Particles from non-linear waves [Was: Solitons...]

From: Gerard Westendorp (westy31_at_xs4all.nl)
Date: 07/22/04 

Date: 22 Jul 2004 05:16:47 -0400

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 = 0 (center of mass frame)
   m_A*v_A^2 = m_B*v_B^2 + m_C*v_C^2
   = 0

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,..) =
       (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=0
   M_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.

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