Conformal coupling of a scalar field

Definition

On an $n$ -dimensional pseudo-Riemannian manifold with $n\geq3$ , the conformally coupled scalar operator in the convention $\Box_g=-\operatorname{tr}_g\nabla d$ is

L_g=\Box_g+\frac{n-2}{4(n-1)}\operatorname{Scal}_g,

where $\operatorname{Scal}_g$ is the scalar curvature. The conformally coupled massless scalar equation is $L_g\phi=0$ .

If $\widetilde g=\Omega^2g$ for a positive smooth function $\Omega$ , then

L_{\widetilde g}\!\left(\Omega^{-(n-2)/2}\phi\right) =\Omega^{-(n+2)/2}L_g\phi.

This covariance distinguishes the coefficient

\xi_n=\frac{n-2}{4(n-1)}

from the minimally coupled value $\xi=0$ . A nonzero mass term breaks this conformal covariance.

References

Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press, 1994. Publisher record. Relevant: Chapter 4.