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What is a C*-algebra?
It is a quantization of space.
It is a generalization of space.
It is a way of
speaking about space without referring to "points". So
you are
allowing space to be a little "cloudy" or "fuzzy", very much like the
idea of a wavefunction of an electron in quantum mechanics. The
wavefunction does not tell you at precisely what "point" the electron
is located (which would violate the Heisenberg Uncertaintly Principle),
but gives you a
"probability amplitude" of where it is.
Mathematically, a
space is a topological space (as is studied in
Topology).
So, the study of C*-algebras is a generalization of
Topology---sometimes
referred to as Noncommutative Topology.
It studies noncommutative topological spaces.
What is its history?
The field of
Operator Algebras might not have come into
existence
without modern physics. For it gradually evolved during the
development of quantum theory in the 1920's and in subsequent
pioneering
works of Murray and von Neumann---who studied the mathematical
structure that was
naturally emerging from Quantum Theory in the works of Planck, Bohr,
Einstein,
Schrodinger, Dirac, Heisenberg, Born, and others around the same time.
Operator Algebras emerged from the necessity, in Quantum
Theory,
that important observables do not commute with each other.
For example, position and momentum observables, time and energy, or
angular
momenta or spin observables---all of which came to be best represented
by noncommuting operators or matrices. (Hence the
name
"noncommutative geometry".) It is amazing that these abstract
mathematical
ideas became "necessitated" by numerous experiments in the real world.
Physicists
proved that!
Throughout its development, however, Operator Algebras has
taken a life
of its own and became an independent field of mathematical
investigation.
When you have newly emerging objects, you don't just leave them
unstudied!
So people began to probe the structure of these algebras, without
worrying
about whether they will have any 'practical' application.
Extensive study of them has been made, and continues to be made.
And now, after it has developed significantly over the last seventy
years since its inception, Operator Algebras---mainly through the new
geometry
of Alain
Connes---is now promising to furnish a new mathematical foundation
on which it can be useful to its original ancestor (Physics),
and perhaps help in unifying gravitation
together with Quantum Physics and so eventually leading to Einstein's
long sought
unified field theory (where all the four known forces would be unified
into
a single force).
What is a good example? And
what is its physical relevance?
The most concrete
example of C*-algebras that illustrate the
aforementioned
"generalization" and "quantization" of space comes from the study of
the
rotation C*-algebras. These algebras are generated by two unitaries U
and
V ("position" and "momentum" unitaries of Quantum Mechanics) enjoying
the
Heisenberg commutation relation
where h is the
angle of rotation (Planck's constant). (This
is the
unitary form of the famous relation of Heisenberg px-xp=ih.) A deep
result of Rieffel, Pimsner, and Voiculescu in the early 1980's, is that
if
we perturb Planck's constant however slightly, we get a very different
(non-isomorphic) algebra. Inside this algebra live some important
Hamiltonians, such as that considered by Hofstader of electrons in a
crystal lattice. Granting some scalings, this Hamiltonian can be
written as
and is called the
Harper operator. (When scales are
introduced it is called the
almost Mathieu operator.) The energy spectrum of H as a function of
magnetic
field strength has a beautiful structure, called Hofstader's
"butterfly". (The mysterious fact is that if the magnetic
field strength is an irrational
number, then the energy spectrum is a totally disconnected space,
called a Cantor
set.) Further reference to such connections is found in A. Connes'
classic book, Noncommutative
Geometry
(Academic Press, 1994), and especially to the references to J.
Bellisard's work
therein.
The most fascinating contribution of this new geometry is
related to the fabric of spacetime. In physics one often thinks of
space
as being locally Euclidean. (This is assumed in differential geometry
and
in Einstein's General Relativity.)
This locally Euclidean hypothesis is removed by looking instead
at a (noncommutative) C*-algebra.
The noncommutative aspect of space plays a central part in the
formulation of
Quantum Mechanics, as in the Heisenberg uncertainty principle, which
basically
says that the coordinates describing the position and momentum of a
quantum
particle (like the electron) do no commute with each other. In fact, it
is
encouraging (and telling) that some prominant theoretical
physicists---including
Fields medalist Ed Witten
of the Institute
for Advanced Study---have
taken
this approach and used it in their research investigations. It gives
hope
that this new Mathematics could form a more promising basis on which
can be
develop a unified field theory (where quantum phenomena and gravitation
would
be unified, as well as the unification of the four knows forces of
nature).
Another link to mathematical physics arises in the study of
the Quantum
Hall
Effect where, for example, the Hall conductivity of electrons in a
crystal in a magnetic field is shown to be equal to Connes'
noncommutative Chern
character---the Chern character of a projection (in a C*-algebra)
associated to the Fermi energy level. In 1980, von Klitzing showed that
the Hall conductivity of a disordered crystal is quantized and in 1985
he won the Nobel prize in
Physics (at the age of 42) for this discovery.
Connes managed to
reformulate the Standard Model (which unifies the
weak,
strong, and electromagnetic interactions) in terms of Noncommutative
Geometry. This is an extraordinary feat, and is done in the last chapter of his
book (Ibid).
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