Undergrad Why is Scalar Massless Wave Equation Conformally Invariant?

[user1139]

TL;DR

I wish to gain a physical intuitive understanding as to why the scalar massless wave equation is conformally invariant.

It can be shown mathematically that the scalar massless wave equation is conformally invariant. However, doing so is rather tedious and muted in terms of physical understanding. As such, is there a physically intuitive explanation as to why the scalar massless wave equation is conformally invariant?

 

[vanhees71]

It's easier seen in energy-momentum representation, i.e., with
$$\phi(x)=\int_{\mathbb{R}^4} \mathrm{d}^4 p \exp(-\mathrm{i} p_{\mu} x^{\mu}) \tilde{\phi}(p)$$
the solution of the massless wave equation implies that $p_{\mu} p^{\mu}=0$, i.e., $p^{\mu}$ must be light-like.

Obviously the equation is invariant under the Poincare group, i.e., under Lorentz boosts, rotations and translations, but here you only need to preserve the light cone in momentum space not the Minkowski product between all vectors. So besides the Lorentz boosts and rotations (building together the proper orthochronous Lorentz). The corresponding symmetry group mapping light cones to themselves is larger, and that's the conformal group. Wikipedia gives a nice introduction

https://en.wikipedia.org/wiki/Conformal_symmetry

 

[PeterDonis]

is there a physically intuitive explanation as to why the scalar massless wave equation is conformally invariant?

One way to look at it is that a massless field does not have any built-in scale of energy or length; waves of the field of all possible frequencies/wavelengths are possible and "look the same" from the standpoint of the physics of how they propagate. The only invariant about the propagation of the waves is the light cones. So the invariance property you would expect the wave equation to have is the one that only preserves the light cones, i.e., conformal invariance.

A massive field, by contrast, has a built-in energy/length scale, given by its invariant mass. So you would expect its wave equation to only be invariant under the more restricted group of transformations that preserves, not just the light cones, but the invariant mass, i.e., the Poincare group.