615 research outputs found
Dynamical aspects in the Quantizer-Dequantizer formalism
The use of the quantizer-dequantizer formalism to describe the evolution of a
quantum system is reconsidered. We show that it is possible to embed a manifold
in the space of quantum states of a given auxiliary system by means of an
appropriate quantizer-dequantizer system. If this manifold of states is
invariant with respect to some unitary evolution, the quantizer-dequantizer
system provides a classical-like realization of such dynamics, which in general
is non linear. Integrability properties are also discussed. Weyl systems and
generalized coherente states are used as a simple illustration of these ideas.Comment: 15 page
Schwinger's Picture of Quantum Mechanics IV: Composition and independence
The groupoids description of Schwinger's picture of quantum mechanics is
continued by discussing the closely related notions of composition of systems,
subsystems, and their independence. Physical subsystems have a neat algebraic
description as subgroupoids of the Schwinger's groupoid of the system. The
groupoids picture offers two natural notions of composition of systems: Direct
and free products of groupoids, that will be analyzed in depth as well as their
universal character. Finally, the notion of independence of subsystems will be
reviewed, finding that the usual notion of independence, as well as the notion
of free independence, find a natural realm in the groupoids formalism. The
ideas described in this paper will be illustrated by using the EPRB experiment.
It will be observed that, in addition to the notion of the non-separability
provided by the entangled state of the system, there is an intrinsic
`non-separability' associated to the impossibility of identifying the entangled
particles as subsystems of the total system.Comment: 32 pages. Comments are welcome
Covariant Variational Evolution and Jacobi Brackets: Fields
The analysis of the covariant brackets on the space of functions on the
solutions to a variational problem in the framework of contact geometry
initiated in the companion letter Ref.19 is extended to the case of the
multisymplectic formulation of the free Klein-Gordon theory and of the free
Schr\"{o}dinger equation.Comment: 16 page
Covariant Jacobi Brackets for Test Particles
We show that the space of observables of test particles carries a natural
Jacobi structure which is manifestly invariant under the action of the
Poincar\'{e} group. Poisson algebras may be obtained by imposing further
requirements. A generalization of Peierls procedure is used to extend this
Jacobi bracket on the space of time-like geodesics on Minkowski space-time.Comment: 13 pages Submitted to MPL
Geometry from divergence functions and complex structures
Motivated by the geometrical structures of quantum mechanics, we introduce an
almost-complex structure on the product of any parallelizable
statistical manifold . Then, we use to extract a pre-symplectic form and
a metric-like tensor on from a divergence function. These tensors
may be pulled back to , and we compute them in the case of an N-dimensional
symplex with respect to the Kullback-Leibler relative entropy, and in the case
of (a suitable unfolding space of) the manifold of faithful density operators
with respect to the von Neumann-Umegaki relative entropy.Comment: 19 pages, comments are welcome
Lagrangian description of Heisenberg and Landau-von Neumann equations of motion
An explicit Lagrangian description is given for the Heisenberg equation on
the algebra of operators of a quantum system, and for the Landau-von Neumann
equation on the manifold of quantum states which are isospectral with respect
to a fixed reference quantum state.Comment: 13 page
Fixation of genetic variation and optimization of gene expression: The speed of evolution in isolated lizard populations undergoing Reverse Island Syndrome
The ecological theory of island biogeography suggests that mainland populations should be more genetically divergent from those on large and distant islands rather than from those on small and close islets. Some island populations do not evolve in a linear way, but the process of divergence occurs more rapidly because they undergo a series of phenotypic changes, jointly known as the Island Syndrome. A special case is Reversed Island Syndrome (RIS), in which populations show drastic phenotypic changes both in body shape, skin colouration, age of sexual maturity, aggressiveness, and food intake rates. The populations showing the RIS were observed on islets nearby mainland and recently raised, and for this they are useful models to study the occurrence of rapid evolutionary change. We investigated the timing and mode of evolution of lizard populations adapted through selection on small islets. For our analyses, we used an ad hoc model system of three populations: wild-type lizards from the mainland and insular lizards from a big island (Capri, Italy), both Podarcis siculus siculus not affected by the syndrome, and a lizard population from islet (Scopolo) undergoing the RIS (called P. s. coerulea because of their melanism). The split time of the big (Capri) and small (Scopolo) islands was determined using geological events, like sea-level rises. To infer molecular evolution, we compared five complete mitochondrial genomes for each population to reconstruct the phylogeography and estimate the divergence time between island and mainland lizards. We found a lower mitochondrial mutation rate in Scopolo lizards despite the phenotypic changes achieved in approximately 8,000 years. Furthermore, transcriptome analyses showed significant differential gene expression between islet and mainland lizard populations, suggesting the key role of plasticity in these unpredictable environments
Hamilton-Jacobi approach to Potential Functions in Information Geometry
The search for a potential function allowing to reconstruct a given
metric tensor and a given symmetric covariant tensor on a manifold
is formulated as the Hamilton-Jacobi problem associated with a
canonically defined Lagrangian on . The connection between this
problem, the geometric structure of the space of pure states of quantum
mechanics, and the theory of contrast functions of classical information
geometry is outlined.Comment: 16 pages. A discussion on the Kullback-Leibler divergence has been
added. To appear in Journal of Mathematical Physic
A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the - Family
The so-called -z-\textit{R\'enyi Relative Entropies} provide a huge
two-parameter family of relative entropies which includes almost all well-known
examples of quantum relative entropies for suitable values of the parameters.
In this paper we consider a log-regularized version of this family and use it
as a family of potential functions to generate covariant symmetric
tensors on the space of invertible quantum states in finite dimensions. The
geometric formalism developed here allows us to obtain the explicit expressions
of such tensor fields in terms of a basis of globally defined differential
forms on a suitable unfolding space without the need to introduce a specific
set of coordinates. To make the reader acquainted with the intrinsic formalism
introduced, we first perform the computation for the qubit case, and then, we
extend the computation of the metric-like tensors to a generic -level
system. By suitably varying the parameters and , we are able to recover
well-known examples of quantum metric tensors that, in our treatment, appear
written in terms of globally defined geometrical objects that do not depend on
the coordinates system used. In particular, we obtain a coordinate-free
expression for the von Neumann-Umegaki metric, for the Bures metric and for the
Wigner-Yanase metric in the arbitrary -level case.Comment: 50 pages, 1 figur
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