In this paper we will present tha main features of what can be called Schwinger's foundational approach to Quantum Mechanics. The basic ingredients formulation are the \textit{selective measurements}, whose algebraic composition rules define a mathematical structure groupoid, which is associated with any physical system. After introduction axioms concepts observables and states, statistical interpretation evolution derived. An example finally introduced support theoretical description approach.

The groupoid description of Schwinger’s picture quantum mechanics is continued by discussing the closely related notions composition systems, subsystems, and their independence. Physical subsystems have a neat algebraic as subgroupoids system. offers two natural systems: Direct free products groupoids, that will be analyzed in depth well universal character. Finally, notion independence reviewed, finding usual independence, find realm formalism. ideas described this paper illustrated using...

In this paper we shall consider the stratified manifold of quantum states and vector fields which act on it. particular, show that infinitesimal generator GKLS evolution is composed a unitary transformations plus gradient field along with Kraus transversal to strata defined by involutive distribution generated former ones.

The search for a potential function $S$ allowing to reconstruct given metric tensor $g$ and symmetric covariant $T$ on manifold $\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with canonically defined Lagrangian $T\mathcal{M}$. connection between this problem, geometric structure of space pure states quantum mechanics, theory contrast functions classical information geometry outlined.

In this paper, we review some geometrical aspects pertaining to the world of monotone quantum metrics in finite dimensions. Particular emphasis is given an unfolded perspective for states that built out spectral theorem and naturally suited investigate comparison with classical case probability distributions.

We show that the space of observables test particles has a natural Jacobi structure which is manifestly invariant under action Poincaré group. Poisson algebras may be obtained by imposing further requirements. A generalization Peierls procedure used to extend this bracket time-like geodesics on Minkowski spacetime.

The basic ingredients of the groupoid interpretation Quantum Mechanics are discussed. Starting with motivation to consider measure groupoids as appropriate statistical and kinematical setting describe quantum mechanical systems, algebra observables system is constructed von Neumann groupoid. theory for classical systems briefly analysed. As a final application these ideas, Feynman's sum-over-histories discussed in picture.

This paper contains a set of lecture notes on manifolds with boundary and corners, particular attention to the space quantum states. A geometrically inspired way dealing these kind is presented, explicit examples are given in order clearly illustrate main ideas.

Using the recently developed groupoidal description of Schwinger's picture Quantum Mechanics, a new approach to Dirac's fundamental question on role Lagrangian in Mechanics is provided. It shown that function $\ell$ groupoid configurations (or kinematical groupoid) quantum system determines state von Neumann algebra histories system. This function, which we call {\itshape q-Lagrangian}, can be described terms $\mathcal{L}$ Lie algebroid theory. When pair smooth manifold $M$, quadratic...

A novel derivation of Feynman’s sum-over-histories construction the quantum propagator using groupoidal description Schwinger picture Quantum Mechanics is presented. It shown that such corresponds to GNS representation a natural family states called Dirac–Feynman–Schwinger (DFS) states. Such are obtained from q-Lagrangian function [Formula: see text] on groupoid configurations system. The histories system constructed and allows us define DFS state algebra groupoid. particular instance pairs...

It is well known that certain features of a quantum theory cannot be described in the standard picture on Hilbert space. In particular, this happens when we try to formally frame field theory, or thermodynamic system with finite density. This forces us introduce different types algebras, more general than ones usually encounter course mechanics. We show how these algebras naturally arise Schwinger description mechanics an infinite spin chain. use machinery Dirac-Feynman-Schwinger (DFS)...

The formulation of covariant brackets on the space solutions to a variational problem is analyzed in framework contact geometry. It argued that Poisson algebra functionals fields should be read as subalgebra within an functions equipped with Jacobi bracket suitable manifold.

Schr\"{o}dinger's famous Gedankenexperiment involving a cat is used as motivation to discuss the evolution of states composition classical and quantum systems in groupoid formalism for physical theories introduced recently. It shown that notion system, sense Birkhoff von Neumann, equivalent, case with countable number outputs, totally disconnected Abelian Neumann algebra. In accordance Raggio's theorem, impossibility evolving product state composite system made up one into an entangled by...