arXiv subjects
Publications and source records attributed to Giuseppe Marmo.
We go on with the program started in the companion paper [CDI+] of defining a Poisson bracket structure on the space of solutions of the equations of motion of first order Hamiltonian field theories. The case of non-Abelian gauge theories is addressed by using a suitable version of the coisotropic embedding theorem.
Starting from the canonical symmetroid $\mathcal{S}(G)$ associated with a groupoid $G$, the issue of describing dynamical maps in the groupoidal approach to Quantum Mechanics is addressed. After inducing a Haar measure on the canonical symmetroid $\mathcal{S}(G)$, the associated von-Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid $G$ and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.
In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwinger's picture of Quantum Mechanics in which a groupoid is associated with every quantum system. The attention is focused on spin tomography: In this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system. Using the group of bisections of this groupoid, which are special subsets in one-to-one correspondence with the outcomes, a frame is defined and it is used to prove the validity of the tomographic reconstruction. The special case of the set of outcomes being the set of integers modulo n, with n odd prime, is considered in detail. In this case the subgroup of discrete affine linear transformations, whose graphs are linear subspaces of the groupoid, provides a \textit{quorum} in close analogy with the continuos case.
We study the isospectrality problem for a free quantum particle confined in a ring with a junction, analyzing all the self-adjoint realizations of the corresponding Hamiltonian in terms of a boundary condition at the junction. In particular, by characterizing the energy spectrum in terms of a spectral function, we classify the self-adjoint realizations in two classes, identifying all the families of isospectral Hamiltonians. These two classes turn out to be discerned by the action of parity (i.e. space reflection), which plays a central role in our discussion.
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.
We derive one- and two-dimensional models for classical electromagnetism by making use of Hadamard's method of descent. Low-dimensional electromagnetism is conceived as a specialization of the higher dimensional one, in which the fields are uniform along the additional spatial directions. This strategy yields two independent electromagnetisms in two spatial coordinates, and four independent electromagnetisms in one spatial coordinate.
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 $L$.
Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem -- connecting symmetries with constants of the motion -- within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic 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.
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.
We present a mathematical construction of new quantum information measures that generalize the notion of logarithmic negativity. Our approach is based on formal group theory. We shall prove that this family of generalized negativity functions, due their algebraic properties, is suitable for studying entanglement in many-body systems. Under mild hypotheses, the new measures are computable entanglement monotones. Also, they are composable: their evaluation over tensor products can be entirely computed in terms of the evaluations over each factor, by means of a specific group law. In principle, they might be useful to study separability and (in a future perspective) criticality of mixed states, complementing the role of Rényi's entanglement entropy in the discrimination of conformal sectors for pure states.
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.
In this work it is shown that there is an inherent nonlinear evolution in the dynamics of the so-called generalized coherent states. To show this, the immersion of a classical manifold into the Hilbert space of quantum mechanics is employed. Then one may parametrize the time-dependence of the wave function through the variation of parameters in the classical manifold. Therefore, the immersion allows to consider the so-called principle of analogy, i.e. using the procedures and structures available from the classical setting to employ them in the quantum framework.
We review how an algebraic formulation for the dynamics of a physical system allows to describe a reduction procedure for both classical and quantum evolutions.
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.
We use the general setting for contrast (potential) functions in statistical and information geometry provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. We study the case when the two-form is degenerate and show how in sufficiently regular cases one reduces it to a pseudometric structures. Transversal Levi-Civita connections for Riemannian foliations are generalized to the Lie groupoid/Lie algebroid case.
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.
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.