17,314 research outputs found
Efficient Simulation of Quantum State Reduction
The energy-based stochastic extension of the Schrodinger equation is a rather
special nonlinear stochastic differential equation on Hilbert space, involving
a single free parameter, that has been shown to be very useful for modelling
the phenomenon of quantum state reduction. Here we construct a general closed
form solution to this equation, for any given initial condition, in terms of a
random variable representing the terminal value of the energy and an
independent Brownian motion. The solution is essentially algebraic in
character, involving no integration, and is thus suitable as a basis for
efficient simulation studies of state reduction in complex systems.Comment: 4 pages, No Figur
Schwinger Algebra for Quaternionic Quantum Mechanics
It is shown that the measurement algebra of Schwinger, a characterization of
the properties of Pauli measurements of the first and second kinds, forming the
foundation of his formulation of quantum mechanics over the complex field, has
a quaternionic generalization. In this quaternionic measurement algebra some of
the notions of quaternionic quantum mechanics are clarified. The conditions
imposed on the form of the corresponding quantum field theory are studied, and
the quantum fields are constructed. It is shown that the resulting quantum
fields coincide with the fermion or boson annihilation-creation operators
obtained by Razon and Horwitz in the limit in which the number of particles in
physical states .Comment: 20 pages, Plain Te
Octonionic Version of Dirac Equations
It is shown that a simple continuity condition in the algebra of split
octonions suffices to formulate a system of differential equations that are
equivalent to the standard Dirac equations. In our approach the particle mass
and electro-magnetic potentials are part of an octonionic gradient function
together with the space-time derivatives. As distinct from previous attempts to
translate the Dirac equations into different number systems here the wave
functions are real split octonions and not bi-spinors. To formulate positively
defined probability amplitudes four different split octonions (transforming
into each other by discrete transformations) are necessary, rather then two
complex wave functions which correspond to particles and antiparticles in usual
Dirac theory.Comment: Version accepted by Int. J Mod. Phy
Collapse models with non-white noises
We set up a general formalism for models of spontaneous wave function
collapse with dynamics represented by a stochastic differential equation driven
by general Gaussian noises, not necessarily white in time. In particular, we
show that the non-Schrodinger terms of the equation induce the collapse of the
wave function to one of the common eigenstates of the collapsing operators, and
that the collapse occurs with the correct quantum probabilities. We also
develop a perturbation expansion of the solution of the equation with respect
to the parameter which sets the strength of the collapse process; such an
approximation allows one to compute the leading order terms for the deviations
of the predictions of collapse models with respect to those of standard quantum
mechanics. This analysis shows that to leading order, the ``imaginary'' noise
trick can be used for non-white Gaussian noise.Comment: Latex, 20 pages;references added and minor revisions; published as J.
Phys. A: Math. Theor. {\bf 40} (2007) 15083-1509
Towards Quantum Superpositions of a Mirror: an Exact Open Systems Analysis
We analyze the recently proposed mirror superposition experiment of Marshall,
Simon, Penrose, and Bouwmeester, assuming that the mirror's dynamics contains a
non-unitary term of the Lindblad type proportional to -[q,[q,\rho]], with q the
position operator for the center of mass of the mirror, and \rho the
statistical operator. We derive an exact formula for the fringe visibility for
this system. We discuss the consequences of our result for tests of
environmental decoherence and of collapse models. In particular, we find that
with the conventional parameters for the CSL model of state vector collapse,
maintenance of coherence is expected to within an accuracy of at least 1 part
in 10^{8}. Increasing the apparatus coupling to environmental decoherence may
lead to observable modifications of the fringe visibility, with time dependence
given by our exact result.Comment: 4 pages, RevTeX. Substantial changes mad
Breaking quantum linearity: constraints from human perception and cosmological implications
Resolving the tension between quantum superpositions and the uniqueness of
the classical world is a major open problem. One possibility, which is
extensively explored both theoretically and experimentally, is that quantum
linearity breaks above a given scale. Theoretically, this possibility is
predicted by collapse models. They provide quantitative information on where
violations of the superposition principle become manifest. Here we show that
the lower bound on the collapse parameter lambda, coming from the analysis of
the human visual process, is ~ 7 +/- 2 orders of magnitude stronger than the
original bound, in agreement with more recent analysis. This implies that the
collapse becomes effective with systems containing ~ 10^4 - 10^5 nucleons, and
thus falls within the range of testability with present-day technology. We also
compare the spectrum of the collapsing field with those of known cosmological
fields, showing that a typical cosmological random field can yield an efficient
wave function collapse.Comment: 13 pages, LaTeX, 3 figure
Correction-to-scaling exponent for two-dimensional percolation
We show that the correction-to-scaling exponents in two-dimensional
percolation are bounded by Omega <= 72/91, omega = D Omega <= 3/2, and Delta_1
= nu omega <= 2, based upon Cardy's result for the critical crossing
probability on an annulus. The upper bounds are consistent with many previous
measurements of site percolation on square and triangular lattices, and new
measurements for bond percolation presented here, suggesting this result is
exact. A scaling form evidently applicable to site percolation is also found
Multi-particle Correlations in Quaternionic Quantum Systems
We investigate the outcomes of measurements on correlated, few-body quantum
systems described by a quaternionic quantum mechanics that allows for regions
of quaternionic curvature. We find that a multi-particle interferometry
experiment using a correlated system of four nonrelativistic, spin-half
particles has the potential to detect the presence of quaternionic curvature.
Two-body systems, however, are shown to give predictions identical to those of
standard quantum mechanics when relative angles are used in the construction of
the operators corresponding to measurements of particle spin components.Comment: REVTeX 3.0, 16 pages, no figures, UM-P-94/54, RCHEP-94/1
Proof of Jacobi identity in generalized quantum dynamics
We prove that the Jacobi identity for the generalized Poisson bracket is
satisfied in the generalization of Heisenberg picture quantum mechanics
recently proposed by one of us (SLA). The identity holds for any combination of
fermionic and bosonic fields, and requires no assumptions about their mutual
commutativity.Comment: 9 pages, plain tex file, IASSNS-HEP-93/4
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