x\toλx
In a scale invariant quantum field theory, by definition each operator
O
x\toλx
λ-\Delta
\Delta
O
\langleO(x)O(0)\rangle
(x2)-\Delta
\langleO1(λx1)O2(λ
-\Delta1-\Delta2-\ldots | |
x | |
2)\ldots\rangle= λ |
\langleO1(x1)O2(x2)\ldots\rangle
Most scale invariant theories are also conformally invariant, which imposes further constraints on correlation functions of local operators.[1]
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
\Delta1
\Delta2
\Delta1+\Delta2
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
\sigma
Operator multiplication is subtle in interacting theories compared to free theories. The operator product expansion of two operators with dimensions
\Delta1
\Delta2
\Delta1+\Delta2
\sigma x \sigma
\epsilon
\sigma
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
\Delta=\Delta0+\gamma(g)
\Delta0
\gamma(g)
g
\gamma(g)
Generally, due to quantum mechanical effects, the couplings
g
\gamma(g)
It may happen that the evolution of the couplings will lead to a value
g=g*
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.
M3,4
\sigma=\phi1,2
\epsilon=\phi1,3