I Why is Scalar Massless Wave Equation Conformally Invariant?

user1139
Messages
71
Reaction score
8
TL;DR Summary
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?
 
Last edited:
Physics news on Phys.org
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
 
Thomas1 said:
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.
 
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
I started reading a National Geographic article related to the Big Bang. It starts these statements: Gazing up at the stars at night, it’s easy to imagine that space goes on forever. But cosmologists know that the universe actually has limits. First, their best models indicate that space and time had a beginning, a subatomic point called a singularity. This point of intense heat and density rapidly ballooned outward. My first reaction was that this is a layman's approximation to...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...
Back
Top