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origin of mass, other dimensions, etc.



Hugh Logan wrote:
It is stated,"In particular, a massless state in the higher
dimensional theory will show up in the lower dimensional theory as a
tower of equally spaced massive states." .... I would like to see
more specific examples.

Those of us who are not string theorists will, unfortunately, have to
rely on explanations intended for non-specialists.

Abject reliance is not necessary in this case. This can
be understood in terms understandable to undergraduates, if
you're willing to take a reasonably broad view of things.

1) Graph the energy-versus-momentum relationship for an
ordinary massive particle such as an alpha particle.
Be sure the ordinate includes the rest energy as well
as the kinetic energy. Be sure the abscissa covers
the range from zero up to relativistic velocities.
Nice hyperbola, right?

2a) Write down the equation for waves in a waveguide.
For definiteness, let the electric field be polarized
in the X direction, and let the wave propagate in the
Z direction. Graph the frequency as a function of k_z,
the wavevector-component in the direction of propagation.
Nice hyperbola, right?

2b) On the waveguide graph, relabel the axes as follows:
Multiply the ordinate by hbar and call it energy.
Multiply the abscissa by hbar and call it projected
momentum.

*) Why did I bring this up? Imagine you live in a D=2
flatland corresponding to a plane in the ZX direction
bisecting the waveguide. You see an electromagnetic
wave moving through flatland, with an energy-versus-
momentum relationship that is identical to what we
expect for a massive particle. "The same equations
have the same solutions".

The EM field was massless in D=3 but has mass in D=2,
so this is an example of the type requested.

It may be worth emphasizing that the flatlanders have
no access to the Y dimension ... and it is k_y that
sets the cutoff frequency. So the origin of the mass
is a deep mystery to the flatlanders.

The following seemingly-disparate bits of terminology
mean the same thing:
-- D=3 wave terminology: k_y, the wavevector-component
in the non-propagating direction.
-- D=2 wave terminology: waveguide cutoff frequency.
-- D=2 particle terminology: mass gap.

Also the following are synonymous:
-- Wave terminology: dispersion relation.
-- Particle terminology: energy-versus-momentum relationship.

==============

Pedagogical note: This trick was shown to me at an early
age. I had no trouble understanding the physics involved;
I figured it was just another case of "the same equations
have the same solutions." Years later, when I got to grad
school, I was shocked to find that although everybody had
heard of the trick, many of them didn't like it. In particular '
I overheard some faculty standing around saying
"That Ken Wilson is such an eccentric genius; he can do
quantum physics and classical physics at the same time.
He thinks of his particles in terms of waves in waveguides."

Later on I tried explaining this to some junior-level physics
majors. I knew they had all the prerequisite knowledge,
including
-- energy = rest-energy plus kinetic energy
-- waves, dispersive waves, and waveguides
-- energy = hbar omega; momentum = hbar k
-- flatland; higher and lower dimensions
But when I tried to put that all together, they had a
hard time with it.

I have a theory, unsubstantiated but plausible. The theory
says that they had *compartmentalized* their knowledge.
They had a book on special relativity that told them about
mass and kinetic energy, but there were no waveguides in
that book and no hbars either. Similarly the book that
discussed classical waves didn't talk about QM and didn't
talk about relativity (although it did talk about D=1,
D=2, and D=3 waves). There was evidently some unwritten
11th commandment that said that problems in one subfield
were to be solved using only the methods of that subfield.
Apparently all their previous books and all their previous
teachers had adhered to this commandment.

The scary thing is that this waveguide/mass trick doesn't
even count as multidisciplinary or interdisciplinary,
since it only involves sub-disciplines within the
overall discipline of physics. If the students have a
hard time with inter-sub-disciplinary thinking, what are
they going to do when faced with real-world problems,
which generally require real interdisciplinary thinking???

This needs fixing in a big way. It needs fixing at every
level from third grade up through every course in every
subject in highschool and college. It can't be fixed
overnight, but that's no excuse for not getting started.