153 research outputs found
Structure of nonlinear gauge transformations
Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations
(NGT) defined in terms of a wave function do not form a group. To get
a group property one has to consider transformations that act differently on
different branches of the complex argument function and the knowledge of the
value of is not sufficient for a well defined NGT. NGT that are well
defined in terms of form a semigroup parametrized by a real number
and a nonzero which is either an integer or . An extension of NGT to projectors and general density matrices
leads to NGT with complex . Both linearity of evolution and Hermiticity
of density matrices are gauge dependent properties.Comment: Final version, to be published in Phys.Rev.A (Rapid Communication),
April 199
Comment on "Consistency, amplitudes, and probabilities in quantum theory"
In a recent article [Phys. Rev. A 57, 1572 (1998)] Caticha has concluded that
``nonlinear variants of quantum mechanics are inconsistent.'' In this note we
identify what it is that nonlinear quantum theories have been shown to be
inconsistent with.Comment: LaTeX, 5 pages, no figure
On classical models of spin
The reason for recalling this old paper is the ongoing discussion on the
attempts of circumventing certain assumptions leading to the Bell theorem
(Hess-Philipp, Accardi). If I correctly understand the intentions of these
Authors, the idea is to make use of the following logical loophole inherent in
the proof of the Bell theorem: Probabilities of counterfactual events A and A'
do not have to coincide with actually measured probabilities if measurements of
A and A' disturb each other, or for any other fundamental reason cannot be
performed simulaneously. It is generally believed that in the context of
classical probability theory (i.e. realistic hidden variables) probabilities of
counterfactual events can be identified with those of actually measured events.
In the paper I give an explicit counterexample to this belief. The "first
variation" on the Aerts model shows that counterfactual and actual problems
formulated for the same classical system may be unrelated. In the model the
first probability does not violate any classical inequality whereas the second
does. Pecularity of the Bell inequality is that on the basis of an in principle
unobservable probability one derives probabilities of jointly measurable random
variables, the fact additionally obscuring the logical meaning of the
construction. The existence of the loophole does not change the fact that I was
not able to construct a local model violating the inequality with all the other
loopholes eliminated.Comment: published as Found. Phys. Lett. 3 (1992) 24
Geometric Phases and Mielnik's Evolution Loops
The cyclic evolutions and associated geometric phases induced by
time-independent Hamiltonians are studied for the case when the evolution
operator becomes the identity (those processes are called {\it evolution
loops}). We make a detailed treatment of systems having equally-spaced energy
levels. Special emphasis is made on the potentials which have the same spectrum
as the harmonic oscillator potential (the generalized oscillator potentials)
and on their recently found coherent states.Comment: 11 pages, harvmac, 2 figures available upon request; CINVESTAV-FIS
GFMR 11/9
Nonlinear Quantum Mechanics and Locality
It is shown that, in order to avoid unacceptable nonlocal effects, the free
parameters of the general Doebner-Goldin equation have to be chosen such that
this nonlinear Schr\"odinger equation becomes Galilean covariant.Comment: 10 pages, no figures, also available on
http://www.pt.tu-clausthal.de/preprints/asi-tpa/012-97.htm
Derivation of the Rules of Quantum Mechanics from Information-Theoretic Axioms
Conventional quantum mechanics with a complex Hilbert space and the Born Rule
is derived from five axioms describing properties of probability distributions
for the outcome of measurements. Axioms I,II,III are common to quantum
mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first
noted by Turing and von Neumann, in which the increase in entropy resulting
from a measurement is reduced by a suitable intermediate measurement. This is
shown to be impossible for local hidden variable theories. Axiom IV, together
with the first three, almost suffice to deduce the conventional rules but allow
some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom
V recognizes a property of the distribution of outcomes of random measurements
on qubits which holds only in the complex Hilbert space model. It is then shown
that the five axioms also imply the conventional rules for all dimensions.Comment: 20 pages, 6 figure
Magnetic operations: a little fuzzy physics?
We examine the behaviour of charged particles in homogeneous, constant and/or
oscillating magnetic fields in the non-relativistic approximation. A special
role of the geometric center of the particle trajectory is elucidated. In
quantum case it becomes a 'fuzzy point' with non-commuting coordinates, an
element of non-commutative geometry which enters into the traditional control
problems. We show that its application extends beyond the usually considered
time independent magnetic fields of the quantum Hall effect. Some simple cases
of magnetic control by oscillating fields lead to the stability maps differing
from the traditional Strutt diagram.Comment: 28 pages, 8 figure
Nonlocal looking equations can make nonlinear quantum dynamics local
A general method for extending a non-dissipative nonlinear Schr\"odinger and
Liouville-von Neumann 1-particle dynamics to an arbitrary number of particles
is described. It is shown at a general level that the dynamics so obtained is
completely separable, which is the strongest condition one can impose on
dynamics of composite systems. It requires that for all initial states
(entangled or not) a subsystem not only cannot be influenced by any action
undertaken by an observer in a separated system (strong separability), but
additionally that the self-consistency condition is fulfilled. It is shown that a correct
extension to particles involves integro-differential equations which, in
spite of their nonlocal appearance, make the theory fully local. As a
consequence a much larger class of nonlinearities satisfying the complete
separability condition is allowed than has been assumed so far. In particular
all nonlinearities of the form are acceptable. This shows that
the locality condition does not single out logarithmic or 1-homeogeneous
nonlinearities.Comment: revtex, final version, accepted in Phys.Rev.A (June 1998
The Classical Schrodinger's Equation
A non perturbative numerical method for determining the discrete spectra is
deduced from the classical analogue of the Schrodinger's equation. The energy
eigenvalues coincide with the bifurcation parameters for the classical orbits.Comment: UUEncoded Postscript, 18 pages, 4 figures inserted in tex
On a complex differential Riccati equation
We consider a nonlinear partial differential equation for complex-valued
functions which is related to the two-dimensional stationary Schrodinger
equation and enjoys many properties similar to those of the ordinary
differential Riccati equation as, e.g., the famous Euler theorems, the Picard
theorem and others. Besides these generalizations of the classical
"one-dimensional" results we discuss new features of the considered equation
like, e.g., an analogue of the Cauchy integral theorem
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