beyond the top quark
Hunting the Snark
implication that the discovery of the Top (or, as I prefer, Truth) quark would
in effect "complete" the standard model. My question is why should this be so?
Why couldn't there be an "infinite" series of quarks at ever increasing energy
levels? I realize that this is probably the first question one asks in QCD,
but could some kind soul please give me the standard answer? Thanks.
/>
( //------------------------------------------------------------(
(*)OXOXOXOXO(*>=S=T=O=R=M=B=R=I=N=G=E=R-------- \
( \\--------------------------------------------------------------)
\> Steven Marshall
"Hard to say Ma'am. I think my cerebellum just fused" -- Calvin
Christopher Neufeld
>Another "physics lite" question inspired by NOVA. There seemed to be the
>implication that the discovery of the Top (or, as I prefer, Truth) quark would
>in effect "complete" the standard model. My question is why should this be so?
>Why couldn't there be an "infinite" series of quarks at ever increasing energy
>levels? I realize that this is probably the first question one asks in QCD,
>but could some kind soul please give me the standard answer? Thanks.
>
family. This was shown on NOVA. For example, we associate the up and down
quarks with the electron and the electron neutrino.
Current experimentation shows the electron and muon neutrinos to be very
light (last I'd heard the mass limit on the tau neutrino was rather high),
possibly massless.
Cosmology constrains the number of low mass neutrinos to not more than
four, five if you really stretch the model.
The Z0 lifetime decreases with the number of low mass neutrinos into
which it can decay. If the neutrinos are much lighter than half the Z0 mass
(ie much lighter than about 40GeV), they provide a large phase space into
which the Z0 can decay. Recent measurements at LEP and SLAC have
established the lifetime of the Z0 to enough accuracy that the number of
lepton families with light neutrinos is known to be three, to within two
sigmas. I should point out that these numbers are being refined even as I
write this, as the data is only weeks old, so the two sigma value may
already be out of date.
So, it is unlikely that there is a fourth flavour of light neutrino, and
hence under the standard model there are believed to be only three families
of quarks. Of course, if there is a fourth lepton family with a 100GeV
neutrino, this argument is invalid. This is also considered unlikely,
though not impossible.
> \> Steven Marshall
--
Christopher Neufeld....Just a graduate student | "Give me ten men like
neu...@helios.physics.utoronto.ca | Clouseau and I could
cneu...@pro-generic.cts.com | destroy the world."
"Don't edit reality for the sake of simplicity" |
john baez
Hey, compared to the ESP stuff this is heeeavy, man.
>Why couldn't there be an "infinite" series of quarks at ever increasing energy
>levels? I realize that this is probably the first question one asks in QCD,
>but could some kind soul please give me the standard answer? Thanks.
According to standard mythology (the standard model, not
QCD by itself) the leptons come in "generations" paralleling
those of the quarks. Why? Don't ask. :-) If one assumes
that all the neutrinos are massless, or at least quite light,
too many generations means too many light particles around,
which would screw up the thermodynamics of the early universe,
or, even if you don't believe in the details of the big bang
scenario, supernovae (neutrinos carry out most of the
energy of a supernova initially, apparently).
More recently, people have started mass manufacturing Z's
which can decay into neutrinos. Observations indicate that
there are exactly 3 (light) neutrinos, hence (IF ONE BELIEVES
THE GENERATION IDEA) exactly three generations of quarks.
I don't know if there's a more compelling argument for the
top being the last quark. The "symmetry" between leptons and
quarks is to my mind one of those great mysteries of physics
that awaits a good explanation.
Soren G. Frederiksen -- Ohio State University
> Another "physics lite" question inspired by NOVA. There seemed to be the
> implication that the discovery of the Top (or, as I prefer, Truth) quark would
> in effect "complete" the standard model. My question is why should this be so?
> Why couldn't there be an "infinite" series of quarks at ever increasing energy
> levels? I realize that this is probably the first question one asks in QCD,
> but could some kind soul please give me the standard answer? Thanks.
>
It's possible that there is an "infinite" series of quarks, but NOT if the
standard model as we now know it is correct. In the standard model for every
pair of quarks that you have you must also have a corresponding pair of
leptons, a "heavy" electron plus a massless neutrino. So the number of
different types of neutrino's is a measure of how many generations (or series)
of quarks that one has. However, recent results from CERN have shown that the
number of neutrino's that can exist is about 3.0 +/- 0.2, which is obviously
less than 4 or much much much less than infinity.
This experiment was done at the LEP accelerator (an electron-positron
collider), and the basic idea was to study the decay of the Z0 (the neutral
partner of the W). One simply counts the number of events produced as a
function of energy, close to the mass of the Z0, looks at the distribution and
the width of the distribution will tell you how many neutrinos with a mass less
than half that of the Z0 exist. Considering that the Z0 has a mass of 90
GeV/c**2, that's pretty massive. Of course that is the limit in this
experiment, if neutrino's are not massive and the neutrino in the fourth
generation has a mass above 45 GeV/c**2 then the LEP limit does not work.
Of course, then I don't think you'd be talking about the "standard model"
anymore, or at least a modified version of the "standard model".
I hope that answers your question.
Soren Frederiksen
Eric G. Stern
> Another "physics lite" question
> inspired by NOVA. There seemed to be the implication that
> the discovery of the Top (or, as I prefer, Truth) quark
> would in effect "complete" the standard model. My question
> is why should this be so? Why couldn't there be an
> "infinite" series of quarks at ever increasing energy
> levels? I realize that this is probably the first question
> one asks in QCD, but could some kind soul please give me the
> standard answer? Thanks.
There is no particular reason why there couldn't be an infinite number
of quark generations. However, it appears not to be so for no good reason.
The recent turn-on of LEP has produced enough Z particles so that we can get
a very good measurement of the Z width. The Z width contains a term which is
proportional to the number of neutrinos that the Z can decay into. In the
standard model, neutrinos are massless so the Z could decay into any neutrino
that exists. Even if they aren't massless, it appears that they are quite
light on the scale of the Z mass. Using the measured width of the Z, we now
know that there are only three generations of leptons, which are the electron
family, the muon family, and the tau family. For symmetry reasons, we think
that the number of lepton generations is the same as the number of
quark generations which is why the standard model is complete once thee top
quark has been found. Why three generations is not well understood, although
we know why it is not more that 15 or so.
Eric Stern
Dept. of Physics and Astronomy
University of Pittsburgh
st...@unix.cis.pitt.edu
H. de Bruijn
> ... the discovery of the Top (or, as I prefer, Truth) quark ...
^^^^^
Or, as I prefer, Bull, with its well known anti-particle, finally resulting in
the Grand Unified Theory of Nature called BSD ...
Before reading the .signature:
A little bit of Sense (and good taste) would be NO idleness in Physics. (HdB)
--
* Han de Bruijn; computer graphics | "A little bit of Physics *
* TU Computing Centre; P.O. Box 354 | would be NO idleness in *
* 2600 AJ Delft; The Netherlands | Mathematics" (HdB). *
* Fax: +31 15 78 37 87 ============================================
Loren Petrich
>In article <8...@arkham.enet.dec.com> mars...@elric.dec.com (Hunting the Snark) writes:
>>Why couldn't there be an "infinite" series of quarks at ever increasing energy
>>levels? I realize that this is probably the first question one asks in QCD,
>>but could some kind soul please give me the standard answer? Thanks.
>
>According to standard mythology (the standard model, not
>QCD by itself) the leptons come in "generations" paralleling
>those of the quarks. Why? Don't ask. :-)
>
>top being the last quark. The "symmetry" between leptons and
>quarks is to my mind one of those great mysteries of physics
>that awaits a good explanation.
This whole question of why there is this symmetry between
leptons and quarks and why there are three generations of elementary
fermions are questions that Grand Unified Theories (GUT's) are
intended to answer. Why the symmetry between quarks and leptons?
Apparently, this is something that prevents the occurrence of
"anomalies", quantum-mechanical violations of certain classical
symmetries. For one type of anomaly of the electro-weak interactions
not to appear, the charges of all elementary fermions must add up to
zero. For each generation, we have 3*(2/3) + 3*(-1/3) + 0 + (-1) = 0.
Note the counting of quark colors (3 each). It is for that reason that
it is thought that the top quark must exist.
The occurrence of three "light" generations is still not well
understood. The most likely answer to this question may come from
Superstring theory. Strings have an infinite hierarchy of states; with
spacing between mass^2 values of order (Planck mass)^2 expected. The
elementary particles that we observe are expected to be some subset of
the massless superstring states. These states would remain massless if
(1) space-time was flat in all 10 dimensions and (2) there was no
symmetry breaking of the "gauge" fields. However, both (1) and (2) are
expected to be violated, making a large fraction of the states highly
massive, with masses only a few orders of magnitude less than the
Planck mass, at least. Only a certain subset of them would have the
relatively low masses of the observed elementary particles. Which
subset? There are some promising results from superstring theory, but
I'm not sure I'd want to bet on that.
^
Loren Petrich, the Master Blaster \ ^ /
lo...@moonzappa.llnl.gov \ ^ /
One may need to route through any of: \^/
sunlight.llnl.gov <<<<<<<<+>>>>>>>>
lll-lcc.llnl.gov /v\
lll-crg.llnl.gov / v \
star.stanford.edu / v \
v
"I'm supposed to keep my mouth shut" -- Madonna
Jeff Boscole
There is no top quark.
.
Eric G. Stern
>
> There is no top quark.
> .
Interesting theory. Do you have supporting evidence?
Does the top quark violate some fundamental symmetry in your theory?
What is this symmetry and how is it observed?
Jeff Boscole
>> There is no top quark.
>
There is no supporting evidence.
>Does the top quark violate some fundamental symmetry in your theory?
There is no violation. There is no symmetry. There is no theory.
>What is this symmetry and how is it observed ?
There is no observation.
.
John Logajan
>There is no top quark.
What is this, some sort of egalitarian physics?
--
- John Logajan @ Network Systems; 7600 Boone Ave; Brooklyn Park, MN 55428
- log...@ns.network.com, jo...@logajan.mn.org, 612-424-4888, Fax 424-2853
Robert Firth
A: There is no top quark.
B: Interesting theory. Do you have supporting evidence?
A: There is no supporting evidence.
The Tao is silent! However, here's my theory: the structure of
matter is infinitely complex, like fractals. The more energy you
smash into it, the more meaningless fundamental particles you will
discover.
Eventually you may discover that Nature should be enticed, not
tortured. Such is the way of the Tao.
Loren Petrich
>
>There is no top quark.
>
Why do you say that?
If it was shown not to exist, then a lot of particle
physicists will empathize with Albert Einstein, who commented that, if
the bending of starlight due to the Sun's gravity was not the right
size, that "I would feel very sorry for the dear Lord -- the theory is
correct."
Loren Petrich
Say what?
Seriously, folx.
According to the "Standard Model", which has been well-tested
so far, the top quark has to exist. Why?
First of all, let us consider the known elementary particles.
These are the left-handed lepton doublet (electrons + neutrinos), the
right-handed electron, the left-handed quark doublet (up+down), the
right-handed up quark and the right-handed down quark. Here,
"electron" includes the mu and the tau, "up" includes charmed and top
(if it exists), and "down" includes strange and bottom.
Dirac mass terms connect the left- and right-handed components
of a particle field; electrons, ups, and downs all have Dirac masses.
These matrices have the number of generations as their dimension;
their eigenvalues determine the mass values and their eigenvectors
determine the generation mixing. For the up and the down quark sets,
the eigenvectors are not quite parallel, giving rise to
cross-generation decays.
A charged-weak interaction turns one member of a left-handed
doublet into the other, thus making decays and whatnot.
In this theoretical scheme, the top quark _has_ to exist.
Otherwise, the bottom quark would have a hard time decaying into
lighter quarks -- as it has been observed to do. Furthermore, it
behaves the same way under the weak interaction as do the other
quarks.
Any comment on this theoretical argument?
Ranjan Samuel Muttiah
>But I think it's quite likely that something like what you're
>saying is right. The notion of "fundamental particles" seems
>to be leading us down an infinite corridor... with no light at
>the end in sight (despite the crowings of the superstring crowd,
That means by a simple logical argument atoms don't exist at all!
For collisions of matter each one of the "infinite particles" of atoms
has to be affected. Thus, an infinite time. This contradicts observation,
therefore atoms don't exist.
john baez
>The Tao is silent!
Now you've gone and told everyone. Go ahead, shout it from
the rooftops.
>However, here's my theory: the structure of
>matter is infinitely complex, like fractals. The more energy you
>smash into it, the more meaningless fundamental particles you will
>discover.
>Eventually you may discover that Nature should be enticed, not
>tortured. Such is the way of the Tao.
Actually, it's more likely that we'll simply run out of
money. I bet the SSC will be a gigantic fiscal fiasco.
But I think it's quite likely that something like what you're
saying is right. The notion of "fundamental particles" seems
to be leading us down an infinite corridor... with no light at
the end in sight (despite the crowings of the superstring crowd,
who you may notice are lapsing into an embarassed silence lately).
The only serious attempts I know to find a different paradigm
are Heisenberg's unified field theory and Geoffrey Chew's
concept of "particle democracy" (not to be confused with
the "there is no top dog" theory), both of which try to explain
particles as resonances in a complex field theory which, however,
is not built from "fundamental particle fields" in the way that,
say, the standard model is. Chew had the audacity to hope
that the analyticity, unitarity, and a few other properties
of the S-matrix would uniquely determine the field theory, and
the particles would fall out as consequences. The guy who
wrote the Tao of Physics was all gung-ho about this "bootstrap
theory", precisely because of its philosophical charms, but it
was cast aside in favor of the quark theory.
I wish I knew what a good approach was. I've always been
inclined to think that the clue somehow lies in: 1) the
"generations" of increasing mass leptons (and quarks, those
more dubious entities), with their mysterious mass ratios,
and 2) gravity, always the odd man out among the four fundamental
forces.
But if particle accelerators "torture" nature, you'd think that
supernovae would be illegal. When the animal rights groups
work their way down past bacteria, viruses, etc. we can expect
to see particle rights becoming a major issue.
alan herbst
>>Why couldn't there be an "infinite" series of quarks at ever increasing
>>energy levels?
>I don't know if there's a more compelling argument for the
>top being the last quark. The "symmetry" between leptons and
>quarks is to my mind one of those great mysteries of physics
>that awaits a good explanation.
For further reading on the above subject of elementary particle
generations, the following references are provided:
1. Scientific American, June 1988, page 66, "Particle Accelerators Test
Cosmological Theory" by David N. Schramm and Gary Steigman.
Subtitle: "Is there a limit to the number of families of elementary
particles? Debris from the big-bang origin of the universe suggests
there is, and accelerators are reaching the energies required to confirm
the limit".
2. Scientific American, August 1988, page 60, "Beyond Truth and
Beauty: A Fourth Family of Particles" by David B. Cline.
Subtitle: "Three families of the fundamental particles called quarks
and leptons are known. Recent experiments hint that there is one more
family, but there are probably no more than five".
Happy Top Hunting!
Sir Six
>That means by a simple logical argument atoms don't exist at all!
>For collisions of matter each one of the "infinite particles" of atoms
>has to be affected. Thus, an infinite time. This contradicts observation,
>therefore atoms don't exist.
This is merely an absurd variation on Zeno's "paradox": That
to cover a finite region of space, on must cover an infinite number
of points in space, and since one is only at one point at any given
time, one requires an infinite number of points in time, and
therefore an infinite amount of time.
Of course, the flaw in this reasoning is that an infinite
number of points in time can be contained in a finite interval of
time just as an infinite number of points in space can be contained
in a finite region of space.
There are at least two obvious problems with Ranjan's
reasoning: 1) An infinite number of points in time exists in a
finite interval. Thus, suppose that the first particle is affected
in a half a bleem (an Orkian unit of time), the second particle is
affected a quarter of a bleem later, the third an eighth of a bleem
later, etc. Thus, after one bleem has elapsed, all particles will
have been affected, even if there are an infinite number of them.
2) Besides, we haven't ruled out the possibility of an infinite
number of particles being affected simultaneously.
Then again, maybe Ranjan's article was intended to be humorous.
If so, please regard this article as such as well, so I don't look
like such an idiot. Thank you.
Sir Six
Kurt Sonnenmoser
writes:
>According to the "Standard Model", which has been well-tested
>so far, the top quark has to exist. Why?
[ ... explanation deleted ... ]
>Any comment on this theoretical argument?
Of course, the best conceivable argument in favour of the existence of a
yet unobserved particle is its theoretical necessity in an otherwise
successful theory. The list of particles postulated this way is long:
the positron (Dirac), the pi-meson (Yukawa), the W and the Z (Glashow,
Salam, Weinberg), etc. In fact, considering the tremendous success of
the Standard Model, it would be a much bigger miracle if the top quark
wouldn't exist.
However, the Standard Model does not *have* to be the right theory.
So the top quark does not *have* to exist. You might say: It has to
exist, because otherwise current theoretical particle physics would be
in serious trouble :-)
--
Kurt Sonnenmoser, Institut fuer Theoretische Physik, Universitaet Zuerich,
Switzerland. Standard disclaimer applies. UUCP:...mcvax!cernvax!forty2!ks
Tony Wallis
| > The notion of "fundamental particles" seems to be leading us down an
| That means by a simple logical argument atoms don't exist at all! For
| collisions of matter each one of the "infinite particles" of atoms has
| to be affected. Thus, an infinite time. This contradicts observation,
| therefore atoms don't exist.
The sum of an infinite number of terms can be finite.
E.g. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1
i.e. you *can* do an infinite number of things in finite time -
*mathematically* there is no inherent problem. Physically, however,
current theory breaks down around 10^-43 secs. = 10^-33 cms. (the
Planck time and length).
--
--
Tony Wallis to...@yunexus.UUCP/to...@nexus.yorku.ca (York U. Toronto Canada)
john baez
HUH?? "That means....." First, what's "that"? The only thing
vaguely sensible is that your "that" is "the existence of an
infinite sequence of more and more fundamental particles". Is
this what "that" is??? If it is, your argument is baloney.
(People usually say "a simple logical argument" when it's baloney.)
Or maybe you're joking. You're joking, right?
Greg
>saying is right. The notion of "fundamental particles" seems
>to be leading us down an infinite corridor... with no light at
I find this hard to believe, given the history of physics. Even
200 years ago there were zillions of "basic" kinds of matter and
dozens of "fundamental" forces. There were flesh, wood, fibers,
metals, salts, fluids, gases, and rocks, for example. Each of these
had very different properties and came in many different varieties.
The "forces" were less in number but more mysterious: Heat, gravity,
pressure, inertia, light, sound, and chemical reactions, for example.
Since then we've seen great simplifications. By 1900 there was a list
of elements: Only about 100 fundamental kinds of matter which explained
many things. Although they were complicated, they fell into patterns.
By 1950 almost all that had been observed up to the 20th century could
be explained with four particles: Electrons, neutrons, protons, and
photons. The only force involved was the electromagnetic force, implicit
in the description of the particles. But alas, there were still a
few phenomena that had yet to be explained.
And what do we have now? Four forces; EM, the weak force, the strong
force, and gravity; which physicists claim explain absolutely
everything at accessible energies. There are 16 particles: The W, the
Z, the photon, the gluon, three electrons, three neutrinos, and six
quarks. And that's supposed to be all! And there are patterns:
Counting the electroweak theory, the four forces should be counted as
three, the W,Z, and photon as one, the six quarks as three, and each
electron the same as its neutrino. Not only that, nature has repeated
itself: You have an electron-neutrino pair and a quark pair three
times. The whole theory can be stated on one page.
So the tunnel looks far too short to me to be infinite. There is dim
light already. Of course, there are still mysteries, principally
that gravity remains aloof, and there is the difficulty that we
can't afford to go to much higher energies. But I believe brain
power will eventually prevail, even if fiscal power will not.
Theoretical physics would live even if the SSC was killed.
>Actually, it's more likely that we'll simply run out of
>money. I bet the SSC will be a gigantic fiscal fiasco.
It already is. But then, Government funding in general is now based
on a philosophy of big science over good science.
---
Greg
Tim Maroney
>>But I think it's quite likely that something like what you're
>>saying is right. The notion of "fundamental particles" seems
>>to be leading us down an infinite corridor... with no light at
>>the end in sight (despite the crowings of the superstring crowd,
In article <96...@medusa.cs.purdue.edu> mut...@cs.purdue.edu (Ranjan
Samuel Muttiah) writes:
>That means by a simple logical argument atoms don't exist at all!
>For collisions of matter each one of the "infinite particles" of atoms
>has to be affected. Thus, an infinite time. This contradicts observation,
>therefore atoms don't exist.
What is this, Zeno's Paradox Considered as a Helix of Semiprecious
Stones? It don't wash, Ran. Who says the elements of the infinite set
get affected one at a time? If an infinite number of people all step
left at the same time, then it only takes the duration of a single step
for them all to finish. Even if they do get affected one at a time,
then if each transition takes infinitesimal time, the whole sequence
can complete in finite time. The reciprocal of infinity times infinity
is one.
The fractal infinity model of quantum physics is interesting; it might
make sense of the prenormalization infinities, anyway. But I'm unclear
on whether it's something that a lot of people seem to feel intuitively
drawn towards, or it's an actual current physical theory in competition
with the Standard Model.
I also have some problems with the Firth/Baez idea of an infinity of
particle types at descending scales. The number of particles does seem
to drop as you descend size scales; there are uncountably many possible
molecules, an integral infinity of possible atoms, a finite number of
apparent baryons (yes, I'm oversimplifying, but the point stands), only
a few quarks, and presumably even fewer quark components. The laws of
combinatorial mathematics seem to require this. It seems hard to get
around the idea that at some point, we'll hit a level where there's
only one kind of particle and one kind of force. That doesn't mean we
won't also find fractal infinities at that level, of course.
--
Tim Maroney, Mac Software Consultant, sun!hoptoad!tim, t...@toad.com
"But don't you see, the color of wine in a crystal glass can be spiritual.
The look in a face, the music of a violin. A Paris theater can be infused
with the spiritual for all its solidity."
-- Lestat, THE VAMPIRE LESTAT, Anne Rice
Ranjan Muttiah
>In article <96...@medusa.cs.purdue.edu> mut...@cs.purdue.edu (Ranjan
>Samuel Muttiah) writes:
>>That means by a simple logical argument atoms don't exist at all!
>>For collisions of matter each one of the "infinite particles" of atoms
>>has to be affected. Thus, an infinite time. This contradicts observation,
>>therefore atoms don't exist.
>
>What is this, Zeno's Paradox Considered as a Helix of Semiprecious
>Stones? It don't wash, Ran. Who says the elements of the infinite set
>get affected one at a time? If an infinite number of people all step
>left at the same time, then it only takes the duration of a single step
>for them all to finish. Even if they do get affected one at a time,
>then if each transition takes infinitesimal time, the whole sequence
>can complete in finite time. The reciprocal of infinity times infinity
>is one.
ahem. Since when have dancing partners come into the physical
picture Tim ? :) Actually, the argument isn't mine. It's Leibniz',
in a letter to Huygens. Who said that a little bit of letter.plagarism
is illegal ?
john baez
>make sense of the prenormalization infinities, anyway. But I'm unclear
>on whether it's something that a lot of people seem to feel intuitively
>drawn towards, or it's an actual current physical theory in competition
>with the Standard Model.
As far as I know there is no real "theory" (equations & all) which
gives some solid form to the idea of nested particles within
particles ad infinitum.
>I also have some problems with the Firth/Baez idea of an infinity of
>particle types at descending scales.
This wasn't really what I had in mind; I was proposing a model
in which one might THINK there were infinitely many levels
of more and more `fundamental' particles, if the only theories
one considered were those built out of elementary particles,
but in fact there were NO really elementary particles. For
example, the observed mesons and baryons could all be excitations
of some field theory whose Lagrangian was not a sum of free-particle
Lagrangians and interaction terms. The Heisenberg and Chew models
were of this type.
The number of particles does seem
>to drop as you descend size scales; there are uncountably many possible
>molecules, an integral infinity of possible atoms, a finite number of
>apparent baryons (yes, I'm oversimplifying, but the point stands), only
>a few quarks, and presumably even fewer quark components. The laws of
>combinatorial mathematics seem to require this. It seems hard to get
>around the idea that at some point, we'll hit a level where there's
>only one kind of particle and one kind of force. That doesn't mean we
>won't also find fractal infinities at that level, of course.
I agree that renouncing the fundamental particle concept at this
point would be premature, since it seems like progress is
being made. I must point out that you're exaggerating the
progress a bit here. Unless you're considering molecules made
of infinitely many atoms, or counting the continuum of possible
ways there are to configure the atoms in some flexible molecules,
there are countably many molecules, rather like there are countably
many possible finite-length words made out of a finite alphabet.
There are finitely many possible atoms as far as we now believe;
not even counting nuclear instability, when there are enough
electrons there will be vacuum "sparking" causing spontaneous
formation of positron/electron pairs which reduce the atomic
number by one. Naive calculations give element 137 (fine structure
constant) as the biggest possible one. It seems that there are
infinitely many possible baryons, though one could call many of
these excited states. But anyway, this is just nitpicking. If
we get one kind of particle and one kind of force, especially if
the particle is the carrier of the force, I'll be very content.
Six quarks, six leptons (one of each unobserved), a photon,
a W, a Z, some gluons, an observed Higgs, and an elusive
graviton make me very unhappy. So I just wanted to raise again
the possibility that the notion of "free" particles which
"interact" to form bound states might be philosophically naive
and misleading. Time will tell.
As for the SSC, someone pointed out that it's already a fiscal
fiasco. My point was that I bet it'll get a lot worse.
Ranjan Muttiah
>HUH?? "That means....." First, what's "that"? The only thing
>vaguely sensible is that your "that" is "the existence of an
>infinite sequence of more and more fundamental particles". Is
>this what "that" is??? If it is, your argument is baloney.
>(People usually say "a simple logical argument" when it's baloney.)
>Or maybe you're joking. You're joking, right?
Yes, and let's hear you tell us how funny things can get.
Ranjan Muttiah
> There are at least two obvious problems with Ranjan's
>reasoning: 1) An infinite number of points in time exists in a
>finite interval. Thus, suppose that the first particle is affected
>in a half a bleem (an Orkian unit of time), the second particle is
>affected a quarter of a bleem later, the third an eighth of a bleem
>later, etc. Thus, after one bleem has elapsed, all particles will
>have been affected, even if there are an infinite number of them.
>2) Besides, we haven't ruled out the possibility of an infinite
>number of particles being affected simultaneously.
The picture I have is to have an infinite number of marbles laid on
the ground one behind the other. Now roll a marble to the head
of the line. The time it takes one marble to strike another to the
right (local time) will be the same. On a global time scale, it
should take an infinite amount of time.
This argument assumes of course that atoms can be extended to infinity.
BONER !?!?
I have a theory (if it could be called that). It goes like this.....
One sunday afternoon I was taking a break from Hawking's latest book
and decided to flip on the t.v. I came across a PBS show that was was very
good, and at the end of the Nova-type show the conclusion you had to come
to was that since science worked for most civilizations before ours and proved
to be "incorrect" ours science must also be
What does this have to do with anything? well in the show a certain
people (I can't recall who) thought that everything in the universe was
circular or spherical, i.e. planets are spheres, orbits are circles etc.
Well it was soon discovered that orbits were not spherical so to keep the
theory they said that while the moon or whatever was going in a big circle
it was also going in little circles. Turns out that this theory gets ellipticalorbits of planets (sorry I didn't explain it to well).
Point (finally) being that it seems to me that science keeps creating
smaller and smaller particles to explain the make-up of matter. Could it be
that we are using smaller circles to explain matter when it is really our
theory that needs revamping? I realize this is a run-of-the-mill question
being that the top quark has not yet been found despite fermilab's efforts,
but doesn't the smaller circles theory seem remarkably similar to our
"smaller particles" theory?
P.S. I'm sure or hope this is going to generate lots of mail slashing
my posting which is what I want.
john baez
>In article <41...@ucrmath.UCR.EDU> ba...@x.UUCP (john baez) writes:
>
Again, you leave me bewildered. What do you mean,
how funny things can get? Things in general?
Anyway, here's omething I've been studying that's an
example of how funny things can get. People are currently
interested in the idea that high-Tc superconductivity
might be a two-dimensional effect due to the layered
structures of the crystals involved. Funny things can
happen in two dimensions. If we think of a two-dimensional
magnet as having a spin of the same length at each point
but varying direction, so that the spin vector is a smoothly
varying unit vector, and if this vector approaches a constant
value at spatial infinity, then what we have is a map from
the (2-)sphere to itself. (The 2-sphere is the sphere
of unit vectors in 3-dimensional space, and similarly
for the n-sphere.) It's known that there are various homotopy
classes of such maps, parametrized by an index known as
the degree, which is very much like a "winding number"
in that it counts how many times the 2-sphere is wrapped
around itself (with a sign!). These "twists" can act
like particles, and are known as skyrmions (and anti-
skyrmions). From topological reasons total skyrmion
number is conserved if things change smoothly with time.
Are these skyrmions bosons or fermions? Well, it
turns out that if there is a "topological term" in
the Lagrangian, they can be bosons, or fermions, or
"anyons" (things in-between bosons and fermions,
which smoothly interpolate between the two, and can
only exist in 2 dimensional space!), depending on the
constant.
If I could draw pictures, it would all seem much clearer.
Mike Rogers
>left at the same time, then it only takes the duration of a single step
>for them all to finish. Even if they do get affected one at a time,
Ah yes. But how to communicate to this infinite amt. of people your
desire that they should all jump "at the same time" within a finite amount
of time?
Ah paradox!
--
Mike Rogers, 6.3.3 TCD, D2, Eire. | You wish to make love to me? You are
mi...@maths.tcd.ie (UNIX => preferred) | very wrinkled.It would be interesting.
mi...@tcdmath.uucp (UUCP=>oldie/goodie) | Jharek Carnelian
msro...@vax1.tcd.ie(VMS => blergh) |
H. de Bruijn
In article <90032.0...@PSUVM.BITNET> a poor fellow writes:
> Please take it easy on me. I've had a total of two Physics Courses,
> and walked away from them with little understanding.
He is not to be blamed.
Richard P. Feynman - yes, the Nobel prize winner on Physics - once said in an
interview that could'nt understand several of these recent Physical theories.
If a professor of mine said "cannot understand" I knew he meant: "It's wrong".
Re: The Standard Model == Ptolemaic Epicycles? (the Master Blaster's)
Very much worse! :->
In article <8...@arkham.enet.dec.com> the Stormbringer writes:
> ... the discovery of the Top (or, as I prefer, Truth) quark ...
the Grand Unified Theory of Nature called BSD ...
In article <48...@lll-winken.LLNL.GOV> the Master Blaster disagrees:
> Any comment on this theoretical argument?
In article <12...@forty2.UUCP> Kurt Sonnenmoser writes:
> ..., because otherwise current theoretical physics would be
> in serious trouble :-)
No smilies. It *IS* in serious trouble already.
Before reading the .signature:
A little bit of Sense would be NO idleness in Physics. (HdB)
john baez
>In article <90032.0...@PSUVM.BITNET> a poor fellow writes:
>> Please take it easy on me. I've had a total of two Physics Courses,
>> and walked away from them with little understanding.
>He is not to be blamed.
>
>Richard P. Feynman - yes, the Nobel prize winner on Physics - once said in an
>interview that could'nt understand several of these recent Physical theories.
>If a professor of mine said "cannot understand" I knew he meant: "It's wrong".
De Bruijn is fond of this story, and has told it before, and
no doubt will tell it again. If Feynman didn't understand
something, it was probably wrong. If your average student
doesn't understand something, chances are that he just doesn't
understand it.
Along the general lines of this subject, however, I would
like to know FROM PHYSICISTS IN THE KNOW what the current
mood is concerning string theory. I feel what began with
a bang is ending with a whimper (after the perpetrators have
all gotten tenure). I would like this impression confirmed
or rebutted. Please don't answer if you get your info from
OMNI, or even Science News; I want to know what the "insiders"
think.
Mikel Lechner
> Point (finally) being that it seems to me that science keeps creating
> smaller and smaller particles to explain the make-up of matter. Could it be
> that we are using smaller circles to explain matter when it is really our
> theory that needs revamping? I realize this is a run-of-the-mill question
> being that the top quark has not yet been found despite fermilab's efforts,
> but doesn't the smaller circles theory seem remarkably similar to our
> "smaller particles" theory?
Actually, these new predicted particles are larger than the previous particles.
There large masses are what make them diffucult to create and observe.
As far as an infinite regression of levels (or circles as you describes),
it would appear that there is a limit here. The uncertainty principle
seems to set lower bounds on the sizes of particles (or at least their
mass-energy). Higher mass particles would merely be compositions of
smaller mass quarks.
Of course the current theory may be wrong and there are yet smaller components
making up the quarks and leptons. But this would be pure speculation.
Quark and lepton theory so far has a fairly solid basis.
I think to say that an old theory is wrong is to overstate the case.
I prefer to think that the old theory was incomplete or deficient in
some way. I don't really believe in an ultimate theory. Science is
a process of continually refining our understanding of the universe.
Newton's theory of gravity wasn't wrong, it was incomplete because it
did not account for relativistic effects. And General Relativity
does not explain gravity at the quantum mechanical level. This does not
make GR wrong, just incomplete.
If it is discovered in a few years that there are more fundamental
consitutents of matter than quarks and leptons, so be it. In meantime
we've learned a lot of useful things. Such a discovery does not
totally invalidate what we learn about quarks and leptons.
--
If you explain so clearly that nobody
can misunderstand, somebody will.
Mikel Lechner
Teradyne EDA, Inc. UUCP: mi...@teraida.UUCP
Warren G. Anderson
>In article <10...@hoptoad.uucp> t...@hoptoad.UUCP (Tim Maroney) writes:
> >left at the same time, then it only takes the duration of a single step
> >for them all to finish. Even if they do get affected one at a time,
>
>desire that they should all jump "at the same time" within a finite amount
>of time?
> Ah paradox!
If the people are all point-people, and they were, say, in a line segment
of finite or infinitesimal length, and they passed the message left, say,
immediately upon getting it from the right, why, it would take no time at
all to get the message to everyone once the rightmost person got it. Of
course, in our universe, we would suspect that the message would be passed
along no more quickly than a photon could traverse the length of the line
segment, but that is still a finite time.
########################### _`|'_ #############################################
## Warren G. Anderson |o o| You have a mind to tell you what's true, ##
## Dept. of Applied Math ( ^ ) and a heart to tell you what's right, and ##
## University of Waterloo /\-/\ a lifetime to figure out which is which. ##
Ranjan Muttiah
>Again, you leave me bewildered. What do you mean,
>how funny things can get?
That if atoms can be extended out to infinity that would lead to
a contradiction with observation. Refute this!
>It's known that there are various homotopy
>classes of such maps, parametrized by an index known as
>the degree, which is very much like a "winding number"
>in that it counts how many times the 2-sphere is wrapped
>around itself (with a sign!). These "twists" can act
>like particles, and are known as skyrmions (and anti-
>skyrmions).
Yep, I do sniff something funny here. Since we are on the
topic of superconductors, tell us something about the Meissner
effect(flux exclusion) along similar lines.
>Are these skyrmions bosons or fermions? Well, it
>turns out that if there is a "topological term" in
>the Lagrangian, they can be bosons, or fermions, or
>"anyons" (things in-between bosons and fermions,
>which smoothly interpolate between the two, and can
>only exist in 2 dimensional space!), depending on the
>constant.
Could you run this by again ? Only this time in very slow
motion.
Peter S. Shenkin
>If a professor of mine said "cannot understand" I knew he meant: "It's wrong".
I recently heard a story about a conversation between Kramers and a Cardinal.
Kramers reportedly said, "The trouble is, when you don't understand
something, you think it's a miracle. For me, when I *do* understand
something, I think it's a miracle!"
-P.
************************f*u*cn*rd*ths*u*cn*gt*a*gd*jb**************************
Peter S. Shenkin, Department of Chemistry, Barnard College, New York, NY 10027
(212)854-1418 she...@cunixc.cc.columbia.edu(Internet) shenkin@cunixc(Bitnet)
john baez
>
>That if atoms can be extended out to infinity that would lead to
>a contradiction with observation. Refute this!
No thanks.
>>Are these skyrmions bosons or fermions? Well, it
>>turns out that if there is a "topological term" in
>>the Lagrangian, they can be bosons, or fermions, or
>>"anyons" (things in-between bosons and fermions,
>>which smoothly interpolate between the two, and can
>>only exist in 2 dimensional space!), depending on the
>>constant.
>
>Could you run this by again ? Only this time in very slow
>motion.
It's a fascinating story but really works better with
pictures. I'll just say this. If you rotate a boson
by 2pi it's just like it was before you started, whereas
a fermion acquires a phase of -1 (that is, it's wave-function
does) and only gets back after a rotation of 4pi. (To
understand this, note that if you hold a coffee cup and
turn it around by 2pi your arm will have a twist in it,
but if you turn it around 2pi in the SAME DIRECTION again
your arm will have NO twist in it.) Mathematically, we say
the rotation group in three dimensions has Z mod 2 as its
fundamental group. But in 2 dimensions the fundamental group
of the rotation group is Z (the integers), which allows for
any phase e^(i theta) to appear when one rotates a particle
by 2 pi. If theta is zero one has bosons, if theta is pi
one has fermions, but there are lots of types of "anyons"
in between. In a way more surprising is that a boson can
become an anyon through it's interactions with other stuff.
This comes from a "topological term in the Lagrangian".
It seems to be important in the fractional quantum Hall effect,
and people hope it has to do with high-Tc.
Paul McCann
[stuff re anyons deleted. Someone asks..(sorry, rubbed out the name)
for a somewhat clearer explanation of the theory. To which John replies]
> It's a fascinating story but really works better with
> pictures. I'll just say this. If you rotate a boson
> by 2pi it's just like it was before you started, whereas
> a fermion acquires a phase of -1 (that is, it's wave-function
> does) and only gets back after a rotation of 4pi.
Forget about "rotating" the individual particles, just consider the phase
change of the wavefunction under the interchange of (identical) particles.
(Which is, of course, 1 for bosons and -1 for fermions). For a system of
n anyons there is, in general, no fixed complex number of unit modulus
(ie phase) that relates the wavefunctions before and after the interchange.
Because the phase change depends upon the path that is taken in performing
the interchange. Specifically the way in which the two particles wind around
any of the other particles that are present.
That is, given a wavefunction for a certain configuration of identical
particles, say psi(z1,z2,...,zn), and a new configuration
(z_(p1),z_(p2),...,z_(pn)) (where (p1,p2,..,.pn) is a permutation of
(1,2,...,n)), we do not in general know the phase of the wavefunction
psi(z_(p1),z_(p2),...,z_(pn)).
For the mathematicians in the audience the wavefunctions of the anyons
are acted upon by a 1-D representation of the braid group, rather than the
usual situation where the relevant group is the symmetric group.
In the latter case the only
one dimensional representations are the trivial one (this
gives bosons, no phase change under any interchanges) and the one in which
the phase changes by (-1)^(sign p), where sign p is the sign of the
permutation giving the change in configuration (the case of fermions).
The braid group representations are somewhat more complex (ha ha), but
reasonably easy to describe (I'll undertake this if anybody is interested,
requires a little knowledge of the group itself, but is reasonably
straightforward)
> Mathematically, we say
> the rotation group in three dimensions has Z mod 2 as its
> fundamental group. But in 2 dimensions the fundamental group
> of the rotation group is Z (the integers), which allows for
> any phase e^(i theta) to appear when one rotates a particle
> by 2 pi. If theta is zero one has bosons, if theta is pi
> one has fermions, but there are lots of types of "anyons"
> in between. In a way more surprising is that a boson can
> become an anyon through it's interactions with other stuff.
> This comes from a "topological term in the Lagrangian".
> It seems to be important in the fractional quantum Hall effect,
> and people hope it has to do with high-Tc.
True, but not particularly illuminating. The first two sentences really
relate the idea that the rotation operators in more than two dimensions
are constrained by their commutation relations to deliver integer and
half integer eigenvalues (recall the "ladder operators" stuff from
the first quantum mechanics course). For 2 dimensions of course there are
no such constraints, and the rotation may have any real number as its
eigenvalue.
A boson "becomes an anyon" by the attachment of a magnetic flux tube; so a
gas of anyons may be considered as a gas of bosons with infinitely thin
magnetic solenoids attached (all perpendicular to the plane of motion).
In this way the phase change
under interchanges of particles is easily understood as the good old
Aharonov-Bohm effect.
The above is relevant to the fractional quantum Hall effect because
of the two dimensional nature of the effect. The accepted explanation
of the effect is based upon a trial wavefunction proposed by Robert Laughlin
in 1983, the elementary excitations of which obey fractional statistics
(ie are anyons).
As mentioned above, I will rave some more if there is any interest. In
particular the way in which the braid group arises in two dimensions is
rather fascinating. (There is an article which deals with many of the
issues raised here in a recent (Jan) Physics Today (Search and Discovery),
which does not however invoke the braid group.)
Hope the above is useful ; Paul McCann.
john baez
>Pardon me. I was under the misconception that I was posting to sci.physics,
>and thus couched things in appropriate language. Aah, now I can wax
>mathematical.
Great!
>Of course I have restricted myself to 2-d here without
>explanation. In fact for dim M>2 Birman showed that
> pi_1(C/) is isomorphic to pi_1(M)^n x S_n , and hence
>uninteresting (essentially, removing the diagonal points is not going
>to make the configuration space any more interesting topologically; see below).
This isn't completely uninteresting if M is not simply connected.Do you know if this has been studied? Could it be related to the
fact that spinors are not uniquely defined on non-simply-connected
manifolds (a manifold admits a spin structure if the second Stiefel
-Whitney class vanishes, and then admits as many spin structures
as there are elements in H^1(M,Z/2), I believe. I guess this means
that spinors can pick up an extra minus sign if you take them
around a noncontractible loop. But pi_1(M) can have more
representations - or even characters, so it seems like there should
be other statistics possible on non-simply-connected manifolds.
>This post is already far too long, so I'll just mention that there is a
>wonderful example of how the topology of the manifold can influence the
>statistics of the particles moving on it involving the 2-sphere, which
>I'll post if requested (this is a test to see if anybody is still awake
>this far into an article). Hope this helps, Paul.
I'm awake and interested, I don't know about anyone else.
(By the way, do you know my friend Varghese Mathai... he's
doing some work with Alan Carey).
As I said, I'm curious about the relation between spin & statistics
in these stranger contexts. Are there zillions of representations
of the braid group (I guess I mean 1-dim reps)?
Myself, I'm working with Rossen Dandoloff trying to come
up with a rationale for the existence of Hopf terms in
2+1-dimensional sigma-models. I've essentially blundered into
this field 'cause I like him and he's a condensed matter
theorist.
Ranjan Muttiah
>>Could you run this by again ? Only this time in very slow
>>motion.
>become an anyon through it's interactions with other stuff.
>This comes from a "topological term in the Lagrangian".
>It seems to be important in the fractional quantum Hall effect,
>and people hope it has to do with high-Tc.
John, my email to you is bouncing back.
Thanks for the clarification. There is pretty nice article in
a Physics Today dated about October 1989 I think.
Bill Jefferys
#As mentioned above, I will rave some more if there is any interest. In
#particular the way in which the braid group arises in two dimensions is
#rather fascinating. (There is an article which deals with many of the
#issues raised here in a recent (Jan) Physics Today (Search and Discovery),
#which does not however invoke the braid group.)
Rave on. This is interesting. (Could not reach you by mail).
Bill Jefferys
Ranjan Muttiah
>In article <2...@spam.ua.oz> pmc...@spam.oz.au (Paul McCann) writes:
>#As mentioned above, I will rave some more if there is any interest. In
>
>Bill Jefferys
Ditto that.
Paul McCann
>>Of course I have restricted myself to 2-d here without
>>explanation. In fact for dim M>2 Birman showed that
>> pi_1(C/) is isomorphic to pi_1(M)^n x S_n , and hence
>>uninteresting (essentially, removing the diagonal points is not going
>>to make the configuration space any more interesting topologically; see below).
>
> This isn't completely uninteresting if M is not simply connected.Do you know if this has been studied? Could it be related to the
> fact that spinors are not uniquely defined on non-simply-connected
> manifolds (a manifold admits a spin structure if the second Stiefel
> -Whitney class vanishes, and then admits as many spin structures
> as there are elements in H^1(M,Z/2), I believe. I guess this means
> that spinors can pick up an extra minus sign if you take them
> But pi_1(M) can have more
> representations - or even characters, so it seems like there should
> be other statistics possible on non-simply-connected manifolds.
Hmm, this is a very shady area, the whole notion of quantum mechanics on
multiply connected manifolds is quite nebulous. The only pointers that I
can offer are related to my research intersest, the quantum Hall effect.
When I wrote that things were "not interesting" I was more than a little
hasty. Perhaps in the sense that the braid group is trivially
specified in this case. But as far as
formulating things in this framework goes the best I can do is
refer to the work of Grumm, Thirring and Narnhofer that has appeared in
Acta Physica Austriaca around 1984 and 1985, in which the situation
amounts to the quantum Hall effect on multiply connected (2)-manifolds.
Alan Carey suggested a paper of Segal's from the early seventies which
discusses this type of thing. Should anybody be able to clarify this
sidestepping I would be most grateful.
Anyway, the way I tend to picture things is that the possible statistics on a
multiply connected manifold of dimension greater than three are still
just Fermi and Bose (the "local" generators sigma_i still satisfy the
relation sigma_i^2=1, so if X denotes a 1-d rep we require
X(sigma_i)=+1 or -1.) But of course this ignores the "global generators".
For example, what if we leave particles 2,...,n unchanged but take
particle 1 for a ride around a non-contractible loop? The above mentioned
papers consider the case when the transport is defined by the Hamiltonian.
But I am still not even sure how to formulate things here, so whereof one
cannot speak (thereof one must gesticulate wildly).More positive stuff follows!
>>there is a
>>wonderful example of how the topology of the manifold can influence the
>>statistics of the particles moving on it involving the 2-sphere.
Representations stuff; well, remember the defining relations which are
common ("local" ones above);
sigma_i sigma_j = sigma_j sigma_i for |i-j|>1
sigma_i sigma_(i+1) sigma_i =sigma_(i+1) sigma_i sigma_(i+1)
Applying X to the second of these yields that X(sigma_i)=X(sigma_(i+1)),
for i=1,...,n. That is, a 1-d rep is specified by a single phase exp(i theta).
On the plane any phase is allowed, but I show below that this is not true
for the sphere (this is stuff done by Thouless and Wu, using Birman's (?)
calculation of the braid group of the 2-sphere).
Anyway, for the sphere there is an additional defining relation for the
sigma_i's. It is
sigma_1 sigma_2 ...sigma_(n-1) sigma_(n-1) sigma_(n-2) ... sigma_1 = 1.
1---------2----3- Perhaps not the clearest diagram, but
1 2 3 I think you can see what is going on.
1 3 The first particle is looping around
(n=3 case) 2 1 3 all of the others. On the sphere this
2 1 is of course contractible (you just let
2 31 slip the loop of the "back" of the
2 3 sphere). As regards representations, well
2 1 3 of course the relation must be satisfied.
2 3 That is, we require that
1 2 3 X(sigma_i)^2(n-1)=1
1--------2----3- ie exp(i theta 2(n-1))=exp(2 pi i m), m in Z.
So the allowed values of the representations can be read off as
theta=m pi/(n-1) for m in {1,2,...,2(n-1)}
And we see that the topology of the (simply connected!) manifold is
limiting the possible statistics of particles moving on it,an outcome I
find intriguing in spite of its evident nature.
> Myself, I'm working with Rossen Dandoloff trying to come
> up with a rationale for the existence of Hopf terms in
> 2+1-dimensional sigma-models.
I'm very vague on sigma models, but have heard that the only real rationale
for adding a topological term to the sigma model is to reduce the symmetry
of the system (this is Witten's excuse; it's too symmetric to act as a
baby QCD).
> (By the way, do you know my friend Varghese Mathai... he's
> doing some work with Alan Carey).
Mathai, Alan and myself have been doing some work on the fractional quantum
Hall effect (without much success as yet), attempting to construct a
mathematical framework that will accommodate fractional quantum numbers.
Big problems arise due to the non-compactness of the product manifold after
the diagonal points have been removed. (Alan is my "Doktorvater".)
(Apologies for the delay in getting this posted. A first version was nigh
on complete when our computer fainted in the heat,eating my post).
john baez
single phase exp(i theta).
Aha. So presumably there lurks in here a "spin-statistics
theorem" relating this phase to the phase parametrizing the
1-d reps of the rotation group. Pardon me while I think aloud.
Say you have n particles satisfying the exp(i theta)
representation of the braid group. Then one should be able to
determine the spin of the whole system (modulo integers) by
rotating a configuration around by 2pi, thinking of this as a
braid, and calculating the phase. If n = 2, one has
sigma_1^2 :
1 2
1 2
1 2
2
2 1
2 1
2 1
2 1
2 1
1
1 2
1 2
1 2
so the total spin is 2theta (mod 1).
Hmm. But this argument in the degenerate case n = 1
would always give spin zero (trivial braid), so it's probably
neglecting "intrinsic spin". I recall an article in which
some ideas like this were used to show, quite heuristically,
that spins of anyons didn't add the way spins add in 3d.
I think it's Wilczek and Zee Phys. Rev. Lett. 51 p. 2250.
>On the plane any phase is allowed, but I show below that this is not true
>for the sphere[....]
>And we see that the topology of the (simply connected!) manifold is
>limiting the possible statistics of particles moving on it,an outcome I
>find intriguing in spite of its evident nature.
Nice!!!
>I'm very vague on sigma models, but have heard that the only real rationale
>for adding a topological term to the sigma model is to reduce the symmetry
>of the system (this is Witten's excuse; it's too symmetric to act as a
>baby QCD).
We're looking for a natural rationale, and there should be
one related to Berry's-phase ideas. I may post the idea
once I get some details hammered out.
>Mathai, Alan and myself have been doing some work on the fractional quantum
>Hall effect (without much success as yet), attempting to construct a
>mathematical framework that will accommodate fractional quantum numbers.
>Big problems arise due to the non-compactness of the product manifold after
>the diagonal points have been removed. (Alan is my "Doktorvater".)
Say hi to Mathai!! I've sent him email concerning various
articles I have on Witten theory (using the expectation of
Wilson loops in Chern-Simons gauge theories on 3-manifolds
to come up with knot invariants), but he's never replied!!
If he/you/Alan want copies let me know.
The "baby version" of Witten theory seems to me to be 2-dim
Yang-Mills QFT, and some sort of braid relations show up here
when one slips circles off each other in 2d.
Paul McCann
> In article <2...@spam.ua.oz> pmc...@spam.oz.au (Paul McCann) writes:
>>That is, a 1-d rep [of the braid group] is specified by a
> single phase exp(i theta).
>
> Aha. So presumably there lurks in here a "spin-statistics
> theorem" relating this phase to the phase parametrizing the
> 1-d reps of the rotation group. Pardon me while I think aloud.
> Say you have n particles satisfying the exp(i theta)
> representation of the braid group. Then one should be able to
> determine the spin of the whole system (modulo integers) by
> rotating a configuration around by 2pi, thinking of this as a
> braid, and calculating the phase.
>
> Hmm. But this argument in the degenerate case n = 1
> would always give spin zero (trivial braid), so it's probably
> neglecting "intrinsic spin".
Exactly; the spin of an anyon is (mod Z) theta/2pi + s, where s is the
intrinsic spin of the particle. That is, the phase change upon interchange
of such anyons is exp(2pi(theta/2pi + s)). In this sense there is a
generalized spin-statistics theorem.
>
> We're looking for a natural rationale, and there should be
> one related to Berry's-phase ideas. I may post the idea
> once I get some details hammered out.
Please do, even embryonic ideas often prove stimulating (particularly
embryonic ideas (?)). Wu (P.R.L. vol 53 no 2 p111) implements fractional
statistics in the O(3) non-linear sigma model, and explains the use of
the multivalued wavefunctions that I mentioned in my first post on this tack.
It's a very neat way of absorbing the fractional statistics into a phase term
i.e. it eliminates the long range forces resulting from the non-zero
vector potential describing the "flux tubes".
A paper linking this approach with that of Wilczek and Zee in the paper
that you (John) cited is Wu and Zee, Physics Letters 147B p325.
> articles I have on Witten theory (using the expectation of
> Wilson loops in Chern-Simons gauge theories on 3-manifolds
> to come up with knot invariants).
Is this the paper that appeared in CommMathsPhys last year? Anyway
there was a paper describing the situation in which the representations of the
braid group need not be one dimensional, and the word "phase" takes on
a somewhat generalized meaning. A lot of work has been done on this
aspect of statistics, and it seems that generalized phases are allowed.
But the papers I know of use a very abstract setting, looking at
axiomatic quantum field theory, which is somewhat out of my realm.
I think Witten's stuff is the first hint
that such ideas may be implemented physically.
Douglas E Feather
stty erase \^H
trap clear 0
biff y
EDPROFILE=.edpro export EDPROFILE
echo -n "Are you using a $TERM terminal? "
read ans
case $ans in
y*|Y*) ;;
*) echo -n "Please type terminal type (eg wy85,beeb,...) "
read TERM
export TERM
esac