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Quantum
Optics
Quantum Optics is the study of radiation and matter
in the optical wavelength domain, where sophisticated
advances in laser technology enable tests of fundamental
physical questions with unprecedented precision. Optical probes
of coherent states of atoms and photons permit new insights into
questions about the basic foundations of quantum
mechanics and are leading to concrete realization
of futuristic applications such as quantum computing, cryptography
and even tele-portation. They are also permitting existing advances
in laser cooling and trapping (CAT) techniques that were highlighted
by the 1997 Nobel Prize in Physics.
Tachyons
They are hypothetical subatomic particle whose
velocity always exceeds that of light. The existence
of the Tachyon, though not experimentally established,
appears consistent with the theory of relativity, which was originally
thought to apply only to particles traveling at or less than the
speed of light. Just an ordinary particle such
as an electron can exist only at speeds less than
that of light, so a Tachyon could exist only at speeds above that
of light, at which point its mass would be real
and positive. Upon losing energy, a Tachyon would
accelerate; the faster it traveled, the less energy it would have.
Elementary Particles
The most basic physical constituents of the universe.
Atoms are the basic units of the chemical elements
but are themselves composed of smaller particles.
The first subatomic particle to be discovered was the electron,
identified in 1897 by Joseph John Thomson. The
nucleus of ordinary hydrogen was subsequently
recognized as a single particle and was named the proton. The
third basic particle in an atom, the neutron was discovered in 1932.
Although models of the atom consisting of just
these three particles are sufficient to account
for all forms of chemical behavior of matter, Quantum mechanics
predicted the existence of additional elementary particles.
Quantum Field Theory
Body of physical principles designed to account
for subatomic phenomena. The theory also has found
applications in other branches of Physics. The theory
arises from the attempt to combine the principles of Quantum Mechanics
with those of relativity in an effort to describe processes such
as high-energy collisions in which particles may
be created or destroyed. The prototype of quantum
field theories is Quantum electrodynamics (QED), which describes
the interaction of electrically charged particles via electromagnetic
fields. Here, electric and magnetic forces are
regarded as arising from the emission and absorption
of exchange particles or photons. These can be represented
as disturbances of electromagnetic fields, much as ripples on a
lake are disturbances of the water. Under suitable
conditions, photons may become entirely free of
charged particles; they are then detectable as light and
other forms of electromagnetic radiation. Similarly, particles such
as electrons are themselves regarded as disturbances
of their own quantified fields. Numerical predictions
based on QED agree with experimental data to within
one part in 10,000,000 in some cases.
There is a widespread conviction among physicists
that other forces in nature--the weak force responsible
for radioactive beta-decay; the strong force,
which binds together the constituents of atomic nuclei; and perhaps
also gravitational forces--can be described by
theories similar to QED. These theories are known
collectively as gauge theories. Each of the forces is mediated
by its own set of exchange particles, and differences between the
forces are reflected in the properties of these
particles. For example, electromagnetic and gravitational
forces operate over long distances, and their
exchange particles (the photon and the graviton) have no mass. The
weak and strong forces operate only over distances
shorter than the size of an atomic nucleus. They
are mediated by massive particles, which can travel only
short distances during the exchange process.
It is also hoped that all the forces can be encompassed
in a single gauge field theory. In such a unified
theory, all the forces would have a common origin and
would be related by mathematical symmetries. The simplest result
would be that all the forces had identical properties.
A mechanism called spontaneous symmetry breaking
is used to account for the observed differences.
A unified theory of electromagnetic and weak forces already has
considerable experimental support; it is likely
that this theory can be extended to include the
strong force. There also exist theories that include the
gravitational force, but these are more speculative.
Spectral value sets. We discuss semi-algebraicity,
continuous dependence on parameters and show some
numerical examples. In particular, we discuss the difference
between real and complex spectral value sets of normal matrices.
While the normal case is trivial if complex perturbations
are considered, the problem turns out to be more
complicated in the real case.
A further topic of the talk is a quantification
of the often-remarked phenomenon, that spectral
value sets tend to be large for matrices with great
departure from normality. We discuss a bound for spectral value
sets based on the departure from normality in
Henrici's sense. Finally, the relation of departure
from normality and the transient behavior of dynamic systems will
be considered. In particular, we introduce the notion of stability
with normal transient behavior and give a sufficient
condition for this property in terms of departure
from normality.
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