Restricted Weyl invariance in four-dimensional curved spacetime

Abstract

We discuss the physics of {\it restricted Weyl invariance}, a symmetry of dimensionless actions in four dimensional curved space time. When we study a scalar field nonminimally coupled to gravity with Weyl(conformal) weight of $-1$ (i.e. scalar field with the usual two-derivative kinetic term), we find that dimensionless terms are either fully Weyl invariant or are Weyl invariant if the conformal factor $\Omega (x)$ obeys the condition $g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }\Omega =0$. We refer to the latter as {\it restricted Weyl invariance}. We show that all the dimensionless geometric terms such as $R^2$, $R_{\mu \nu }{R}^{\mu \nu }$ and $R_{\mu \nu \sigma \tau }{R}^{\mu \nu \sigma \tau }$ are restricted Weyl invariant. Restricted Weyl transformations possesses nice mathematical properties such as the existence of a composition and an inverse in four dimensional space-time. We exemplify the distinction among rigid Weyl invariance, restricted Weyl invariance and the full Weyl invariance in dimensionless actions constructed out of scalar fields and vector fields with Weyl weight zero.

arXiv comment19 pages, no figures, references added, to appear in Phys. Rev. D

References in corpus (8)

• On Renormalization Group Flows in Four Dimensions
• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• What Maxwell Theory in D<>4 teaches us about scale and conformal invariance
• Scale and conformal invariance in field theory: a physical counterexample
• CP-violating CFT and trace anomaly
• Generalised Scale Invariant Theories
• Local Couplings and Sl(2,R) Invariance for Gauge Theories at One Loop
• Inflationary behavior of $R^2$ gravity in a conformal framework

Cited by in corpus (24)

• Scale-Invariant Quadratic Gravity and Inflation in the Palatini Formalism
• Palatini formulation of pure $R^2$ gravity yields Einstein gravity with no massless scalar
• Weyl versus Conformal Invariance in Quantum Field Theory
• Traversable Morris-Thorne-Buchdahl wormholes in quadratic gravity
• Generating Einstein gravity, cosmological constant and Higgs mass from restricted Weyl invariance
• Inflation in Weyl-invariant Einstein-Cartan gravity
• Gravitating magnetic monopole via the spontaneous symmetry breaking of pure $R^2$ gravity
• Critical gravity from four dimensional scale invariant gravity
• Exact quantum conformal symmetry, its spontaneous breakdown, and gravitational Weyl anomaly
• Anomaly-free scale symmetry and gravity
• On Restricted Weyl Symmetry
• Quantum Scale Invariant Gravity with de Donder Gauge
• Note on Weyl versus Conformal Invariance in Field Theory
• Quadratic Gravity and Restricted Weyl Symmetry
• Restricted Weyl Symmetry and Spontaneous Symmetry Breakdown of Conformal Symmetry
• Spacetime and Planck mass generation from scale-invariant degenerate gravity
• Higgs Potential from Weyl Conformal Gravity
• Critical reassessment of the restricted Weyl symmetry
• Emerging Newtonian potential in pure $R^2$ gravity on a de Sitter background
• Cancelling Weyl anomaly from position dependent coupling
• A mechanism to generate varying speed of light via Higgs-dilaton coupling: Theory and cosmological applications
• Dynamically induced spin-2 mass in a Weyl-invariant framework
• CFTs on curved spaces
• Non-triviality of asymptotically flat Buchdahl-inspired metrics in pure $R^2$ gravity