16,819 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
Global unitary fixing and matrix-valued correlations in matrix models
We consider the partition function for a matrix model with a global unitary
invariant energy function. We show that the averages over the partition
function of global unitary invariant trace polynomials of the matrix variables
are the same when calculated with any choice of a global unitary fixing, while
averages of such polynomials without a trace define matrix-valued correlation
functions, that depend on the choice of unitary fixing. The unitary fixing is
formulated within the standard Faddeev-Popov framework, in which the squared
Vandermonde determinant emerges as a factor of the complete Faddeev-Popov
determinant. We give the ghost representation for the FP determinant, and the
corresponding BRST invariance of the unitary-fixed partition function. The
formalism is relevant for deriving Ward identities obeyed by matrix-valued
correlation functions.Comment: Tex, 22 page
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
No Eigenvalue in Finite Quantum Electrodynamics
We re-examine Quantum Electrodynamics (QED) with massless electron as a
finite quantum field theory as advocated by Gell-Mann-Low, Baker-Johnson,
Adler, Jackiw and others. We analyze the Dyson-Schwinger equation satisfied by
the massless electron in finite QED and conclude that the theory admits no
nontrivial eigenvalue for the fine structure constant.Comment: 13 pages, Late
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
Structure and Properties of Hughston's Stochastic Extension of the Schr\"odinger Equation
Hughston has recently proposed a stochastic extension of the Schr\"odinger
equation, expressed as a stochastic differential equation on projective Hilbert
space. We derive new projective Hilbert space identities, which we use to give
a general proof that Hughston's equation leads to state vector collapse to
energy eigenstates, with collapse probabilities given by the quantum mechanical
probabilities computed from the initial state. We discuss the relation of
Hughston's equation to earlier work on norm-preserving stochastic equations,
and show that Hughston's equation can be written as a manifestly unitary
stochastic evolution equation for the pure state density matrix. We discuss the
behavior of systems constructed as direct products of independent subsystems,
and briefly address the question of whether an energy-based approach, such as
Hughston's, suffices to give an objective interpretation of the measurement
process in quantum mechanics.Comment: Plain Tex, no figure
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
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