Editing Scale invariance (section)

==Scale invariance in classical field theory==
[[Classical field theory]] is generically described by a field, or set of fields,  ''φ'', that depend on coordinates, ''x''. Valid field configurations are then determined by solving [[differential equations]] for ''φ'', and these equations are known as [[field equation]]s.

For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields,
:<math>x\rightarrow\lambda x~,</math>
:<math>\varphi\rightarrow\lambda^{-\Delta}\varphi~.</math>

The parameter Δ is known as the [[scaling dimension]] of the field, and its value depends on the theory under consideration.  Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is '''not''' scale-invariant.

A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution,  ''φ''(''x''), one always has other solutions of the form

:<math>\lambda^\Delta \varphi(\lambda x).</math>

===Scale invariance of field configurations===
For a particular field configuration,  ''φ''(''x''),  to be scale-invariant, we require that
:<math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math>

where Δ is, again, the [[scaling dimension]] of the field.

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will '''not''' be scale-invariant, and in such cases the symmetry is said to be [[spontaneously broken]].

===Classical electromagnetism===

An example of a scale-invariant classical field theory is [[electromagnetic field|electromagnetism]] with no charges or currents. The fields are the electric and magnetic fields, '''E'''('''x''',''t'') and '''B'''('''x''',''t''), while their field equations are [[Maxwell's equations]].

With no charges or currents, [[electromagnetic field#Light as an electromagnetic disturbance|these field equations]] take the form of [[wave equation]]s
: <math>
\begin{align}
\nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} \\[6pt]
\nabla^2\mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2}
\end{align}
</math>
where ''c'' is the speed of light.

These field equations are invariant under the transformation
:<math>
\begin{align}
x\rightarrow\lambda x, \\[6pt]
t\rightarrow\lambda t.
\end{align}
</math>

Moreover, given solutions of Maxwell's equations, '''E'''('''x''', ''t'') and '''B'''('''x''', ''t''),  it holds that 
'''E'''(''λ'''''x''', ''λt'') and '''B'''(''λ'''''x''', ''λt'')  are also solutions.

===Massless scalar field theory===
Another example of a scale-invariant classical field theory is the massless [[scalar field theory|scalar field]] (note that the name [[scalar (physics)|scalar]] is unrelated to scale invariance). The scalar field, {{math|''φ''('''''x''''', ''t'')}} is a function of a set of spatial variables, '''''x''''', and a time variable, {{mvar|t}}.

Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,
:<math>\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0,</math>
and is invariant under the transformation
:<math>x\rightarrow\lambda x,</math>
:<math>t\rightarrow\lambda t.</math>

The name massless refers to the absence of a term <math>\propto m^2\varphi</math> in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. In [[relativistic field theory|relativistic field theories]], a mass-scale, {{mvar|m}}  is physically equivalent to a fixed length scale through
:<math>L=\frac{\hbar}{mc},</math>
and so it should not be surprising that massive scalar field theory is ''not'' scale-invariant.

====''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.