Gerard Westendorp
Jul22-04, 05:16 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOz wrote:\n\n[..]\n\n> What puzzles me is why research into solitons and their possible use\n> for explaining particles seemed to have lapsed. I would admit that\n> the maths might be arcane (hey most maths seems arcane to me) but\n> that doesn\'t seem to have stopped people in the past.\n\nI think Einstein worked on this too, so you are in good company.\n\nTo avoid distracting Tessel too much from explaining the inverse\nscattering transform (which is according to the article by Palais that\nhe is recommending hugely important!) I diverted this topic into\na parrallel thread. I agree that is at least interesting to\nwonder how one could let elementary particles emerge from\nsome classical wave theory, or, if this impossible, see why.\n\nOf course, I don\'t know how to do this, but that doesn\'t stop me\nfrom writing...\n\nA very hard thing that must be explained is that all spin 1/2\nparticles, like electron, muon, tauon, and also the quarks, and\neven *composite* particles like helium nuclei all have exactly\nh_bar/2 angular momentum. As shown in an article by Ohanian, which\nhas already been discussed a few times in spr, this spin is\nactually equal to the integral of the naturally present angular\nmomentum density of the wave packet, not added later.\nAngular momentum is always h_bar/2, and the\nprobability of finding the particle integrates to unity over space.\nThese 2 are related, because if a Dirac wave is normalized\nand goes to zero rapidly before infinity (i.e. a hump), then the\nangular momentum will be h_bar/2, as shown by Ohanian. It has to\ndo with the way a wave packet solution must behave if it is to\ngo to zero before x->inf.\n\n\nThe only way I can explain such a single stringent requirement on\notherwize very dissimilar particles is if *interactions* can only\ninvolve a totality of a particle. As I though about it a little,\nI found some surprising restrictions on what interactions are\nactually allowable.\n\nAs we are studying particle-like wave structures, we can more\nor less assign a momentum (px, py,..) and an energy (E) to\nour humps, which should be reasonable localized in space. This\nmakes them kind of solitons, although they may not fit certain\ndefinitions of solititons.\n\nAnyway, if these humps have energy and momentum, then interactions\nbetween them should conserve this energy, momentum, and of course\nalso angular momentum and charge.\n\nConsider a 3-vertex event:\n\n\n\n/B\n/\nA -----<\n\\\n\\C\n\n\nIn other words, B and C collide to form A, or A splits up\ninto B and C. If we take 1 dimension, and classical physics,\nand operate in the center of momentum frame, then we get:\n\nm_A*v_A = m_B*v_B + m_C*v_C = 0 (center of mass frame)\nm_A*v_A^2 = m_B*v_B^2 + m_C*v_C^2\n= 0\n\nHey, so the only solution is zero velocity. In other words,\nnon-trial cases of such collisions like this would not be\nallowed. In classical\nparticle collision, such a collision would be inelastic, and\nthus not conserve energy.\n1 dimensional solutions like the KdV solutions solve this\nproblem by moving through each other, rather than\nmerging. (disintegrating into a splash could have been\nan option also maybe)\n\nI did 1 dimension in the center of mass frame , but it is the\nsame in any dimension, and in any reference frame; you cannot\nhave elastic merging or splitting of classical particles.\n\nHow about interactions like this:\n\nA B\n\\ /\n\\ /\n\\ /\n\\----/\n/ \\\n/ \\\n/ \\\n\nThey contain 3 vertices too. However, the exchanged particle could\nbe a virtual particle, i.e. it need not be a well-defined hump,\nwhich therefor need not have a well-defined momentum and\nkinetic energy.\nSo we will certainly need virtual particles to get any\nmeaningful theory.\n\nIf we apply conservation requirements on to our legged\ndiagram, which looks much\nlike a Feymnan graph, we can go on to derive that in\nthe center of mass frame, we can only get a *deflection*\nof the particles, not a change in kinetic energies. This\nis also shown somewhere in the Feynman Lectures.\n\n\nAnd now look at angular momentum conservation. This puts also\na severe restriction on the possible outcomes. If angular\nmomentum is contained in the wave motion around itself of each\nhump, then angular an angular momentum change would also\nimply an energy change. But in the center of mass frame, an\nenergy change is not allowed! One possibility is that angular\nmomentum flips sign on both A and B. Then a similar wave\npattern before and after the collision would be there,\nbut turning the other way, which would presumably have the\nsame energy.\nBut that is precisely what we want! Only a flip of AM is\nallowed, not gradual exchange of small chunks!\n\n\nOf course, I cheated a bit. I mixed classical physics\nwith relativistic stuff in a way that suits me. The\nrelativistic part is where I said that AM is part of\nthe energy, but I restricted to classical physics when\nI assumed that all energy is kinetic, and that mass is\nseparately conserved.\n\nBut in relativity, ot seems that we *can* have conservation\nof energy and momentum in a merging of particles:\n\n(sqrt[m_A^2 + p_A^2], px_A, py_A,..) =\n(sqrt[m_B^2 + p_B^2], px_B, py_B,..)\n+(sqrt[m_C^2 + p_C^2], px_C, py_C,..)\n\nIn the center of 3-momentum, we get that:\n\np_A=0\nM_A = sqrt(M_B^2 + p_B^2) + sqrt(M_C^2 + p_C^2)\n\nSo instead of an inelastic collision, we get that kinetic\nenergy is convered into mass, of the stationary\nparticle A.\n\nBut looked upon as a wave again, the particle A must have\nsome way of storing energy without moving (i.e. have mass)\n\nOK, so relativity makes things more complicated, and my\nresults using classical kinetic energy are cheating. But they\nstill seem a bit intriguing though. Maybe we can build on\nit a bit further in this thread.\n\n\nGerard\n\n\nI will be away for 2 weeks, so I my replies will be late.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz wrote:
[..]
> What puzzles me is why research into solitons and their possible use
> for explaining particles seemed to have lapsed. I would admit that
> the maths might be arcane (hey most maths seems arcane to me) but
> that doesn't seem to have stopped people in the past.
I think Einstein worked on this too, so you are in good company.
To avoid distracting Tessel too much from explaining the inverse
scattering transform (which is according to the article by Palais that
he is recommending hugely important!) I diverted this topic into
a parrallel thread. I agree that is at least interesting to
wonder how one could let elementary particles emerge from
some classical wave theory, or, if this impossible, see why.
Of course, I don't know how to do this, but that doesn't stop me
from writing...
A very hard thing that must be explained is that all spin 1/2
particles, like electron, muon, tauon, and also the quarks, and
even *composite* particles like helium nuclei all have exactly
h_{bar}/2 angular momentum. As shown in an article by Ohanian, which
has already been discussed a few times in spr, this spin is
actually equal to the integral of the naturally present angular
momentum density of the wave packet, not added later.
Angular momentum is always h_{bar}/2, and the
probability of finding the particle integrates to unity over space.
These 2 are related, because if a Dirac wave is normalized
and goes to zero rapidly before infinity (i.e. a hump), then the
angular momentum will be h_{bar}/2, as shown by Ohanian. It has to
do with the way a wave packet solution must behave if it is to
go to zero before x->inf.
The only way I can explain such a single stringent requirement on
otherwize very dissimilar particles is if *interactions* can only
involve a totality of a particle. As I though about it a little,
I found some surprising restrictions on what interactions are
actually allowable.
As we are studying particle-like wave structures, we can more
or less assign a momentum (px, py,..) and an energy (E) to
our humps, which should be reasonable localized in space. This
makes them kind of solitons, although they may not fit certain
definitions of solititons.
Anyway, if these humps have energy and momentum, then interactions
between them should conserve this energy, momentum, and of course
also angular momentum and charge.
Consider a 3-vertex event:
/B
/
A -----<
\
\C
In other words, B and C collide to form A, or A splits up
into B and C. If we take 1 dimension, and classical physics,
and operate in the center of momentum frame, then we get:
m_A*v_A = m_B*v_B + m_C*v_C =[/itex] (center of mass frame)
m_A*v_A^2 = m_B*v_B^2 + m_C*v_C^2
=
Hey, so the only solution is zero velocity. In other words,
non-trial cases of such collisions like this would not be
allowed. In classical
particle collision, such a collision would be inelastic, and
thus not conserve energy.
1 dimensional solutions like the KdV solutions solve this
problem by moving through each other, rather than
merging. (disintegrating into a splash could have been
an option also maybe)
I did 1 dimension in the center of mass frame , but it is the
same in any dimension, and in any reference frame; you cannot
have elastic merging or splitting of classical particles.
How about interactions like this:
A B
\ /\ /\ /
\----/
/ \/ \/ \
They contain 3 vertices too. However, the exchanged particle could
be a virtual particle, i.e. it need not be a well-defined hump,
which therefor need not have a well-defined momentum and
kinetic energy.
So we will certainly need virtual particles to get any
meaningful theory.
If we apply conservation requirements on to our legged
diagram, which looks much
like a Feymnan graph, we can go on to derive that in
the center of mass frame, we can only get a *deflection*
of the particles, not a change in kinetic energies. This
is also shown somewhere in the Feynman Lectures.
And now look at angular momentum conservation. This puts also
a severe restriction on the possible outcomes. If angular
momentum is contained in the wave motion around itself of each
hump, then angular an angular momentum change would also
imply an energy change. But in the center of mass frame, an
energy change is not allowed! One possibility is that angular
momentum flips sign on both A and B. Then a similar wave
pattern before and after the collision would be there,
but turning the other way, which would presumably have the
same energy.
But that is precisely what we want! Only a flip of AM is
allowed, not gradual exchange of small chunks!
Of course, I cheated a bit. I mixed classical physics
with relativistic stuff in a way that suits me. The
relativistic part is where I said that AM is part of
the energy, but I restricted to classical physics when
I assumed that all energy is kinetic, and that mass is
separately conserved.
But in relativity, ot seems that we *can* have conservation
of energy and momentum in a merging of particles:
(\sqrt[m_A^2 + p_A^2], px_A, py_A,.[itex].) =(\sqrt[m_B^2 + p_B^2], px_B, py_B,..)+(\sqrt[m_C^2 + p_C^2], px_C, py_C,..)
In the center of 3-momentum, we get that:
p_A=0M_A = \sqrt(M_B^2 + p_B^2) + \sqrt(M_C^2 + p_C^2)
So instead of an inelastic collision, we get that kinetic
energy is convered into mass, of the stationary
particle A.
But looked upon as a wave again, the particle A must have
some way of storing energy without moving (i.e. have mass)
OK, so relativity makes things more complicated, and my
results using classical kinetic energy are cheating. But they
still seem a bit intriguing though. Maybe we can build on
it a bit further in this thread.
Gerard
I will be away for 2 weeks, so I my replies will be late.
[..]
> What puzzles me is why research into solitons and their possible use
> for explaining particles seemed to have lapsed. I would admit that
> the maths might be arcane (hey most maths seems arcane to me) but
> that doesn't seem to have stopped people in the past.
I think Einstein worked on this too, so you are in good company.
To avoid distracting Tessel too much from explaining the inverse
scattering transform (which is according to the article by Palais that
he is recommending hugely important!) I diverted this topic into
a parrallel thread. I agree that is at least interesting to
wonder how one could let elementary particles emerge from
some classical wave theory, or, if this impossible, see why.
Of course, I don't know how to do this, but that doesn't stop me
from writing...
A very hard thing that must be explained is that all spin 1/2
particles, like electron, muon, tauon, and also the quarks, and
even *composite* particles like helium nuclei all have exactly
h_{bar}/2 angular momentum. As shown in an article by Ohanian, which
has already been discussed a few times in spr, this spin is
actually equal to the integral of the naturally present angular
momentum density of the wave packet, not added later.
Angular momentum is always h_{bar}/2, and the
probability of finding the particle integrates to unity over space.
These 2 are related, because if a Dirac wave is normalized
and goes to zero rapidly before infinity (i.e. a hump), then the
angular momentum will be h_{bar}/2, as shown by Ohanian. It has to
do with the way a wave packet solution must behave if it is to
go to zero before x->inf.
The only way I can explain such a single stringent requirement on
otherwize very dissimilar particles is if *interactions* can only
involve a totality of a particle. As I though about it a little,
I found some surprising restrictions on what interactions are
actually allowable.
As we are studying particle-like wave structures, we can more
or less assign a momentum (px, py,..) and an energy (E) to
our humps, which should be reasonable localized in space. This
makes them kind of solitons, although they may not fit certain
definitions of solititons.
Anyway, if these humps have energy and momentum, then interactions
between them should conserve this energy, momentum, and of course
also angular momentum and charge.
Consider a 3-vertex event:
/B
/
A -----<
\
\C
In other words, B and C collide to form A, or A splits up
into B and C. If we take 1 dimension, and classical physics,
and operate in the center of momentum frame, then we get:
m_A*v_A = m_B*v_B + m_C*v_C =[/itex] (center of mass frame)
m_A*v_A^2 = m_B*v_B^2 + m_C*v_C^2
=
Hey, so the only solution is zero velocity. In other words,
non-trial cases of such collisions like this would not be
allowed. In classical
particle collision, such a collision would be inelastic, and
thus not conserve energy.
1 dimensional solutions like the KdV solutions solve this
problem by moving through each other, rather than
merging. (disintegrating into a splash could have been
an option also maybe)
I did 1 dimension in the center of mass frame , but it is the
same in any dimension, and in any reference frame; you cannot
have elastic merging or splitting of classical particles.
How about interactions like this:
A B
\ /\ /\ /
\----/
/ \/ \/ \
They contain 3 vertices too. However, the exchanged particle could
be a virtual particle, i.e. it need not be a well-defined hump,
which therefor need not have a well-defined momentum and
kinetic energy.
So we will certainly need virtual particles to get any
meaningful theory.
If we apply conservation requirements on to our legged
diagram, which looks much
like a Feymnan graph, we can go on to derive that in
the center of mass frame, we can only get a *deflection*
of the particles, not a change in kinetic energies. This
is also shown somewhere in the Feynman Lectures.
And now look at angular momentum conservation. This puts also
a severe restriction on the possible outcomes. If angular
momentum is contained in the wave motion around itself of each
hump, then angular an angular momentum change would also
imply an energy change. But in the center of mass frame, an
energy change is not allowed! One possibility is that angular
momentum flips sign on both A and B. Then a similar wave
pattern before and after the collision would be there,
but turning the other way, which would presumably have the
same energy.
But that is precisely what we want! Only a flip of AM is
allowed, not gradual exchange of small chunks!
Of course, I cheated a bit. I mixed classical physics
with relativistic stuff in a way that suits me. The
relativistic part is where I said that AM is part of
the energy, but I restricted to classical physics when
I assumed that all energy is kinetic, and that mass is
separately conserved.
But in relativity, ot seems that we *can* have conservation
of energy and momentum in a merging of particles:
(\sqrt[m_A^2 + p_A^2], px_A, py_A,.[itex].) =(\sqrt[m_B^2 + p_B^2], px_B, py_B,..)+(\sqrt[m_C^2 + p_C^2], px_C, py_C,..)
In the center of 3-momentum, we get that:
p_A=0M_A = \sqrt(M_B^2 + p_B^2) + \sqrt(M_C^2 + p_C^2)
So instead of an inelastic collision, we get that kinetic
energy is convered into mass, of the stationary
particle A.
But looked upon as a wave again, the particle A must have
some way of storing energy without moving (i.e. have mass)
OK, so relativity makes things more complicated, and my
results using classical kinetic energy are cheating. But they
still seem a bit intriguing though. Maybe we can build on
it a bit further in this thread.
Gerard
I will be away for 2 weeks, so I my replies will be late.