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 formula_1. 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" acquires under a dilatation formula_1 a factor formula_3, where formula_4 is a number called the scaling dimension of "O". This implies in particular that the two point correlation function  formula_5 depends on the distance as formula_6. More generally, correlation functions of several local operators must depend on the distances in such a way that formula_7 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" 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 formula_8 and formula_9 is a new operator whose dimension is the sum formula_10. 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 formula_11 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 formula_8 and formula_9 will generally give not a unique operator but infinitely many operators, and their dimension will not generally be equal to formula_10. In the  above two-dimensional Ising model example, the operator product formula_15 gives an operator formula_16 whose dimension is 1 and not twice the dimension of formula_11. 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 formula_18, where formula_19 is the dimension when all couplings are set to zero (i.e. the classical dimension), while formula_20 is called the anomalous dimension, and is expressed as a power series in the couplings collectively denoted as formula_21. Such a separation of scaling dimensions into the classical and anomalous part is only meaningful when couplings are small, so that formula_20 is a small correction. Generally, due to quantum mechanical effects, the couplings formula_21 do not remain constant, but vary (in the jargon of quantum field theory, "run") with the distance scale according to their beta-function. Therefore the  anomalous dimension formula_20 also depends on the distance scale in such theories. In particular correlation functions of local operators are no longer simple powers but have a more complicated dependence on the distances, generally with logarithmic corrections. It may happen that the evolution of the couplings will lead to a value formula_25 where the beta-function vanishes. Then at long distances the theory becomes scale invariant, and the anomalous dimensions stop running. Such a behavior is called an infrared fixed point. 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.