Covariant De-Donder Weyl Formalism

• Covariant De-Donder Weyl Formalism is a geometric framework that unifies scale invariance and gauge theories in gravity using a Weyl-covariant derivative.
• It employs quadratic curvature invariants and a dynamic Weyl gauge field to derive manifestly covariant equations of motion and conservation laws.
• The formalism facilitates spontaneous symmetry breaking, bridging high-energy conformal phases with Einstein gravity and offering insights into quantum cosmology.

The Covariant De-Donder Weyl Formalism is a geometric and operator framework for field theory and gravity, distinguished by its manifest spacetime covariance and its capacity to incorporate not only the metric but also local Weyl (scale/dilatation) symmetries within a gauge-theoretic setting. By blending the multisymplectic or polysymplectic Hamiltonian formulations with Weyl geometry, the theory unifies the treatment of conformal invariance, canonical quantization, and higher-derivative (quadratic) dynamical equations for gravity. This framework serves as a foundation for constructing manifestly covariant equations of motion, conservation laws, and a consistent approach to quantization—crucial for advances in gravitational physics and quantum field theory.

1. Weyl Gauge Covariant Formulation and Fundamental Geometric Structures

The Weyl gauge covariant formalism regards conformal geometry as an active gauge theory of the Weyl group, locally gauging dilatations alongside Poincaré symmetries. The fundamental dynamical objects are the metric $g_{\mu\nu}$, the Weyl gauge field $w_\mu$ (or equivalently $A_\mu$), and, as necessary, compensating scalars associated with mass-generating mechanisms. The key geometric construct is the Weyl gauge covariant derivative $V$, defined to act on any tensor with Weyl scaling weight $w$ as

$V_\mu T = \nabla_\mu T - w\,w_\mu T,$

where $\nabla_\mu$ is the Levi-Civita or affine connection. Through the inclusion of $w_\mu$, parallel transport defined by $V_\mu$ preserves the length of vectors, ensuring a "metric" connection ($Vg = 0$) in a physical sense and avoiding the non-metricity issues (such as the "second clock effect") intrinsic to original Weyl geometry proposals.

The field strengths associated with local dilatations (dilatation curvature, e.g., $$F_{\mu\nu} = \partial_\mu w_\nu - \partial_\nu w_\mu$$) and the quadratic invariants constructed from the Weyl tensor are central ingredients in action principles.

2. Covariant Equations of Motion and Conservation Laws

Employing the most general quadratic action—built from curvature invariants such as $R^2$, $(Weyl)_{\mu\nu\rho\sigma}^2$, and $F_{\mu\nu}^2$—variation yields field equations that are explicit in their Weyl-gauge-covariant structure. For example, the generalized Einstein tensor (denoted $W^{\text{grav}}$ or $W^{\mu\nu}$) incorporates contributions from all quadratic operators: $W^{\text{grav}} = 0.$ The variation with respect to the Weyl gauge field leads to a current conservation law: $V_\mu j^\mu = 0,$ where $j^\mu$ stands for the Weyl (dilatation) Noether current.

A central result is the dual validity of these conservation laws: both the energy-momentum tensor and the dilatation current satisfy conservation under the Weyl covariant derivative $V$ in the full gauge picture, and under the Levi-Civita connection in the Riemannian context once on-shell. Moreover, the trace of the generalized Einstein tensor is directly proportional to the divergence of the Weyl current: $2\,\text{tr}(W^{\text{grav}}) = V_\mu j^\mu,$ encapsulating the trace anomaly structure in a manifestly gauge-covariant language (Condeescu et al., 21 Dec 2024).

3. Comparison with Riemannian (Metric) Formulation

In the Weyl-covariant approach, diffeomorphisms are gauged in tandem with local scale transformations, so that all geometrical and dynamical quantities transform appropriately under the full symmetry. When recast into the equivalent Riemannian (metric) formulation, the Weyl gauge field appears as an extra dynamical "matter" field, coupled non-minimally to gravity. Standard diffeomorphism invariance is preserved, and Weyl invariance emerges as a hidden or spontaneously broken symmetry. Conservation laws remain valid under the Riemannian connection, a direct consequence of the underlying gauged diffeomorphism invariance present in the Weyl picture (Condeescu et al., 21 Dec 2024).

4. Dimensional Extension and Analytical Continuation

While much of the formalism is developed in four dimensions (where, e.g., the Gauss–Bonnet topological invariant simplifies the structure of equations), the theory is analytically continued to arbitrary $d$ dimensions. Ultraviolet regulator scales commonly appearing in field theory (such as those introduced in dimensional regularization) are traded for covariant expressions involving the (Weyl-covariant) scalar curvature, such as $R^{(d-4)/2}$, ensuring that all couplings retain proper Weyl weight and that gauge invariance is manifest across dimensions. The structure of the equations of motion and conservation laws generalizes accordingly.

5. Gauge Fixing, Spontaneous Symmetry Breaking, and Connection to Einstein Gravity

In the unbroken phase, the theory is fully scale invariant with no manifest mass scales. The Planck scale and cosmological constant arise only after the Weyl symmetry is spontaneously broken: the Weyl gauge field acquires a mass via an analog of the Higgs–Stueckelberg mechanism (absorbing a scalar dilaton), and the action reduces to Einstein–Hilbert gravity with additional massive vector and (potentially) higher-curvature corrections. This dynamical generation of mass scales links the high-energy conformal phase with observable low-energy gravitational dynamics and offers a structurally geometric solution to hierarchy problems (Condeescu et al., 21 Dec 2024).

6. Physical Implications and Applications

The manifestly Weyl gauge covariant formalism has several physical applications:

• Ultraviolet Completion: Quadratic gravity theories of this form are power-counting renormalizable and, in some versions, asymptotically free.
• Inflationary Cosmology: The $R^2$ sector is connected to Starobinsky-like inflation, and the scale invariance leads to distinctive predictions for early-universe phenomenology.
• Dark Matter Phenomenology: The massive Weyl gauge field and emergent geometric structures in the spontaneously broken phase could provide a geometric foundation for modifications of gravity at galactic scales.
• Conformally Invariant Gauge-Fixing: The structure of the Eastwood–Singer gauge for Maxwell theory and De Donder gauge for gravity finds a natural home in the Weyl-covariant setting, ensuring invariance under both diffeomorphisms and dilatations (Faci, 2011).
• Anomaly Structure: The manifestly gauge-covariant formulation allows consistent computation and tracking of Weyl and diffeomorphism anomalies, critical for quantum consistency.

7. Synthesis and Broader Context

This instantiation of the Covariant De-Donder Weyl Formalism recasts gravity as a gauge theory of the Weyl group, uniting scale invariance, diffeomorphism invariance, and metric covariance. Its formalism is robust across dimensions and supports unbroken and spontaneously broken phases. Conservation laws and dynamical equations hold covariantly in both Weyl and Riemannian geometries, reflecting a deep symmetry structure. The framework underpins current approaches to high-energy gravity, quantum cosmology, and geometric unification, providing the geometric and operator-theoretic foundation for the paper of scale-invariant field theories and quantum gravity (Condeescu et al., 21 Dec 2024).