Scaling dimension

In theoretical physics, scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes rescaling properties of the operator under spacetime dilations ${\displaystyle x\to \lambda x}$. If the quantum field theory is scale invariant, scaling dimensions of operators are fixed numbers, otherwise they are functions depending on the distance scale.

In a scale invariant quantum field theory, by definition each operator O acquires under a dilatation ${\displaystyle x\to \lambda x}$ a factor ${\displaystyle \lambda ^{-\Delta }}$, where ${\displaystyle \Delta }$ is a number called the scaling dimension of O. This implies in particular that the two point correlation function ${\displaystyle \langle O(x)O(0)\rangle }$ depends on the distance as ${\displaystyle (x^{2})^{-\Delta }}$. More generally, correlation functions of several local operators must depend on the distances in such a way that ${\displaystyle \langle O_{1}(\lambda x_{1})O_{2}(\lambda x_{2})\ldots \rangle =\lambda ^{-\Delta _{1}-\Delta _{2}-\ldots }\langle O_{1}(x_{1})O_{2}(x_{2})\ldots \rangle }$

...
Wikipedia