22,822 research outputs found
Bordism, rho-invariants and the Baum-Connes conjecture
Let G be a finitely generated discrete group. In this paper we establish
vanishing results for rho-invariants associated to
(i) the spin-Dirac operator of a spin manifold with positive scalar curvature
(ii) the signature operator of the disjoint union of a pair of homotopy
equivalent oriented manifolds with fundamental group G.
The invariants we consider are more precisely
- the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite
dimensional unitary representations.
- the L2-rho invariant of Cheeger-Gromov
- the delocalized eta invariant of Lott for a finite conjugacy class of G.
We prove that all these rho-invariants vanish if the group G is torsion-free
and the Baum-Connes map for the maximal group C^*-algebra is bijective. For the
delocalized invariant we only assume the validity of the Baum-Connes conjecture
for the reduced C^*-algebra.
In particular, the three rho-invariants associated to the signature operator
are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov
rho-invariants the latter result had been established by Navin Keswani. Our
proof re-establishes this result and also extends it to the delocalized
eta-invariant of Lott. Our method also gives some information about the
eta-invariant itself (a much more saddle object than the rho-invariant).Comment: LaTeX2e, 60 pages; the gap pointed out by Nigel Higson and John Roe
is now closed and all statements of the first version of the paper are proved
(with some small refinements
The strange case of Dr. Petit and Mr. Dulong
The Dulong-Petit limiting law for the specific heats of solids, one of the
first general results in thermodynamics, has provided Mendeleev with a powerful
tool for devising the periodic table and gave an important support to
Boltzmann's statistical mechanics. Even its failure at low temperature,
accounted for by Einstein, paved the way to the the quantum mechanical theory
of solids. These impressive consequences are even more surprising if we bear in
mind that, when this law was announced, thermal phenomena were still explained
using Lavoisier's concept of caloric and Dalton's atomic theory was in its
infancy. Recently, however, bitter criticisms charging Dulong and Petit of
`data fabrication' and fraud, have been raised. This work is an attempt to
restore a more balanced view of the work performed by these two great
scientists and to give them back the place they deserve in the framework of the
development of modern science.Comment: Submitted to "Quaderni di Storia della Fisica", SIF (Italian Physical
Society) publishe
Quantum degrees of freedom of a region of spacetime
The holographic principle and the thermodynamics of de Sitter space suggest
that the total number of fundamental degrees of freedom associated with any
finite-volume region of space may be finite. The naive picture of a short
distance cut-off, however, is hardly compatible with the dynamical properties
of spacetime, let alone with Lorentz invariance. Considering the regions of
space just as general ``subsystems'' may help clarifying this problem. In usual
QFT the regions of space are, in fact, associated with a tensor product
decomposition of the total Hilbert space into ``subsystems'', but such a
decomposition is given a priori and the fundamental degrees of freedom are
labelled, already from the beginning, by the spacetime points. We suggest a new
strategy to identify ``localized regions'' as ``subsystems'' in a way which is
intrinsic to the total Hilbert-space dynamics of the quantum state of the
fields.Comment: 6 pages, talk given at the XXVIII Spanish Relativity Meeting, Oviedo
Spain, to appear in the proceeding
Ultrarelativistic electron states in a general background electromagnetic field
The feasibility of obtaining exact analytical results in the realm of QED in
the presence of a background electromagnetic field is almost exclusively
limited to a few tractable cases, where the Dirac equation in the corresponding
background field can be solved analytically. This circumstance has restricted,
in particular, the theoretical analysis of QED processes in intense laser
fields to within the plane-wave approximation even at those high intensities,
achievable experimentally only by tightly focusing the laser energy in space.
Here, within the Wentzel-Kramers-Brillouin (WKB) or eikonal approximation, we
construct analytically single-particle electron states in the presence of a
background electromagnetic field of general space-time structure in the
realistic assumption that the initial energy of the electron is the largest
dynamical energy scale in the problem. The relatively compact expression of
these states opens, in particular, the possibility of investigating
analytically strong-field QED processes in the presence of spatially focused
laser beams, which is of particular relevance in view of the upcoming
experimental campaigns in this field.Comment: 7 pages, 1 figur
Nonlinear Breit-Wheeler pair production in a tightly focused laser beam
The only available analytical framework for investigating QED processes in a
strong laser field systematically relies on approximating the latter as a plane
wave. However, realistic high-intensity laser beams feature much more complex
space-time structures than plane waves. Here, we show the feasibility of an
analytical framework for investigating strong-field QED processes in laser
beams of arbitrary space-time structure by determining the energy spectrum of
positrons produced via nonlinear Breit-Wheeler pair production as a function of
the background field in the realistic assumption that the energy of the
incoming photon is the largest dynamical energy in the problem. A numerical
evaluation of the angular resolved positron spectrum shows significant
quantitative differences with respect to the analogous result in a plane wave,
such that the present results will be also important for the design of upcoming
strong laser facilities aiming at measuring this process.Comment: 6 pages, 1 figur
First-order strong-field QED processes in a tightly focused laser beam
In [Phys. Rev. Lett. \textbf{117}, 213201 (2016)] we have determined the
angular resolved and the total energy spectrum of a positron produced via
nonlinear Breit-Wheeler pair production by a high-energy photon
counterpropagating with respect to a tightly focused laser beam. Here, we first
generalize the results in [Phys. Rev. Lett. \textbf{117}, 213201 (2016)] by
including the possibility that the incoming photon is not exactly
counterpropagating with respect to the laser field. As main focus of the
present paper, we determine the photon angular resolved and total energy
spectrum for the related process of nonlinear Compton scattering by an electron
impinging into a tightly-focused laser beam. Analytical integral expressions
are obtained under the realistic assumption that the energy of the incoming
electron is the largest dynamical energy of the problem and that the electron
is initially almost counterpropagating with respect to the laser field. The
crossing symmetry relation between the two processes in a tightly focused laser
beam is also elucidated.Comment: 24 pages, no figure
Groups with torsion, bordism and rho-invariants
Let G be a discrete group, and let M be a closed spin manifold of dimension
m>3 with pi_1(M)=G. We assume that M admits a Riemannian metric of positive
scalar curvature. We discuss how to use the L2-rho invariant and the
delocalized eta invariant associated to the Dirac operator on M in order to get
information about the space of metrics with positive scalar curvatur1e.
In particular we prove that, if G contains torsion and M is congruent 3 mod 4
then M admits infinitely many different bordism classes of metrics with
positive scalar curvature. We show that this is true even up to diffeomorphism.
If G has certain special properties then we obtain more refined information
about the ``size'' of the space of metric of positive scalar curvature, and
these results also apply if the dimension is congruent to 1 mod 4. For example,
if G contains a central element of odd order, then the moduli space of metrics
of positive scalar curvature has infinitely many components, if it is not
empty.
Some of our invariants are the delocalized eta-invariants introduced by John
Lott. These invariants are defined by certain integrals whose convergence is
not clear in general, and we show, in effect, that examples exist where this
integral definitely does not converge, thus answering a question of Lott.
We also discuss the possible values of the rho invariants of the Dirac
operator and show that there are certain global restrictions (provided the
scalar curvature is positive).Comment: 21 pages; comma in metadata (author field) added. final version to
appear in Pacific Journal of Mathematic
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