Scaling dimension

Scale-invariant quantum field theory

In a scale invariant quantum field theory, by definition each operator ${\displaystyle O}$ acquires under a dilation ${\displaystyle x\to \lambda x}$ a factor ${\displaystyle {\lambda }^{-\mathrm{\Delta }}}$, where ${\displaystyle \mathrm{\Delta }}$ is a number called the scaling dimension of ${\displaystyle O}$. This implies in particular that the two point correlation function ${\displaystyle ⟨O(x)O(0)⟩}$ depends on the distance as ${\displaystyle (x^2{)}^{-\mathrm{\Delta }}}$. More generally, correlation functions of several local operators must depend on the distances in such a way that ${\displaystyle ⟨O_1(\lambda x_1)O_2(\lambda x_2)\dots ⟩={\lambda }^{-{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2-\dots }⟨O_1(x_1)O_2(x_2)\dots ⟩}$

Most scale invariant theories are also conformally invariant, which imposes further constraints on correlation functions of local operators.

Free field theories

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 ${\displaystyle O}$ 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 ${\displaystyle {\mathrm{\Delta }}_1}$ and ${\displaystyle {\mathrm{\Delta }}_2}$ is a new operator whose dimension is the sum ${\displaystyle {\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2}$.

When interactions are turned on, the scaling dimension receives a correction called the anomalous dimension (see below).

Interacting field theories

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. For example, in the scale (and conformally) invariant theory describing the critical points of the two-dimensional Ising model there is an operator ${\displaystyle \sigma }$ whose dimension is 1/8.

Operator multiplication is subtle in interacting theories compared to free theories. The operator product expansion of two operators with dimensions ${\displaystyle {\mathrm{\Delta }}_1}$ and ${\displaystyle {\mathrm{\Delta }}_2}$ will generally give not a unique operator but infinitely many operators, and their dimension will not generally be equal to ${\displaystyle {\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2}$. In the above two-dimensional Ising model example, the operator product ${\displaystyle \sigma \times \sigma }$ gives an operator ${\displaystyle ϵ}$ whose dimension is 1 and not twice the dimension of ${\displaystyle \sigma }$.