104 research outputs found
The Classical Limit of Quantum Mechanics and the Fejer Sum of the Fourier Series Expansion of a Classical Quantity
In quantum mechanics, the expectation value of a quantity on a quantum state,
provided that the state itself gives in the classical limit a motion of a
particle in a definite path, in classical limit goes over to Fourier series
form of the classical quantity. In contrast to this widely accepted point of
view, a rigorous calculation shows that the expectation value on such a state
in classical limit exactly gives the Fej\'{e}r's arithmetic mean of the partial
sums of the Fourier series
Large times off-equilibrium dynamics of a particle in a random potential
We study the off-equilibrium dynamics of a particle in a general
-dimensional random potential when . We demonstrate the
existence of two asymptotic time regimes: {\it i.} stationary dynamics, {\it
ii.} slow aging dynamics with violation of equilibrium theorems. We derive the
equations obeyed by the slowly varying part of the two-times correlation and
response functions and obtain an analytical solution of these equations. For
short-range correlated potentials we find that: {\it i.} the scaling function
is non analytic at similar times and this behaviour crosses over to
ultrametricity when the correlations become long range, {\it ii.} aging
dynamics persists in the limit of zero confining mass with universal features
for widely separated times. We compare with the numerical solution to the
dynamical equations and generalize the dynamical equations to finite by
extending the variational method to the dynamics.Comment: 70 pages, 7 figures included, uuencoded Z-compressed .tar fil
A Hopf laboratory for symmetric functions
An analysis of symmetric function theory is given from the perspective of the
underlying Hopf and bi-algebraic structures. These are presented explicitly in
terms of standard symmetric function operations. Particular attention is
focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and
twisted products (Rota cliffordizations) induced by branching operators in the
symmetric function context. The latter are shown to include the algebras of
symmetric functions of orthogonal and symplectic type. A commentary on related
issues in the combinatorial approach to quantum field theory is given.Comment: 29 pages, LaTeX, uses amsmat
Schroedinger equation for joint bidirectional motion in time
The conventional, time-dependent Schroedinger equation describes only
unidirectional time evolution of the state of a physical system, i.e., forward
or, less commonly, backward. This paper proposes a generalized quantum dynamics
for the description of joint, and interactive, forward and backward time
evolution within a physical system. [...] Three applications are studied: (1) a
formal theory of collisions in terms of perturbation theory; (2) a
relativistically invariant quantum field theory for a system that kinematically
comprises the direct sum of two quantized real scalar fields, such that one
field evolves forward and the other backward in time, and such that there is
dynamical coupling between the subfields; (3) an argument that in the latter
field theory, the dynamics predicts that in a range of values of the coupling
constants, the expectation value of the vacuum energy of the universe is forced
to be zero to high accuracy. [...]Comment: 30 pages, no figures. Related material is in quant-ph/0404012.
Differs from published version by a few added remarks on the possibility of a
large-scale-average negative energy density in spac
Continuous Symmetries of Difference Equations
Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.Comment: Review articl
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