650 research outputs found
Dirac monopole with Feynman brackets
We introduce the magnetic angular momentum as a consequence of the structure
of the sO(3) Lie algebra defined by the Feynman brackets. The Poincare momentum
and Dirac magnetic monopole appears as a direct result of this framework.Comment: 10 page
From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics
In 1990, Dyson published a proof due to Feynman of the Maxwell equations
assuming only the commutation relations between position and velocity. With
this minimal assumption, Feynman never supposed the existence of Hamiltonian or
Lagrangian formalism. In the present communication, we review the study of a
relativistic particle using ``Feynman brackets.'' We show that Poincar\'e's
magnetic angular momentum and Dirac magnetic monopole are the consequences of
the structure of the Lorentz Lie algebra defined by the Feynman's brackets.
Then, we extend these ideas to the dual momentum space by considering
noncommutative quantum mechanics. In this context, we show that the
noncommutativity of the coordinates is responsible for a new effect called the
spin Hall effect. We also show its relation with the Berry phase notion. As a
practical application, we found an unusual spin-orbit contribution of a
nonrelativistic particle that could be experimentally tested. Another practical
application is the Berry phase effect on the propagation of light in
inhomogeneous media.Comment: Presented at the 3rd Feynman Festival (Collage Park, Maryland,
U.S.A., August 2006
Monopole and Berry Phase in Momentum Space in Noncommutative Quantum Mechanics
To build genuine generators of the rotations group in noncommutative quantum
mechanics, we show that it is necessary to extend the noncommutative parameter
to a field operator, which one proves to be only momentum dependent.
We find consequently that this field must be obligatorily a dual Dirac monopole
in momentum space. Recent experiments in the context of the anomalous Hall
effect provide for a monopole in the crystal momentum space. We suggest a
connection between the noncommutative field and the Berry curvature in momentum
space which is at the origine of the anomalous Hall effect.Comment: 4 page
Semiclassical Dynamics of Electrons in Magnetic Bloch Bands: a Hamiltonian Approach
y formally diagonalizing with accuracy the Hamiltonian of electrons
in a crystal subject to electromagnetic perturbations, we resolve the debate on
the Hamiltonian nature of semiclassical equations of motion with Berry-phase
corrections, and therefore confirm the validity of the Liouville theorem. We
show that both the position and momentum operators acquire a Berry-phase
dependence, leading to a non-canonical Hamiltonian dynamics. The equations of
motion turn out to be identical to the ones previously derived in the context
of electron wave-packets dynamics.Comment: 4 page
Interrupt Timed Automata: verification and expressiveness
We introduce the class of Interrupt Timed Automata (ITA), a subclass of
hybrid automata well suited to the description of timed multi-task systems with
interruptions in a single processor environment. While the reachability problem
is undecidable for hybrid automata we show that it is decidable for ITA. More
precisely we prove that the untimed language of an ITA is regular, by building
a finite automaton as a generalized class graph. We then establish that the
reachability problem for ITA is in NEXPTIME and in PTIME when the number of
clocks is fixed. To prove the first result, we define a subclass ITA- of ITA,
and show that (1) any ITA can be reduced to a language-equivalent automaton in
ITA- and (2) the reachability problem in this subclass is in NEXPTIME (without
any class graph). In the next step, we investigate the verification of real
time properties over ITA. We prove that model checking SCL, a fragment of a
timed linear time logic, is undecidable. On the other hand, we give model
checking procedures for two fragments of timed branching time logic. We also
compare the expressive power of classical timed automata and ITA and prove that
the corresponding families of accepted languages are incomparable. The result
also holds for languages accepted by controlled real-time automata (CRTA), that
extend timed automata. We finally combine ITA with CRTA, in a model which
encompasses both classes and show that the reachability problem is still
decidable. Additionally we show that the languages of ITA are neither closed
under complementation nor under intersection
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