4,652 research outputs found
The Chern-Simons Action in Non-Commutative Geometry
A general definition of Chern-Simons actions in non-commutative geometry is
proposed and illustrated in several examples. These are based on
``space-times'' which are products of even-dimensional, Riemannian spin
manifolds by a discrete (two-point) set. If the *algebras of operators
describing the non-commutative spaces are generated by functions over such
``space-times'' with values in certain Clifford algebras the Chern-Simons
actions turn out to be the actions of topological gravity on the
even-dimensional spin manifolds. By contrasting the space of field
configurations in these examples in an appropriate manner one is able to
extract dynamical actions from Chern-Simons actions.Comment: 40 page
Magnetism and the Weiss Exchange Field - A Theoretical Analysis Inspired by Recent Experiments
The huge spin precession frequency observed in recent experiments with
spin-polarized beams of hot electrons shot through magnetized films is
interpreted as being caused by Zeeman coupling of the electron spins to the
so-called Weiss exchange field in the film. A "Stern-Gerlach experiment" for
electrons moving through an inhomogeneous exchange field is proposed. The
microscopic origin of exchange interactions and of large mean exchange fields,
leading to different types of magnetic order, is elucidated. A microscopic
derivation of the equations of motion of the Weiss exchange field is presented.
Novel proofs of the existence of phase transitions in quantum XY-models and
antiferromagnets, based on an analysis of the statistical distribution of the
exchange field, are outlined.Comment: 36 pages, 3 figure
Absence of Embedded Mass Shells: Cerenkov Radiation and Quantum Friction
We show that, in a model where a non-relativistic particle is coupled to a
quantized relativistic scalar Bose field, the embedded mass shell of the
particle dissolves in the continuum when the interaction is turned on, provided
the coupling constant is sufficiently small. More precisely, under the
assumption that the fiber eigenvectors corresponding to the putative mass shell
are differentiable as functions of the total momentum of the system, we show
that a mass shell could exist only at a strictly positive distance from the
unperturbed embedded mass shell near the boundary of the energy-momentum
spectrum.Comment: Revised version: a remark added at the end of Section
Spin - or, actually: Spin and Quantum Statistics
The history of the discovery of electron spin and the Pauli principle and the
mathematics of spin and quantum statistics are reviewed. Pauli's theory of the
spinning electron and some of its many applications in mathematics and physics
are considered in more detail. The role of the fact that the tree-level
gyromagnetic factor of the electron has the value g = 2 in an analysis of
stability (and instability) of matter in arbitrary external magnetic fields is
highlighted. Radiative corrections and precision measurements of g are
reviewed. The general connection between spin and statistics, the CPT theorem
and the theory of braid statistics are described.Comment: 50 pages, no figures, seminar on "spin
On the Atomic Photoeffect in Non-relativistic QED
In this paper we present a mathematical analysis of the photoelectric effect
for one-electron atoms in the framework of non-relativistic QED. We treat
photo-ionization as a scattering process where in the remote past an atom in
its ground state is targeted by one or several photons, while in the distant
future the atom is ionized and the electron escapes to spacial infinity. Our
main result shows that the ionization probability, to leading order in the
fine-structure constant, , is correctly given by formal time-dependent
perturbation theory, and, moreover, that the dipole approximation produces an
error of only sub-leading order in . In this sense, the dipole
approximation is rigorously justified.Comment: 25 page
The Gravitational Sector in the Connes-Lott Formulation of the Standard Model
We study the Riemannian aspect and the Hilbert-Einstein gravitational action
of the non-commutative geometry underlying the Connes-Lott construction of the
action functional of the standard model. This geometry involves a two-sheeted,
Euclidian space-time. We show that if we require the space of forms to be
locally isotropic and the Higgs scalar to be dynamical, then the Riemannian
metrics on the two sheets of Euclidian space-time must be identical. We also
show that the distance function between the two sheets is determined by a
single, real scalar field whose VEV sets the weak scale.Comment: Latex file, 29 page
Multi-Particle Anderson Localisation: Induction on the Number of Particles
This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09}
covering the two-particle Anderson model. Here we establish the phenomenon of
Anderson localisation for a quantum -particle system on a lattice
with short-range interaction and in presence of an IID external potential with
sufficiently regular marginal cumulative distribution function (CDF). Our main
method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS},
\cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an
induction on the number of particles, as was proposed in our earlier manuscript
\cite{CS07}. Similar results have been recently obtained in an independent work
by Aizenman and Warzel \cite{AW08}: they proposed an extension of the
Fractional-Moment Method (FMM) developed earlier for single-particle models in
\cite{AM93} and \cite{ASFH01} (see also references therein) which is also
combined with an induction on the number of particles.
An important role in our proof is played by a variant of Stollmann's
eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved
earlier in \cite{C08}, admits a straightforward extension covering the case of
multi-particle systems with correlated external random potentials: a subject of
our future work. We also stress that the scheme of our proof is \textit{not}
specific to lattice systems, since our main method, the MSA, admits a
continuous version. A proof of multi-particle Anderson localization in
continuous interacting systems with various types of external random potentials
will be published in a separate papers
On the flux phase conjecture at half-filling: an improved proof
We present a simplification of Lieb's proof of the flux phase conjecture for
interacting fermion systems -- such as the Hubbard model --, at half filling on
a general class of graphs. The main ingredient is a procedure which transforms
a class of fermionic Hamiltonians into reflection positive form. The method can
also be applied to other problems, which we briefly illustrate with two
examples concerning the model and an extended Falicov-Kimball model.Comment: 23 pages, Latex, uses epsf.sty to include 3 eps figures, to appear in
J. Stat. Phys., Dec. 199
- âŠ