Giuseppe Marmo

In this paper we develop a general framework for potentials on Lauritzen manifolds, namely smooth manifolds equipped with a pseudo-Riemannian metric and a pair of conjugate affine connections that may have non-vanishing torsion. We show how the theory of bi-forms accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions in a cohomological framework. Within this formalism, we construct a canonical contrast bi-form on dually curvature-free Lauritzen manifolds and establish its principal structural properties. Several illustrative examples are analysed.

We evolve the analysis presented in [CDI+24], where we approached the problem of defining a Poisson structure on the space of solutions of the equations of motions for point particle dynamics and for the field equations describing the free and sourceless Maxwell field within a first order Hamiltonian approach. In this paper, we consider the space of solutions of Yang - Mills field equations with non-Abelian gauge symmetries. Using the coisotropic embedding theorem in an infinite dimensional setting, we present a Poisson structure on a space which suitably enlarges the pre-symplectic space of classical solutions.

In this work, we show that extending the standard description of space-time symmetries from groups of isometries to the more flexible framework of kinematical groupoids allows for the extension of Wigner’s program to curved space-times. We propose a new definition of elementary particles as irreducible projective representations of the kinematical groupoids supporting the theory. By choosing a natural kinematical groupoid associated with any space-time, called the Wigner groupoid, we demonstrate that such irreducible projective representations are characterized by quantum numbers similar to those characterizing the irreducible projective representations of the Poincaré group. Describing the irreducible projective representations of groupoids poses its own difficulties. To address this, we develop a suitable extension of Mackey’s theory of induced representations of groups, proving that projective representations of transitive Lie groupoids with connected isotropy groups are in one-to-one correspondence with the projective representations of their isotropy groups. The application of these results provides a classification of elementary particles valid for a large class of space-times. This classification largely reproduces Wigner’s standard classification on Minkowski space-time, while a new family of representations emerges, corresponding to massless particles in the presence of a magnetic-like background field.

It is well known that relativistic invariance introduceds strong constraints in the interactions of classical particles. We generalize the non-interaction theorems for Lorentz violating systems which still preserve a maximal subgroup of the Poincaré group. In particular, we analyze the motion of a particle preserving the very special relativity symmetry introduced by Cohen and Glashow. We also discuss the case of the homogeneous subgroup. We extend the analysis for Galilei invariant multiparticle systems and for the dynamics of a particle symmetric under any maximal subgroup of Galilei group, including the homogeneous subgroup.

Contrast functions play a fundamental role in information geometry, providing a means for generating the geometric structures of a statistical manifold: a pseudo-Riemannian metric and a pair of torsion-free conjugate affine connections. Conventional contrast-based approaches become indeed insufficient within settings where torsion is naturally present, such as quantum information geometry. This paper introduces contrast bi-forms, a generalisation of contrast functions that systematically encode metric and connection data, allowing for arbitrary affine connections regardless of torsion. It will be shown that they provide a unified framework for statistical potentials, offering new insights into the inverse problem in information geometry. As an example, we consider teleparallel manifolds, where torsion is intrinsic to the geometry, and show how bi-forms naturally accommodate these structures.

The conditions under which a given manifold [ ] may be given a tangent bundle or a cotangent bundle structure are analyzed. This is an important property arising in different contexts. For instance, in the study of integrability of a given dynamics the existence of alternative compatible structures is very relevant, as well as in the geometric approach to Classical Mechanics. On the other hand in the quantum-to-classical transition, a Weyl system plays an important role for it provides (within the so-called Weyl-Wigner formalism) a description of quantum mechanics on a (symplectic) phase-space [ ]. A Lagrangian subspace [ ] of the (linear) phase space determines thus a maximal set of pairwise commuting unitary operators, which is used to parametrize the quantum states. As the choice of this maximal Abelian set of observables is not unique, the different choices make the phase space to become diffeomorphic to different cotangent bundles [ ] corresponding to different choices for the base manifold (and hence the fibers). These motivating ideas are used to study how to define alternative tangent and/or cotangent bundle structures on a phase space.

It is well known that relativistic invariance introduce strong constraints in the interactions of classical particles. We generalize the non-interaction theorems for Lorentz violating systems which still preserve a subgroup of Poincaré symmetry. In particular we analize the case of very special relativity introduced by Cohen and Glashow. We also extend the analysis for Galilei invariant multiparticle systems and for some anisotropic systems which are still invariant under some maximal subgroups of Galilei group.

If η is a contact form on a manifold M such that the orbits of the Reeb vector field R form a simple foliation F on M, then the presymplectic 2-form d η on M induces a symplectic structure ω on the quotient manifold N = M / F . We call ( M , η ) a contactification of the symplectic manifold ( N , ω ) . First, we present an explicit geometric construction of contactifications of some coadjoint orbits of connected Lie groups. Our construction is a far going generalization of the well-known contactification of the complex projective space C P n − 1 , being the unit sphere S 2 n − 1 in C n , and equipped with the restriction of the Liouville 1-form on C n . Second, we describe a constructive procedure for obtaining contactification in the process of the Marsden–Weinstein–Meyer symplectic reduction and indicate geometric obstructions for the existence of compact contactifications. Third, we show that contactifications provide a nice geometrical tool for a Lagrangian description of Hamiltonian systems on compact symplectic manifolds ( N , ω ) , on which symplectic forms never admit a ‘vector potential’.