272 research outputs found
A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source
We consider the following geometric optics problem: Construct a system of two
reflectors which transforms a spherical wavefront generated by a point source
into a beam of parallel rays. This beam has a prescribed intensity
distribution. We give a rigorous analysis of this problem. The reflectors we
construct are (parts of) the boundaries of convex sets. We prove existence of
solutions for a large class of input data and give a uniqueness result. To the
author's knowledge, this is the first time that a rigorous mathematical
analysis of this problem is given. The approach is based on optimal
transportation theory. It yields a practical algorithm for finding the
reflectors. Namely, the problem is equivalent to a constrained linear
optimization problem.Comment: 5 Figures - pdf files attached to submission, but not shown in
manuscrip
Shocks and Universal Statistics in (1+1)-Dimensional Relativistic Turbulence
We propose that statistical averages in relativistic turbulence exhibit
universal properties. We consider analytically the velocity and temperature
differences structure functions in the (1+1)-dimensional relativistic
turbulence in which shock waves provide the main contribution to the structure
functions in the inertial range. We study shock scattering, demonstrate the
stability of the shock waves, and calculate the anomalous exponents. We comment
on the possibility of finite time blowup singularities.Comment: 37 pages, 7 figure
Thermal Quantum Fields without Cut-offs in 1+1 Space-time Dimensions
We construct interacting quantum fields in 1+1 dimensional Minkowski space,
representing neutral scalar bosons at positive temperature. Our work is based
on prior work by Klein and Landau and Hoegh-KrohnComment: 48 page
Quantum Sturm-Liouville Equation, Quantum Resolvent, Quantum Integrals, and Quantum KdV : the Fast Decrease Case
We construct quantum operators solving the quantum versions of the
Sturm-Liouville equation and the resolvent equation, and show the existence of
conserved currents. The construction depends on the following input data: the
basic quantum field and the regularization .Comment: minor correction
Fermionization, Convergent Perturbation Theory, and Correlations in the Yang-Mills Quantum Field Theory in Four Dimensions
We show that the Yang-Mills quantum field theory with momentum and spacetime
cutoffs in four Euclidean dimensions is equivalent, term by term in an
appropriately resummed perturbation theory, to a Fermionic theory with nonlocal
interaction terms. When a further momentum cutoff is imposed, this Fermionic
theory has a convergent perturbation expansion. To zeroth order in this
perturbation expansion, the correlation function of generic components
of pairs of connections is given by an explicit, finite-dimensional integral
formula, which we conjecture will behave as \noindent for where is a positive integer depending
on the gauge group In the case where we conjecture that \noindent so that the rate
of decay of correlations increases as Comment: Minor corrections of notation, style and arithmetic errors;
correction of minor gap in the proof of Proposition 1.4 (the statement of the
Proposition was correct); further remark and references adde
On the mixing property for a class of states of relativistic quantum fields
Let be a factor state on the quasi-local algebra of
observables generated by a relativistic quantum field, which in addition
satisfies certain regularity conditions (satisfied by ground states and the
recently constructed thermal states of the theory). We prove that
there exist space and time translation invariant states, some of which are
arbitrarily close to in the weak* topology, for which the time
evolution is weakly asymptotically abelian
Quantum Gowdy Model: Schrodinger Representation with Unitary Dynamics
The linearly polarized Gowdy model is paradigmatic for studying
technical and conceptual issues in the quest for a quantum theory of gravity
since, after a suitable and almost complete gauge fixing, it becomes an exactly
soluble midisuperspace model. Recently, a new quantization of the model,
possessing desired features such as a unitary implementation of the gauge group
and of the time evolution, has been put forward and proven to be essentially
unique. An appropriate setting for making contact with other approaches to
canonical quantum gravity is provided by the Schr\"odinger representation,
where states are functionals on the configuration space of the theory. Here we
construct this functional description, analyze the time evolution in this
context and show that it is also unitary when restricted to physical states,
i.e. states which are solutions to the remaining constraint of the theory.Comment: 21 pages, version accepted for publication in Physical Review
Fock representations from U(1) holonomy algebras
We revisit the quantization of U(1) holonomy algebras using the abelian C*
algebra based techniques which form the mathematical underpinnings of current
efforts to construct loop quantum gravity. In particular, we clarify the role
of ``smeared loops'' and of Poincare invariance in the construction of Fock
representations of these algebras. This enables us to critically re-examine
early pioneering efforts to construct Fock space representations of linearised
gravity and free Maxwell theory from holonomy algebras through an application
of the (then current) techniques of loop quantum gravity.Comment: Latex file, 30 pages, to appear in Phys Rev
Quantum field theory on manifolds with a boundary
We discuss quantum theory of fields \phi defined on (d+1)-dimensional
manifold {\cal M} with a boundary {\cal B}. The free action W_{0}(\phi) which
is a bilinear form in \phi defines the Gaussian measure with a covariance
(Green function) {\cal G}. We discuss a relation between the quantum field
theory with a fixed boundary condition \Phi and the theory defined by the Green
function {\cal G}. It is shown that the latter results by an average over \Phi
of the first. The QFT in (anti)de Sitter space is treated as an example. It is
shown that quantum fields on the boundary are more regular than the ones on
(anti) de Sitter space.Comment: The version to appear in Journal of Physics A, a discussion on the
relation to other works in the field is adde
Path Integral Approach to the Non-Relativistic Electron Charge Transfer
A path integral approach has been generalized for the non-relativistic
electron charge transfer processes. The charge transfer - the capture of an
electron by an ion passing another atom or more generally the problem of
rearrangement collisions is formulated in terms of influence functionals. It
has been shown that the electron charge transfer process can be treated either
as electron transition problem or as elastic scattering of ion and atom in the
some effective potential field. The first-order Born approximation for the
electron charge transfer cross section has been reproduced to prove the
adequacy of the path integral approach for this problem.Comment: 19 pages, 1 figure, to appear in Journal of Physics B: Atomic,
Molecular & Optical, vol.34, 200
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