Flatland is not dark (was: Photons in other dimensions)

[John Denker <jsd@AV8N.COM>]

J. Green wrote:
> "photons from another dimension that still act as photons in ours?"
....
> It seems I'm not asking the right question.

Actually there are two questions here
  a) photons _from_ another dimension, as just mentioned, and
  b) photons _in_ other dimensions, as the Subject: line says.

Version (a) is definitely not the right question, as various
folks have pointed out.  A photon is what it is.  Either it's
in our universe or it isn't.  If it's a photon in our universe,
it plays by *exactly* the same rules as all the other photons
in our universe.  There is no provision in the theory for any
new dynamical variables or new initial/boundary conditions to
express where the photon came "from" -- and not the slightest
experimental reason to believe anything new is needed.

Version (a) is a Rorschach test.  It cannot possibly mean
anything, although it may give an imaginative person an
opportunity to go off on an interesting tangent.

Hugh Logan wisely ignored version (a) and wrote a nice essay
tangentially related to version (b).  I won't quote the
details here, but interested parties should take a look at
  http://lists.nau.edu/cgi-bin/wa?A2=ind0404&L=phys-l&P=R6053

If I may be permitted to take a tangent from a tangent, I
would like to pursue the remark:

> > According to one version (3) of string theory, photons can exist only
> > in this 3-d brane, the remaining compactified dimensions being "dark
> > dimensions." I don't know how well this interpretation is accepted
> > among string theorists.

I'm not a string maven myself, but I'm pretty sure this
business of D=3 being the only home for electromagnetism
is a mistake.  It's an understandable and rather interesting
mistake, but a mistake none the less.

The simplest way I know to explain this requires drawing a
distinction between *truth* and *knowledge*.  The way I
use the words, certain truths are universal and long-lasting:
e.g. the planet Jupiter has had moons for many thousands of
years.  In contrast, human understanding of this truth has
been very unevenly distributed in time and space.  Before
Galileo's time, nobody had any knowledge whatsoever of this
truth ... and even afterward the knowledge was somewhat
spotty.

Truths may be discovered, but quite a lot of what I call
knowledge is manufactured.

In the case of electromagnetism, the truth is what it is.
The truth hasn't changed and isn't likely to change.  But
in the 1860s Maxwell manufactured a theory that allowed
people for the first time to catch a glimpse of how
electromagnetism works.

Maxwell's construction depends on the magnetic field vector,
B.  In 20th-century notation we write B in terms of del
cross E, which means that B is necessarily in a _third_
spatial direction, perpendicular to the polarization of E
and perpendicular to the direction of propagation.

Now the crux of my argument is that the magnetic field vector
is not the universal eternal truth.  It is an invention that
was cobbled up to represent our knowledge of certain things.
And it most certainly is not the only possible representation.

In particular, what some people call a magnetic field vector
in the Z direction can be represented at least as well by
a *bivector* in the XY plane.  That is, when I look through
the telescope I borrowed from William Clifford and Hermann
Grassmann, I see a different and grander truth:  I see a
version of electromagnetism that does not require any
cross products and does not require any right-hand rule.
Whereas the cross-product machinery only makes sense in
D=3+1 (three space dimensions plus time), the geometric-
algebra machinery makes perfect sense in D=2+1, D=3+1,
and indeed D=N+1 for all N>=2.

You need one spatial dimension for the direction of propagation,
plus another spatial dimension for the polarization of the
electric field, and you need the time dimension, and that's
it.  Any additional dimensions just go along for the ride.

Flatland is not dark.

For the proof of this, and other details, see
  http://www.av8n.com/physics/maxwell-ga.htm

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

If you want to pursue this down to one spatial dimension,
consider the electromagnetic signal on a coax cable or an
optical fiber.  If I stretch out a few kilometers of coax,
there is a range of length-scales covering about six
orders of magnitude where the Hausdorff dimentionality
  http://mathworld.wolfram.com/HausdorffMeasure.html
is one.  That is, the number of disks of size x I need
to cover the cable scales inversely with x to the first
power.  With optical fiber I can get one-dimensional
scaling over a range of 8 or 9 orders of magnitude.  So
for a wide range of practical purposes (but not all
purposes) we can demonstrate electromagnetism in D=1+1,
i.e. a single spatial dimension.  This provides a nice
way to understand what the stringfellows are talking
about when they talk about some small dimensions being
"compactified" or "curled up".  Just think of the
magnetic field lines in the coax.  They are curled up
and close on themselves, entirely inside the coax, on
a length-scale tiny compared to the long dimension of
the stretched-out coax.