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The diffeomorphism invariance of scalars is often written as: $$ \phi'(x') = \phi(x).\tag{1}$$ However, while scaling transformation is a type of diffeomorphism, in many places (say Di Francesco, Matthieu and Senechal page 38), you see the following for scalar fields: $$\phi'(\lambda x)=\lambda^{-\Delta} \phi(x).\tag{2.121}$$ This is taken to define the scaling dimension $\Delta$. Aren't these two definitions incongruent unless the scaling dimension is 0? I'm guessing no, I just have trouble seeing what is supposed to be happening here.

EDIT: To clarify a little bit, I don't fully understand why the prime appears in the second equation.

Andreas Christophilopoulos
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2 Answers2

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  1. Eq. (1) is a passive general coordinate transformation of a scalar in GR, or is an active Lorentz transformation of a scalar in SR.

  2. In contrast, an active infinitesimal diffeomorphism generated by a vector field $X$ of a scalar $\phi$ in GR satisfies $${\cal L}_X\phi = X[\phi].$$

  3. Eq. (2.121) is an active scale transformation.

Qmechanic
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I finally figured it out. The first one is just a passive coordinate change, as explained by @Qmechanic in their answer. But the other one is something completely different. It has nothing to do with how the field behaves in different points in spacetime, or with any coordinate change. Equation (2.121) is a symmetry of the action.

To be a bit clearer, DiFrancesco et al. are working in their book with a scale invariant action. What they want to say with this relation is that if you substitute $\phi'(x)$ for $\phi(x)$ in the action functional, it doesn't change:

$$ S[\phi']=S[\phi] $$

That's all there is to it really. In other words, classically one could say that if the field is a solution of the EOM, the transformed one also is.

Andreas Christophilopoulos
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