(a)Qm\nQm\' = -sin(a)Qe + cos(a) Qm\n\ninduces an equivalent transformation on the Electric and Magnetic\nfields due to Maxwell equations such as:\n\nE\' = cos(a)E + sin(a)B\nB\' = -sin(a)E + cos(a)B\n\n\nin this case, the Maxwell equations are invariant upon arbitrary\nglobal changes of the a-angle, which makes one thinks that if at least\none monopole exists, this gauge invariance of Maxwell equations upon\na-angle variations, then the electric charge must be simply an\na-rotation of the magnetic monopole charge\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0408180826.71e002b8@posting.google.com>...
> Why would the existence of magnetic monopoles mean that electric
> charge is quantized as Paul Dirac thought?

if you consider Qe an electric charge object, and Qm an magnetic
charge object (aka monopole) then the transformation

Qe' = cos(a)Qe + sin(a)Qm
Qm' = -sin(a)Qe + cos(a) Qm

induces an equivalent transformation on the Electric and Magnetic
fields due to Maxwell equations such as:

E' = cos(a)E + sin(a)BB' = -sin(a)E + cos(a)B

in this case, the Maxwell equations are invariant upon arbitrary
global changes of the a-angle, which makes one thinks that if at least
one monopole exists, this gauge invariance of Maxwell equations upon
a-angle variations, then the electric charge must be simply an
a-rotation of the magnetic monopole charge

Franz Heymann

Aug19-04, 12:35 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"alistair" <alistair@goforit64.fsnet.co.uk> wrote in message\nnews:861c1b21.0408180826.71e002b8@posting .google.com...\n>\n> Why would the existence of magnetic monopoles mean that electric\n> charge is quantized as Paul Dirac thought?\n\nThat is actually not what Dirac thought.\nWhat he proved was that a system consisting of a monopole and a charge\nwould have a one-valued wave-function only if a certain relationship\nwas obeyed between the magnitudes of the charge and the pole strength.\n\nFranz\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"alistair" <alistair@goforit64.fsnet.co.uk> wrote in message
news:861c1b21.0408180826.71e002b8@posting.google.c om...
>
> Why would the existence of magnetic monopoles mean that electric
> charge is quantized as Paul Dirac thought?

That is actually not what Dirac thought.
What he proved was that a system consisting of a monopole and a charge
would have a one-valued wave-function only if a certain relationship
was obeyed between the magnitudes of the charge and the pole strength.

Franz

Danny Ross Lunsford

Aug24-04, 04:40 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>disposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0408181400.3dc47b63@posting.google. com>...\n\n> if you consider Qe an electric charge object, and Qm an magnetic\n> charge object (aka monopole) then the transformation\n>\n> (duality)\n>\n> in this case, the Maxwell equations are invariant upon arbitrary\n> global changes of the a-angle, which makes one thinks that if at least\n> one monopole exists, this gauge invariance of Maxwell equations upon\n> a-angle variations, then the electric charge must be simply an\n> a-rotation of the magnetic monopole charge\n\nYou don\'t need any poles, just a global duality transformation. I can\nconvert all the charges to poles, or to some fixed ratio of charges\nand poles (dyons). Maxwell is duality-covariant. But that\'s only half\nthe problem - the Lorentz dynamics have to be checked...\n\nInterestingly, the energy tensor+ itself is globally duality\n*invariant*, not just covariant. So it doesn\'t care if I use charges\nor poles. But, QED definitely cares, because the fine structure\nconstant (naively) goes from 1/137 to 137. So QMD is unusable. Also,\ngetting poles to work in a Dirac-like equation is not easy.\n\nFiguring this out is probably key to the entire problem of getting a\nsensible QED.\n\n+Note that to make a duality-covariant energy expression, you have to\nadd driving terms on the right that vanish for theta(duality)=0,\nnamely\n\nTmn,n = Fmn Jn - F*mn Kn = Re (F + iF*)mn(J + iK)n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>disposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0408181400.3dc47b63@posting.google.com>...

> if you consider Qe an electric charge object, and Qm an magnetic
> charge object (aka monopole) then the transformation
>
> (duality)
>
> in this case, the Maxwell equations are invariant upon arbitrary
> global changes of the a-angle, which makes one thinks that if at least
> one monopole exists, this gauge invariance of Maxwell equations upon
> a-angle variations, then the electric charge must be simply an
> a-rotation of the magnetic monopole charge

You don't need any poles, just a global duality transformation. I can
convert all the charges to poles, or to some fixed ratio of charges
and poles (dyons). Maxwell is duality-covariant. But that's only half
the problem - the Lorentz dynamics have to be checked...

Interestingly, the energy tensor+ itself is globally duality
*invariant*, not just covariant. So it doesn't care if I use charges
or poles. But, QED definitely cares, because the fine structure
constant (naively) goes from 1/137 to 137. So QMD is unusable. Also,
getting poles to work in a Dirac-like equation is not easy.

Figuring this out is probably key to the entire problem of getting a
sensible QED.

+Note[/itex] that to make a duality-covariant energy expression, you have to
add driving terms on the right that vanish for \theta(duality)=0,
namely

Tmn,n = Fmn [itex]Jn - F*mn Kn = Re (F + iF*)mn(J + iK)n

Gerard Westendorp

Aug24-04, 04:56 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber wrote:\n\n[..]\n\n\n> By the Poincare lemma a locally closed form is locally exact on a\nstarshaped\n> region. This means that there is a 1-form "vector potential" A defined\n> everywhere except on a ray originating at the monopole and going out to\n> infinity (or outside the coordinate patch that we are working in, anyway)\n> such that\n>\n> F = dA\n>\n> where A is defined.\n>\n> The line on which A is not defined is called the "Dirac string".\n\n\nI\' ll try to visualize this. A monopole whould have a B-field that\nlooks the same as a coulomb E-field, ie lines coming outof a point.\nTry to construct a vector potential (A) which has this as\nits curl:\nYou start at the North pole of a spherical surface. To produce\nthe B-arrow poking through the North pole, The A-field could be a\nsmall circle around the North Pole. Then, to get teh other B\'s,\nthe A-field on the sphere surface would have\nto form circles concentric to the Noth pole, with ever increasing\nmagnitude as we get further from the North pole. This is OK until\nwe get to the South pole. Here we get a kind of ill-defined point.\n\nSo yeh, I can see the Dirac string now...\n\nThe rest of the derivation [thanks for writing it in a form\nthat I can understand btw] seems\n\n\n[..]\n\n\n>\n> But this is true if and only if the product of the electric and the\nmagnetic\n> charge satisfies\n>\n> e g = n 2pi\n>\n> for any integer n.\n>\n> This is the famous Dirac quantization condition. Solved for e it says\nthat\n> given any magnetic monopole with magnetic charge g in the world, we\n> immediately know that the electric charge must be quantized as\n>\n> e = n 2 pi/g\n>\n> for some integer n.\n\n\nThis seems quite weird. Suppose we put in a few extra monopoles into\nthe universe. This would slightly alter the total g, and thus slightly\nalter the minimum quantum of e.\n\nGerard\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber wrote:

[..]

> By the Poincare lemma a locally closed form is locally exact on a
starshaped
> region. This means that there is a 1-form "vector potential" A defined
> everywhere except on a ray originating at the monopole and going out to
> infinity (or outside the coordinate patch that we are working in, anyway)
> such that
>
> F = dA
>
> where A is defined.
>
> The line on which A is not defined is called the "Dirac string".

I' ll try to visualize this. A monopole whould have a B-field that
looks the same as a coulomb E-field, ie lines coming outof a point.
Try to construct a vector potential (A) which has this as
its curl:
You start at the North pole of a spherical surface. To produce
the B-arrow poking through the North pole, The A-field could be a
small circle around the North Pole. Then, to get teh other B's,
the A-field on the sphere surface would have
to form circles concentric to the Noth pole, with ever increasing
magnitude as we get further from the North pole. This is OK until
we get to the South pole. Here we get a kind of ill-defined point.

So yeh, I can see the Dirac string now...

The rest of the derivation [thanks for writing it in a form
that I can understand btw] seems

[..]

>
> But this is true if and only if the product of the electric and the
magnetic
> charge satisfies
>
> e g = n 2pi
>
> for any integer n.
>
> This is the famous Dirac quantization condition. Solved for e it says
that
> given any magnetic monopole with magnetic charge g in the world, we
> immediately know that the electric charge must be quantized as
>
> e = n 2 \pi/g
>
> for some integer n.

This seems quite weird. Suppose we put in a few extra monopoles into
the universe. This would slightly alter the total g, and thus slightly
alter the minimum quantum of e.

Gerard

Urs Schreiber

Aug24-04, 06:19 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Gerard Westendorp" <westy31@xs4all.nl> schrieb im Newsbeitrag\nnews:412872C7.40304@xs4all.nl...\n\n> Urs Schreiber wrote:\n\n> > This is the famous Dirac quantization condition. Solved for e it says\n> > that given any magnetic monopole with magnetic charge g in the world,\n> > we immediately know that the electric charge must be quantized as\n> >\n> > e = n 2 pi/g\n> >\n> > for some integer n.\n\n> This seems quite weird. Suppose we put in a few extra monopoles into\n> the universe. This would slightly alter the total g, and thus slightly\n> alter the minimum quantum of e.\n\nGood point. The relation really holds for elementary charges only:\n\nSuppose you\'d have two magnetic monopoles, one of charge g1 and the other of\ncharge g2, both a finite distance apart from each other (otherwise they\nwould be just one monopole with elementary charge g = g1+g2).\n\nYou could still compute their total charge by integrating F over a surface\nenclosing both of them.\n\nBut then the discussion changes: Now dF = 0 expect at the two locations of\nthe monopoles and using Poincare\'s lemma again we can find A such that dF =\nA on a star shaped region excluding these two points.\n\nStar shaped means there is at least one point from which each other point of\nthe region can be reached on a straight line without leaving the region\nanywhere in between. This means that now both monopoles are "throwing a\nshadow" and we are left with two Dirac strings and find that the total\nmagnetic charge is the sum of the integral of A around both of them.\n\nBut no particle worldline is equal to that joint contour, because the latter\nis disconnected. Hence the only way to recover the previous argument is to\ninstead pick two single surfaces around each of the monopoles sepereately.\nThis reproduces the result from before that\n\ne = n 2pi / g1 = m 2pi / g2 .\n\n\nBTW, all this has a pretty straightforward generalization to higher\ndimensional charged objects, where it is called the\n\nDirac-Nepomechie-Teitelboim quantization condition\n\ndue to the authors of\n\nPhys. Rev. D31 (1985) 1921\nPhys. Lett. B167 (1986) 69\n\nwhich is reviewed in equation (14.1.7) of hep-th9709062 .\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Gerard Westendorp" <westy31@xs4all.nl> schrieb im Newsbeitrag
news:412872C7.40304@xs4all.nl...

> Urs Schreiber wrote:

> > This is the famous Dirac quantization condition. Solved for e it says
> > that given any magnetic monopole with magnetic charge g in the world,
> > we immediately know that the electric charge must be quantized as
> >
> > e = n 2 \pi/g
> >
> > for some integer n.

> This seems quite weird. Suppose we put in a few extra monopoles into
> the universe. This would slightly alter the total g, and thus slightly
> alter the minimum quantum of e.

Good point. The relation really holds for elementary charges only:

Suppose you'd have two magnetic monopoles, one of charge g1 and the other of
charge g2, both a finite distance apart from each other (otherwise they
would be just one monopole with elementary charge g = g1+g2).

You could still compute their total charge by integrating F over a surface
enclosing both of them.

But then the discussion changes: Now dF = expect at the two locations of
the monopoles and using Poincare's lemma again we can find A such that dF =
A on a star shaped region excluding these two points.

Star shaped means there is at least one point from which each other point of
the region can be reached on a straight line without leaving the region
anywhere in between. This means that now both monopoles are "throwing a
shadow" and we are left with two Dirac strings and find that the total
magnetic charge is the sum of the integral of A around both of them.

But no particle worldline is equal to that joint contour, because the latter
is disconnected. Hence the only way to recover the previous argument is to
instead pick two single surfaces around each of the monopoles sepereately.
This reproduces the result from before that

e = n[/itex] 2pi / g1 = m 2pi [itex]/ g2 .

BTW, all this has a pretty straightforward generalization to higher
dimensional charged objects, where it is called the

Dirac-Nepomechie-Teitelboim quantization condition

due to the authors of

Phys. Rev. D31 (1985) 1921
Phys. Lett. B167 (1986) 69

which is reviewed in equation (14.1.7) of hep-th9709062 .

Urs Schreiber

Aug25-04, 04:45 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Gerard Westendorp" <westy31@xs4all.nl> schrieb im Newsbeitrag\nnews:412872C7.40304@xs4all.nl...\n> Urs Schreiber wrote:\n\n> > e = n 2 pi/g\n> >\n> > for some integer n.\n>\n>\n> This seems quite weird. Suppose we put in a few extra monopoles into\n> the universe. This would slightly alter the total g, and thus slightly\n> alter the minimum quantum of e.\n\n\nIt occured to me that in my first reply to this I may have missed your\npoint. Maybe you were concerned about what happens when we start with a\nmagnetic monopole of charge g, find\n\ne = n 2pi /g\n\nfrom that and then assume that there are moreover magnetic monopoles with\ncharges that are integrer multiples of g?\n\nIn that case choose n=1 for the smallest charge and increase n appropriately\nfor the others.\n\nFor instance if the smallest magnetic charge is g and another monopole has\ncharge\n\ng\' = 3g\n\nthen we need to satisfy the quantization conditions\n\ne = n 2pi /g = m 2pi / g\' = (m/3) 2pi / g .\n\nBut these can be solved for setting the arbitrary integers n,m to\n\nn = 1\nm = 3 .\n\nSo from this, or from just inverting the equation to obtain g = 2pi / e\ntimes an integer, we find that both e and g must come in integer multiples\nof some fundamental unit of electric or magnetic charge, respectively.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Gerard Westendorp" <westy31@xs4all.nl> schrieb im Newsbeitrag
news:412872C7.40304@xs4all.nl...
> Urs Schreiber wrote:

> > e = n 2 \pi/g
> >
> > for some integer n.
>
>
> This seems quite weird. Suppose we put in a few extra monopoles into
> the universe. This would slightly alter the total g, and thus slightly
> alter the minimum quantum of e.

It occured to me that in my first reply to this I may have missed your
point. Maybe you were concerned about what happens when we start with a
magnetic monopole of charge g, find

e = n[/itex] 2pi /g

from that and then assume that there are moreover magnetic monopoles with
charges that are integrer multiples of g?

In that case choose n=1 for the smallest charge and increase n appropriately
for the others.

For instance if the smallest magnetic charge is g and another monopole has
charge

g' = 3g

then we need to satisfy the quantization conditions

e = n 2pi /g = m 2pi / g' = (m/3) 2pi [itex]/ g .

But these can be solved for setting the arbitrary integers n,m to

n = 1m = 3 .

So from this, or from just inverting the equation to obtain g = 2pi / e
times an integer, we find that both e and g must come in integer multiples
of some fundamental unit of electric or magnetic charge, respectively.

Charles J. Quarra

Aug27-04, 02:39 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>antimatter33@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.0408190933.4ac8414e@posting.google. com>...\n>\n> You don\'t need any poles, just a global duality transformation. I can\n> convert all the charges to poles, or to some fixed ratio of charges\n> and poles (dyons). Maxwell is duality-covariant. But that\'s only half\n> the problem - the Lorentz dynamics have to be checked...\n>\n> Interestingly, the energy tensor+ itself is globally duality\n> *invariant*, not just covariant. So it doesn\'t care if I use charges\n> or poles. But, QED definitely cares, because the fine structure\n> constant (naively) goes from 1/137 to 137. So QMD is unusable. Also,\n> getting poles to work in a Dirac-like equation is not easy.\n>\n> Figuring this out is probably key to the entire problem of getting a\n> sensible QED.\n>\n> +Note that to make a duality-covariant energy expression, you have to\n> add driving terms on the right that vanish for theta(duality)=0,\n> namely\n>\n> Tmn,n = Fmn Jn - F*mn Kn = Re (F + iF*)mn(J + iK)n\n\n\nSupposedly, the vector potential A_mu associated with\nElectromagnetism arises when one asks for covariance under U(1) local\nvariations of the field, which simplest solution is to define the\ngauge covariant derivative d/_mu = d_mu- i*e*A_mu.\n\nCan you apply the trick again and ask for Maxwell-Dirac equations to\nbe covariant under _-*local*-_ duality transformations (arbitrary dyon\nangle as a scalar field), or is there a constraint relation between\nthis gauge angle and the traditional gauge scalar that "produces" the\nA_mu?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>antimatter33@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.0408190933.4ac8414e@posting.google.com>...
>
> You don't need any poles, just a global duality transformation. I can
> convert all the charges to poles, or to some fixed ratio of charges
> and poles (dyons). Maxwell is duality-covariant. But that's only half
> the problem - the Lorentz dynamics have to be checked...
>
> Interestingly, the energy tensor+ itself is globally duality
> *invariant*, not just covariant. So it doesn't care if I use charges
> or poles. But, QED definitely cares, because the fine structure
> constant (naively) goes from 1/137 to 137. So QMD is unusable. Also,
> getting poles to work in a Dirac-like equation is not easy.
>
> Figuring this out is probably key to the entire problem of getting a
> sensible QED.
>
> +Note that to make a duality-covariant energy expression, you have to
> add driving terms on the right that vanish for \theta(duality)=0,
> namely
>
> Tmn,n = Fmn Jn - F*mn Kn = Re (F + iF*)mn(J + iK)n

Supposedly, the vector potential A_{mu} associated with
Electromagnetism arises when one asks for covariance under U(1) local
variations of the field, which simplest solution is to define the
gauge covariant derivative d/_mu = d_{mu}- i*e*A_{mu}.

Can you apply the trick again and ask for Maxwell-Dirac equations to
be covariant under _-*local*-_ duality transformations (arbitrary dyon
angle as a scalar field), or is there a constraint relation between
this gauge angle and the traditional gauge scalar that "produces" the
A_{mu}?

Danny Ross Lunsford

Aug31-04, 03:55 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\ndisposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0408260030.f1f4fce@posting.google.c om>...\n\n> Can you apply the trick again and ask for Maxwell-Dirac equations to\n> be covariant under _-*local*-_ duality transformations (arbitrary dyon\n> angle as a scalar field), or is there a constraint relation between\n> this gauge angle and the traditional gauge scalar that "produces" the\n> A_mu?\n\nYes you can gauge classical duality, and the result is interesting - a\nsort of theory of torsion as the gauge field. This is satisfying in\nthe following sense - it\'s possible to interpret torsion as a sort of\nintrinsic angular momentum coming from geometry, and we know that a\ncharge-pole system has a built-in angular momentum not coming from\nspin as such. But it\'s unsatisfying because it\'s reducible with\nrespect to gravity.\n\nThe gauge as applied to a Dirac spinor should look something like (in\nthe Dirac rep)\n\npsi -> exp(iy5 w) psi = ( cos w + iy5 sin w ) psi\n\nso the "large" and "small" parts of the spinor go as\n\npsi_L -> psi_L cos w + i sin w psi_S\n\npsi_S -> i sin w psi_L + cos w psi_S\n\nand in particular for w=pi/2 they are exchanged up to a phase. This is\nvery intriguing, as it indicates a relationship to positive vs.\nnegative energy, and we know that the energy tensor itself is\nduality-invariant.\n\nYou can do electrodynamics with a "dual potential" such that\n\nE = -curl V - dA/dt - grad a\nB = curl A - dV/dt - grad v\n\nso that with global duality one has decoupled wave equations for the\nmagentic and electric sources\n\nD^2 A = -J d.A = 0\nD^2 V = -K d.V = 0\n\nOf course the decoupling makes it vacuous - just globally\nduality-rotate away the V.\n\nThe problem now is, if I try to make a covariant derivative it has to\nlook something like\n\ndm + ie Am + g y5 Bm\n\nwith g real, and you\'ll quickly see that you\'re in trouble with the\nDirac equation, trying to get a conserved current from this. And\nyou\'ll see that the form is ambiguous - do I mean\n\n(e + iy5 g) (Am + other vector)\n\nor\n\ne (Am + igy5 other vector)\n\nwhat\'s complex and what\'s not (i.e. what lives in the even subalgebra\nof the Dirac algebra and what lives in the complex plane).\n\nI think this can be fixed but for now I\'ve put it aside for other\nthings. In short - I think gauged duality is a very interesting idea\nthat\'s not yet needed. The connection with energy is too interesting\nto be wrong.\n\n-drl\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>disposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0408260030.f1f4fce@posting.google.com>...

> Can you apply the trick again and ask for Maxwell-Dirac equations to
> be covariant under _-*local*-_ duality transformations (arbitrary dyon
> angle as a scalar field), or is there a constraint relation between
> this gauge angle and the traditional gauge scalar that "produces" the
> A_{mu}?

Yes you can gauge classical duality, and the result is interesting - a
sort of theory of torsion as the gauge field. This is satisfying in
the following sense - it's possible to interpret torsion as a sort of
intrinsic angular momentum coming from geometry, and we know that a
charge-pole system has a built-in angular momentum not coming from
spin as such. But it's unsatisfying because it's reducible with
respect to gravity.

The gauge as applied to a Dirac spinor should look something like (in
the Dirac rep)

\psi -> \exp(iy5 w) \psi = ( cos w +[/itex] iy5 sin w ) \psi

so the "large" and "small" parts of the spinor go as

\psi_L -> \psi_L cos w + i sin w \psi_S\psi_S -> i sin w \psi_L + cos w \psi_S

and in particular for w=\pi/2 they are exchanged up to a phase. This is
very intriguing, as it indicates a relationship to positive vs.
negative energy, and we know that the energy tensor itself is
duality-invariant.

You can do electrodynamics with a "dual potential" such that

E = -curl V - dA/dt - grad a
B = curl A - dV/dt - grad v

so that with global duality one has decoupled wave equations for the
magentic and electric sources

D^2 A = -J d.A =
D^2 V = -K d.V =

Of course the decoupling makes it vacuous - just globally
duality-rotate away the V.

The problem now is, if I try to make a covariant derivative it has to
look something like

[itex]dm + ie Am + g y5 Bm

with g real, and you'll quickly see that you're in trouble with the
Dirac equation, trying to get a conserved current from this. And
you'll see that the form is ambiguous - do I mean

(e + iy5 g) (Am + other vector)

or

e (Am + igy5 other vector)

what's complex and what's not (i.e. what lives in the even subalgebra
of the Dirac algebra and what lives in the complex plane).

I think this can be fixed but for now I've put it aside for other
things. In short - I think gauged duality is a very interesting idea
that's not yet needed. The connection with energy is too interesting
to be wrong.

-drl

Oz

Sep19-04, 06:56 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nUrs Schreiber <Urs.Schreiber@uni-essen.de> writes\n\n>This is very easy to see once you know the differential geometric\n>formulation of the electromagnetic field.\n>\n>Let me first give the answer in that language and then comment on some\n>details.\n\n<cough>\n\nAny chance of a handwavy version of the answer for dummies?\nSounds interesting.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> writes

>This is very easy to see once you know the differential geometric
>formulation of the electromagnetic field.
>
>Let me first give the answer in that language and then comment on some
>details.

<cough>

Any chance of a handwavy version of the answer for dummies?
Sounds interesting.

--
Oz
This post is worth absolutely nothing and is probably fallacious.

BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.

Oz

Sep22-04, 03:45 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nUrs Schreiber <Urs.Schreiber@uni-essen.de> writes\n>"alistair" <alistair@goforit64.fsnet.co.uk> schrieb im Newsbeitrag\n>news:861c1b21.0408180826.71e002b8@po sting.google.com...\n>>\n>> Why would the existence of magnetic monopoles mean that electric\n>> charge is quantized as Paul Dirac thought?\n>\n>This is very easy to see once you know the differential geometric\n>formulation of the electromagnetic field.\n\nOK, I admit I don\'t fully follow your argument, but I believe you.\n\nHowever one wonders if the corollary is worth considering.\n\nJust for fun, like?\n\nSince electric charge IS quantised, and there exists at least one\nelectron, does this mean there must be at least one magnetic monopole.\n\nOf course it may be \'at infinity\'.\n\nHmmm....\n\nWhat would an electron at the edge of the observable universe look like\nto us? Its travelling at c, and so presumably (flaps hands furiously)\nlooks to us like a magnetic monopole?\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> writes
>"alistair" <alistair@goforit64.fsnet.co.uk> schrieb im Newsbeitrag
>news:861c1b21.0408180826.71e002b8@posting.google.c om...
>>
>> Why would the existence of magnetic monopoles mean that electric
>> charge is quantized as Paul Dirac thought?
>
>This is very easy to see once you know the differential geometric
>formulation of the electromagnetic field.

OK, I admit I don't fully follow your argument, but I believe you.

However one wonders if the corollary is worth considering.

Just for fun, like?

Since electric charge IS quantised, and there exists at least one
electron, does this mean there must be at least one magnetic monopole.

Of course it may be 'at infinity'.

Hmmm....

What would an electron at the edge of the observable universe look like
to us? Its travelling at c, and so presumably (flaps hands furiously)
looks to us like a magnetic monopole?

--
Oz
This post is worth absolutely nothing and is probably fallacious.

BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.

Arnold Neumaier

Sep23-04, 04:47 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOz wrote:\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> writes\n>\n>>"alistair" <alistair@goforit64.fsnet.co.uk> schrieb im Newsbeitrag\n>>news:861c1b21.0408180826.71e002b8@p osting.google.com...\n>>\n>>>Why would the existence of magnetic monopoles mean that electric\n>>>charge is quantized as Paul Dirac thought?\n>>\n>>This is very easy to see once you know the differential geometric\n>>formulation of the electromagnetic field.\n>\n>\n> OK, I admit I don\'t fully follow your argument, but I believe you.\n>\n> However one wonders if the corollary is worth considering.\n>\n> Just for fun, like?\n>\n> Since electric charge IS quantised, and there exists at least one\n> electron, does this mean there must be at least one magnetic monopole.\n\nNo, it doesn\'t. You are not allowed to conclude from \'A implies B\'\nand a knowledge of B that A must hold. So your statement is not a\ncorollary.\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz wrote:
> Urs Schreiber <Urs.Schreiber@uni-essen.de> writes
>
>>"alistair" <alistair@goforit64.fsnet.co.uk> schrieb im Newsbeitrag
>>news:861c1b21.0408180826.71e002b8@posting.google.c om...
>>
>>>Why would the existence of magnetic monopoles mean that electric
>>>charge is quantized as Paul Dirac thought?
>>
>>This is very easy to see once you know the differential geometric
>>formulation of the electromagnetic field.
>
>
> OK, I admit I don't fully follow your argument, but I believe you.
>
> However one wonders if the corollary is worth considering.
>
> Just for fun, like?
>
> Since electric charge IS quantised, and there exists at least one
> electron, does this mean there must be at least one magnetic monopole.

No, it doesn't. You are not allowed to conclude from 'A implies B'
and a knowledge of B that A must hold. So your statement is not a
corollary.

Arnold Neumaier

Charles J. Quarra

Sep24-04, 08:11 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>antimatter33@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.0408301338.1a0ac84a@posting.google. com>...\n> disposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0408260030.f1f4fce@posting.google.c om>...\n>\n> > Can you apply the trick again and ask for Maxwell-Dirac equations to\n> > be covariant under _-*local*-_ duality transformations (arbitrary dyon\n> > angle as a scalar field), or is there a constraint relation between\n> > this gauge angle and the traditional gauge scalar that "produces" the\n> > A_mu?\n>\n> Yes you can gauge classical duality, and the result is interesting - a\n> sort of theory of torsion as the gauge field. This is satisfying in\n> the following sense - it\'s possible to interpret torsion as a sort of\n> intrinsic angular momentum coming from geometry, and we know that a\n> charge-pole system has a built-in angular momentum not coming from\n> spin as such. But it\'s unsatisfying because it\'s reducible with\n> respect to gravity.\n\n\nI Would be very interested in following the steps to this derivation,\nat least the beginning of it? When you put a space-time variable theta\n(theta being the dyon angle) then in the Maxwell equations the\ndivergence and curls of E and B bring some theta gradient terms. So i\nassume that the next step would be either to redefine the divergences\nand curls, together with a redefinition of the Dirac current Psi*\ngamma_nu Psi with in total cancels out, but no idea about what\npossibilities one has here to modify these quantities in the search of\nsuch magic cancelation.\n\nAnyhow one should take into account that whatever modification to the\ndivergence/curl derivatives should be consistent with changes to the\ngauge covariant derivative of the fermion wavefunction d_mu - i e A_mu\n\nBut first of all; What is the argument to understand why the duality\ntransform is of the form exp(iy5w)?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>antimatter33@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.0408301338.1a0ac84a@posting.google.com>...
> disposablemailaccountfornews@yahoo.com.ar (Charles J. Quarra) wrote in message news:<bc979c06.0408260030.f1f4fce@posting.google.com>...
>
> > Can you apply the trick again and ask for Maxwell-Dirac equations to
> > be covariant under _-*local*-_ duality transformations (arbitrary dyon
> > angle as a scalar field), or is there a constraint relation between
> > this gauge angle and the traditional gauge scalar that "produces" the
> > A_{mu}?
>
> Yes you can gauge classical duality, and the result is interesting - a
> sort of theory of torsion as the gauge field. This is satisfying in
> the following sense - it's possible to interpret torsion as a sort of
> intrinsic angular momentum coming from geometry, and we know that a
> charge-pole system has a built-in angular momentum not coming from
> spin as such. But it's unsatisfying because it's reducible with
> respect to gravity.

I Would be very interested in following the steps to this derivation,
at least the beginning of it? When you put a space-time variable \theta(\theta being the dyon angle) then in the Maxwell equations the
divergence and curls of E and B bring some \theta gradient terms. So i
assume that the next step would be either to redefine the divergences
and curls, together with a redefinition of the Dirac current \Psi*\gamma_nu \Psi with in total cancels out, but no idea about what
possibilities one has here to modify these quantities in the search of
such magic cancelation.

Anyhow one should take into account that whatever modification to the
divergence/curl derivatives should be consistent with changes to the
gauge covariant derivative of the fermion wavefunction d_{mu} - i e A_{mu}

But first of all; What is the argument to understand why the duality
transform is of the form \exp(iy5w)?

Gerard Westendorp

Sep29-04, 02:25 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOz wrote:\n\n\n>\n> Any chance of a handwavy version of the answer for dummies?\n> Sounds interesting.\n>\n\n\nI\'ll try, I am something like a dummy myself..\n\nEver tried constructing the B-field for a star that\nejects charged particles? The current would be in spherically\nsymmetric rays.\nUsing curl (B) = j, you find that you cannot build the\nB-field! There is no solution, and certainly not a specially\nsymmetric one.\n\nLuckily, you need to add dE/dt to the equation:\n\ncurl (B) = j + dE/dt\n\nAnd, because charge is conserved, you get\n\nB = 0.\n\nSo charge *has* to be conserved for the whole Maxwell\ncircus to work.\n\nRelated to this is:\nTry to construct the vector potential (A) for a\nmagnetic monopole field. The B field has the same shape\nas the j-field discussed above. Again, you cannot find\na solution that works everywhere in space. You can get\nit to work on all points except the "Dirac string", which\nis a kind of umbilical cord from infinity to the monopole,\non which the A-field is discontinuous and infinite.\n(It is not so hard to figure this out, just try\nconstructing A on a sphere, starting at the North pole)\nBut because A is not directly detectable, this is not such\na problem.\n\nUnless you consider the Aharanov Bohm effect. This effect\nis caused by the phase shift that an A-field induces on a\nwave function of a charged particle. Phase shifs are not\ndirectly detecable, but if a particle takes 2 different paths\nwith different net phase shifts, then interference between\nthe 2 paths *would* be detectable. And the A-field of a\nmagnetic monopole *would* give detectable phase shifts\naround the Dirac string.\nThe problem is that these phase shifts depend on arbitrary\nchoices we make when we chose the unphysical Dirac string.\n\nIt is a bit like choosing a coordinate system and finding\nthat physical quantities depend explicitly on the choice\nof origin. That would be too weird to be true.\n\nThe only escape would be if the magnetic monopoles would\nhave phase shifts that are exact multiples of 2pi. This\nis possible, if e*m = N*2pi.\n\n\nOf course, another escape would be to give up B = curl(A).\nBut that might upset people. It would mean giving up\nF = dA, which is kind of a nice equation that everyone\nlikes.\n\nOh yes, and another escape would be that magnetic monopoles\ndo not exist.\n\nGerard\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz wrote:

>
> Any chance of a handwavy version of the answer for dummies?
> Sounds interesting.
>

I'll try, I am something like a dummy myself..

Ever tried constructing the B-field for a star that
ejects charged particles? The current would be in spherically
symmetric rays.
Using curl (B) = j, you find that you cannot build the
B-field! There is no solution, and certainly not a specially
symmetric one.

Luckily, you need to add dE/dt to the equation:

curl (B) = j + dE/dt

And, because charge is conserved, you get

B = .

So charge *has* to be conserved for the whole Maxwell
circus to work.

Related to this is:
Try to construct the vector potential (A) for a
magnetic monopole field. The B field has the same shape
as the j-field discussed above. Again, you cannot find
a solution that works everywhere in space. You can get
it to work on all points except the "Dirac string", which
is a kind of umbilical cord from infinity to the monopole,
on which the A-field is discontinuous and infinite.
(It is not so hard to figure this out, just try
constructing A on a sphere, starting at the North pole)
But because A is not directly detectable, this is not such
a problem.

Unless you consider the Aharanov Bohm effect. This effect
is caused by the phase shift that an A-field induces on a
wave function of a charged particle. Phase shifs are not
directly detecable, but if a particle takes 2 different paths
with different net phase shifts, then interference between
the 2 paths *would* be detectable. And the A-field of a
magnetic monopole *would* give detectable phase shifts
around the Dirac string.
The problem is that these phase shifts depend on arbitrary
choices we make when we chose the unphysical Dirac string.

It is a bit like choosing a coordinate system and finding
that physical quantities depend explicitly on the choice
of origin. That would be too weird to be true.

The only escape would be if the magnetic monopoles would
have phase shifts that are exact multiples of 2pi. This
is possible, if e*m = N*2pi.

Of course, another escape would be to give up B = curl(A).
But that might upset people. It would mean giving up
F = dA, which is kind of a nice equation that everyone
likes.

Oh yes, and another escape would be that magnetic monopoles
do not exist.

Gerard