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from
real numbers to octonions
Regardless
of the kind of numbers used in a particular natural
science hypo- thesis or theory, the testable result is a
real number. Girolamo Cardano, who in 1545 boldly
introduced the square root of -1 (now abbreviated as i
for ‘imaginary’) into his calculations giving ordinary
real number as result, was not sure why this trick worked
- all he knew was that it gave him the right result.
Mathematicians began working with complex numbers of the
form a+bi, where a and b are ordinary
real numbers, and natural science calculations became
flooded with them. We readily accept complex numbers in
grade school because of the geometrical visualization (see
on the right): the set of all real numbers forms a line
(one-dimensional set) while any complex number is a point
in the plane (two-dimensional set). Addition, subtraction,
multiplication, and division could be presented as
geometric manipulation on the line for real numbers while
in the plane for complex numbers.
The
reason we use complex numbers is that we can solve more
equations with complex numbers than with real numbers. If
a two-dimensional number system gives the user added
calculating power, what about even higher- dimensional
systems? In 1835, W.R. Hamilton learned the hard way that
a three-dimensional number system is not in accord with
division. By 1958 it was proved that any division algebra
must have dimensions one (the real numbers), two (the
complex numbers), four (the quaternions), or eight (the
octonions).
In
a good popular writing The Strangest Numbers in String
Theory, J.C. Baez and J. Huerta contemplate on higher
meaning and role of quaternions and octonions.
Interestingly enough, theoretical physicists promoting the
hypo- thesis of supersymmetry (a symmetry between matter
and the forces of nature) found the octonions to suit
unified description of matter and forces. In string theory
(which requires supersymmetry) two dimensions are added:
one for the string and one for time. Most string theorists
consider only 10-dimensi- onal versions of the theory to
be self-consistent. The M-theory adds one more dimension,
up to 11, replacing one-dimensional string with a
two-dimensional membrane. |
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an
excerpt from The Strangest Numbers in String Theory
by J.C. Baez
and J. Huerta, in The Best Writings on Mathematics 2012
At
this point, we should emphasize that string theory and
M-theory have not yet made experimentally testable
predictions. They are beautiful dreams - but so far only
dreams. The universe we leave in does not look 10- or
11-dimensional, and we have not seen any symmetry between
matter and force particles. A while ago, David Gross, one
of the world’s leading experts on string theory, put the
odds of seeing some evidence for supersymmetry at CERN’s
Large Hadron Collider at 50 percent. Skeptics say they are
much less. Only time will tell.
Because
of this uncertainty, we are still a long way from knowing
if the strange octonions are of fundamental importance in
understanding the world we see around us or merely a piece
of beautiful mathematics. Of course, mathematical beauty
is a worthy end in itself, but it would be even more
delightful if the octonions turned out to be built into
the fabric of nature. As the story of the complex numbers
and countless other mathematical developments
demonstrates, it would hardly be the first time that
purely mathematical inventions later provided the tool
that physicists need. |
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The
usage of octonions in a consistent string theory will
prove neither the theory nor the deeper essence of
octonions. Both, the physical laws and the mathematical
objects, are product of our mind. So-called Laws of Nature
are actually laws of our thinking about Nature. |
