Hiding in the Mirror (12 page)

Nevertheless, Kaluza was not immune to the
seductions of mathematical beauty. He found the remarkable
connection between the mathematical form of electromagnetism and
general relativity too compelling to resist, and in the conclusion
of his paper he wrote hopefully: Even in the face of all the
physical and epistemological difficulties which we have seen piling
up against the conception presented here, it is still hard to
believe that all these relations in their virtually unsurpassed
formal unity, should amount to the mere alluring play of a
capricious accident. Should more than an empty formalism be found
to reside behind these presumed connections, we would then face a
new triumph of Einstein’s general relativity, whose appropriate
application to a five-dimensional world is our main concern
here.

It would fall to later investigators to begin
to ascribe possible physical meaning to Kaluza’s fifth dimension, to
attempt to explain why it might be invisible, and to explore the
other possible physical consequences of this idea.

The first person to seriously take up this task
was physicist Oskar Klein (the son of Sweden’s first rabbi), who in
1926 independently discovered the mathematical relations earlier
demonstrated by Kaluza. (Somewhat later, even Einstein himself
became sufficiently enamored by the idea that in 1938 he and
colleague Peter Bergmann essentially reproduced Klein’s ideas in a
paper that represents Einstein’s own continuing search for a unified
theory of all interactions.) Klein, who studied with one of the
fathers of quantum mechanics, Niels Bohr, was motivated in his
investigations to try to understand the underlying nature of
various strange phenomena predicted in quantum theory, where
particles can sometimes act like waves, and probability appears to
replace certainty in physical predictions. Indeed, the developments
associated with this possible unification of electromagnetism and
gravity were taking place even as the top theoretical physicists of
the time were wrestling with the implications and mathematics of
the emerging quantum mechanical understanding of atomic phenomena.
The strange nature of atomic spectra—the discrete set of colors of
light emitted by different gases as you heat them up—and the nature
of the radiation emitted by so-called black bodies (i.e., objects
that very nearly absorb all colors of radiation equally and thus
appear black) as you heat them were considered to be much more
urgent problems than the more esoteric possible unification of two
theories that on their own held up remarkably well. Between 1913
and 1918, Niels Bohr had developed the first quantum theory of
atomic spectra by developing a series of rather ad hoc and unusual
rules to “explain” the energy levels of hydrogen. It was not until
1925–26—coincident with Klein’s work on extra dimensions—that
Werner Heisenberg and Erwin Schrödinger independently developed
selfconsistent formulations of quantum mechanics, which also
implied a host of associated “spooky” phenomena, to use a phrase of
Einstein’s, who never fully bought into the whole picture.

Still, having a solid mathematical formulation
of the rules of quantum mechanics and having a full physical
understanding of the theory are two different things. Unlike both
special relativity and general relativity, where comprehending the
mathematics can provide one with a more or less complete physical
picture, quantum mechanics defies all classical intuition.

For example, in the quantum world, subatomic
particles such as electrons behave at the same time as if they are
both waves and particles. That is, while individual electrons may
seem like particles, they can nevertheless do things baseballs
never do, such as being partially transmitted through and reflected
by objects simultaneously. While the equations of quantum mechanics
are themselves completely deterministic, the results of experiments
are not. Rather, the equations allow one to calculate the
probability
that an experiment will yield a certain
result. In 1927 Werner Heisenberg discovered one of the most
frustrating aspects of the newly developed quantum theory, which
has become known as the Heisenberg uncertainty principle, and which
stated that, independent of one’s measuring apparatus, there were
certain combinations of physical quantities, such as a particle’s
position and velocity, that could never be known with an accuracy
beyond some fundamental limit, no matter how long or hard one
tried. Einstein was not the only one who was repelled by the
thought of inherent indeterminacy in our knowledge of the physical
world. Perhaps this uncertainty arose because of our lack of
experimental knowledge of some

“hidden variables” that, if we had access to
them, would allow precise and arbitrarily accurate predictions of
experimental phenomena. Klein thus rediscovered Kaluza’s
five-dimensional unification scheme, but his motivation was somewhat
different than Kaluza’s. Klein, the student of Bohr, hoped that
this higher-dimensional framework might explain the basis of weird
quantum mechanical phenomena, like the uncertainty principle, which
he thought might be understood somehow as being due to our
experiencing only a four-dimensional projection of a
five-dimensional universe. This was the scientific equivalent, in a
very loose sense, of explaining weird paranormal phenomena by means
of the agency of an invisible fourth dimension (the difference, of
course, being that weird quantum phenomena actually have been
experimentally shown to exist!). It is also somewhat ironic that
Klein’s motivation for rediscovering Kaluza’s model came from
quantum mechanics, because this was precisely what Kaluza worried
about as possibly killing the whole idea. As he somewhat poetically
stated at the end of his 1919 paper: “In any case, every Ansatz
(i.e., postulate) which claims universal validity is threatened by
the sphinx of modern physics, quantum theory.”

In any case, not only did Klein reproduce
Kaluza’s mathematics, but because he took the possible physical
existence of a fifth dimension more seriously, as a physicist rather
than a mathematician, he was able to examine the physical
consequences, in particular for quantum theory and also for
electromagnetism, of such a fifth dimension. He also addressed the
question of why it might not be observable.

His solution, which was later reproduced by
Einstein and Bergmann, was to argue that this extra dimension was
curled in a small circle—so small, in fact, that it could not be
probed with existing experiments. In this scenario one could
imagine the four dimensions of space as follows: “Above” every
single point in our visible three-dimensional space a small circle
“sticks out” into the fourth dimension. If one suppresses one
dimension and represents our three-dimensional universe as a plane,
the extra dimension could therefore be pictured by lining up an
infinite number of infinitely long soda straws side by side. At each
point in the plane one could travel in a circle around the side of
the soda straw lying on the plane, returning back to where one
started.

In fact, this analogy of the soda straw is
useful from another point of view. Seen from a distance, a straw
looks as if it has no thickness—as if it were a simple
one-dimensional line. However, upon closer examination, one sees
that the straw is actually a cylinder: a two-dimensional object
(two-dimensional because one can move up and down along the length
of the straw, or travel in a perpendicular direction around the
side of the straw). If the diameter of the straw was small
enough—say, the size of a human hair—one might not be able to
perceive its thickness in the second dimension without a
microscope. If it was really small, even a microscope might not
reveal this second dimension. And so it could be with our universe:
An extra curled-up dimension lying above every point in space would
be invisible if it was curled up on a subatomic scale. While I have
presented this example by appealing to our classical intuition,
Klein’s argument actually relied instead on the wavelike nature of
elementary particles arising out of quantum mechanics. It is well
known that waves are not significantly disturbed by obstacles that
are much smaller than their wavelength. A water wave in the ocean,
for example, moves around a small pebble without any problem, but a
large rock will protect the water behind it from the disturbance
produced by an oncoming wave. The French physicist Louis de Broglie
had shown in 1924 that quantum mechanics implied that a
“wavelength” could be ascribed to every particle, that would be
inversely proportional to the particle’s “momentum” (which in turn
depends upon its mass times its velocity). The higher the momentum,
the smaller the wavelength. Indeed, this is why objects that are
much more massive than atoms tend to behave classically: Their
quantum mechanical wavelengths are so small as to be invisible, so
that these objects behave, for all intents and purposes, as if they
were simply particles, like billiard balls.

In order for an experiment to probe some scale,
the wavelengths of the particles that one sends in as probes—be
they the elementary particles associated with electromagnetic
radiation called photons, or some other particles, such as
electrons—must be smaller than the scale that one wishes to
explore. (Otherwise, the incoming wave will not be disturbed by the
object one wishes to probe.) This in turn means that the momentum,
and thus the energy imparted to our particle probes, must be larger
than a certain amount.

As a result, Klein, and later Einstein and
Bergmann, argued that if the radius of the fifth dimension was
smaller than a certain amount, then in order to send particles into
this extra dimension to even resolve it one would need more energy
than was then currently available in existing experiments. Because
of this property, the fifth dimension could exist, yet remain
effectively invisible in all existing experiments.

At the same time as providing this physical
mechanism to keep the fifth dimension phenomenologically viable,
Klein argued that the existence of an extra curled-up dimension
might explain why all electric charges come in integral multiples
of the charge on an electron (i.e., why we have never discovered
any object with a charge equal to, say, 1.33 or minus 2.4 times the
charge on an electron). Every object has a charge equal to . .
.−3,−2,−1,0,1,2,3 . . . times an electron’s charge.

Remember that in the Kaluza theory,
electromagnetism is a four-dimensional remnant of what one would,
if one had five-dimensional sensibilities, feel as part of a
five-dimensional gravitational field. Also remember that general
relativity provides a relation between the underlying energy of
objects moving through space with the curvature of space they thus
produce.

All of this together implies that if some
particle can move in the direction of the circular fifth dimension,
it will have an impact upon the geometry of the fifth dimension. To
this, Klein added one last feature of the quantum world—namely,
that every particle also has a wavelike character. For particles
whose motion in the fifth dimension is fast enough so that their
quantum-mechanical wavelength is small enough to allow them to “fit”
within it, then some familiar features of wave phemonena will take
over. Now, on a vibrating string only certain wavelengths are
allowed, which explains why longer strings, when plucked, produce
lower notes than shorter strings. On a vibrating string, only
certain harmonics can survive—waves whose wavelength has a specific
relationship to the length of the string (that is, is equal to the
length of the string, half the length of the string, one-third the
length of the string, etc.). (For those of you who are getting
excited at the mention of the word
string
,
you may calm down. This has nothing to do with superstring theory,
which we shall get to later.)

Now, if this held true for particles moving
around the circular extra dimension, since a particle’s quantum
mechanical wavelength is determined by its velocity, then only
particles with certain fixed velocities would be able to propagate
all the way around the extra dimension. A fixed set of velocities
implies a fixed set of energies associated with the particle. But
since energy affects geometry in general relativity, then if this
theory applies in the full five-dimensional space, it means that the
geometry of the fourth spatial dimension will be affected in
specific, discrete ways by the presence of such particles.

Remarkably, in the Kaluza theory the effect of
this change in the geometry of the fourth spatial dimension would
be measured in our three-dimensional space as the existence of an
electric field. Since the energies are only allowed in discrete
values, the resulting electric fields, which arise from electric
charges that we would view as emanating from the location in our
three-dimensional space where these particles start their voyages
around the extra dimension, must also come in discrete steps. Thus,
all charged particles would have electric charges that are discrete
multiples of some basic charge. In this way, Klein proposed that an
extra dimension could explain not only the existence of both
gravity and electromagnetism, but also the nature of all charged
objects we measure in our universe!

With so much going for it, one might wonder why
the Kaluza-Klein theory (as it is now known) did not become the
next big thing in physics in the 1920s and ’30s, and why these
physicists are not now household names, like Einstein. There are a
number of reasons. First, it became clear in these decades that the
laws of quantum mechanics developed by Schrödinger, Heisenberg, and
later by Dirac and others, while weird in the extreme, were
nevertheless perfectly consistent with all experiments and,
moreover, were inconsistent with the existence of extra “hidden
variables” that might somehow lead to the apparent probabilistic
nature of the theory. Thus, there was no apparent need for extra
dimensions in which to hide these variables.