Editing Scale invariance (section)

====''φ''<sup>4</sup> theory====
The field equations in the examples above are all [[linear]] in the fields, which has meant that the [[scaling dimension]],&nbsp;Δ, has not been so important. However, one usually requires that the scalar field [[action (physics)|action]] is dimensionless, and this fixes the [[scaling dimension]] of {{mvar|φ}}. In particular,
:<math>\Delta=\frac{D-2}{2},</math>
where {{mvar|D}} is the combined number of spatial and time dimensions.

Given this scaling dimension for {{mvar|φ}}, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless [[Phi to the fourth|φ<sup>4</sup> theory]] for {{mvar|D}}&nbsp;=&nbsp;4. The field equation is
:<math>\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.</math>

(Note that the name {{mvar|φ}}<sup>4</sup> derives from the form of the [[Phi to the fourth#The Lagrangian|Lagrangian]], which contains the fourth power of {{mvar|φ}}.)

When {{mvar|D}}&nbsp;=&nbsp;4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is Δ&nbsp;=&nbsp;1. The field equation is then invariant under the transformation
:<math>x\rightarrow\lambda x,</math>
:<math>t\rightarrow\lambda t,</math>
:<math>\varphi (x)\rightarrow\lambda^{-1}\varphi(x).</math>

The key point is that the parameter {{mvar|g}} must be dimensionless, otherwise one introduces a fixed length scale into the theory: For  {{mvar|φ}}<sup>4</sup>  theory, this is only the case in {{mvar|D}}&nbsp;=&nbsp;4.
Note that under these transformations the argument of the function {{mvar|φ}} is unchanged.