 Scaling dimension

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 $x^{\prime }=\lambda x$, ${\mathcal{O}}^{\prime }(x^{\prime })={\lambda }^{-\Delta }\mathcal{O}(x)$, 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.

The term "scaling dimension" can refer to different concepts depending on the context in which it is used, particularly in physics and mathematics. Here are a couple of relevant interpretations: 1. **In Physics (Statistical Mechanics and Quantum Field Theory)**: The scaling dimension is a property of operators in conformal field theories (CFTs). It describes how the correlation functions of those operators change under rescaling of the coordinates.