Jordan Francois

We treat the mechanics of point particles as a one-dimensional general-relativistic gauge field theory, which may be referred to as mechanical field theory (MFT), exploiting the bundle geometry of mechanical field space (MFS). The diffeomorphism covariance of MFT encodes its relational character, arising—as in all general-relativistic physics—via the conjunction of a hole and a point-coincidence argument. Any putative “boundary problem,” meaning the claim that “spacetime” boundaries break diffeomorphism and gauge symmetries, thereby dissolves. It is highlighted that the standard path integral (PI) on the MFS, the exact analog of the PI used in gauge field theory, is conceptually and technically from the standard PI of quantum mechanics. We then use the dressing field method to give a manifestly invariant and relational reformulation of MFT, which reproduces the standard textbook formulation when a clock field is chosen as a (natural) dressing field. The dressed, or basic, PI on the MFS, defining i.e., the quantization of invariant relational degree of freedom—is shown to reproduce the standard PI of quantum mechanics. This establishes the soundness of relational quantization as a general guiding principle: We outline it for general-relativistic gauge field theories.

Transformer models contain substantial internal redundancy arising from coordinate-dependent representations and continuous symmetries, in model space and in head space, respectively. While recent approaches address this by explicitly breaking symmetry, we propose a complementary framework based on symmetry reduction. We reformulate representations, attention mechanisms, and optimization dynamics in terms of invariant relational quantities, eliminating redundant degrees of freedom by construction. This perspective yields architectures that operate directly on relational structures, providing a principled geometric framework for reducing parameter redundancy and analyzing optimization.

Réduction des symétries de jauges : une nouvelle approche géométrique: Le principe de symétrie locale, ou symétrie de jauge, est à la base de notre compréhension des interactions fondamentales. Le language naturelle des théories de jauge est la théorie des connections sur les espaces fibrés, une branche de la géométrie différentielle. En dépit de son importance, la symétrie de jauge pose deux difficultés qui méritent d'être mises en exergue: 1) L'invariance de jauge interdit les termes de masses pour les champs d'interactions, ce qui est en conflit avec la phénoménologie de l'interaction faible. 2) La quantification des théories de jauge est délicate puisque l'intégrale fonctionnelle est a priori mal définie. La symétrie de jauge doit donc être réduite. Essentiellement trois stratégies se présentent, répondant à l'un ou l'autre des deux problèmes. Le fixage de jauge répond à 2 (méthode de Faddeev-Popov). La brisure spontanée de symétrie répond à 1 (méchanisme de Higgs). Enfin, le théorème de réduction des fibrés répond à 1.On propose ici une nouvelle stratégie de réduction des symétries de jauge: la méthode du `dressing field'. C'est un résultat de géométrie différentielle qui se trouve être à la base de la notion de `variables de Dirac'. On montre que cette méthode éclaire certains travaux récents en physique hadronique. Le secteur électrofaible du Modèle Standard est traité ce qui induit une nouvelle interprétation. L'extension de la méthode aux G-structure d'ordre supérieur, ainsi qu'une application à la géométrie conforme, est donnée. Enfin on montre comment la méthode modifie l'algèbre BRS d'une théorie de jauge, et une analyse préliminaire de son impact sur la question des anomalies en Théorie Quantique des Champs est proposée.

In general relativity and gauge field theory, one often encounters a claim, which may be called the , according to which “boundaries break diffeomorphism and gauge symmetries”. We argue that this statement has the same conceptual structure as the , and is thus likewise defused by the : We show that the boundary problem dissolves once it is understood that a , thus its boundary, is and . This insight can be technically implemented via the dressing field method, a systematic tool to exhibit the gauge-invariant content of general-relativistic gauge field theories, whereby physical field-theoretical degrees of freedom co-define each other and define, coordinatize, the . We illustrate our claim with a simple application to the case of general relativity.

We reassess foundational aspects of Metric-Affine Gravity (MAG) in light of the Dressing Field Method, a tool allowing to systematically build gauge-invariant field variables. To get MAG started, one has to deal with the problem of “gauge translations”. We first recall that Cartan geometry is the proper mathematical foundation for gauge theories of gravity, and that this problem never arises in that framework, which still allows to clarify the geometric status of gauge translations. Then, we show how the MAG kinematics is obtained via dressing in a technically streamlined way, which highlights that it reduces to a Cartan-geometric kinematics.

We reconstrue , a notion introduced by Alvarez-Valenzuela-Zanelli, as an attempt to use the framework of supersymmetric field theory to describe fermionic matter fields and bosonic gauge fields in a unified way, as parts of a single superconnection. It hinges upon the so-called . Unfortunately, the formal and conceptual status of the ansatz has remained unclear, preventing unconventional supersymmetry to be used in a principled way as a general approach beyond the model in which it was first considered. In this paper, we lift this restriction by showing that the ansatz is a special case of the , a new systematic tool to exhibit the gauge-invariant content of general-relativistic gauge field theories.

It is recalled how relationality arises as the core insight of general‐relativistic gauge field theories from the articulation of the generalized hole and point‐coincidence arguments. Hence, a compelling case for a manifestly relational framework ensues naturally. A formulation for such a framework is proposed, based on a significant development of the dressing field method of symmetry reduction. A version for the group of automorphisms of a principal bundle over a manifold is first developed, as it is the most natural and elegant, and as hosts all the mathematical structures relevant to general‐relativistic gauge field theory. However, as the standard formulation is local, on , the relational framework for local field theory is then developed. The generalized point‐coincidence argument is manifestly implemented, whereby the physical field‐theoretical degrees of freedoms co‐define each other and define, coordinatize, the physical spacetime itself. Applying the framework to General Relativity, relational Einstein equations are obtained, encompassing various notions of “scalar coordinatization” à la Kretschmann–Komar and Brown–Kuchař.

The group of vertical diffeomorphisms of a principal bundle forms the 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, induced 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 of 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 a bundle. Identities relating it to the usual operations of Cartan calculus — inner product, exterior and (Nijenhuis-) Lie derivative — are immediately read as special cases of general results. We also consider 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.

The dressing field method (DFM) is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff ( M ) -invariant objects, which has a natural relational interpretation. Its precise formulation relies on a clear understanding of the bundle geometry of field space. By detailing it, among other things we stress the geometric nature of field-independent and field-dependent diffeomorphisms, and highlight that the heuristic ‘extended bracket’ for field-dependent vector fields often featuring in the covariant phase space literature can be understood as arising from the Frölicher-Nijenhuis bracket. Furthermore, by articulating this bundle geometry with the covariant phase space approach, we give a streamlined account of the elementary objects of the (pre)symplectic structure of a Diff ( M ) -theory: Noether charges and their bracket, as induced by the standard prescription for the presymplectic potential and 2-form. We give conceptually transparent expressions allowing to read the integrability conditions and the circumstances under which the bracket of charge is Lie, and the resulting Poisson algebras of charges are central extensions of the Lie algebras of field-independent ( diff ( M ) ) and field-dependent vector fields. We show that, applying the DFM, one obtains a Diff ( M ) -invariant and manifestly relational formulation of a general relativistic field theory. Relying on results just mentioned, we easily derive the ‘dressed’ (relational) presymplectic structure of the theory. This reproduces or extends results from the gravitational edge modes and gravitational dressings literature. In addition to simplified technical derivations, the conceptual clarity of the framework supplies several insights and allows us to dispel misconceptions.