[Paper Review] Conformal Decomposition of the Effective Action and Covariant Curvature Expansion

A. O. Barvinsky, A. G. Mirzabekian+1 more|ArXiv.org|Oct 18, 1995

Geophysics and Sensor TechnologyEngineering17 citations

TL;DR

This paper presents a conformal decomposition of the one-loop effective action in 4D quantum field theory, separating it into an anomalous part (Riegert action) and a conformal-invariant part $\overline{W}$, using a covariant curvature expansion up to third order. The method fixes gauge freedom via conformal gauge choice and rewrites $\overline{W}$ in a new conformal basis, drastically simplifying its structure and enabling applications in quantum black hole physics and gravitational collapse.

ABSTRACT

The class of effective actions exactly reproducing the conformal anomaly in 4D is considered. It is demonstrated that the freedom within this class can be fixed by the choice of the conformal gauge. The conformal invariant part of the generic one-loop effective action expanded in the covariant series up to third order in the curvature is rewritten in the new conformal basis. The possible applications of the obtained results are discussed.

Motivation & Objective

• To establish a class of conformal decompositions of the one-loop effective action in 4D, fixing gauge freedom via conformal gauge choice.
• To identify the Riegert action as a representative of a broader class of non-local effective actions generating the conformal anomaly.
• To compute the conformal-invariant part $\overline{W}$ of the effective action using covariant curvature expansion up to third order in curvature.
• To reformulate $\overline{W}$ in a new conformal basis, achieving significant simplification of the non-local invariants.
• To explore applications in quantum black hole physics and gravitational collapse, particularly via conformal equivalence to constant-curvature spacetimes near horizons.

Proposed method

• Integrate the conformal anomaly along conformal group orbits to derive the anomalous part $W_{\rm R}$, which is non-local and involves the inverse of a fourth-order differential operator $\mathcal{D}$.
• Use the $\alpha$-representation and spectral integral techniques to convert the covariant curvature expansion into a manifestly conformal basis.
• Apply the conformal gauge condition to fix the residual freedom in the conformal decomposition, ensuring uniqueness.
• Express the conformal-invariant part $\overline{W}$ in terms of Green's functions and inverse operators $1/\Box_i$, with explicit integral representations.
• Utilize algorithms from [8] to convert $\alpha$-representations into spectral, Laplace, and explicit integral forms for $\overline{W}$.
• Derive explicit expressions for the 3-point vertex functions $\overline{\Gamma}_{26}, \overline{\Gamma}_{28}, \overline{\Gamma}_{29}$ in momentum space, involving $\alpha_i$-dependent coefficients and inverse differential operators.

Experimental results

Research questions

• RQ1 How can the one-loop effective action in 4D be decomposed into conformal-anomalous and conformal-invariant parts, and what fixes the freedom in this decomposition?
• RQ2 What is the structure of the conformal-invariant part $\overline{W}$ of the effective action when expanded covariantly up to third order in curvature?
• RQ3 How does rewriting $\overline{W}$ in the new conformal basis simplify its non-local structure compared to the standard covariant expansion?
• RQ4 Can the conformal decomposition method be applied to non-conformal field theories, and under what conditions does it remain valid?
• RQ5 What are the implications of this decomposition for quantum black hole physics, particularly near the horizon where spacetime is conformally equivalent to $R \times H^3$?

Key findings

• The conformal decomposition of the one-loop effective action is uniquely fixed by choosing the conformal gauge, eliminating residual freedom in the decomposition.
• The conformal-invariant part $\overline{W}$ is explicitly computed up to third order in curvature using covariant expansion and re-expressed in a new conformal basis, leading to a significant simplification of the non-local structure.
• The anomalous part $W_{\rm R}$ is shown to be a special case of a broader class of non-local actions generating the conformal anomaly, with the Riegert action being the minimal representative.
• The 3-point vertex functions $\overline{\Gamma}_{26}, \overline{\Gamma}_{28}, \overline{\Gamma}_{29}$ are derived in momentum space with explicit dependence on $\alpha_i$ and inverse differential operators $1/\Box_i$, confirming the structure of $\overline{W}$.
• The conformal basis representation allows for a natural extension to curved spacetimes with constant negative curvature, such as near black hole horizons, where the metric is conformally equivalent to $R \times H^3$.
• The method remains valid even for generic operators $H$, not just conformal-invariant models, broadening its applicability beyond strictly conformal theories.