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This book covers the theory and applications of the Wigner phase
space distribution function and its symmetry properties. The book
explains why the phase space picture of quantum mechanics is
needed, in addition to the conventional Schroedinger or Heisenberg
picture. It is shown that the uncertainty relation can be
represented more accurately in this picture. In addition, the phase
space picture is shown to be the natural representation of quantum
mechanics for modern optics and relativistic quantum mechanics of
extended objects.
This book covers the theory and applications of the Wigner phase
space distribution function and its symmetry properties. The book
explains why the phase space picture of quantum mechanics is
needed, in addition to the conventional Schroedinger or Heisenberg
picture. It is shown that the uncertainty relation can be
represented more accurately in this picture. In addition, the phase
space picture is shown to be the natural representation of quantum
mechanics for modern optics and relativistic quantum mechanics of
extended objects.
Einstein's energy-momentum relation is applicable to particles of
all speeds, including the particle at rest and the massless
particle moving with the speed of light. If one formula or
formalism is applicable to all speeds, we say it is
'Lorentz-covariant.' As for the internal space-time symmetries,
there does not appear to be a clear way to approach this problem.
For a particle at rest, there are three spin degrees of freedom.
For a massless particle, there are helicity and gauge degrees of
freedom. The aim of this book is to present one Lorentz-covariant
picture of these two different space-time symmetries. Using the
same mathematical tool, it is possible to give a Lorentz-covariant
picture of Gell-Mann's quark model for the proton at rest and
Feynman's parton model for the fast-moving proton. The mathematical
formalism for these aspects of the Lorentz covariance is based on
two-by-two matrices and harmonic oscillators which serve as two
basic scientific languages for many different branches of physics.
It is pointed out that the formalism presented in this book is
applicable to various aspects of optical sciences of current
interest.
This book explains the Lorentz mathematical group in a language
familiar to physicists. While the three-dimensional rotation group
is one of the standard mathematical tools in physics, the Lorentz
group of the four-dimensional Minkowski space is still very strange
to most present-day physicists. It plays an essential role in
understanding particles moving at close to light speed and is
becoming the essential language for quantum optics, classical
optics, and information science. The book is based on papers and
books published by the authors on the representations of the
Lorentz group based on harmonic oscillators and their applications
to high-energy physics and to Wigner functions applicable to
quantum optics. It also covers the two-by-two representations of
the Lorentz group applicable to ray optics, including cavity,
multilayer and lens optics, as well as representations of the
Lorentz group applicable to Stokes parameters and the Poincar
sphere on polarization optics.
This book explains the Lorentz mathematical group in a language
familiar to physicists. While the three-dimensional rotation group
is one of the standard mathematical tools in physics, the Lorentz
group of the four-dimensional Minkowski space is still very strange
to most present-day physicists. It plays an essential role in
understanding particles moving at close to light speed and is
becoming the essential language for quantum optics, classical
optics, and information science. The book is based on papers and
books published by the authors on the representations of the
Lorentz group based on harmonic oscillators and their applications
to high-energy physics and to Wigner functions applicable to
quantum optics. It also covers the two-by-two representations of
the Lorentz group applicable to ray optics, including cavity,
multilayer and lens optics, as well as representations of the
Lorentz group applicable to Stokes parameters and the Poincare
sphere on polarization optics.
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