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massless Klein-Gordon equation not conformally invariant??
Wald discusses conformal transformations in appendix D. He shows that the source-free Maxwell's equations in four dimensions are conformally invariant, and this makes sense to me, since with photons all you can do is measure the light-cone structure of spacetime, which is conformally invariant. But he also shows that the massless Klein-Gordon equation for a field [itex]\phi[/itex] is not conformally invariant unless you modify it by adding in a term proportional to [itex]R\phi[/itex]. This confuses me. If it's a massless field, then shouldn't the same considerations apply to it as to light? In other words, if you can't build a clock out of light waves, why should you be able to build a clock out of massless scalar waves? In both cases, they propagate along lightlike geodesics...?? What is the physical significance of adding in the [itex]R\phi[/itex] term? Is the idea that this is the only physically correct way to generalize to curved spacetime? What physically motivated examples are there? The Brans-Dicke scalar field? The Higgs?
Wald discusses conformal transformations in appendix D. He shows that the source-free Maxwell's equations in four dimensions are conformally invariant, and this makes sense to me, since with photons all you can do is measure the light-cone structure of spacetime, which is conformally invariant. But he also shows that the massless Klein-Gordon equation for a field [itex]\phi[/itex] is not conformally invariant unless you modify it by adding in a term proportional to [itex]R\phi[/itex]. This confuses me. If it's a massless field, then shouldn't the same considerations apply to it as to light? In other words, if you can't build a clock out of light waves, why should you be able to build a clock out of massless scalar waves? In both cases, they propagate along lightlike geodesics...?? What is the physical significance of adding in the [itex]R\phi[/itex] term? Is the idea that this is the only physically correct way to generalize to curved spacetime? What physically motivated examples are there? The Brans-Dicke scalar field? The Higgs?