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Concept# Scaling dimension

Summary

In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations . If the quantum field theory is scale invariant, scaling dimensions of operators are fixed numbers, otherwise they are functions of the distance scale.
In a scale invariant quantum field theory, by definition each operator O acquires under a dilation a factor , where is a number called the scaling dimension of O. This implies in particular that the two point correlation function depends on the distance as . More generally, correlation functions of several local operators must depend on the distances in such a way that
Most scale invariant theories are also conformally invariant, which imposes further constraints on correlation functions of local operators.
Free theories are the simplest scale-invariant quantum field theories. In free theories, one makes a distinction between the elementary operators, which are the fields appearing in the Lagrangian,
and the composite operators which are products of the elementary ones. The scaling dimension of an elementary operator O is determined by dimensional analysis from the Lagrangian (in four spacetime dimensions, it is 1 for elementary bosonic fields including the vector potentials, 3/2 for elementary fermionic fields etc.). This scaling dimension is called the classical dimension (the terms canonical dimension and engineering dimension are also used). A composite operator obtained by taking a product of two operators of dimensions and is a new operator whose dimension is the sum .
When interactions are turned on, the scaling dimension receives a correction called the anomalous dimension (see below).
There are many scale invariant quantum field theories which are not free theories; these are called interacting. Scaling dimensions of operators in such theories may not be read off from a Lagrangian; they are also not necessarily (half)integer.

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