34 research outputs found
Small oscillations and the Heisenberg Lie algebra
The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems
for quadratic Hamiltonians of on coadjoint orbits of the
Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie
algebra that admits an ad-invariant metric. Its quadratic induces
the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that
one on . This system is a Lax pair equation whose solution can
be computed with help of the Adjoint representation. For a certain class of
functions, the Poisson commutativity on the coadjoint orbits in
is related to the commutativity of a family of derivations of the
2n+1-dimensional Heisenberg Lie algebra . Therefore the complete
integrability is related to the existence of an n-dimensional abelian
subalgebra of certain derivations in . For instance, the motion
of n-uncoupled harmonic oscillators near an equilibrium position can be
described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of
the AKS schem
The gravity-related decoherence master equation from hybrid dynamics
Canonical coupling between classical and quantum systems cannot result in
reversible equations, rather it leads to irreversible master equations.
Coupling of quantized non-relativistic matter to gravity is illustrated by a
simplistic example. The heuristic derivation yields the theory of
gravity-related decoherence proposed longtime ago by Penrose and the author.Comment: 9pp, extended version of invited talk at Fifth International Workshop
DICE2010 (Castello Pasquini/Castiglioncello/Tuscany, Sept. 13-17, 2010
Higher spin quaternion waves in the Klein-Gordon theory
Electromagnetic interactions are discussed in the context of the Klein-Gordon
fermion equation. The Mott scattering amplitude is derived in leading order
perturbation theory and the result of the Dirac theory is reproduced except for
an overall factor of sixteen. The discrepancy is not resolved as the study
points into another direction. The vertex structures involved in the scattering
calculations indicate the relevance of a modified Klein-Gordon equation, which
takes into account the number of polarization states of the considered quantum
field. In this equation the d'Alembertian is acting on quaternion-like plane
waves, which can be generalized to representations of arbitrary spin. The
method provides the same relation between mass and spin that has been found
previously by Majorana, Gelfand, and Yaglom in infinite spin theories
Mixing Quantum and Classical Mechanics
Using a group theoretical approach we derive an equation of motion for a
mixed quantum-classical system. The quantum-classical bracket entering the
equation preserves the Lie algebra structure of quantum and classical
mechanics: The bracket is antisymmetric and satisfies the Jacobi identity, and,
therefore, leads to a natural description of interaction between quantum and
classical degrees of freedom. We apply the formalism to coupled quantum and
classical oscillators and show how various approximations, such as the
mean-field and the multiconfiguration mean-field approaches, can be obtained
from the quantum-classical equation of motion.Comment: 31 pages, LaTeX2