Lucrezia Ravera

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.

We propose a first-order geometric Lagrangian for four-dimensional conformal gravity within the Cartan formulation, which yields, dynamically, the standard constraints on the fields, expected for conformal gravity. Upon imposing the dynamical constraints, together with the request of conformal invariance of the off-shell Lagrangian, the theory reduces to the standard expression for conformal gravity, in terms of quadratic curvature invariants. Our results clarify the geometric status of conformal gravity as a gauge theory and open the way to a similar dynamical implementation of the constraints in higher dimensions and supersymmetric extensions.

The ultrarelativistic (Carrollian) regime of gravity has recently emerged as a fertile framework for exploring holography, non-Lorentzian symmetries, and geometric limit of general relativity. In this Letter, we establish the presence of a nonvanishing torsion within three-dimensional Carrollian gravity by constructing the Carrollian Mielke-Baekler (C-MB) gravity theory in its Chern-Simons formulation, obtained as the ultrarelativistic limit of the relativistic Mielke-Baekler model. The resulting C-MB theory features nonzero temporal torsion and curvature, together with spatial curvature, providing the most general three-dimensional Carrollian gravity model with these properties. Temporal torsion affects nonaffinity of null generators and boundary dynamics. Several known ultrarelativistic gravity theories arise as particular limits of this framework, highlighting its unifying character.

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.

We propose a first-order geometric Lagrangian for four-dimensional conformal gravity within the Cartan formulation, which yields, dynamically, the standard constraints on the fields, expected for conformal gravity. Upon imposing the dynamical constraints, together with the request of conformal invariance of the off-shell Lagrangian, the theory reduces to the standard expression for conformal gravity, in terms of quadratic curvature invariants. Our results clarify the geometric status of conformal gravity as a gauge theory and open the way to a similar dynamical implementation of the constraints in higher dimensions and supersymmetric extensions.

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.

A bundle geometric formulation of non‐relativistic many‐particles Quantum Mechanics is presented. A wave function is seen to be a ‐valued cocyclic tensorial 0‐form on configuration space‐time seen as a principal bundle, while the Schrödinger equation flows from its covariant derivative, with the action functional supplying a (flat) cocyclic connection 1‐form on the configuration bundle. In line with the historical motivations of Dirac and Feynman, ours is thus a Lagrangian geometric formulation of QM, in which the Dirac–Feynman path integral arises in a geometrically natural way. Applying the dressing field method , we obtain a relational reformulation of this geometric non‐relativistic QM: a relational wave function is realised as a basic cocyclic 0‐form on the configuration bundle. In this relational QM, any particle position can be used as a dressing field, i.e., as a “physical reference frame.” The dressing field method naturally accounts for the freedom in choosing the dressing field, which is readily understood as a covariance of the relational formulation under changes of physical reference frame.

We present a manifestly diffeomorphism-invariant simple model of galaxy dynamics obtained by applying the Dressing Field Method (DFM) to a general-relativistic system comprising the metric and four scalar fields, phenomenologically representing the four-velocity of a cosmological fluid or dust field. The DFM, a systematic tool for extracting the gauge-invariant content in general-relativistic theories, provides a physical coordinatization that yields corrective terms to the rotational velocity profile. These corrections produce galaxy rotation curves that combine a Keplerian and a constant velocity terms, effectively emulating a Dark Matter contribution. We compare DFM-derived rotation curves to observed data for spiral galaxies, from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database, showing that the DFM allows to fit them well.