conformally coupled scalar field

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S = \frac{1}{2}\int\!d^Dx\,\sqrt{-g} \left( g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi - \frac{D-2}{4(D-1)}R \phi^2 \right)\,.

Weyl transformation

Under

g_{\mu\nu} \to \Omega^2 g_{\mu\nu} = e^{2\omega} g_{\mu\nu}\,
\phi \to \Omega^{-\Delta} \phi\,,

where \Delta = \tfrac{D-2}{2}\,,

\frac{1}{2}\int\!d^Dx\,\sqrt{-g} \left( g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi + m^2\phi^2\right)\, \to \frac{1}{2}\int\!d^dx\, \sqrt{-g} \Omega^D \left( \Omega^{-2} g^{\mu\nu} \partial_\mu (\Omega^{-\Delta}\phi) \partial_\nu (\Omega^{-\Delta}\phi) + m^2 \Omega^{-2\Delta}\phi^2\right)\,,
=\frac{1}{2}\int\!d^dx\, \sqrt{-g} \left( g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi + 2\Omega^{d-2-\Delta}g^{\mu\nu} \partial_\mu \phi (\partial_\nu \Omega^{-\Delta})\phi + \Omega^d \Omega^{-2} g^{\mu\nu} \phi^2 \partial_\mu \Omega^{-\Delta} \partial_\nu \Omega^{-\Delta} + m^2 \Omega^{-2\Delta}\phi^2\right)\,,
=\frac{1}{2}\int\!d^dx\, \sqrt{-g} \left( g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi+ \Delta (\nabla^2\omega) \phi^2 + \Delta^2 (\nabla\omega )^2 \phi^2 + m^2 \Omega^{-2\Delta}\phi^2\right)\,,

where we have dropped a surface term. On the other hand, the Ricci scalar transforms as

\frac{D-2}{4(D-1)} \sqrt{-g} R \phi^2 \to \frac{D-2}{4(D-1)} \sqrt{-g} \Omega^{D-2} \left[ R - 2(D-1)\nabla^2 \omega - (D-1)(D-2) (\nabla \omega)^2 \right] \Omega^{-2\Delta} \phi^2\,,

which cancels all but the mass term. Therefore the conformally coupled scalar field is invariant under Weyl transformations if m^2 = 0\,.

References

[1]

  1. ↑ Blau, Steve; Visser, Matt; Wipf, Andreas (1988). "Determinants Of Conformal Wave Operators In Four- Dimensions". Phys. Lett. B209: 209. DOI:10.1016/0370-2693(88)90934-3. 
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