Weyl invariance

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Suppose we have an action that is invariant under general coordinate transformations, i.e., diffeomorphisms, written symbolically as

S[g,\Phi] = \int\!d^dx\, \mathcal{L}(g,x,\Phi,\partial\Phi)\,,

where g_{\mu\nu}\, is the metric and \Phi\, represents some set of fields. We take a Weyl transformation to mean a transformation which replaces the metric g_{\mu\nu}\, by g'_{\mu\nu} = \Omega^2 g_{\mu\nu}\, where the conformal factor \Omega^2 = e^{2\omega}\, represents some local rescaling of distances. Note that we transform neither the coordinates nor the fields. Infinitesimally, then,

g_{\mu\nu} \to \Omega^2 g_{\mu\nu} \approx g_{\mu\nu} + 2\omega g_{\mu\nu}\,.

If S\, is invariant under a Weyl transformation, i.e.,

S[\Omega^2 g,\Omega^{-\Delta}\Phi] = S[g,\Phi]\,,

then we say the theory is Weyl invariant. Weyl invariance is related to, but not the same as, conformal invariance. In particular, Weyl invariance requires that \Phi\, be coupled to gravity.

Stress-energy tensor

One definition of the stress-energy tensor is given by the variation of the action with respect to the metric:

\delta S \equiv \frac{1}{2} \int\!d^dx\, \sqrt{g} T^{\mu\nu}\delta g_{\mu\nu}\,,

or, alternatively,

T^{\mu\nu} \equiv \frac{2}{\sqrt{g}}\frac{\delta S}{\delta g_{\mu\nu}}\,

where g = |\det g_{\mu\nu}|\,. Therefore, if \delta g_{\mu\nu} = 2\omega g_{\mu\nu}\,, then

\delta S \equiv \int\!d^dx\, \sqrt{g} T^{\mu}_\mu \omega\,.

Therefore, with the proper boundary conditions, Weyl invariance requires that T^\mu_\mu = 0\,, i.e., that the stress-energy tensor be traceless.

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