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
is a number called the scaling dimension of
This implies in particular that the two point correlation function
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.
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
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.).
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.
For example, in the scale (and conformally) invariant theory describing the critical points of the two-dimensional Ising model there is an operator
will generally give not a unique operator but infinitely many operators, and their dimension will not generally be equal to
In the above two-dimensional Ising model example, the operator product
[2][1] 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.
Scaling dimensions of operators in such theories can be expressed schematically as
is called the anomalous dimension, and is expressed as a power series in the couplings collectively denoted as
[3] Such a separation of scaling dimensions into the classical and anomalous part is only meaningful when couplings are small, so that
Generally, due to quantum mechanical effects, the couplings
do not remain constant, but vary (in the jargon of quantum field theory, run) with the distance scale according to their beta-function.
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
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.
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