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Alberto Ibort

Publications and source records attributed to Alberto Ibort.

At least 19 recordsLinked to original sources
arXiv2026

A groupoidal approach to quantum reference frames

We develop the kinematical and operator-algebraic foundations of a groupoid-based relational quantum field theory (RQFT) on curved spacetimes. Indeed, the usual group-based quantum reference frame (QRF) formalism is not directly suited to generic curved Lorentzian backgrounds as global symmetry groups are typically absent or too small. We formulate a notion of QRF for a continuous groupoid. This yields a groupoid relativization map and relational observables. We show that a localization limit recovers the ordinary non-relational description. We construct canonical sharp groupoid QRFs, which form the groupoidal counterpart of the ideal group QRFs based on $L^2(G)$. We further prove that the groupoid QRF construction reduces to the standard operational QRF formalism for locally compact groups. The action groupoid QRFs are torsor QRFs only for specific classes of fields of positive operator-valued measures (POVMs), and the torsor relativization map only applies to constant operator fields of system observables. We review the foundations of RQFT in Minkowski spacetime. We prove new results that further link RQFT to Wightman QFT. We show that covariant POVMs are $\mu$-continuous with respect to quasi-invariant $\sigma$-finite positive Borel measures $\mu$. Thus, relational quantum fields can be understood as the smearing of pointwise-defined kernels with respect to the QRF's statistics. We develop RQFT in curved spacetime, where we argue that the correct replacement for the Poincar\'e group is the Poincar\'e groupoid of the spacetime. We also indicate how the framework extends further to internal gauge symmetry and relational gauge-covariant quantum field theory via Atiyah groupoids, providing a first step towards formulating a relational quantum Yang-Mills field theory.

quant-ph↗
arXiv2026

Metric tensors and two-forms in information geometry from the GNS construction

We develop a GNS-based construction of geometric tensors on smooth parametric statistical models over $C^*$-algebras. Since the state space of a $C^*$-algebra is generally not a smooth manifold, the construction does not rely on pulling back tensors from an ambient state manifold. Instead, the GNS Hilbert spaces and their duals are organized into non-locally-trivial Hilbert fibrations over the state space. For models satisfying a compatibility condition expressing derivatives of expectation values as continuous functionals on the realified GNS fibers, each tangent vector admits a canonical dual GNS representative. Pulling back the dual GNS Hermitian product along the corresponding canonical lift produces a Hermitian tensor $K$ on the complexified tangent bundle of the model, whose real and imaginary parts define, under suitable regularity assumptions, a smooth weak Riemannian metric tensor $G$ and a smooth two-form $\Omega$. In finite-dimensional parameter manifolds the metric is, of course, strong. The construction recovers the Fisher--Rao metric in the commutative dominated case, the Fubini--Study geometry for pure states up to the normalization and sign convention imposed by the dual GNS pairing, and the SLD metric for faithful quantum states. In finite-dimensional faithful models, the two-form $\Omega$ is proportional, up to convention, to the expected commutator of the SLD representatives, equivalently to the mean Uhlmann curvature. We show through faithful qubits and displaced thermal states that $\Omega$ need not be closed. For bundle-regular models, the associated fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, while closedness of $\Omega$ on the parameter manifold is controlled by the covariant exterior derivative of the canonical real dual GNS lift.

math-ph↗
arXiv2026

Schwarz maps with symmetry

The theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory: CPTP, PPT and Schwarz maps. First, we develop the general structure that equivariant maps $\Phi:\mathcal A \to \mathcal B$ between $C^\ast$-algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality $\Phi(X^\ast X) \geq \Phi(X)^\ast \Phi(X)$ and form an intermediate class between positive and completely positive maps. We completely classify $U(n)$-equivariant on $M_n(\mathbb C)$ and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker $DU(n)$-equivariance (diagonal unitary symmetry) and for tensor-product symmetries $U(n_1) \otimes U(n_2)$. In each case, the parameter regions where $\Phi$ is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the $U(n)$-equivariant family satisfies $\mathrm{PPT} \iff \mathrm{EB}$, while the $DU(2)$, symmetric $DU(3)$, $U(2) \otimes U(2)$ and $U(2) \otimes U(3)$, families obey the $\mathrm{PPT}^2$ conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the $\mathrm{PPT}^2$ property holds.

quant-ph↗
arXiv2025

Smooth sets of fields: A pedagogical introduction

In order to provide a good categorical setting to the many different spaces of fields arising in the description of physical theories, a pedagogical introduction to the categorical notion of smooth sets is provided and some simple properties of the topos of smooth sets are discussed. The introduction of geometrical structures into such spaces is illustrated via the specific examples of the tangent functor and the variational bicomplex.

math-ph↗
arXiv2025

"Fields" in classical and quantum field theories

The challenges posed by the development of field theories, both classical and quantum, force us to question their most basic and foundational ideas like the role and origin of space-time, the meaning of physical states, etc. Among them the notion of ``field'' itself is notoriously difficult to address. These notes aim to analyze such notion from the perspective offered by the groupoid description of quantum mechanics inspired by Schwinger's picture of quantum mechanics. Then, a natural interpretation of the notion of physical fields as functors among appropriate groupoids will emerge. The domain of a field in this new picture is a groupoid that describes ``test particles'', and its codomain is a groupoid that describes the intrinsic nature of the system being probed. Such a space of functors carries some natural structures, which are best described in a categorical language. Some illustrative examples will be presented that could help clarify the various abstract notions discussed in the text.

math-ph↗
arXiv2025

A groupoidal description of elementary particles

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 \textit{Wigner groupoid}, we demonstrate that such irreducible projective representations are characterized by quantum numbers similar to those characterizing the irreducible projective representations of the Poincar\'e 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.

math-ph↗
arXiv2024

Quantum Tomography and the Quantum Radon Transform

A general framework in the setting of $C^*$-algebras for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, is presented. Given a $C^*$-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on $C^*$-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform. The abstract theory is realized by using dynamical systems, that is, groups represented on $C^*$-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.

quant-ph↗
arXiv2023

Symplectic realizations and Lie groupoids in Poisson Electrodynamics

We define the gauge potentials of Poisson electrodynamics as sections of a symplectic realization of the spacetime manifold and infinitesimal gauge transformations as a representation of the associated Lie algebroid acting on the symplectic realization. Finite gauge transformations are obtained by integrating the sections of the Lie algebroid to bisections of a symplectic groupoid, which form a one-parameter group of transformations, whose action on the fields of the theory is realized in terms of an action groupoid. A covariant electromagnetic two-form is obtained, together with a dual two-form, invariant under gauge transformations. The duality appearing in the picture originates from the existence of a pair of orthogonal foliations of the symplectic realization, which produce dual quotient manifolds, one related with space-time, the other with momenta.

hep-th↗
arXiv2023

The Geometry of the solution space of first order Hamiltonian field theories III: Palatini's formulation of General Relativity

We complete the program started in two companion papers 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 General Relativity is addressed by looking at it as a particular non-Abelian gauge theory in a suitable low-energy limit and via a technique related to the coisotropic embedding theorem.

math-ph↗
arXiv2023

On the categorical foundations of quantum information theory: Categories and the Cramer-Rao inequality

An extension of Cencov's categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theory in a natural way, while also allowing to formalise the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer-Rao inequality in this setting.

quant-ph↗
arXiv2023

Groupoid and algebra of the infinite quantum spin chain

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

quant-ph↗
arXiv2022

The Geometry of the solution space of first order Hamiltonian field theories I: from particle dynamics to free Electrodynamics

We analyse the problem of defining a Poisson bracket structure on the space of solutions of the equations of motions of first order Hamiltonian field theories. The cases of Hamiltonian mechanical point systems (as a (0 + 1)-dimensional field) and more general field theories without gauge symmetries are addressed by showing the existence of a symplectic (and, thus, a Poisson) structure on the space of solutions. Also the easiest case of gauge theory, namely free electrodynamics, is considered: within this problem, a pre-symplectic tensor on the space of solutions is introduced, and a Poisson structure is induced in terms of a flat connection on a suitable bundle associated to the theory.

math-ph↗
arXiv2022

G-dual teleparallel connections in Information Geometry

Given a real, finite-dimensional, smooth parallelizable Riemannian manifold $(\mathcal{N},G)$ endowed with a teleparallel connection $\nabla$ determined by a choice of a global basis of vector fields on $\mathcal{N}$, we show that the $G$-dual connection $\nabla^{*}$ of $\nabla$ in the sense of Information Geometry must be the teleparallel connection determined by the basis of $G$-gradient vector fields associated with a basis of differential one-forms which is (almost) dual to the basis of vector fields determining $\nabla$. We call any such pair $(\nabla,\nabla^{*})$ a $G$-dual teleparallel pair. Then, after defining a covariant $(0,3)$ tensor $T$ uniquely determined by $(\mathcal{N},G,\nabla,\nabla^{*})$, we show that $T$ being symmetric in the first two entries is equivalent to $\nabla$ being torsion-free, that $T$ being symmetric in the first and third entry is equivalent to $\nabla^{*}$ being torsion free, and that $T$ being symmetric in the second and third entries is equivalent to the basis vectors determining $\nabla$ ($\nabla^{*}$) being parallel-transported by $\nabla^{*}$ ($\nabla$). Therefore, $G$-dual teleparallel pairs provide a generalization of the notion of Statistical Manifolds usually employed in Information Geometry, and we present explicit examples of $G$-dual teleparallel pairs arising both in the context of both Classical and Quantum Information Geometry.

math-ph↗
arXiv2022

Dynamical maps and symmetroids

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.

math-ph↗
arXiv2022

Quantum Tomography and Schwinger's Picture of Quantum Mechanics

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.

quant-ph↗
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.

math-ph↗
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 $L$.

math-ph↗