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
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Scale invariant quantum field theory
In a scale invariant quantum field theory, by definition each operator O acquires under a dilatation
It should be noted that 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 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
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
Operator multiplication is subtle in interacting theories compared to free theories. The operator product expansion of two operators with dimensions
Non-scale invariant quantum field theory
There are many quantum field theories which, while not being exactly scale invariant, remain approximately scale invariant over a long range of distances. Such quantum field theories can be obtained by adding to free field theories interaction terms with small dimensionless couplings. For example, in four spacetime dimensions one can add quartic scalar couplings, Yukawa couplings, or gauge couplings. Scaling dimensions of operators in such theories can be expressed schematically as
Generally, due to quantum mechanical effects, the couplings
It may happen that the evolution of the couplings will lead to a value
In very special cases, it may happen when the couplings and the anomalous dimensions do not run at all, so that the theory is scale invariant at all distances and for any value of the coupling. For example, this occurs in the N=4 supersymmetric Yang-Mills theory.