102 research outputs found
Normal transport properties for a classical particle coupled to a non-Ohmic bath
We study the Hamiltonian motion of an ensemble of unconfined classical
particles driven by an external field F through a translationally-invariant,
thermal array of monochromatic Einstein oscillators. The system does not
sustain a stationary state, because the oscillators cannot effectively absorb
the energy of high speed particles. We nonetheless show that the system has at
all positive temperatures a well-defined low-field mobility over macroscopic
time scales of order exp(-c/F). The mobility is independent of F at low fields,
and related to the zero-field diffusion constant D through the Einstein
relation. The system therefore exhibits normal transport even though the bath
obviously has a discrete frequency spectrum (it is simply monochromatic) and is
therefore highly non-Ohmic. Such features are usually associated with anomalous
transport properties
Spectral statistics for quantized skew translations on the torus
We study the spectral statistics for quantized skew translations on the
torus, which are ergodic but not mixing for irrational parameters. It is shown
explicitly that in this case the level--spacing distribution and other common
spectral statistics, like the number variance, do not exist in the
semiclassical limit.Comment: 7 pages. One figure, include
Classical motion in force fields with short range correlations
We study the long time motion of fast particles moving through time-dependent
random force fields with correlations that decay rapidly in space, but not
necessarily in time. The time dependence of the averaged kinetic energy and
mean-squared displacement is shown to exhibit a large degree of universality;
it depends only on whether the force is, or is not, a gradient vector field.
When it is, p^{2}(t) ~ t^{2/5} independently of the details of the potential
and of the space dimension. Motion is then superballistic in one dimension,
with q^{2}(t) ~ t^{12/5}, and ballistic in higher dimensions, with q^{2}(t) ~
t^{2}. These predictions are supported by numerical results in one and two
dimensions. For force fields not obtained from a potential field, the power
laws are different: p^{2}(t) ~ t^{2/3} and q^{2}(t) ~ t^{8/3} in all dimensions
d\geq 1
On a semiclassical formula for non-diagonal matrix elements
Let be a Schr\"odinger operator on the real
line, be a bounded observable depending only on the coordinate and
be a fixed integer. Suppose that an energy level intersects the potential
in exactly two turning points and lies below
. We consider the semiclassical limit
, and where is the th
eigen-energy of . An asymptotic formula for , the
non-diagonal matrix elements of in the eigenbasis of , has
been known in the theoretical physics for a long time. Here it is proved in a
mathematically rigorous manner.Comment: LaTeX2
Quantum Iterated Function Systems
Iterated functions system (IFS) is defined by specifying a set of functions
in a classical phase space, which act randomly on an initial point. In an
analogous way, we define a quantum iterated functions system (QIFS), where
functions act randomly with prescribed probabilities in the Hilbert space. In a
more general setting a QIFS consists of completely positive maps acting in the
space of density operators. We present exemplary classical IFSs, the invariant
measure of which exhibits fractal structure, and study properties of the
corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include
Crystal properties of eigenstates for quantum cat maps
Using the Bargmann-Husimi representation of quantum mechanics on a torus
phase space, we study analytically eigenstates of quantized cat maps. The
linearity of these maps implies a close relationship between classically
invariant sublattices on the one hand, and the patterns (or `constellations')
of Husimi zeros of certain quantum eigenstates on the other hand. For these
states, the zero patterns are crystals on the torus. As a consequence, we can
compute explicit families of eigenstates for which the zero patterns become
uniformly distributed on the torus phase space in the limit . This
result constitutes a first rigorous example of semi-classical equidistribution
for Husimi zeros of eigenstates in quantized one-dimensional chaotic systems.Comment: 43 pages, LaTeX, including 7 eps figures Some amendments were made in
order to clarify the text, mainly in the 4 first sections. Figures are
unchanged. To be published in: Nonlinearit
Coherent state quantization of a particle in de Sitter space
We present a coherent state quantization of the dynamics of a relativistic
test particle on a one-sheet hyperboloid embedded in a three-dimensional
Minkowski space. The group SO_0(1,2) is considered to be the symmetry group of
the system. Our procedure relies on the choice of coherent states of the motion
on a circle. The coherent state realization of the principal series
representation of SO_0(1,2) seems to be a new result.Comment: Journal of Physics A: Mathematical and General, vol. 37, in pres
Gupta-Bleuler quantization for minimally coupled scalar fields in de Sitter space
We present in this paper a fully covariant quantization of the
minimally-coupled massless field on de Sitter space. We thus obtain a formalism
free of any infrared (e.g logarithmic) divergence. Our method is based on a
rigorous group theoretical approach combined with a suitable adaptation (Krein
spaces) of the Wightman-G\"{a}rding axiomatic for massless fields
(Gupta-Bleuler scheme). We make explicit the correspondence between unitary
irreducible representations of the de Sitter group and the field theory on de
Sitter space-time. The minimally-coupled massless field is associated with a
representation which is the lowest term of the discrete series of unitary
representations of the de Sitter group. In spite of the presence of negative
norm modes in the theory, no negative energy can be measured: expressions as
\le n_{k_1}n_{k_2}...|T_{00}|n_{k_1}n_{k_2}...\re are always positive.Comment: 20 pages, appear in class. quantum gra
Weyl's law and quantum ergodicity for maps with divided phase space
For a general class of unitary quantum maps, whose underlying classical phase
space is divided into several invariant domains of positive measure, we
establish analogues of Weyl's law for the distribution of eigenphases. If the
map has one ergodic component, and is periodic on the remaining domains, we
prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the
equidistribution of eigenfunctions with respect to the ergodic component of the
classical map (quantum ergodicity). We apply our main theorems to quantised
linked twist maps on the torus. In the Appendix, S. Zelditch connects these
studies to some earlier results on `pimpled spheres' in the setting of
Riemannian manifolds. The common feature is a divided phase space with a
periodic component.Comment: Colour figures. Black & white figures available at
http://www2.maths.bris.ac.uk/~majm. Appendix by Steve Zelditc
Simple model of big-crunch/big-bang transition
We present classical and quantum dynamics of a test particle in the
compactified Milne space. Background spacetime includes one compact space
dimension undergoing contraction to a point followed by expansion. Quantization
consists in finding a self-adjoint representation of the algebra of particle
observables. Our model offers some insight into the nature of the cosmic
singularity.Comment: 17 pages, no figures, RevTeX4, accepted for publication in Class.
Quantum Gra
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