22 research outputs found
New Interpretation of the Wigner Function
I define a two-sided or forward-backward propagator for the pseudo-diffusion equation of the 'squeezed' Q function. This propagator leads to squeezing in one of the phase-space variables and anti-squeezing in the other. By noting that the Q function is related to the Wigner function by a special case of the above propagator, I am led to a new interpretation of the Wigner function
Wave and pseudo-diffusion equations from squeezed states
We show that the probability distributions P(sub n)(q,p;y) := the absolute value squared of (n(p,q;y), which are obtained from squeezed states, obey an interesting partial differential equation, to which we give two intuitive interpretations: as a wave equation in one space dimension; and as a pseudo-diffusion equation. We also study the corresponding Wehrl entropies S(sub n)(y), and we show that they have minima at zero squeezing, y = 0
Matrix Representation of Octonions and Generalizations
We define a special matrix multiplication among a special subset of 2N\x 2N
matrices, and study the resulting (non-associative) algebras and their
subalgebras. We derive the conditions under which these algebras become
alternative non-associative and when they become associative. In particular,
these algebras yield special matrix representations of octonions and complex
numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix
representation of octonions also yields elegant insights into Dirac's equation
for a free particle. A few other results and remarks arise as byproducts.Comment: 18 printed page
Contraction of broken symmetries via Kac-Moody formalism
I investigate contractions via Kac-Moody formalism. In particular, I show how
the symmetry algebra of the standard 2-D Kepler system, which was identified by
Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was
denoted by , gets reduced by the symmetry breaking term,
defined by the Hamiltonian For this I
define two symmetry loop algebras , by
choosing the `basic generators' differently. These
can be mapped isomorphically onto subalgebras of , of
codimension 2 or 3, revealing the reduction of symmetry. Both factor algebras
, relative to the corresponding
energy-dependent ideals , are isomorphic to
and for , respectively, just as for the
pure Kepler case. However, they yield two different non-standard contractions
as , namely to the Heisenberg-Weyl algebra or to an abelian Lie algebra, instead of the Euclidean algebra
for the pure Kepler case. The above example suggests a
general procedure for defining generalized contractions, and also illustrates
the {\em `deformation contraction hysteresis'}, where contraction which involve
two contraction parameters can yield different contracted algebras, if the
limits are carried out in different order.Comment: 21 pages, 1 figur
Quantum Bound States with Zero Binding Energy
After reviewing the general properties of zero-energy quantum states, we give
the explicit solutions of the \seq with for the class of potentials
, where , these
solutions are normalizable and correspond to bound states, if the angular
momentum quantum number . [These states are normalizable, even for ,
if we increase the space dimension, , beyond 4; i.e. for .] For the above solutions, although unbound, are normalizable. This is true even
though the corresponding potentials are repulsive for all . We discuss the
physics of these unusual effects.Comment: 11 pages, 4 figures which can be generated from Mathematica commands
given at the end of the file. Latex, REVISED--Fig. 4 Mathematica command
improved and correcte