Weyl Geometry: Gauge Theory & Conformal Invariance

• Weyl geometry is a gauge-theoretic extension of Riemannian geometry that introduces a non-metricity one-form, making length path-dependent and encoding local scale invariance.
• It provides the mathematical framework for constructing conformally invariant field equations and gravitational actions using extended affine connections.
• This geometric approach has significant implications across gravitation, cosmology, quantum theory, and unified field models in modern physics.

Weyl geometry is a gauge-theoretic generalization of Riemannian geometry in which the notion of length becomes path-dependent under parallel transport, governed by a non-metricity one-form. Originally introduced by Hermann Weyl in 1918 to unify gravity and electromagnetism, the framework has developed into a central structure in modern mathematical physics, underpinning developments in conformal geometry, gravitation, particle physics, cosmology, and the geometric formulation of quantum theory. Weyl geometry rigorously encodes local scale (dilatation) invariance, allows for an extended space of connections, and supplies the gauge-covariant tools necessary for constructing conformally invariant actions and field equations.

1. Fundamental Structures

A Weyl manifold is a triple $(M, g, \omega)$, where $M$ is a smooth $n$-dimensional manifold, $g$ is a (pseudo-)Riemannian metric, and $\omega$ is a one-form called the Weyl, scale, or gauge connection. The defining geometric property is the non-metricity condition: $$\nabla_\lambda g_{\mu\nu} = -2 \omega_\lambda g_{\mu\nu}$$ for the unique, torsion-free affine connection $\nabla$, the Weyl connection. Under local scale transformations (Weyl gauge transformations): $$g_{\mu\nu} \mapsto e^{2\alpha(x)} g_{\mu\nu}, \qquad \omega_\mu \mapsto \omega_\mu - \partial_\mu \alpha(x)$$ the structure is preserved; metrics related in this manner are said to belong to the same Weyl-conformal class (Scholz, 2011, Wheeler, 2018, Ghilencea, 2024). In any local coordinates, the Weyl connection coefficients are: $$\Gamma^\rho_{\mu\nu} = \{^{\,\rho}_{\mu\nu}\} + \delta^\rho_\mu \omega_\nu + \delta^\rho_\nu \omega_\mu - g_{\mu\nu} \omega^\rho$$ where $\{^{\,\rho}_{\mu\nu}\}$ are the Levi–Civita symbols of $M$0.

The Weyl-covariant derivative acting on a tensor $M$1 of Weyl-weight $M$2 is: $M$3 which satisfies $M$4.

2. Curvature, Non-metricity, and Conformal Structure

The geometric curvature objects in the Weyl setting generalize their Riemannian analogs:

• Riemann–Weyl tensor:

$M$5

• Ricci tensor and scalar:

$M$6

• Weyl field strength (length curvature):

$M$7

• Decomposition (for $M$8):

$M$9

where $n$0 is the Riemannian scalar. All tensors built from the Weyl connection are strictly invariant under Weyl transformations, except for the scalar curvature, which rescales as $n$1 (Wheeler, 2018, Ghilencea, 2024, Scholz, 2019).

The Weyl tensor (the conformally invariant part of the Riemann curvature) retains its conformal invariance, and integrable Weyl geometries—where $n$2 is exact—reduce locally to Riemannian geometry after a suitable gauge transformation.

3. Gauge Principle, Physical Invariance, and Spontaneous Symmetry Breaking

Weyl geometry implements local scale invariance as an abelian gauge symmetry. Observables built from the affine connection or geodesic structure are gauge-invariant. In field theory, this underpins the construction of strictly scale-invariant gravitational actions, notably Weyl quadratic gravity: $n$3 where each term is manifestly Weyl-invariant and the constants $n$4, $n$5, and $n$6 are built from the Weyl connection (Ghilencea, 2024). Spontaneous breaking of Weyl symmetry (e.g., via a Stueckelberg scalar auxiliary field $n$7 linearizing the $n$8 term) generates all relevant mass scales, with Einstein-Hilbert gravity and standard Riemannian structures recovered at low energies.

4. Weyl Geometry and Gravitational, Matter, and Gauge Fields

Weyl geometry provides an architecture for constructing conformally invariant scalar-tensor, scalar, and gauge field actions. For example, the scale-invariant scalar-tensor action in $n$9 dimensions: $g$0 where $g$1 is a scalar of Weyl weight $g$2, and $g$3 is the Weyl-covariant derivative (Scholz, 2011, Scholz, 2019). After gauge fixing, such actions reduce to Brans–Dicke-type scalar-tensor theories with a fixed "frame," and multiple physical "pictures" (Weyl frame, Riemann frame, Einstein frame) become implementable through local gauge choices (Romero et al., 2012, Romero et al., 2011).

Matter couplings, such as Higgs and gauge fields, may be rendered Weyl-covariant by appropriate minimal coupling to the Weyl connection, and the Standard Model (SM) admits a Weyl-invariant embedding (Ghilencea, 2024, Scholz, 2012). The conformal structure naturally incorporates the Higgs kinetic term and its non-minimal coupling.

5. Quantum Corrections, Anomaly Structure, and Quantum Gravity

Weyl geometry accommodates a geometric formulation of quantum theory in which the quantum potential appears as a manifestation of non-metricity. Notably, the geometric re-expression of the de Broglie–Bohm quantum potential, quantum corrections to the Klein–Gordon equation, and modifications to Maxwell equations have been formulated directly in terms of the Weyl one-form (Joseph, 2021, Liang et al., 2023). In three spatial dimensions, the Weyl scalar curvature becomes directly proportional to the quantum potential, facilitating a bridge between geometry and quantum entanglement (Liang et al., 2023).

Within the context of quantum gravity, the Weyl gauge field and its anomalies have been shown to structure the trace anomalies (Weyl anomalies) of boundary quantum field theories in the holographic framework, with the so-called Weyl-obstruction tensors arising in the Fefferman–Graham ambient metric construction. Weyl-invariant regularization schemes are anomaly-free, with the anomaly reproduced when symmetry is spontaneously broken (Jia, 2024, Ghilencea, 2024).

6. Cosmological Solutions and Applications

Weyl geometry allows new homogeneous and isotropic cosmological solutions, including new vacuum and radiation-dominated universes with nontrivial integration constants absent in Riemannian general relativity. In suitable Weyl gauges (e.g., with zero Weyl vector), redshift and cosmological "constants" such as the effective cosmological constant and dark radiation can be interpreted as emerging from geometric structures, with the Weyl gauge field supplying a mechanism for dark matter or dark energy (Berezin et al., 2021, Scholz, 2012). The cosmological redshift admits an interpretation as a purely geometric (scale connection) effect in non-expanding spacetimes.

7. Extensions, Generalizations, and Physical Interpretations

Recent generalizations, such as the generalized Weyl integrable geometry (GWIG), introduce affine bundle morphisms or dark fields, yielding frameworks capable of regularizing classical field singularities and providing singularity-free models for point particles (Sabetghadam, 2020). In such settings, gauge, metric, and connection transformations acquire expanded structural roles, and the dual view of physics as geometry, topology, and statistics emerges from topological-defect and fluctuating-metric perspectives (Tiwari, 2020).

Weyl geometry continues to offer open directions: the geometric origin of masses, mechanisms for anomaly cancellation, embedding quantum interactions fully in affine connection structures, elucidating the dimensionality of space, and probing the physical consequences of local scale invariance in cosmological and particle physics phenomena (Scholz, 2019, Ghilencea, 2024, Liang et al., 2023).

---

Selected Table: Core Structures of Weyl Geometry

Object	Notation/Expression	Transformation Property
Metric	$g$4	$g$5
Weyl 1-form	$g$6	$g$7
Weyl-covariant derivative	$g$8	$g$9 (weight $\omega$0)
Connection coefficients	See above	Strictly invariant under Weyl transformations
Field strength	$\omega$1	Gauge-invariant

---

References

• "Weyl Geometry and Quantum Corrections" (Joseph, 2021)
• "Weyl geometry in late 20th century physics" (Scholz, 2011)
• "Weyl Geometries and Timelike Geodesics" (Fatibene et al., 2011)
• "Weyl Geometry as Characterization of Space-Time" (Poulis et al., 2011)
• "Quantum gravity from Weyl conformal geometry" (Ghilencea, 2024)
• "Weyl geometry" (Wheeler, 2018)
• "Gauging the spacetime metric -- looking back and forth a century later" (Scholz, 2019)
• "Dark Fields do Exist in Weyl Geometry" (Sabetghadam, 2020)
• "Einstein, Weyl and Asymmetric Geometries" (Siqueira, 2018)
• "Constructing conformally invariant equations by using Weyl geometry" (Faci, 2012)
• "On the cosmological solutions in Weyl geometry" (Berezin et al., 2021)
• "Weyl geometry, topology of space-time and reality of electromagnetic potentials, and new perspective on particle physics" (Tiwari, 2020)
• "General Relativity and Weyl Frames" (Romero et al., 2011)
• "Topics in Weyl Geometry and Quantum Anomalies" (Jia, 2024)
• "General Relativity and Weyl Geometry" (Romero et al., 2012)
• "Paving the way for transitions --- a case for Weyl geometry" (Scholz, 2012)
• "Connections between Weyl geometry, quantum potential and quantum entanglement" (Liang et al., 2023)