). Each monomial of the Lagrangian thus leads to a homogeneous linear equation for the exponents . If there are (inequivalent) coordinates and fields in the Lagrangian, then such equations constitute a square matrix. If this matrix were invertible then there only would be the trivial solution . The condition for a nontrivial solution gives an equation between the space dimensions, and this determines the upper critical dimension (provided there is only one variable dimension in the Lagrangian). A redefinition of the coordinates and fields now shows that determining the scaling exponents is equivalent to a dimensional analysis with respect to the wavevector , with all coupling constants occurring in the Lagrangian rendered dimensionless. Dimensionless coupling constants are the technical hallmark for the upper critical dimension. Naive scaling at the level of the Lagrangian does not directly correspond to physical scaling because a cutoff is required to give a meaning to the field theory and the path integral. Changing the length scale also changes the number of degrees of freedom. This complication is taken into account by the renormalization group. The main result at the upper critical dimension is that scale invariance remains valid for large factors , but with additional factors in the scaling of the coordinates and fields. What happens below or above depends on whether one is interested in long distances (statistical field theory) or short distances (quantum field theory). Quantum field theories are trivial (convergent) below and not renormalizable above . Statistical field theories are trivial (convergent) above and renormalizable below . In the latter case there arise "anomalous" contributions to the naive scaling exponents . These anomalous contributions to the effective critical exponents vanish at the upper critical dimension. It is instructive to see how the scale invariance at the upper critical dimension becomes a scale invariance below this dimension. For small external wave vectors the vertex functions acquire additional exponents, for example . If these exponents are inserted into a matrix (which only has values in the first column) the condition for scale invariance becomes . This equation only can be satisfied if the anomalous exponents of the vertex functions cooperate in some way. In fact, the vertex functions depend on each other hierarchically. One way to express this interdependence are the Dyson–Schwinger equations. Naive scaling at thus is important as zeroth order approximation. Naive scaling at the upper critical dimension also classifies terms