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Alberto Ibort

Publications and source records attributed to Alberto Ibort.

At least 37 records · Page 2Linked to original sources
arXiv2021

Symmetries and Covariant Poisson brackets on pre-symplectic manifolds

Noticing that the space of the solutions of a first order Hamiltonian field theory has a pre-symplectic structure, we describe a class of conserved charges on it associated to the momentum map determined by any symmetry group of transformations. Gauge theories are dealt with by using a symplectic regularization based on an application of Gotay's coisotropic embedding theorem. The analysis of Electrodynamics and of the Klein-Gordon theory illustrates the main results of the theory as well as the emergence of the energy-momentum tensor algebra of conserved currents.

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arXiv2021

Feynman's Propagator in Schwinger's picture of Quantum Mechanics

A novel derivation of Feynman's sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac-Feynman-Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function $\ell$ on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian $\ell$ allow to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman's original derivation of the propagator for a point particle described by a classical Lagrangian $L$.

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arXiv2021

A quantum route to the classical Lagrangian formalism

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

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arXiv2021

Schwinger's picture of quantum mechanics: 2-groupoids and symmetries

Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced.It is shown that, given a groupoid $G\rightrightarrows Ω$ associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global".The microscopic point of view leads to the introduction of an additional layer over the grupoid $G$, giving rise to a suitable algebraic structure of 2-groupoid.On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid.The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.

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arXiv2020

Schrödinger's problem with cats: measurements and states in the Groupoid Picture of Quantum Mechanics

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

quant-ph↗
arXiv2020

Representation of non-semibounded quadratic forms and orthogonal additivity

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail.

math.FA↗
arXiv2020

Covariant reduction of classical Hamiltonian Field Theories: From D'Alembert to Klein-Gordon and Schrödinger

A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a submanifold of the manifold of solutions of the equations describing a field theory. Then, the covariance of the geometrical objects involved, will allow to define equations of motion on a reduced space. The computation of the canonical geometrical structure is performed neatly by using the geometrical framework provided by the multisymplectic description of covariant field theories. The procedure is illustrated by reducing the D'Alembert theory on a five-dimensional Minkowski space-time to a massive Klein-Gordon theory in four dimensions and, more interestingly, to the Schrödinger equation in 3+1 dimensions.

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arXiv2020

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 Schrödinger equation.

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arXiv2020

Covariant Variational Evolution and Jacobi Brackets: Particles

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

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arXiv2020

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.

quant-ph↗
arXiv2020

Schwinger's picture of Quantum Mechanics

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

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arXiv2019

Groupoids and Coherent states

Schwinger's algebra of selective measurements has a natural interpretation in terms of groupoids. This approach is pushed forward in this paper to show that the theory of coherent states has a natural setting in the framework of groupoids. Thus given a quantum mechanical system with associated Hilbert space determined by a representation of a groupoid, it is shown that any invariant subset of the group of invertible elements in the groupoid algebra determines a family of generalized coherent states provided that a completeness condition is satisfied. The standard coherent states for the harmonic oscillator as well as generalized coherent states for f-oscillators are exemplified in this picture.

quant-ph↗
arXiv2019

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $C^{*}$-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $C^{*}$-algebra $\mathscr{A}$ which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $\mathscr{S}$ of a possibly infinite-dimensional, unital $C^{*}$-algebra $\mathscr{A}$ is partitioned into the disjoint union of the orbits of an action of the group $\mathscr{G}$ of invertible elements of $\mathscr{A}$. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $\mathcal{H}$ are smooth, homogeneous Banach manifolds of $\mathscr{G}=\mathcal{GL}(\mathcal{H})$, and, when $\mathscr{A}$ admits a faithful tracial state $τ$ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $τ$ is a smooth, homogeneous Banach manifold for $\mathscr{G}$.

math-ph↗
arXiv2019

Descriptions of Relativistic Dynamics with World Line Condition

In this paper a generalized form of relativistic dynamics is presented: A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in $\mathbb{R}^4$. The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincaré algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the ``eleventh generator'' formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many proposals which evaded the no-interaction theorem.

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arXiv2019

Schwinger's Picture of Quantum Mechanics III: The statistical interpretation

Schwinger's algebra of selective measurements has a natural interpretation in the formalism of groupoids. Its kinematical foundations, as well as the structure of the algebra of observables of the theory, was presented in two previous papers (arXiv:1905.12274 and arXiv:1907.03883). In this paper, a closer look to the statistical interpretation of the theory is taken and it is found that an interpretation in terms of Sorkin's quantum measure emerges naturally. It is proven that a suitable class of states of the algebra of virtual transitions of the theory allows to define quantum measures by means of the corresponding decoherence functionals. Quantum measures satisfying a reproducing property are described and a class of states, called factorizable states, possessing the Dirac-Feynman `exponential of the action' form are characterized. Finally, Schwinger's transformation functions are interpreted similarly as transition amplitudes defined by suitable states. The simple examples of the qubit and the double slit experiment are described in detail, illustrating the main aspects of the theory.

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arXiv2019

Schwinger's Picture of Quantum Mechanics II: Algebras and Observables

The kinematical foundations of Schwinger's algebra of selective measurements were discussed in a previous paper (arXiv:1905.12274) and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analysed. For that, the algebra generated by the observables, as well as the notion of state, are dicussed, and the structure of the transition functions, that plays an instrumental role in Schwinger's picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac-Schrödinger and Born-Jordan-Heisenberg pictures is discussed.

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