1,054 research outputs found
Path integrals on manifolds by finite dimensional approximation
Let M be a compact Riemannian manifold without boundary and let H be a
self-adjoint generalized Laplace operator acting on sections in a bundle over
M. We give a path integral formula for the solution to the corresponding heat
equation. This is based on approximating path space by finite dimensional
spaces of geodesic polygons. We also show a uniform convergence result for the
heat kernels. This yields a simple and natural proof for the
Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of
the heat operator.Comment: 23 page
Generalized Cylinders in Semi-Riemannian and Spin Geometry
We use a construction which we call generalized cylinders to give a new proof
of the fundamental theorem of hypersurface theory. It has the advantage of
being very simple and the result directly extends to semi-Riemannian manifolds
and to embeddings into spaces of constant curvature. We also give a new way to
identify spinors for different metrics and to derive the variation formula for
the Dirac operator. Moreover, we show that generalized Killing spinors for
Codazzi tensors are restrictions of parallel spinors. Finally, we study the
space of Lorentzian metrics and give a criterion when two Lorentzian metrics on
a manifold can be joined in a natural manner by a 1-parameter family of such
metrics.Comment: 29 pages, 2 figure
Twisted-mass QCD, O(a) improvement and Wilson chiral perturbation theory
We point out a caveat in the proof for automatic O(a) improvement in twisted
mass lattice QCD at maximal twist angle. With the definition for the twist
angle previously given by Frezzotti and Rossi, automatic O(a) improvement can
fail unless the quark mass satisfies m_q >> a^2 Lambda_QCD^3. We propose a
different definition for the twist angle which does not require a restriction
on the quark mass for automatic O(a) improvement. In order to illustrate
explicitly automatic O(a) improvement we compute the pion mass in the
corresponding chiral effective theory. We consider different definitions for
maximal twist and show explicitly the absence or presence of the leading O(a)
effect, depending on the size of the quark mass.Comment: 27 pages, no figure
Generic metrics and the mass endomorphism on spin three-manifolds
Let be a closed Riemannian spin manifold. The constant term in the
expansion of the Green function for the Dirac operator at a fixed point is called the mass endomorphism in associated to the metric due to
an analogy to the mass in the Yamabe problem. We show that the mass
endomorphism of a generic metric on a three-dimensional spin manifold is
nonzero. This implies a strict inequality which can be used to avoid
bubbling-off phenomena in conformal spin geometry.Comment: 8 page
Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations
Pattern formation often occurs in spatially extended physical, biological and
chemical systems due to an instability of the homogeneous steady state. The
type of the instability usually prescribes the resulting spatio-temporal
patterns and their characteristic length scales. However, patterns resulting
from the simultaneous occurrence of instabilities cannot be expected to be
simple superposition of the patterns associated with the considered
instabilities. To address this issue we design two simple models composed by
two asymmetrically coupled equations of non-conserved (Swift-Hohenberg
equations) or conserved (Cahn-Hilliard equations) order parameters with
different characteristic wave lengths. The patterns arising in these systems
range from coexisting static patterns of different wavelengths to traveling
waves. A linear stability analysis allows to derive a two parameter phase
diagram for the studied models, in particular revealing for the Swift-Hohenberg
equations a co-dimension two bifurcation point of Turing and wave instability
and a region of coexistence of stationary and traveling patterns. The nonlinear
dynamics of the coupled evolution equations is investigated by performing
accurate numerical simulations. These reveal more complex patterns, ranging
from traveling waves with embedded Turing patterns domains to spatio-temporal
chaos, and a wide hysteretic region, where waves or Turing patterns coexist.
For the coupled Cahn-Hilliard equations the presence of an weak coupling is
sufficient to arrest the coarsening process and to lead to the emergence of
purely periodic patterns. The final states are characterized by domains with a
characteristic length, which diverges logarithmically with the coupling
amplitude.Comment: 9 pages, 10 figures, submitted to Chao
Dirac-harmonic maps from index theory
We prove existence results for Dirac-harmonic maps using index theoretical
tools. They are mainly interesting if the source manifold has dimension 1 or 2
modulo 8. Our solutions are uncoupled in the sense that the underlying map
between the source and target manifolds is a harmonic map.Comment: 26 pages, no figur
The Dirac operator on untrapped surfaces
We establish a sharp extrinsic lower bound for the first eigenvalue of the
Dirac operator of an untrapped surface in initial data sets without apparent
horizon in terms of the norm of its mean curvature vector. The equality case
leads to rigidity results for the constraint equations with spherical boundary
as well as uniqueness results for constant mean curvature surfaces in Minkowski
space.Comment: 16 page
Pion scattering in Wilson ChPT
We compute the scattering amplitude for pion scattering in Wilson chiral
perturbation theory for two degenerate quark flavors. We consider two different
regimes where the quark mass m is of order (i) a\Lambda_QCD^2 and (ii)
a^2\Lambda_QCD^3. Analytic expressions for the scattering lengths in all three
isospin channels are given. As a result of the O(a^2) terms the I=0 and I=2
scattering lengths do not vanish in the chiral limit. Moreover, additional
chiral logarithms proportional to a^2\ln M_{\pi}^2 are present in the one-loop
results for regime (ii). These contributions significantly modify the familiar
results from continuum chiral perturbation theory.Comment: 20 pages, 4 figures. V3: Comments on finite size effects and the
axial vector current added, one more reference. To be published in PR
Applying chiral perturbation to twisted mass Lattice QCD
We have explored twisted mass LQCD (tmLQCD) analytically using chiral
perturbation theory, including discretization effects up to O(a^2), and working
at next-to-leading (NLO) order in the chiral expansion. In particular we have
studied the vacuum structure, and calculated the dependence of pion masses and
decay constants on the quark mass, twisting angle and lattice spacing. We give
explicit examples for quantities that both are and are not automatically
improved at maximal twisting.Comment: 3 pages, 1 figure. Talk given at Lattice2004(spectrum), Fermi
National Accelerator Laboratory, June 21 - 26, 2004. v2: Minor typos fixed,
slight page format adjustment for generating 3 page postscript at the arXiv.
v3: Change to meta-data field only. No change to actual pape
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
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