= Scaling dimension
{title2=$\Delta$}
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The scaling dimension of a local <local field operator> specifies how it rescales under a <dilation>. To avoid mixing active and passive conventions, write a scalar transformation as $x'=\lambda x$, $\mathcal O'(x')=\lambda^{-\Delta}\mathcal O(x)$, where $\Delta$ is the scaling dimension.
Consequently, an identical-scalar <two-point function> in a <conformal field theory> has distance dependence $|x-y|^{-2\Delta}$ up to normalization. The classical dimension follows from the <action>'s units; quantum effects can add an <anomalous dimension>. Scaling dimensions also classify deformations: coupling an operator of dimension $\Delta$ in $d$ dimensions gives a coupling of dimension $d-\Delta$. Whether that coupling grows or shrinks under coarse graining controls its relevance near a <renormalization-group fixed point>.
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