Scaling dimension

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

x

→

λ

x

{\displaystyle x\to \lambda x}

. If the quantum field theory is scale invariant, scaling dimensions of operators are fixed numbers, otherwise they are functions of the distance scale.

Scale-invariant quantum field theory

In a scale invariant quantum field theory, by definition each operator

O

{\displaystyle O}

acquires under a dilation

x

→

λ

x

{\displaystyle x\to \lambda x}

a factor

λ

−

Δ

{\displaystyle \lambda ^{-\Delta }}

, where

Δ

{\displaystyle \Delta }

is a number called the scaling dimension of

O

{\displaystyle O}

. This implies in particular that the two point correlation function

⟨

O

(

x

)

O

(

0

)

⟩

{\displaystyle \langle O(x)O(0)\rangle }

depends on the distance as

(

x

2

)

−

Δ

{\displaystyle (x^{2})^{-\Delta }}

. More generally, correlation functions of several local operators must depend on the distances in such a way that

⟨

O

1

(

λ

x

1

)

O

2

(

λ

x

2

)

…

⟩

=

λ

−

Δ

1

−

Δ

2

−

…

⟨

O

1

(

x

1

)

O

2

(

x

2

)

…

⟩

{\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 }

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

O

{\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

Δ

1

{\displaystyle \Delta _{1}}

and

Δ

2

{\displaystyle \Delta _{2}}

is a new operator whose dimension is the sum

Δ

1

+

Δ

2

{\displaystyle \Delta _{1}+\Delta _{2}}

.