Magnetic monopoles?
bl...@husc15.harvard.edu
and as-of-yet undiscovered?
It seems to me they are utterly impossible, and yet people talk about them
anyway.
Eric Blom
Harvard U
Emory F. Bunn
Why would you say that they are impossible? I would certainly agree
that there is very strong evidence that they are extraordinarily
rare, but that's very different from saying they are impossible.
There may be some reason why they cannot exist, but I have
never heard it. If there is such an argument, I'd be interested in
seeing it.
My understanding is that one fairly generic consequence of grand unified
theories is the existence of very massive magnetic monopoles. In fact,
one of the original motivations for the inflationary model of the Universe
is that it reduces the relic density of monopoles to nearly zero. If
this theory is right, then monopoles are not impossible, just very
rare. In fact, a sufficiently powerful accelerator could even produce
monopole-antimonopole pairs (although the required energy would be
prohibitively large in practice).
-Ted
Daniel Seeman
Pardon my ignorance but would someone be so kind as to briefly explain (if pos-
sible) what a magnetic monopole would "look" like. Or rather, how would a
magnetic monopole "act?" I know theories say they are possible and such, but
I wouldn't know how to look for one to test theory.
dks
Douglas S. Miller
The Maxwell eqns don't have a monopole term but there is no reason you
couldn't add one (div B would then not be zero but equal to the magnetic
charge, exactly like div E = 4 pi rho, you get the idea). I believe it
was Fermi who pointed out that you need just one monopole, anywhere in the
universe, to quantize charge. Since we observe that charge is indeed
quantized, this makes searching for monopoles an interesting thing to do.
So far, no luck.
Doug Miller do...@retzlaff.llnl.gov
"Now if I were a monopole, where would I hide?"
Matt McIrvin
They're not impossible at all. If they existed they would require a
modification of Maxwell's equations, of course.
Some grand unified theories predict that they ought to be around.
People have been interested ever since, by a beautiful mathematical
argument, Dirac showed that their existence would explain the
discreteness of electric charge.
--
Matt McIrvin
Vaughan R. Pratt
>
>I believe it
>was Fermi who pointed out that you need just one monopole, anywhere in the
>universe, to quantize charge. Since we observe that charge is indeed
>quantized, this makes searching for monopoles an interesting thing to do.
If there are no monopoles then searching for them would indeed be
pointless. But if charge quantization only increases the known lower
bound on the number of monopoles in the universe by one, why should the
knowledge that charge is quantized change our mind about searching for
them? (Not objecting, just want to learn how to think like an
experimentalist. :-)
--
Vaughan Pratt
Mike Gross
>Pardon my ignorance but would someone be so kind as to briefly explain (if pos-
>sible) what a magnetic monopole would "look" like. Or rather, how would a
>magnetic monopole "act?" I know theories say they are possible and such, but
>I wouldn't know how to look for one to test theory.
>dks
If you take a compass needle and note its direction anywhere in space around
the monopole, you would notice that the needle would always point toward (or
away from, if the "magnetic charge" is negative) the monopole, regardless of
where you put it. This is analogous to the *electric* field of a point charge.
No phenomena conforming to this field configuration have been observed. A
standard bar magnet will attract the north end of the compass needle at one
end, and repel the needle at the other, for example.
A monopole may be thought of as a "north" end without an associated "south"
end, if you like.
I hope this helps
Mike Gross
Physics Board and Lick Observatory
Univ of California GO SLUGS!!!!
Santa Cruz, CA 95064
gr...@lick.ucsc.edu
Leigh Palmer
do...@retzlaff.llnl.gov writes:
>The Maxwell eqns don't have a monopole term but there is no reason you
>couldn't add one (div B would then not be zero but equal to the magnetic
>charge, exactly like div E = 4 pi rho, you get the idea). I believe it
>was Fermi who pointed out that you need just one monopole, anywhere in
the
>universe, to quantize charge. Since we observe that charge is indeed
>quantized, this makes searching for monopoles an interesting thing to
do.
>So far, no luck.
It's good to see that some folks still have taste in their choice of
units!
That aside, I'd like to share a memory of the fifties with you. I took a
couple of "Nuclear Physics" courses from Luis Alvarez back in about 1957.
He first introduced me to the proof that, if one monopole existed, and
intrinsic angular momentum of an electromagnetic field was quantized
according to the Bohr-Sommerfeld criterion, then electric charge was
quantized, and the strength of the monopole was constrained.
He was actively recruiting students in the class to participate in a
magnetic monopole hunt. The idea was that we would all go down to
Macquarie Island (near the south geomagnetic pole) and lie on the beach
at night looking skyward through polaroids. If an extreme relativistic
cosmic monopole came through the atmosphere it should produce
circumferentially polarized Cerenkov radiation. Ordinary electric charges
produce radially polarized Cerenkov radiation. As lying around on a beach
at night (in the southern winter, to boot) was not considered the thing
to do in my Southern Californian religion, and since I had a good
$310/mo. job with the Division of Highways, I decided not to do it.
The trip never materialized, and so my incorrect choice never came back
to haunt me. While we would not have discovered the monopole that way
(physicists used to think that several events were necessary to establish
the existence of a particle!), I would have missed the opportunity to
work with the best experimental physicist I ever knew. I feel I must add
that Alvarez was also both the best and the most inspiring teacher I ever
had.
Alvarez went on to become the cleverest scientific critic and debunker of
our time. He continued to pursue the monopole experimentally, and his
work in critically examining the claims of others who "discovered"
monopoles is greatly responsible for our present view that, if they exist
at all, they must be extremely rare.
For those who would be inspired by knowing the work of such a fine
scientist I recommend his autobiography, "Alvarez: Adventures of a
Physicist", and also a collection of his works, with commentary by his
colleagues, "Discovering Alvarez", which my wife gave me for Christmas,
and which I have only briefly examined.
Leigh
Sean Merritt
Changes to Maxwell's Eqs:
div E = 4* pi rho(e)
div B = 4* pi rho(m)
curl E = 4*pi/c Jm - 1/c d/dt(B)
curl B = 4*pi/c Je + 1/c d/dt(E) [d/dt is a partial deriv.]
Mr. Gauss:
e = intdv [rho(e)] = 1/4*pi surface-intE*dS E & S are vectors * denotes
dot prod.
g = intdv [rho(m)] = 1/4*pi surface-intB*dS
Eq of continuity:
div Je + 1/c d/dt rho[e] = 0
div Jm + 1/c d/dt rho[m] = 0 d/dt is a partial deriv.
Lorentz Force:
Fe = e(E + 1/c ve ^ B) ^ is cross prod
Fm = g(B - 1/c vm ^ E)
The Dirac quantization Condition (Not Fermi!):
electron in the field of a magnetic monopole eq of motion,
me d2/dt2 (re) = e/c [d/dt (r) ^ B]
re, is electron position.
r, is field point.
J total angular momentum = J mech + J em
J mech = re ^ me (d/dt re)
Jem = 1/4*pi*c Int d3r {r ^ (E ^ B)}
E = e (r- re)/ | r - re |^3 ^3 = R cubed
B = g r/|r|^3
Jem = -eg/c re-hat (unit vector in the re direction)
J = re ^ me (d/dt re) - eg/c re-hat
the radial component of J, Jr is conserved
Jr = re-hat * J again * is dot product
the motion of the electron is within a conical surface
the semi-angle of the cone is Theta-e = arc cos(Jr/J) arc cos (eg/cJ)
the quantization condition requires Jr = (hbar)n/2 or eg/c = 1/2 (hbar)n
Dirac's staement was that if there was just one mag monopole
implied that all electric charges be quantized with the basic
electric unit :
e = (hbar)c/2g .
You can have monopolic atoms, they can induce nuclear reactions,
monopoles moving near atoms exite the atoms(Drell effect),
energy loss in a Fermi gas can be predicted, and you could
make a really efficient motor if you could get enough to form
a magnetic current.
-sjm
--
Sean J. Merritt |"Road-kill has it's seasons just like
Dept of Physics Boston University|anything, there's possums in the autumn
mer...@macro.bu.edu |and farm cats in the spring." T. Waits
Daniel Seeman
>In <1993Jan20.0...@novell.com> dse...@novell.com (Daniel Seeman) writes:
>
>>Pardon my ignorance but would someone be so kind as to briefly explain (if pos-
>>sible) what a magnetic monopole would "look" like. Or rather, how would a
>>magnetic monopole "act?" I know theories say they are possible and such, but
>>I wouldn't know how to look for one to test theory.
>
>>dks
>
>If you take a compass needle and note its direction anywhere in space around
>the monopole, you would notice that the needle would always point toward (or
>away from, if the "magnetic charge" is negative) the monopole, regardless of
>where you put it. This is analogous to the *electric* field of a point charge.
>
>No phenomena conforming to this field configuration have been observed. A
>standard bar magnet will attract the north end of the compass needle at one
>end, and repel the needle at the other, for example.
>
>A monopole may be thought of as a "north" end without an associated "south"
>end, if you like.
>
>I hope this helps
>
Thanks for the reply. Certainly the compass would always point "north" (or what-
ever direction) but what "thing" is producing the field? I see that my original
question was poorly worded. Sorry about that.
Magnets are made of a bunch of charged "things" that spin therefore creating
the magnetic field (using the "right-hand rule," current path and such). But
this classical model predicts dipole magnets. The image I get in mind concern-
ing this is a bunch of charged spheres all spinning with directional arrows
pointing out the "top" and in to the "bottom." The areas where the arrows point
into and out of the sphere are labled (rather arbitrarily it seems) the "South
Pole" and "North Pole" respectively.
So, for a monopole, what are these little (classical looking) spheres doing?
I mean are they now hemispheres with arrows only pointing out of
the top? Certainly the FIELD points in one direction and never "wraps back" to
form the "South Pole." But what "thing" could produce a field like this? How
could I see such a thing? Absolutely, a compass could "see" it, but wouldn't
there be some other local effects that would be "visible" or apparent?
Obviously, the "little hemisphere" idea cannot be the case (for many reasons).
So, what produces a unidirectional field? That is what I was trying to get at
with my first posting.
Sorry for the poorly worded original question...
dks.
GO SLUGS IN DEED!;-)
Daniel Seeman
Thanks for the info! I will take a moment to consider it so I can "picture"
things...;-)
dks.
Andrew Poutiatine
>sible) what a magnetic monopole would "look" like. Or rather, how would a
>magnetic monopole "act?" I know theories say they are possible and such, but
>I wouldn't know how to look for one to test theory.
>
>dks
If I am not mistaken, it would be the analog of the electron, for example. Just
a source of magnetic north or south, without the other. As far as what other
properties it might have, that is way beyond me :^)
-AIP
SCOTT I CHASE
>In article <1jhqnr$7...@agate.berkeley.edu> ted@physics2 (Emory F. Bunn) writes:
>>In article <1993Jan19.1...@husc15.harvard.edu> bl...@husc15.harvard.edu writes:
>>>Are magnetic monopoles theoretically impossible, or are they just improbable
>>>and as-of-yet undiscovered?
>>>
>>>It seems to me they are utterly impossible, and yet people talk about them
>>>anyway.
>>
>>Why would you say that they are impossible? I would certainly agree
>>that there is very strong evidence that they are extraordinarily
>>rare, but that's very different from saying they are impossible.
I wouldn't say that we know that they are extraordinarily rare. The
best experiments done to date, using superconducting rings, have
effective sensitive areas less than, or at best just barely, one
square meter, and have been active on the order of one to two years.
To get no signal in such an experiment sets a hard limit, but the
Universe is a very big place. Stars are not extraordinarily rare, by
my estimate, but none of them have passed through anyone's monopole
detectors recently. :=)
In any event, rarity is not such a big issue, IMHO. For example, how many
B mesons would fly through such a detector in the course of a year? Yet
we can easily make them in the laboratory, and do good physics with them.
Whether there are relic monopoles from some GUT epoch is an important
issue for cosmology. But even if there are none, this does not in any
strong way address whether the things exist, i.e., whether they at one
time in the past were common, and whether we can make them under
suitable experimental circumstances today.
>Pardon my ignorance but would someone be so kind as to briefly explain (if pos-
>sible) what a magnetic monopole would "look" like. Or rather, how would a
>magnetic monopole "act?" I know theories say they are possible and such, but
>I wouldn't know how to look for one to test theory.
Monopoles act like, well... monopoles. For example, if you have a loop
of superconducting current, and a monopole flys through the interior of the
loop, then there will be an permanent, steplike increase in the current
circulating in the loop. Just draw the field lines and think about how
the current would be induced according to Faraday's Law. I'm sure that
experts could tell you more.
-Scott
--------------------
Scott I. Chase "It is not a simple life to be a single cell,
SIC...@CSA2.LBL.GOV although I have no right to say so, having
been a single cell so long ago myself that I
have no memory at all of that stage of my
life." - Lewis Thomas
Benjamin Weiner
>If there are no monopoles then searching for them would indeed be
>pointless. But if charge quantization only increases the known lower
>bound on the number of monopoles in the universe by one, why should the
>knowledge that charge is quantized change our mind about searching for
>them? (Not objecting, just want to learn how to think like an
>experimentalist. :-)
Thinking like an experimentalist would lead you to the conclusion
that there are no monopoles, since none have been discovered. (Unless
you think like a sloppy experimentalist.) Monopoles are a theorist's
dream, however.
The situation is this: Electric charge is quantized (unit e). Why? I dunno.
But if a magnetic monopole existed (anywhere!) with some "magnetic charge"
g, then in order to keep the interaction between it and any electric charge
consistent with some reasonable assumptions of quantum mechanics,
(either quantization of angular mom., or gauge-invariance of the wavefn.),
it is possible to show that
g * e = hbar * c * n / 2 , where n is some integer.
I believe Sean Merritt has posted a derivation of this. There is a
discussion in Chapter 6 of the infamous J.D. Jackson's _Classical
Electrodynamics_, and references. (this book reminds me faintly of
Borges's Library of Babel ... it has everything in it, *somewhere*)
So if a monopole existed, then we would answer the question "why is
charge quantized" with "because it has to be." I guess we would turn
two mysteries, "why is charge quantized?" and "why is angular momentum
quantized?" into one, by showing that the second implies the first.
Sean Merritt
pr...@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>If there are no monopoles then searching for them would indeed be
>pointless. But if charge quantization only increases the known lower
>bound on the number of monopoles in the universe by one, why should the
>knowledge that charge is quantized change our mind about searching for
>them? (Not objecting, just want to learn how to think like an
>experimentalist. :-)
> Thinking like an experimentalist would lead you to the conclusion
> that there are no monopoles, since none have been discovered. (Unless
> you think like a sloppy experimentalist.) Monopoles are a theorist's
> dream, however.
I always thought the best theorist DO think like experimentalist. Are
they not always waiting to grab the results off the griddle and offer
an explination? And do they not always(in the case of the best) offer
an experiment that can prove thier theory?
>
> The situation is this: Electric charge is quantized (unit e). Why? I dunno.
> But if a magnetic monopole existed (anywhere!) with some "magnetic charge"
> g, then in order to keep the interaction between it and any electric charge
> consistent with some reasonable assumptions of quantum mechanics,
> (either quantization of angular mom., or gauge-invariance of the wavefn.),
> it is possible to show that
> g * e = hbar * c * n / 2 , where n is some integer.
> I believe Sean Merritt has posted a derivation of this. There is a
> discussion in Chapter 6 of the infamous J.D. Jackson's _Classical
> Electrodynamics_, and references. (this book reminds me faintly of
> Borges's Library of Babel ... it has everything in it, *somewhere*)
>
> So if a monopole existed, then we would answer the question "why is
> charge quantized" with "because it has to be." I guess we would turn
> two mysteries, "why is charge quantized?" and "why is angular momentum
> quantized?" into one, by showing that the second implies the first.
Conversely all magnetic charges should be integral multiples of
g = hbar*c
-------
2*e
= e/2*alpha alpha = fine. str con. = 68.5 * e.
in the literature g is sometimes g-sub D, D for Dirac.
Benjamin Weiner
>I always thought the best theorist DO think like experimentalist. Are
>they not always waiting to grab the results off the griddle and offer
>an explination? And do they not always(in the case of the best) offer
>an experiment that can prove thier theory?
In a perfect world ...
But also, the problems facing an experimentalist are not quite the
same as those facing a theorist ... where theorists may require
facility with complicated mathematics, or a sort of unchecked
creativity that leads them to invent things (like quarks, say),
a great experimentalist is someone whose creativity lies in thinking
up *how* to observe something, that possibly everyone assumed
simply could not be done - like Rabi's molecular beam experiments,
or Glaser and Alvarez for the bubble chamber, for example. (In _A
Random Walk in Science_ Glaser reminisces that the APS scheduled one
of his first talks about the bubble chamber in the Saturday "crackpot
session.")
Of course the above is a caricature and not to be taken too seriously ...
I think one problem in physics is that physicists tend to be bowled
over by theory and assume that experiments are cut-and-dried things
which are done out of cookbooks and simply return a "yes" or "no"
for a theory, and that they don't need to worry about knowing both
theory and experiment.
Gary Lipton
Grand Unified Theories popular in the early 80's predicted the
existence of *very* massive monopoles - sorry I don't have figures,
but I recall that the mass was on the order of micrograms - pretty
hefty. Some GUTs also predicted that monopoles could "catalyze"
proton decay - needless to say, none of this was verified experimentally.
In 1983 (I think) Cabrera of Stanford ran an experiment to detect
monopoles. The experiment detected one event which had a virtually
perfect signature for a monopole. This event occurred on a Sunday
when no one was monitoring the apparatus. The event was, as far as
I know, never explained and never repeated.
--
***************************************
* Gary Lipton, Dept. of EE *
* Lafayette College, Easton, PA 18042 *
***************************************
SCOTT I CHASE
>
>In 1983 (I think) Cabrera of Stanford ran an experiment to detect
>monopoles. The experiment detected one event which had a virtually
>perfect signature for a monopole. This event occurred on a Sunday
>when no one was monitoring the apparatus. The event was, as far as
>I know, never explained and never repeated.
Closer to early 1982. In the spring on 1982 I was in a freshman EM class
at MIT where Alan Guth declared that div B = 0 (Except at Stanford) as one
of Maxwell's equations.
Although nobody was ever able to explain the mystery event, Cabrera has
since has new ones of the same kind. His new detectors are coincidence
detectors. The original one was a single superconducting loop. The
newer ones are several loops, organized rather like the plastic loops on
a six-pack of soda, with two sets stacked one on top of the other.
If a monopole passes through one loop of the upper set, it will, with
high probability, pass through the corresponding loop in the lower set.
A valid monopole signal thus requires a coincidence of two loops. However,
during several years of running, Cabrera et.al. have had at least one
singles event, i.e., a single loop fired in the same way as the original
mystery signal - only this time there was a lower loop which did not fire
to veto the event as unambiguous, though unexplained, noise.
Based upon this experience, Cabrera has, in a paper (last year, if I
remember correctly) in PRL, officially retracted his original mystery
event as a possible monopole signal.
Daniel Seeman
>Some odds and ends about monopoles:
>
>Grand Unified Theories popular in the early 80's predicted the
>existence of *very* massive monopoles - sorry I don't have figures,
>but I recall that the mass was on the order of micrograms - pretty
>hefty. Some GUTs also predicted that monopoles could "catalyze"
>proton decay - needless to say, none of this was verified experimentally.
>
>In 1983 (I think) Cabrera of Stanford ran an experiment to detect
>monopoles. The experiment detected one event which had a virtually
>perfect signature for a monopole. This event occurred on a Sunday
>when no one was monitoring the apparatus. The event was, as far as
>I know, never explained and never repeated.
>
the project said the "detector" had been found to be defective in some way. But
my memory of this is definately foggy by now.
dks.
john baez
>If there are no monopoles then searching for them would indeed be
>pointless. But if charge quantization only increases the known lower
>bound on the number of monopoles in the universe by one, why should the
>knowledge that charge is quantized change our mind about searching for
>them? (Not objecting, just want to learn how to think like an
>experimentalist. :-)
For starters, you are being way too logical. It goes like this:
monopoles would explain charge quantization, which is otherwise
not easy to explain. So let's look for monopoles!
Yes, of course there might be just ONE monopole. For all we know, that's
what they saw in that mysterious irreproducible experiment - the one
and only monopole! But it goes against the grain in particle physics
to have particles that there's just ONE of. If there were "unique" particles
floating around it would be hard to discover universal laws, so we
optimistically assume there aren't.
Sean Merritt
In article <1993Jan20....@CSD-NewsHost.Stanford.EDU> pr...@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>If there are no monopoles then searching for them would indeed be
>pointless. But if charge quantization only increases the known lower
>bound on the number of monopoles in the universe by one, why should the
>knowledge that charge is quantized change our mind about searching for
>them? (Not objecting, just want to learn how to think like an
>experimentalist. :-)
# For starters, you are being way too logical. It goes like this:
# monopoles would explain charge quantization, which is otherwise
# not easy to explain. So let's look for monopoles!
The percipitating factor for the new searches was more due to the fact
that theorist found that monopoles "appeared" in GUT's. I think it
was in 1974, (for example see G. t'Hooft, Nucl. Phys. B, 79, 276 (1974))
that the subject heated up again. Although there has been a steady
stream of papers dating back to 1931, by both theorist and experimentalist.
The list reads like a "Who's Who" of physics(Alvarez,Bohm,Aharonov,Schwinger
Weinberg,Cabibbo,Berry(the phase guy),Kittlel,Drell,Rubbia, among others
and of course Dirac).
The interest is wide and varied. Berry for example showed that there
is a geometrical phase factor associated with an electron in the
field of a monopole.
I still think that the fact that they would add symmetry to Maxwell's
equations is a valid reason for the searches.
Matt McIrvin
>In article <1993Jan20....@CSD-NewsHost.Stanford.EDU> pr...@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>If there are no monopoles then searching for them would indeed be
>>pointless. But if charge quantization only increases the known lower
>>bound on the number of monopoles in the universe by one, why should the
>>knowledge that charge is quantized change our mind about searching for
>>them? (Not objecting, just want to learn how to think like an
>>experimentalist. :-)
>For starters, you are being way too logical. It goes like this:
>monopoles would explain charge quantization, which is otherwise
>not easy to explain. So let's look for monopoles!
I thought until a few days ago that an *alternative* explanation for
charge quantization would be to embed the electroweak gauge group
(including its abelian U(1) subgroup, which does not require particles
to have discrete weak hypercharge) in some nonabelian GUT gauge group,
which would restrict everything to transform as some discrete
representation of the GUT group and therefore force electric charge
to come in discrete units. However, someone recently told me
that there was a beautiful theorem stating that all such GUTs had
at least the possibility of magnetic monopoles, so one explanation
would, in a sense, imply the other! Does anyone know how true this is?
--
Matt McIrvin
Matt Austern
> Hmm, I don't. First, while naive monopoles change dF = 0, *d*F = J to the
> nicer-looking dF = K, *d*F = J (where F is the field strength, J the
> electric current, and K is the magnetic current), this destroys the
> gauge symmetry of the equations! Why trade a hefty infinite-dimensional
> symmetry group for a measly U(1)?
Is this really true? I've never thought very deeply about this, but
it seems reasonably plausible to me that you can have a version of
electromagnetism with "symmetric" looking Maxwell's equations if you
just change the gauge group from U(1) to U(1)xU(1), with electric
charges as the sources for one vector potential and magnetic charges
as the sources for the other.
Aside from the objection that U(1)xU(1) is an awfully ugly gauge
group, is there something obviously wrong with this idea that I've
overlooked?
--
Matthew Austern Just keep yelling until you attract a
(510) 644-2618 crowd, then a constituency, a movement, a
aus...@lbl.bitnet faction, an army! If you don't have any
ma...@physics.berkeley.edu solutions, become a part of the problem!
John C. Baez
>that theorist found that monopoles "appeared" in GUT's. I think it
>was in 1974, (for example see G. t'Hooft, Nucl. Phys. B, 79, 276 (1974))
>that the subject heated up again.
You're right.
>I still think that the fact that they would add symmetry to Maxwell's
>equations is a valid reason for the searches.
Hmm, I don't. First, while naive monopoles change dF = 0, *d*F = J to the
nicer-looking dF = K, *d*F = J (where F is the field strength, J the
electric current, and K is the magnetic current), this destroys the
gauge symmetry of the equations! Why trade a hefty infinite-dimensional
symmetry group for a measly U(1)? (Of course, one could argue that the
infinite-dimensional group, being "gauge" symmetries, is nonphysical and
worth less than the puniest group of "physical" symmetries. But gauge
theories have considerable charms.) Second, and more importantly, in
the context of GUTs one is not actually toying with Maxwell's equations
by putting in magnetic currents as above: one is going to a wholly more
complicated theory in which the Higgs field determines which gauge field
counts as the "electromagnetic" one by means of spontaneous symmetry
breaking, and monopoles are due to regions of space that can't make up
their minds (so to speak).
Andre Roberge
>In article <MERRITT.93...@macro.bu.edu> mer...@macro.bu.edu (Sean Merritt) writes:
(about monopoles)
>>I still think that the fact that they would add symmetry to Maxwell's
>>equations is a valid reason for the searches.
>Hmm, I don't. First, while naive monopoles change dF = 0, *d*F = J to the
>nicer-looking dF = K, *d*F = J (where F is the field strength, J the
>electric current, and K is the magnetic current), this destroys the
>gauge symmetry of the equations!
Not really... You're able to rewrite them as dF' = 0 and *d*F' = J'
by using a duality transformation (see Jackson Sect. 6.12) and thus still
be able to write F' = dA' (where the ' refers to the transformed
fields). Of course you can only do that if all particles have the same
ratio of magnetic to electric charge. But I bet you (I've never seen it
mentioned) that this is exactly what you get out of any decent GUT.
{it should be related to the fact that the proton charge comes out
to be exactly the same as the positron charge in GUTs. Use that plus
Dirac's quantization condition to prove this...}
Sean Merritt
In article <MERRITT.93...@macro.bu.edu> mer...@macro.bu.edu (Sean Merritt) writes:
>The precipitating factor for the new searches was more due to the fact
>that theorist found that monopoles "appeared" in GUT's. I think it
>was in 1974, (for example see G. t'Hooft, Nucl. Phys. B, 79, 276 (1974))
>that the subject heated up again.
# You're right.
>I still think that the fact that they would add symmetry to Maxwell's
>equations is a valid reason for the searches.
# Hmm, I don't. First, while naive monopoles change dF = 0, *d*F = J to the
# nicer-looking dF = K, *d*F = J (where F is the field strength, J the
# electric current, and K is the magnetic current), this destroys the
# gauge symmetry of the equations! Why trade a hefty infinite-dimensional
# symmetry group for a measly U(1)? (Of course, one could argue that the
# infinite-dimensional group, being "gauge" symmetries, is nonphysical and
# worth less than the puniest group of "physical" symmetries. But gauge
# theories have considerable charms.) Second, and more importantly, in
# the context of GUTs one is not actually toying with Maxwell's equations
# by putting in magnetic currents as above: one is going to a wholly more
# complicated theory in which the Higgs field determines which gauge field
# counts as the "electromagnetic" one by means of spontaneous symmetry
# breaking, and monopoles are due to regions of space that can't make up
# their minds (so to speak).
Yes, what you say about the Higgs is correct.
if we have a Gauge Group G such that:
_
G _) SU(3) x SU(2) X U(1),
The fact that G is simple both implies that there is only a single
gauge coupling constant and the quantization of weak hypercharge
(and hence electric charge).
now a Higgs is responsible for the symmeytry breaking, the gauge group
G is broken to a subgroup H by some Higgs field phi. In any finite
energy the asymptotic value of phi in any direction must minimize the
Higgs potential. There will be topologically non-trivial choices
of phi (phi = infinity,theta,phi) and thus monopole solutions
if the homotopy group is non-trivial pi2(G/H)(has more than one
element).
Theorem: If G is compact semi-simple, and simple connected:
pi2(G/H) = pi1(H)
pi1(H) is the group whose elements are homotopy classes
of functions from the circle S(superscript1).
This is the theorem that implies if there is a Higgs
then there is a monopole.
pi2(G/H) = pi1 (U(1)e-m x SU(3)color) = Z.
All GUTs have monopoles.
Now you want to validate the GUT (and the Standard Model),
1) You build a machine that can reach the energies of the
Higgs sector.
2) You look for monopoles which would be relics of the symmetry
breaking.
3) You do both.
For the details see the above reference by t'Hooft and
H. Georgi and S.L. Glashow , Phys Rev. Lett. 32, 438 (1974)
J. Kim Phys Rev. D23 , 2706 (1981).
Matt McIrvin
>In <1993Jan27.0...@galois.mit.edu> jb...@riesz.mit.edu (John C. Baez) writes:
>>First, while naive monopoles change dF = 0, *d*F = J to the
>>nicer-looking dF = K, *d*F = J (where F is the field strength, J the
>>electric current, and K is the magnetic current), this destroys the
>>gauge symmetry of the equations!
>Not really... You're able to rewrite them as dF' = 0 and *d*F' = J'
>by using a duality transformation (see Jackson Sect. 6.12) and thus still
>be able to write F' = dA' (where the ' refers to the transformed
>fields). Of course you can only do that if all particles have the same
>ratio of magnetic to electric charge. But I bet you (I've never seen it
>mentioned) that this is exactly what you get out of any decent GUT.
No. If that were the case the "magnetic monopoles" would be no
magnetic monopoles at all-- the duality transformation you've described
would illustrate that the electromagnetic part of the theory was just
monopole-free QED in disguise! If you perform the transformation on
electromagnetism as we know it you can re-identify every charged
particle as a "magnetic monopole" of this variety, and the physics
remains the same as it is with no monopoles.
When people talk about magnetic monopoles, what they mean is a
situation in which particles can have *different* ratios of
magnetic to electric charge. By convention (well, in the context of
the GUT I suppose it's not entirely convention, but if you just look
at the electrodynamics it's totally arbitrary) we say that the kind
of charge electrons and protons have is the electric charge, and
then these monopoles can *still* have magnetic charge-- I think GUT
monopoles are electrically neutral, in fact.
--
Matt McIrvin
John C. Baez
>In <1993Jan27.0...@galois.mit.edu> jb...@riesz.mit.edu (John C. Baez) writes:
>
>>In article <MERRITT.93...@macro.bu.edu> mer...@macro.bu.edu (Sean Merritt) writes:
> (about monopoles)
>>>I still think that the fact that they would add symmetry to Maxwell's
>>>equations is a valid reason for the searches.
>
>>Hmm, I don't. First, while naive monopoles change dF = 0, *d*F = J to the
>>nicer-looking dF = K, *d*F = J (where F is the field strength, J the
>>electric current, and K is the magnetic current), this destroys the
>>gauge symmetry of the equations!
>
>Not really... You're able to rewrite them as dF' = 0 and *d*F' = J'
>by using a duality transformation (see Jackson Sect. 6.12) and thus still
>be able to write F' = dA' (where the ' refers to the transformed
>fields). Of course you can only do that if all particles have the same
>ratio of magnetic to electric charge. But I bet you (I've never seen it
>mentioned) that this is exactly what you get out of any decent GUT.
>{it should be related to the fact that the proton charge comes out
>to be exactly the same as the positron charge in GUTs. Use that plus
>Dirac's quantization condition to prove this...}
Matt Austern (I think) asked if you could save gauge invariance in my
version of Maxwell-with-monopoles by going to U(1) x U(1) gauge group.
I don't see how. Seems to me this would just give "double
electromagnetism" - two copies of the ordinary Maxwell's equations.
I.e., electicity and magnetism with the usual properties (no magnetic
monopoles) and schmelectricity and schmagnetism with the usual
properties (no schmagnetic monopoles).
Roberge's suggestion also fails to sway me. If all particles have the
same ratio of magnetic to electric charge one can simply do a rotation
in the complex (E+iB) plane and say, without any essential difference,
that all particles have no magnetic charge. It sounds like this is
precisely the exercise in Jackson.
Plus, it's NOT true in GUTs that all particles have the same ratio of
magnetic to electric charge. At least not in the GUTs that predict
magnetic monopoles!!
John C. Baez
Theorem: If G is compact semi-simple, and simply connected:
pi2(G/H) = pi1(H)
pi1(H) is the group whose elements are homotopy classes
of functions from the circle S(superscript1).
This is the theorem that implies if there is a Higgs
then there is a monopole.
pi2(G/H) = pi1 (U(1)e-m x SU(3)color) = Z.
All GUTs have monopoles.
------------------------------------------------------------------------
Nice! Maybe to repay my debt I should explain the few tools of homotopy
theory one needs to prove this theorem. There is really not much one
needs to know to prove this sort of thing. First, for *any* Lie group
G, pi_2(G) = 0. Second, the following exact sequence for any bundle E
-> X with fiber F (we will apply this to the bundle G -> G/H with fiber
H):
... -> pi_n(F) -> pi_n(E) -> pi_n(X) -> pi_{n-1}(F) -> ...
This sequence is exact, meaning that the image of each arrow
(homomorphism) is the kernel of the next.
In our case we want to look at
pi_2(G) -> pi_2(G/H) -> pi_1(H) -> pi_1(G).
We know that pi_2(G) is zero, and we're assuming G is simply connected,
which *by definition* means pi(G) = 0. So we get
0 -> pi_2(G/H) -> pi_1(H) -> 0
The middle arrow has kernel zero and image equal to all of pi_1(H), so
it's an isomorphism - QED. (Or should I say QCD?)
Note that unless I've screwed up we never needed G to be compact
semisimple to get pi_2(G/H) = pi_1(H).
Anyone seriously interested in solitons, instantons or spontaneous
symmetry breaking should learn some of this stuff. For physicists, a
pleasant way might be to read Aspects of Symmetry by Sidney Coleman,
especially the section entitled Topological Conservation Laws... check
out footnote 24, where he says "Homotopy experts will recongnize the
preceding eight paragraphs as the world's most ham-handed exposition of
the exact homotopy sequence..." It's actually quite a nice heuristic
proof.
Allan Adler
Now that you have explained the math to the physicists, maybe now you
can explain the physics to the mathematicians?
My standards for such explanations might be more severe than necessary.
I need to know how, starting with the gauge group that describes the
particles, one computes the number of dollars one has to ask the
legislature for to build the accelerator with which one hopes to
observe the particle.
However, I will settle for the following: please confirm or deny, and
in any case elaborate on the assertion that one is still dealing
with the Schroedinger (or Dirac) equation and all this stuff about
gauge groups just goes into describing the potential term.
One physicist told me so but declined to go into any details.
Allan Adler
a...@altdorf.ai.mit.edu
Matt McIrvin
>However, I will settle for the following: please confirm or deny, and
>in any case elaborate on the assertion that one is still dealing
>with the Schroedinger (or Dirac) equation and all this stuff about
>gauge groups just goes into describing the potential term.
>One physicist told me so but declined to go into any details.
In a very real sense, yes-- in the general sense that
H |psi> = i hbar d/dt |psi>
is the Schrodinger equation for any physical system, not just a single
particle moving through space but a set of fields. The problem of
particle physics is to find:
(1) The space in which |psi> lives
(2) H, the Hamiltonian, which includes "potential" and "kinetic" terms
(1) involves the particle content of the theory. That also affects
(2), as does the nature of the interactions between the particles.
The Dirac equation is buried in the details of H, in this view. What
gauge theory gives you is a nice prescription for finding some "potential"
terms in H given certain assumptions about symmetries of the space of states.
--
Matt McIrvin
John C. Baez
>a...@zurich.ai.mit.edu (Allan Adler) writes:
>
>>However, I will settle for the following: please confirm or deny, and
>>in any case elaborate on the assertion that one is still dealing
>>with the Schroedinger (or Dirac) equation and all this stuff about
>>gauge groups just goes into describing the potential term.
>>One physicist told me so but declined to go into any details.
>
>In a very real sense, yes-- in the general sense that
>
> H |psi> = i hbar d/dt |psi>
>
>is the Schrodinger equation for any physical system, not just a single
>particle moving through space but a set of fields.
All quite true, but also, in a very real sense, no. McIrvin above
writes the general abstract "Schroedinger equation" which simply says
that a 1-parameter unitary group is generated by a self-adjoint operator
H. This is different from the good old Schroedinger equation in which H
is essentially the Laplacian. In gauge field theories one seeks to work
within the format of the of QM, hence seeks to use the general abstract
Schroedinger equation with some H. Now there are lots of gauge theories
of different sorts. In the simplest, describing an electron coupled to
electromagnetism, Adler's suspicion is close to the truth; the vector
potential for the EM field simply provides a potential term (albeit one
of a sort more subtle than the basic V(x)). In full-fledge quantum
electrodynamics things are more complicated, since one seeks to describe
the dynamics of the EM field and not just use it as a potential for the
electron. Here one uses the Yang-Mills equations for gauge group U(1)
and the electron is a particular representation of the gauge group. To
treat this quantum-mechanically requires "second quantization". Ditto
for more complicated gauge field theories like the standard model. I
have to run now but perhaps this will point Allan in the direction of
further questions.
Allan Adler
equation and the gauge theories just go into constructing the potential
term, John Baez writes:
In gauge field theories one seeks to work
within the format of the of QM, hence seeks to use the general abstract
Schroedinger equation with some H. Now there are lots of gauge theories
of different sorts. In the simplest, describing an electron coupled to
electromagnetism, Adler's suspicion is close to the truth; the vector
potential for the EM field simply provides a potential term (albeit one
of a sort more subtle than the basic V(x)).
I think that means "Yes".
In full-fledge quantum
electrodynamics things are more complicated, since one seeks to describe
the dynamics of the EM field and not just use it as a potential for the
electron. Here one uses the Yang-Mills equations for gauge group U(1)
and the electron is a particular representation of the gauge group. To
treat this quantum-mechanically requires "second quantization". Ditto
for more complicated gauge field theories like the standard model.
I think that means "No".
I have to run now but perhaps this will point Allan in the direction of
further questions.
Yes, thanks, it does.
(1) Let G be a Lie group, g its Lie algebra. The examples
of gauge theories I have seen suggest that one wants G to be compact.
I have heard rumors that one also likes to let G be noncompact, e.g.
Irving Segal is fond of SU(2,2) and SO(4,2) (which are locally isomorphic)
and others are fond of SO(3,1) and the Poincare group. But I don't know
whether it is ok to have gauge theories based on these noncompact
groups, at least physically. Is it? And instead of using the compact
groups, why not use the complex groups, e.g. SL(n,C) instead of SU(n)?
(2) Let X be a manifold and let E be a principle G bundle and let D be a
connection on E. I suppose one wants X to have some kind of structure on it
too, but I am not sure what. Now, is this enough information to write down
the Schroedinger equation for particles interacting via the force
described by D? And if so, how does one write it down?
(3) I have seen this second quantization in some physics books,
such as one by Haken, others by Amnon Yariv and elsewhere. Inevitably,
I am left without a clue as to how one knows what is going on
anymore and in particular how one is able to compute the number of
dollars to ask the legislature for. On the other hand, I have heard
rumors that second quantization has to do with lifting ideals from
A to U, where U is the universal enveloping algebra of g and A is the
associated graded algebra. Or was that first quantization? At any rate,
is there a concise abstract and general point of view on second
quantization? Well, there are more rumors: with first quantizaiton
one gets a representation of the Lie algebra sl(2,R) or sl(2,C)
on your Hilbert space, while with second quantization one gets
this representation from the Weil representation, with sl(2,R)
normalizing a representation of a Heisenberg Lie algebra, the generators
of the Heisenberg Lie algebra being your basic creation and annihilation
operators. Since I am in touch with all this only by rumor,
will the real second quantization please stand up?
(4) Whether the preceding question is well posed or not, there is
still the uneasy feeling that it is not really sufficiently
general. It seems to me that the formulation is tailored to
Quantum Electrodynamics (QED) and that things may not be so
simple if one wants to talk about second quantization for
forces other than the electromagnetic force. One reason I feel
uneasy about this is that the equations describing the electromagnetic
force are linear and the general equations for other Lie groups
are nonlinear. So I need to be persuaded that this does not make
an essential difference when one wants to talk about second
quantization for forces other than the electromagnetic force. Does it?
Allan Adler
a...@altdorf.ai.mit.edu
Dale Hall
>Sean Merritt writes:
>>
>>Theorem: If G is compact semi-simple, and simply connected:
>>
>>pi2(G/H) = pi1(H)
>>
>>pi1(H) is the group whose elements are homotopy classes
>>of functions from the circle S(superscript1).
>>
>>
>
>theory one needs to prove this theorem. There is really not much one
>needs to know to prove this sort of thing. First, for *any* Lie group
>G, pi_2(G) = 0. Second, the following exact sequence for any bundle E
>-> X with fiber F (we will apply this to the bundle G -> G/H with fiber
>H):
>
>... -> pi_n(F) -> pi_n(E) -> pi_n(X) -> pi_{n-1}(F) -> ...
>
>This sequence is exact, meaning that the image of each arrow
>(homomorphism) is the kernel of the next.
John proceeds to use the vanishing of pi_2(G) and the above
long exact sequence to show that the above theorem is true
even with fewer assumptions. I've cut to the chase:
>
>Note that unless I've screwed up we never needed G to be compact
>semisimple to get pi_2(G/H) = pi_1(H).
>
As I've noted in my summary, this may be a minor nit, but it
could represent a glaring hole in my ever-dwindling command of
topology: there is a potential weak point in writing the
quotient sequence H --> G --> G/H as a fibration. In the case
Sean and John were looking at, H is a closed subgroup of the
Lie group G, and for this, the quotient is a [locally trivial]
fiber bundle. However, I'm not sure that you even get a
fibration (the condition necessary to produce the long exact
sequence above) if H is not a closed subgroup.
For instance, I'm imagining some non-compact H (of positive
codimension) embedded as a dense subgroup of G or of some
closed subgroup. The only real example I've been toying with
is a copy a(R^1) of the line, embedded as a line of irrational
slope in the torus T^2. This example has a quotient space
with the trivial topology, and the quotient mapping does not
even admit path lifting, let alone lifting of homotopies.
Result: no fibration, no long exact sequence, and no fun.
The example seems to allow generalization to R^n --> T^(n+k)
and probably inclusion of the torus as a maximal torus in your
favorite Lie group, but I'm not sure of what other examples
exist (e.g., U(n) --> something compact?). Actually, now that
I think of it, this is all either well-known or wrong.
That's about all I had to say, unless I'm wrong about the
above example, in which case, I'll mumble about going to
listen to some Chopin etudes until I have tears of joy in my
eyes.
Dale.