Quantum Equivalence of $f(R)$ Gravity and Scalar-tensor Theories in the Jordan and Einstein Frames

Abstract

The $f(R)$ gravity and scalar-tensor theory are known to be equivalent at the classical level. We study if this equivalence is valid at the quantum level. There are two descriptions of the scalar-tensor theory in the Jordan and Einstein frames. It is shown that these three formulations of the theories give the same determinant or effective action on shell, and thus they are equivalent at the quantum one-loop level on shell in arbitrary dimensions. We also compute the one-loop divergence in $f(R)$ gravity.

15 pages, v2: minor corrections, refs added

References in corpus (13)

• Ultraviolet properties of f(R)-Gravity
• On the renormalization group flow of f(R)-gravity
• Asymptotic safety in the f(R) approximation
• Search of scaling solutions in scalar-tensor gravity
• The Renormalization Group flow of unimodular f(R) gravity
• Physical non-equivalence of the Jordan and Einstein frames
• Field Parametrization Dependence in Asymptotically Safe Quantum Gravity
• Renormalization Group Equation for $f(R)$ gravity on hyperbolic spaces
• Gauges and functional measures in quantum gravity I: Einstein theory
• Scaling solutions for Dilaton Quantum Gravity
• Frame Covariant Nonminimal Multifield Inflation
• Gauges and functional measures in quantum gravity II: Higher derivative gravity
• Meaning of the field dependence of the renormalization scale in Higgs inflation

Cited by in corpus (32)

• Selected topics in scalar-tensor theories and beyond
• Thermodynamical correspondence of $f(R)$ gravity in Jordan and Einstein frames
• Frame Covariance in Quantum Gravity
• Frame-invariant approach to higher-dimensional scalar-tensor gravity
• Consistent early and late time cosmology from the RG flow of gravity
• Higgs-$R^2$ inflation -- full slow-roll study at tree-level
• Non-minimal (self-)running inflation: metric vs. Palatini formulation
• Distinguishing Jordan and Einstein frames in gravity through entanglement
• Affine quantisation of the Brans-Dicke theory: Smooth bouncing and the equivalence between the Einstein and Jordan frames
• Multiple point criticality principle and Coleman-Weinberg inflation
• On the number of $e$-folds in the Jordan and Einstein frames
• Scale-invariant inflation with one-loop quantum corrections
• Inflation with $R_{(\alpha \beta )}$ terms in the Palatini formulation
• Dynamical equivalence of $f(R)$ gravity in Jordan and Einstein frames
• $f(R,R_{\mu \nu }^2)$ at one loop
• $f\left(R,{\nabla }_{{\mu }_1}R,\dots ,{\nabla }_{{\mu }_1}\dots {\nabla }_{{\mu }_n}R\right)$ theories of gravity in Einstein frame: A higher order modified Starobinsky inflation model in the Palatini approach
• Higgs field in cosmology
• Canonical Analysis of Brans-Dicke Theory Addresses Hamiltonian Inequivalence between Jordan and Einstein Frames
• Equivalence between Scalar-Tensor theories and $f(R)$-gravity: From the action to Cosmological Perturbations
• Towards a unitary, renormalizable and ultraviolet-complete quantum theory of gravity
• Disformal invariance of cosmological observables
• Ultraviolet Unitarity Violations in Non-minimally Coupled Scalar-Starobinsky Inflation
• Renormalization Group for Non-minimal ${\varphi }^{2}R$ Couplings and Gravitational Contact Interactions
• Naked singularities in quadratic f(R) gravity
• Conformal and kinetic couplings as two Jordan frames of the same theory
• Quantum gravitational corrections to the inflationary power spectra in scalar-tensor theories
• Jordan and Einstein Frames from the perspective of $\omega =-3/2$ Hamiltonian Brans-Dicke theory
• Einstein and Jordan frame correspondence in quantum cosmology: Expansion-collapse duality
• Equivalence of scalar-tensor theories and scale-dependent gravity
• Asymptotically Schwarzschild solutions in $f(R)$ extension of general relativity
• Ward identities under the frame transformations in curved spacetime
• What can we learn from the conformal noninvariance of the Klein-Gordon equation?