1,188 research outputs found
The algebra of Wick polynomials of a scalar field on a Riemannian manifold
On a connected, oriented, smooth Riemannian manifold without boundary we
consider a real scalar field whose dynamics is ruled by , a second order
elliptic partial differential operator of metric type. Using the functional
formalism and working within the framework of algebraic quantum field theory
and of the principle of general local covariance, first we construct the
algebra of locally covariant observables in terms of equivariant sections of a
bundle of smooth, regular polynomial functionals over the affine space of the
parametrices associated to . Subsequently, adapting to the case in hand a
strategy first introduced by Hollands and Wald in a Lorentzian setting, we
prove the existence of Wick powers of the underlying field, extending the
procedure to smooth, local and polynomial functionals and discussing in the
process the regularization ambiguities of such procedure. Subsequently we endow
the space of Wick powers with an algebra structure, dubbed E-product, which
plays in a Riemannian setting the same role of the time ordered product for
field theories on globally hyperbolic spacetimes. In particular we prove the
existence of the E-product and we discuss both its properties and the
renormalization ambiguities in the underlying procedure. As last step we extend
the whole analysis to observables admitting derivatives of the field
configurations and we discuss the quantum M{\o}ller operator which is used to
investigate interacting models at a perturbative level.Comment: 35 page
Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory
The perturbative approach to nonlinear Sigma models and the associated
renormalization group flow are discussed within the framework of Euclidean
algebraic quantum field theory and of the principle of general local
covariance. In particular we show in an Euclidean setting how to define Wick
ordered powers of the underlying quantum fields and we classify the freedom in
such procedure by extending to this setting a recent construction of Khavkine,
Melati and Moretti for vector valued free fields. As a by-product of such
classification, we prove that, at first order in perturbation theory, the
renormalization group flow of the nonlinear Sigma model is the Ricci flow.Comment: 24 page
Unified Scenario for Composite Right-Handed Neutrinos and Dark Matter
We entertain the possibility that neutrino masses and dark matter (DM)
originate from a common composite dark sector. A minimal effective theory can
be constructed based on a dark interaction with three flavors of
massless dark quarks; electroweak symmetry breaking gives masses to the dark
quarks. By assigning a charge to one flavor, a stable "dark kaon"
can provide a good thermal relic DM candidate. We find that "dark neutrons" may
be identified as right handed Dirac neutrinos. Some level of
"neutron-anti-neutron" oscillation in the dark sector can then result in
non-zero Majorana masses for light Standard Model neutrinos. A simple
ultraviolet completion is presented, involving additional heavy
-charged particles with electroweak and lepton Yukawa couplings. At
our benchmark point, there are "dark pions" that are much lighter than the
Higgs and we expect spectacular collider signals arising from the UV framework.
This includes the decay of the Higgs boson to ,
where () can be any lepton, with displaced vertices. We discuss
the observational signatures of this UV framework in dark matter searches and
primordial gravitational wave experiments; the latter signature is potentially
correlated with the decay.Comment: 8 pages, 4 figures, 1 table. Version published on PR
Stochastic quantisation of the fractional model in the full subcritical regime
We construct the fractional Euclidean quantum field theory on
in the full subcritical regime via parabolic stochastic quantisation. Our
approach is based on the use of a truncated flow equation for the effective
description of the model at sufficiently small scales and on coercive estimates
for the non-linear stochastic partial differential equation describing the
interacting field.Comment: 64 page
An Algebraic and Microlocal Approach to the Stochastic Non-linear Schr\"odinger Equation
In a recent work [DDRZ20], it has been developed a novel framework aimed at
studying at a perturbative level a large class of non-linear, scalar, real,
stochastic PDEs and inspired by the algebraic approach to quantum field theory.
The main advantage is the possibility of computing the expectation value and
the correlation functions of the underlying solutions accounting for
renormalization intrinsically and without resorting to any specific
regularization scheme. In this work we prove that it is possible to extend the
range of applicability of this framework to cover also the stochastic
non-linear Schroedinger equation in which randomness is codified by an
additive, Gaussian, complex white noise.Comment: 33 pages. Some typos fixed, Section 1 modifie
A Facility Location Model for Air Pollution Detection
We describe mathematical models and practical algorithms for a problem concerned with monitoring the air pollution in a large city. We have worked on this problem within a project for assessing the air quality in the city of Rome by placing a certain number of sensors on some of the city buses. We cast the problem as a facility location model. By reducing the large number of data variables and constraints, we were able to solve to optimality the resulting MILP model within minutes. Furthermore, we designed a genetic algorithm whose solutions were on average very close to the optimal ones. In our computational experiments we studied the placement of sensors on 187 candidate bus routes. We considered the coverage provided by 10 up to 60 sensors
On the stochastic Sine-Gordon model: an interacting field theory approach
We investigate the massive Sine-Gordon model in the finite ultraviolet regime
on the two-dimensional Minkowski spacetime with an
additive Gaussian white noise. In particular we construct the expectation value
and the correlation functions of a solution of the underlying stochastic
partial differential equation (SPDE) as a power series in the coupling
constant, proving ultimately uniform convergence. This result is obtained
combining an approach first devised in [11] to study SPDEs at a perturbative
level with the one discussed in [4] to construct the quantum sine-Gordon model
using techniques proper of the perturbative, algebraic approach to quantum
field theory (pAQFT). At a formal level the relevant expectation values are
realized as the evaluation of suitably constructed functionals over
. In turn, these are elements of a distinguished
algebra whose product is a deformation of the pointwise one, by means of a
kernel which is a linear combination of two components. The first encompasses
the information of the Feynmann propagator built out of an underlying Hadamard,
quantum state, while the second encodes the correlation codified by the
Gaussian white noise. In our analysis, first of all we extend the results
obtained in [3,4] proving the existence of a convergent modified version of the
S-matrix and of an interacting field as elements of the underlying algebra of
functionals. Subsequently we show that it is possible to remove the
contribution due to the Feynmann propagator by taking a suitable -limit, hence obtaining the sought expectation value of the solution and of
the correlation functions of the SPDE associated to the stochastic Sine-Gordon
model.Comment: 48 page
On a Microlocal Version of Young's Product Theorem
A key result in distribution theory is Young's product theorem which states
that the product between two H\"older distributions
and
can be unambiguously defined if
. We revisit the problem of multiplying two H\"older
distributions from the viewpoint of microlocal analysis, using techniques
proper of Sobolev wavefront set. This allows us to establish sufficient
conditions which allow the multiplication of two H\"older distributions even
when .Comment: 18 pages. Section 2 and Section 3 revise
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