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 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

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

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

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

Geometry from divergence functions and complex structures

Motivated by the geometrical structures of quantum mechanics, we introduce an almost-complex structure $J$ on the product $M\times M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a metric-like tensor on $M\times M$ from a divergence function. These tensors may be pulled back to $M$, 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

Aspects of geodesical motion with Fisher-Rao metric: classical and quantum

The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon's Entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.Comment: 15 pages, 5 figure

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

Hamilton-Jacobi approach to Potential Functions in Information Geometry

The search for a potential function $S$ allowing to reconstruct a given metric tensor $g$ and a given symmetric covariant tensor $T$ on a manifold $\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with a canonically defined Lagrangian on $T\mathcal{M}$. 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