Massless Klein-Gordon equation not conformally invariant?

[bcrowell]

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 $\phi$ is not conformally invariant unless you modify it by adding in a term proportional to $R\phi$. 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 $R\phi$ 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?

 

[arkajad]

The mathematical reason for this is the funny fact that EM field is described by a 2-form (antisymmetric covariant tensor of rank 2) and 2 is one half of 4 (spacetime dimension). Then it follows that the scale does not enter Maxwell equations - it cancels out.

On the other hand a conformally invariant Klein-Gordon field, with R/6 term, defines a scale! So, yes, it is conformally invariant and its solutions break conformal invariance - Higgs is around.

But that's mathematics.

 

[bcrowell]

Thanks for the reply, arkajad!

I realized afterward that I muddied the waters by talking about the Higgs, which is a massive field.

On the other hand a conformally invariant Klein-Gordon field, with R/6 term, defines a scale! So, yes, it is conformally invariant and its solutions break conformal invariance [...]

Sorry, I don't follow you here. How does the scale come in?

 

[arkajad]

Well, to explain how the scale comes from it is good to know how to obtain R/6 in an alternative way. Given a metric $$g_{\mu\nu}$$ you can form a tensor density

$$\hat{\gamma}_{\mu\nu}=\frac{g_{\mu\nu}}{|\det\, g_{\mu\nu}|^{1/4}}$$

that is invariant under scaling $$g_{\mu\nu}\rightarrow \lambda^2 g_{\mu\nu}$$

Now take for the scalar field $$\phi$$ a density of the weight half of that of $$|\det g|^{1/4}$$ and define a new metric

$$\tilde{g}_{\mu\nu}=\phi^2 \hat{\gamma}_{\mu\nu}$$

Take the standard Einstein Lagrangian for this auxiliary metric, and do variations with respect to $$\phi$$ only. You get:

$$(\widehat{\Box}+\frac{1}{6}\hat{R})\phi=0$$

where hats refer to the quantities computed from $$\hat{\gamma}_{\mu\nu}$$ - which define angles only but not the scale. This equation depends only on the conformal structure of your original metric.

Thus $$\phi$$ sets the scale. Of course you can take variations of this method, instead of considering densities you can now get rid of the hats by multiplying $$\phi$$ by the appropriate power of the determinant of your original metric.

I know this looks like a clumsy way of looking at things, but it works.

 

[bcrowell]

Interesting! When you talked about defining a scale, I'd thought you meant defining a constant global scale factor, like the mass of the electron or Planck's constant. I guess what you really meant was fixing the conformal factor in one location relative to another, which is essentially the same as fixing the definition of geodesics. So you can sort of think of the conformal factor as a dynamical field propagating on a background where the light-cone geometry is fixed? Aren't there going to be problems in places where $\phi=0$? Are these just coordinate singularities?

 

[arkajad]

Aren't there going to be problems in places where $\phi=0$? Are these just coordinate singularities?

They are true "singularities" but I think they are not deadly dangerous when put in an appropriate context.

 

[samalkhaiat]

Conformal invariance is possible in massless field theories only. However, examples of non-conformal massless theories do exist. Indeed, one can construct massless, spin-zero field theory (from a second-rank antisymmetric tensor field ) and find that it is not a conformal theory even though it is dynamically (on-shell) equivalent to the conformal scalar field theory.
The general rule for a conformal test is the following;
If a massless theory, $S[\Phi]$, can be extended to curved spacetime, $S[\Phi,g]$, in a way consistent with the local Weyl invariance

$$\delta g_{ab} = 2\sigma (x) g_{ab}$$

$$\delta \Phi_{i}= d_{(i)}\sigma (x) \Phi_{i},$$

one finds the action,

$$S[\Phi, g \rightarrow \eta] \equiv S[\Phi],$$

in Minkowski space to be conformally invariant.


Regards

sam