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José M. Gracia-Bondía

Publications and source records attributed to José M. Gracia-Bondía.

At least 19 recordsLinked to original sources
arXiv2026

Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra

The topology of the Moyal $*$-algebra may be defined in three ways: the algebra may be regarded as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself; or one may construct the $*$-algebra via a filtration of Hilbert spaces (or other Banach spaces) of distributions. We prove the equivalence of the three topologies thereby obtained. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, we give new sufficient conditions for a phase-space function to correspond to a trace-class operator via the Weyl correspondence rule.

math.OA↗
arXiv2026

Algebras of distributions suitable for phase-space quantum mechanics. I

The twisted product of functions on $R^{2N}$ is extended to a $*$-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are investigated. A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product.

math-ph↗
arXiv2025

The full electroweak interaction: an autonomous account

The precise renormalizable interactions in the bosonic sector of electroweak theory are intrinsically determined in the autonomous approach to perturbation theory. This proceeds directly on the Hilbert-Fock space built on the Wigner unirreps of the physical particles, with their given masses: those of three massive vector bosons, a photon, and a massive scalar (the "higgs"). Neither "gauge choices" nor an unobservable "mechanism of spontaneous symmetry breaking" is invoked. Instead, to proceed on Hilbert space requires using string-localized fields to describe the vector bosons. In such a framework, the condition of string independence of the S-matrix yields consistency constraints on the coupling coefficients, the essentially unique outcome being the experimentally known one. The analysis can be largely carried out for other configurations of massive and massless vector bosons, paving the way towards consideration of consistent mass patterns beyond those of the electroweak theory.

hep-th↗
arXiv2024

sQFT: an autonomous explanation of the interactions of quantum particles

Successful applications of a conceptually novel setup of Quantum Field Theory, that accounts for all subtheories of the Standard Model (QED, Electroweak Interaction and Higgs, Yang-Mills and QCD) and beyond (Helicity 2), call for a perspective view in a broader conceptual context. The setting is "autonomous" in the sense of being intrinsically quantum. Its principles are: Hilbert space, Poincaré symmetry and causality. Its free quantum fields are obtained from Wigner's unitary representations of the Poincaré group, with only physical and observable degrees of freedom. A "quantization" of an "underlying" classical theory is not needed. It allows renormalizable perturbation theory with interactions whose detailed structure, and in some cases even the particle content, is predicted by internal consistency. The results confirm and extend observable predictions for the interactions of the SM without assuming a "principle" of gauge invariance.

hep-th↗
arXiv2023

Quantum general covariance

The structure of quantum interactions with fields of helicity two ("gravitons") is strongly constrained by three principles: positivity (Hilbert space), covariance, and locality of observables. To fulfil them simultaneously, some (non-observable) fields need to be non-local. We work with string-localized fields. The results then follow from the condition that entities closely related to observables, like the S-matrix, be local and string-independent. They in particular reproduce the interactions dictated by general covariance in classical field theory. Graviton-matter couplings are consistent only when the graviton self-interaction is taken into account as well.

hep-th↗
arXiv2023

Ensuring locality in QFT via string-local fields

String-local fields constitute a relatively new tool for solving quantum field theory, stressing and embodying locality and positivity. We examine here their usefulness -- as well as some drawbacks. Starting from just the physical masses and charges of the known particles, and bringing string independence in the Bogoliubov-Epstein-Glaser (BEG) theory framework, regarded as a means of discovery, one finds the allowed couplings of quantum fields associated to those particles, and thereby recovers all of the Standard Model (SM) without invoking theoretical prejudices. One of the outcomes is the requirement that the fields be governed by reductive Lie algebras. Another is the need for at least one scalar particle. Yet another is chirality of interactions mediated by massive particles. There is no room in this formulation for ``global'' gauge invariance as an a priori construct. Armed with this modern weapon, we reassess here a few classical and recent conundra. In particular, we examine new perspectives in cosmology from the adoption of string-local fields.

hep-th↗
arXiv2022

Faà di Bruno Hopf algebras

This is a short review on the Faà di Bruno formulas, and some Hopf algebras associated to them. These allow, among several other things, a short proof of the Lie-Scheffers theorem, and relating the Lagrange inversion formulas with antipodes. The link of Faà di Bruno formulas with the theory of set partitions is developed in some detail.

math.CO↗
arXiv2021

Revisiting the Okubo-Marshak argument

Modular localization and the theory of string-localized fields have revolutionized several key aspects of quantum field theory. They reinforce the contention that local symmetry emerges directly from quantum theory, but global gauge invariance remains in general an unwarranted assumption, to be examined case by case. Armed with those modern tools, we reconsider here the classical Okubo-Marshak argument on the non-existence of a "strong CP problem" in quantum chromodynamics.

hep-ph↗
arXiv2019

Density functional theory on phase space

Forty-five years after the point de départ [1] of density functional theory, its applications in chemistry and the study of electronic structures keep steadily growing. However, the precise form of the energy functional in terms of the electron density still eludes us -- and possibly will do so forever [2]. In what follows we examine a formulation in the same spirit with phase space variables. The validity of Hohenberg-Kohn-Levy-type theorems on phase space is recalled. We study the representability problem for reduced Wigner functions, and proceed to analyze properties of the new functional. Along the way, new results on states in the phase-space formalism of quantum mechanics are established. Natural Wigner orbital theory is developed in depth, with the final aim of constructing accurate correlation-exchange functionals on phase space. A new proof of the overbinding property of the Mueller functional is given. This exact theory supplies its home at long last to that illustrious ancestor, the Thomas-Fermi model.

physics.chem-ph↗
arXiv2019

On the kinematics of the last Wigner particle

Wigner's particle classification provides for "continuous spin" representations of the Poincaré group, corresponding to a class of (as yet unobserved) massless particles. Rather than building their induced realizations by use of "Wigner rotations" in the textbooks' way, here we exhibit a scalar-like first-quantized form of those (bosonic) Wigner particles directly, by combining wave equations proposed by Wigner long ago with a recent prequantized treatment employing Poisson structures.

hep-th↗
arXiv2017

The chirality theorem

We show how chirality of the weak interactions stems from string independence in the string-local formalism of quantum field theory.

hep-th↗
arXiv2017

Improved Epstein-Glaser renormalization in $x$-space versus differential renormalization

Renormalization of massless Feynman amplitudes in $x$-space is reexamined here, using almost exclusively real-variable methods. We compute a wealth of concrete examples by means of recursive extension of distributions. This allows us to show perturbative expansions for the four-point and two-point functions at several loop order. To deal with internal vertices, we expound and expand on convolution theory for log-homogeneous distributions. The approach has much in common with differential renormalization as given by Freedman, Johnson and Latorre; but differs in important details.

hep-th↗
arXiv2016

Stora's fine notion of divergent amplitudes

Stora and coworkers refined the notion of divergent quantum amplitude, somewhat upsetting the standard power-counting recipe. This unexpectedly clears the way to new prototypes for free and interacting field theories of bosons of any mass and spin.

math-ph↗
arXiv2013

Quasi-pinning and entanglement in the lithium isoelectronic series

The Pauli exclusion principle gives an upper bound of 1 on the natural occupation numbers. Recently there has been an intriguing amount of theoretical evidence that there is a plethora of additional generalized Pauli restrictions or (in)equalities, of kinematic nature, satisfied by these numbers. Here for the first time a numerical analysis of the nature of such constraints is effected in real atoms. The inequalities are nearly saturated, or quasi-pinned. For rank-six and rank-seven approximations for lithium, the deviation from saturation is smaller than the lowest occupancy number. For a rank-eight approximation we find well-defined families of saturation conditions.

quant-ph↗
arXiv2012

Physical Wigner functions

In spite of their potential usefulness, the characterizations of Wigner functions for Bose and Fermi statistics given by O'Connell and Wigner himself almost thirty years ago has drawn little attention. With an eye towards applications in quantum chemistry, we revisit and reformulate them in a more convenient way.

quant-ph↗
arXiv2012

The lowest excited configuration of harmonium

The harmonium model has long been regarded as an exactly solvable laboratory bench for quantum chemistry [Heisenberg, 1926]. For studying correlation energy, only the ground state of the system has received consideration heretofore. This is a spin singlet state. In this work we exhaustively study the lowest excited (spin triplet) harmonium state, with the main purpose of revisiting the relation between entanglement measures and correlation energy for this quite different species. The task is made easier by working with Wigner quasiprobabilities on phase space.

physics.chem-ph↗