arXiv subjects

Daniel Canarutto

Publications and source records attributed to Daniel Canarutto.

18 recordsLinked to original sources

arXiv2023

On mathematics and knowledge

A sketch of some of the fundamental notions related to the nature of knowledge is offered, with special focus on the role of mathematics and my own opinions. No single idea exposed here is entirely original; indeed, this topic has been explored by legions of philosophers, mathematicians, and scientists throughout history -- I listed a few related books among the references. I look forward to observations and criticisms that may serve to improve this exposition.

math.HO↗

arXiv2020

Gauge field theory without groups

Non-standard topics underlying a partly original approach to gauge field theory are concisely introduced, expressing ideas that were broached in several papers and, eventually, exposed in an organized form in a recently published book. By proposing a change of perspective about the roles and relative importance of several notions, this approach seeks to obtain an overall clarification of foundational matters. In particular, by consistently relying on natural differential geometry, the role of groups is shown to be downgradable to secondary.

math-ph↗

arXiv2018

Covariant-differential formulation of Lagrangian field theory

Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be written as a density on a suitable "covariant prolongation bundle"; the related momenta turn out to be natural vector-valued forms, and the field equations can be expressed in terms of covariant exterior differentials of the momenta. Currents and energy-tensors naturally also fit into this formalism. The examples of bosonic fields and spin one-half fields, interacting with non-Abelian gauge fields, are worked out. The "metric-affine" description of the gravitational field is naturally included, too.

math-ph↗

arXiv2018

A first-order Lagrangian theory of fields with arbitrary spin

The bundles suitable for a description of higher-spin fields can be built in terms of a 2-spinor bundle as the basic `building block'. This allows a clear, direct view of geometric constructions aimed at a theory of such fields on a curved spacetime. In particular, one recovers the Bargmann-Wigner equations and the $2(2j+1)$-dimensional representation of the angular-momentum algebra needed for the Joos-Weinberg equations. Looking for a first-order Lagrangian field theory we argue, through considerations related to the 2-spinor description of the Dirac map, that the needed bundle must be a fibered direct sum of a symmetric `main sector'--carrying an irreducible representation of the angular--momentum algebra|and an induced sequence of `ghost sectors'. Then one indeed gets a Lagrangian field theory that, at least formally, can be expressed in a way similar to the Dirac theory. In flat spacetime one gets plane-wave solutions that are characterised by their values in the main sector. Besides symmetric spinors, the above procedures can be adapted to anti-symmetric spinors and to Hermitian spinors (the latter describing integer-spin fields). Through natural decompositions, the case of a spin-2 field describing a possible deformation of the spacetime metric can be treated in terms of the previous results.

math-ph↗

arXiv2017

On the notions of energy tensors in tetrad-affine gravity

We are concerned with the precise modalities by which mathematical constructions related to energy-tensors can be adapted to a tetrad-affine setting. We show that, for fairly general gauge field theories formulated in that setting, two notions of energy tensor--the canonical tensor and the stress-energy tensor--exactly coincide with no need for tweaking. Moreover we show how both notions of energy-tensor can be naturally extended to include the gravitational field itself, represented by a couple constituted by the tetrad and a spinor connection. Then we examine the on-shell divergences of these tensors in relation to the issue of local energy-conservation in the presence of torsion.

gr-qc↗

arXiv2016

Two-spinor tetrad and Lie derivatives of Einstein-Cartan-Dirac fields

An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Carta-Maxwell-Dirac fields based on "minimal geometric data": all the needed underlying structure is geometrically constructed from the unique assumption of a complex vector field $S\to M$ with 2-dimensional fibers. The Lie derivatives of objects of all considered types, with respect to a vector field $X:M\to TM$, are well-defined without making any special assumption about $X$, and fulfill natural mutual relations.

gr-qc↗

arXiv2016

Overconnections and the energy-tensors of gauge and gravitational fields

A geometric construction for obtaining a prolongation of a connection to a connection of a bundle of connections is presented. This determines a natural extension of the notion of canonical energy-tensor which suits gauge and gravitational fields, and shares the main properties of the energy-tensor of a matter field in the jet space formulation of Lagrangian field theory, in particular with regards to symmetries of the Poincaré-Cartan form. Accordingly, the joint energy-tensor for interacting matter and gauge fields turns out to be a natural geometric object, whose definition needs no auxuliary structures. Various topics related to energy-tensors, symmetries and the Einstein equations in a theory with interacting matter, gauge and gravitational fields can be viewed under a clarifying light. Finally, the symmetry determined by the "Komar superpotential" is expressed as a symmetry of the gravitational Poincaré-Cartan form.

math-ph↗

arXiv2015

Special generalized densities and propagators: a geometric account

Starting from a short review of spaces of generalized sections of vector bundles, we give a concise systematic description, in precise geometric terms, of Leray densities, principal value densities, propagators and elementary solutions of field equations in flat spacetime. We then sketch a partly original geometric presentation of free quantum fields and show how propagators arise from their graded commutators in the boson and fermion cases.

math-ph↗

arXiv2015

On the geometry of ghosts

An inspection of the precise geometric constructions underlying fundamental notions in quantum gauge field theories sheds light on various aspects which tend to be obscured in the usual formalisms. Revising the notions of mutually conjugated "internal" bundles we propose a general rule for the constructions of free quantum fields and their conjugates, naturally yielding the needed fundamental properties with regard to contractions, super-commutators, field momentum and Hamiltonian, and other quantities. This scheme applies to fields of all types; in particular we examine consequences in relation to ghosts and anti-ghosts. Finally in the context of observer-independent Frölicher-smooth quantum bundles we show how the antifield sectors naturally arise, and examine the precise relation among these and the BRST symmetry of a gauge field theory.

math-ph↗

arXiv2014

Natural extensions of electroweak geometry and Higgs interactions

We explore the possibility that the Higgs boson of the standard model be actually a member of a larger family, by showing that a more elaborate internal structure naturally arises from geometrical arguments, in the context of a partly original handling of gauge fields which was put forward in previous papers. A possible mechanism yielding the usual Higgs potential is proposed. New types of point interactions, arising in particular from two-spinor index contractions, are shown to be allowed.

math-ph↗

arXiv2014

Frölicher-smooth geometries, quantum jet bundles and BRST symmetry

We attempt a clarification of geometric aspects of quantum field theory by using the notion of smoothness introduced by Frölicher and exploited by several authors in the study of functional bundles. A discussion of momentum and position representations in curved spacetime, in terms of generalized semi-densities, leads to a definition of quantum configuration bundle which is suitable for a treatment of that kind. A consistent approach to Lagrangian field theories, vertical infinitesimal symmetries and related currents is then developed, and applied to a formulation of BRST symmetry in a gauge theory of the Yang-Mills type.

math-ph↗

arXiv2014

Two-spinor geometry and gauge freedom

Gauge freedom in quantum particle physics is shown to arise in a natural way from the geometry of two-spinors (Weyl spinors). Various related mathematical notions are reviewed, and a special ansatz of the kind "the system defines the geometry" is discussed in connection with the stated results.

math-ph↗

arXiv2014

Nature's software

I bring forward some arguments to support the thesis that nature is fundamentally discrete, and present my own thoughts about the direction in which one could look for a possible, consistent "theory of everything" describing gravitation and quantum particles.

gr-qc↗

arXiv2010

Tetrad gravity, electroweak geometry and conformal symmetry

A partly original description of gauge fields and electroweak geometry is proposed. A discussion of the breaking of conformal symmetry and the nature of the dilaton in the proposed setting indicates that such questions cannot be definitely answered in the context of electroweak geometry.

math-ph↗

arXiv2009

Hermitian vector fields and covariant quantum mechanics of a spin particle

In the context of Covariant Quantum Mechanics for a spin particle, we classify the ``quantum vector fields'', i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the Lie algebra of quantum vector fields is naturally isomorphic to a certain Lie algebra of functions of the classical phase space, called ``special phase functions''. This result provides a covariant procedure to achieve the quantum operators generated by the quantum vector fields and the corresponding observables described by the special phase functions.

math-ph↗

arXiv2008

Fermi transport of spinors and free QED states in curved spacetime

Fermi transport of spinors can be precisely understood in terms of 2-spinor geometry. By using a partly original, previously developed treatment of 2-spinors and classical fields, we describe the family of all transports, along a given 1-dimensional timelike submanifold of spacetime, which yield the standard Fermi transport of vectors. Moreover we show that this family has a distinguished member, whose relation to the Fermi transport of vectors is similar to the relation between the spinor connection and spacetime connection. Various properties of the Fermi transport of spinors are discussed, and applied to the construction of free electron states for a detector-dependent QED formalism introduced in a previous paper.

math-ph↗

arXiv2008

"Minimal geometric data" approach to Dirac algebra, spinor groups and field theories

The three first sections contain an updated, not-so-short account of a partly original approach to spinor geometry and field theories introduced by Jadczyk and myself; it is based on an intrisic treatment of 2-spinor geometry in which the needed background structures do not need to be assumed, but rather arise naturally from a unique geometric datum: a vector bundle with complex 2-dimensional fibres over a real 4-dimensional manifold. The two following sections deal with Dirac algebra and 4-spinor groups in terms of two spinors, showing various aspects of spinor geometry from a different perspective. The last section examines particle momenta in 2-spinor terms and the bundle structure of 4-spinor space over momentum space.

math-ph↗

arXiv2006

Quantum bundles and quantum interactions

A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles (`quantum bundles') over the spacetime manifold yield a quantum ``formalism'' along any 1-dimensional timelike submanifold (a `detector'); in the flat, inertial case this turns out to reproduce the basic results of the usual quantum field theory, while in general it could be seen as a local, ``linearized'' description of the actual physics.

math-ph↗