arXiv subjects

Jordan François

Publications and source records attributed to Jordan François.

At least 19 recordsLinked to original sources

arXiv2026

What Price Fiber Bundle Substantivalism? On How to Avoid Holes in Fibers

On a mathematically foundational level, our most successful physical theories (gauge field theories and general-relativistic theories) are formulated in a framework based on the differential geometry of connections on principal bundles. After reviewing the essentials of this framework, we articulate the generalized hole and point-coincidence arguments, examining how they weight on a substantivalist position towards bundle spaces. This question, then, is considered in light of the Dressing Field Method, which allows a manifestly invariant reformulation of gauge field theories and general-relativistic theories, making their conceptual structure more transparent: it formally implements the point-coincidence argument and thus allows to define (dressed) fields and (dressed) bundle spaces immune to hole-type arguments.

physics.hist-ph↗

arXiv2024

Dressing vs. Fixing: On How to Extract and Interpret Gauge-Invariant Content

There is solid consensus among physicists and philosophers that, in gauge field theory, for a quantity to be physically meaningful or real, it must be gauge-invariant. Yet, every "elementary" field in the Standard Model of particle physics is actually gauge-variant. This has led a number of researchers to insist that new manifestly gauge-invariant approaches must be established. Indeed, in the foundational literature, dissatisfaction with standard methods for reducing gauge symmetries has been expressed: Spontaneous symmetry breaking is deemed conceptually dubious, while gauge fixing suffers the same limitations and is subject to the same criticisms as coordinate choices in General Relativity. An alternative gauge-invariant proposal was recently introduced in the literature, the so-called "dressing field method" (DFM). It is a mathematically subtle tool, and unfortunately prone to be confused with simple gauge transformations, hence with standard gauge~fixings. As a matter of fact, in the physics literature the two are often conflated, and in the philosophy community some doubts have been raised about whether there is any substantial difference between them. Clarifying this issue is of special significance for anyone interested in both the foundational issues of gauge theories and their invariant formulation. It is thus our objective to establish as precisely as possible the technical and conceptual distinctions between the DFM and gauge fixing.

physics.hist-ph↗

arXiv2024

Geometric relational framework for general-relativistic gauge field theories

We remind how relationality arises as the core insight of general-relativistic gauge field theories from the articulation of the generalised hole and point-coincidence arguments. Hence, a compelling case for a manifestly relational framework ensues naturally. We propose our formulation for such a framework, based on a significant development of the dressing field method of symmetry reduction. We first develop a version for the group $\text{Aut}(P)$ of automorphisms of a principal bundle $P$ over a manifold $M$, as it is the most natural and elegant, and as $P$ hosts all the mathematical structures relevant to general-relativistic gauge field theory. Yet, as the standard formulation is local, on $M$, we then develop the relational framework for local field theory. It manifestly implements the generalised point-coincidence argument, whereby the physical field-theoretical degrees of freedoms co-define each other and define, coordinatise, the physical spacetime itself. Applying the framework to General Relativity, we obtain relational Einstein equations, encompassing various notions of "scalar coordinatisation" à la Kretschmann-Komar and Brown-Kuchař.

gr-qc↗

arXiv2024

On the Meaning of Local Symmetries: Epistemic-Ontological Dialectics

We propose our account of the meaning of local symmetries. We argue that the general covariance principle and gauge principle both are principles of democratic epistemic access to the law of physics, leading to ontological insights about the objective nature of spacetime. We further argue that relationality is a core notion of general-relativistic gauge field theory, tacitly encoded by its (active) local symmetries.

physics.hist-ph↗

arXiv2024

Note on the group of vertical diffeomorphisms of a principal bundle, and its relation to the Frölicher-Nijenhuis bracket

The group of vertical diffeomorphisms of a principal bundle forms the generalised action Lie groupoid associated to the bundle. The former is generated by the group of maps with value in the structure group, which is also the group of bisections of the groupoid. The corresponding Lie algebra of general vertical vector fields is generated by maps with value in the Lie algebra of the structure group. The bracket on these maps, induces by the bracket of vertical vector fields, is an ``extended" bracket on gauge parameters: it has been introduced heuristically in physics, notably in the study asymptotic symmetries of gravity. Seeing the set of Lie algebra-valued maps as sections of the action Lie algebroid associated to the bundle, the extended bracket is understood to be a Lie algebroid bracket on those sections. Here, we highlight that this bracket can also be seen to arise from the Frölicher-Nijenhuis bracket of vector-valued differential forms. The benefit of this viewpoint is to insert this extended bracket within the general framework of derivations of forms on the bundle. Identities relating it to usual operations -- inner product, exterior and (Nijenhuis-) Lie derivative -- are immediately read as special cases of general results. We also look at the generalised gauge transformations induced by vertical diffeomorphisms, and discuss their peculiar features. In particular, locally, and contrary to standard gauge transformations arising from vertical bundle automorphisms, they are distinguishable from local gluings when iterated. Yet, the gauge principle still holds.

math-ph↗

arXiv2023

Artificial vs Substantial Gauge Symmetries: a Criterion and an Application to the Electroweak Model

To systematically answer the generalized Kretschmann objection, we propose a mean to make operational a criterion widely recognized as allowing to decide if the gauge symmetry of a theory is artificial or substantial. Our proposition is based on the dressing field method of gauge symmetry reduction, a new simple tool from mathematical physics. This general scheme allows in particular to straightforwardly argue that the notion of spontaneous symmetry breaking is superfluous to the empirical success of the electroweak theory. Important questions regarding the context of justification of the theory then arise.

physics.hist-ph↗

arXiv2021

Gauge Symmetries, Symmetry Breaking, and Gauge-Invariant Approaches

Gauge symmetries play a central role, both in the mathematical foundations as well as the conceptual construction of modern (particle) physics theories. However, it is yet unclear whether they form a necessary component of theories, or whether they can be eliminated. It is also unclear whether they are merely an auxiliary tool to simplify (and possibly localize) calculations or whether they contain independent information. Therefore their status, both in physics and philosophy of physics, remains to be fully clarified. In this overview we review the current state of affairs on both the philosophy and the physics side. In particular, we focus on the circumstances in which the restriction of gauge theories to gauge invariant information on an observable level is warranted, using the Brout-Englert-Higgs theory as an example of particular current importance. Finally, we determine a set of yet to be answered questions to clarify the status of gauge symmetries.

physics.hist-ph↗

arXiv2021

Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions

In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlight the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded regions: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of differential geometric tool of gauge symmetry reduction known as the "dressing field method". Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.

math-ph↗

arXiv2021

Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method

We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two restricting hypothesis. In particular, we derive the general field-dependent gauge transformations of the presymplectic potential and presymplectic 2-form in both cases. We point-out that a generalisation of the standard bundle geometry, called twisted geometry, arises naturally in the study of non-invariant gauge theories (e.g. non-Abelian Chern-Simons theory). These results prove that the well-known problem of associating a symplectic structure to a gauge theory over bounded regions is a generic feature of both classes. The edge modes strategy, recently introduced to address this issue, has been actively developed in various contexts by several authors. We draw attention to the dressing field method as the geometric framework underpinning, or rather encompassing, this strategy. The geometric insight afforded by the method both clarifies it and clearly delineates its potential shortcomings as well as its conditions of success. Applying our general framework to various examples allows to straightforwardly recover several results of the recent literature on edge modes and on the presymplectic structure of general relativity.

math-ph↗

arXiv2020

Twisted Gauge Fields

We propose a generalisation of the notion of associated bundles to a principal bundle constructed via group action cocycles rather than via mere representations of the structure group. We devise a notion of connection generalising Ehresmann connection on principal bundles, giving rise to the appropriate covariant derivative on sections of these twisted associated bundles (and on twisted tensorial forms). We study the action of the group of vertical automorphisms on the objects introduced (active gauge transformations). We also provide the gluing properties of the local representatives (passive gauge transformations). The latter are generalised gauge fields: They satisfy the gauge principle of physics, but are of a different geometric nature than standard Yang-Mills fields. We also examine the conditions under which this new geometry coexists and mixes with the standard one. We show that (standard) conformal tractors and Penrose's twistors can be seen as simple instances of this general picture. We also indicate that the twisted geometry arises naturally in the definition and study of anomalies in quantum gauge field theory.

math-ph↗

arXiv2019

Cartan Connections and Atiyah Lie Algebroids

This work extends previous developments carried out by some of the authors on Ehresmann connections on Atiyah Lie algebroids. In this paper, we study Cartan connections in a framework relying on two Atiyah Lie algebroids based on a $H$-principal fiber bundle $\mathcal{P}$ and its associated $G$-principal fiber bundle $\mathcal{Q} := \mathcal{P} \times_H G$, where $H \subset G$ defines the model for a Cartan geometry. The first main result of this study is a commutative and exact diagram relating these two Atiyah Lie algebroids, which allows to completely characterize Cartan connections on $\mathcal{P}$. Furthermore, in the context of gravity and mixed anomalies, our construction answers a long standing mathematical question about the correct geometrico-algebraic setting in which to combine inner gauge transformations and infinitesimal diffeomorphisms.

math-ph↗

arXiv2018

Dilaton from Tractor and Matter Field from Twistor

Tractors and twistors can be obtained via gauge reduction of the conformal Cartan geometry, thanks to the dressing field method. Perhaps surprisingly, it is possible to reduce the gauge symmetry further still. One can indeed erase Weyl symmetry thanks to the tractor field, in a way reminiscent of how the C^2-scalar field allows to erase SU(2) in the electroweak model. This suggests an alternative to the Weyl or conformal spontaneous symmetry breaking (SSB) that some authors proposed as improvement for Standard Model physics or inflationary cosmology. Here, after gauge reduction via dressing, there remains only the Lorentz gauge symmetry and the twistor field becomes - for all practical purposes - a Dirac spinor field. In a very simple and natural toy model, the latter acquires a mass through Lorentz SSB due to the VEV of the Weyl-invariant tractor field.

math-ph↗

arXiv2018

A classification of global conformal invariants

We provide the full classification, in arbitrary even and odd dimensions, of global conformal invariants, i.e., scalar densities in the spacetime metric and its derivatives that are invariant, possibly up to a total derivative, under local Weyl rescalings of the metric. We use cohomological techniques that have already proved instrumental in the classification of Weyl anomalies in arbitrary dimensions. The approach we follow is purely algebraic and borrows techniques originating from perturbative Quantum Field Theory for which locality is crucial.

math-ph↗

arXiv2016

Tractors and Twistors from conformal Cartan geometry: a gauge theoretic approach I. Tractors

Tractors and Twistors bundles both provide natural conformally covariant calculi on $4D$-Riemannian manifolds. They have different origins but are closely related, and usually constructed bottom-up from prolongation of defining differential equations. We propose alternative top-down gauge theoretic constructions starting from the conformal Cartan bundle and its vectorial and spinorial associated bundles. Our key ingredient is the dressing field method of gauge symmetry reduction, which allows to exhibit tractors and twistors and their associated connections as gauge field of a non-standard kind as far as Weyl rescaling symmetry is concerned. By which we mean that they implement the gauge principle but are of a different geometric nature than the well known differential geometric objects usually underlying gauge theories. We provide the corresponding BRST treatment. The present paper deals with the case of tractors while a companion paper deals with twistors.

math-ph↗

arXiv2016

BRST structure for the mixed Weyl-diffeomorphism residual symmetry

In this paper, we show the compatibility of the so-called "dressing field method", which allows a systematic reduction of gauge symmetries, with the inclusion of diffeomorphisms in the BRST algebra of a gauge theory. The robustness of the scheme is illustrated on two examples where Cartan connections play a significant role. The former is General Relativity, while the latter concerns the second-order conformal structure where one ends up with a BRST algebra handling both the Weyl residual symmetry and diffeomorphisms of spacetime. We thereby provide a geometric counterpart to the BRST cohomological treatment used in~\cite{Boulanger1} in the construction of a Weyl covariant tensor calculus.

math-ph↗

arXiv2015

Weyl gravity and Cartan geometry

We point out that the Cartan geometry known as the second-order conformal structure provides a natural differential geometric framework underlying gauge theories of conformal gravity. We are concerned by two theories: the first one will be the associated Yang-Mills-like Lagrangian, while the second, inspired by~\cite{Wheeler2014}, will be a slightly more general one which will relax the conformal Cartan geometry. The corresponding gauge symmetry is treated within the BRST language. We show that the Weyl gauge potential is a spurious degree of freedom, analogous to a Stueckelberg field, that can be eliminated through the dressing field method. We derive sets of field equations for both the studied Lagrangians. For the second one, they constrain the gauge field to be the `normal conformal Cartan connection'. Finally, we provide in a Lagrangian framework a justification of the identification, in dimension $4$, of the Bach tensor with the Yang-Mills current of the normal conformal Cartan connection, as proved in \cite{Korz-Lewand-2003}.

gr-qc↗

arXiv2015

Residual Weyl symmetry out of conformal geometry and its BRS structure

The conformal structure of second order in $m$-dimensions together with the so-called (normal) conformal Cartan connection, is considered as a framework for gauge theories. The dressing field scheme presented in a previous work amounts to a decoupling of both the inversion and the Lorentz symmetries such that the residual gauge symmetry is the Weyl symmetry. On the one hand, it provides straightforwardly the Riemannian parametrization of the normal conformal Cartan connection and its curvature. On the other hand, it also provides the finite transformation laws under the Weyl rescaling of the various geometric objects involved. Subsequently, the dressing field method is shown to fit the BRS differential algebra treatment of infinitesimal gauge symmetry. The dressed ghost field encoding the residual Weyl symmetry is presented. The related so-called algebraic connection supplies relevant combinations found in the literature in the algebraic study of the Weyl anomaly.

math-ph↗

arXiv2015

Nucleon spin decomposition and differential geometry

In the last few years, the so-called Chen et al. approach of the nucleon spin decomposition has been widely discussed and elaborated on. In this letter we propose a genuine differential geometric understanding of this approach. We mainly highligth its relation to the "dressing field method" we advocated in [C. Fournel, J. François, S. Lazzarini, T. Masson, Int. J. Geom. Methods Mod. Phys. 11, 1450016 (2014)]. We are led to the conclusion that the claimed gauge-invariance of the Chen et al. decomposition is actually unreal.

hep-th↗