3,156 research outputs found
Relational time in generally covariant quantum systems: four models
We analize the relational quantum evolution of generally covariant systems in
terms of Rovelli's evolving constants of motion and the generalized Heisenberg
picture. In order to have a well defined evolution, and a consistent quantum
theory, evolving constants must be self-adjoint operators. We show that this
condition imposes strong restrictions to the choices of the clock variables. We
analize four cases. The first one is non- relativistic quantum mechanics in
parametrized form. We show that, for the free particle case, the standard
choice of time is the only one leading to self-adjoint evolving constants.
Secondly, we study the relativistic case. We show that the resulting quantum
theory is the free particle representation of the Klein Gordon equation in
which the position is a perfectly well defined quantum observable. The
admissible choices of clock variables are the ones leading to space-like
simultaneity surfaces. In order to mimic the structure of General Relativity we
study the SL(2R) model with two Hamiltonian constraints. The evolving constants
depend in this case on three independent variables. We show that it is possible
to find clock variables and inner products leading to a consistent quantum
theory. Finally, we discuss the quantization of a constrained model having a
compact constraint surface. All the models considered may be consistently
quantized, although some of them do not admit any time choice such that the
equal time surfaces are transversal to the orbits.Comment: 18 pages, revtex fil
The Newton Polytope of the Implicit Equation
We apply tropical geometry to study the image of a map defined by Laurent
polynomials with generic coefficients. If this image is a hypersurface then our
approach gives a construction of its Newton polytope.Comment: 18 pages, 3 figure
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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