Editing Noether's theorem (section)

=== Example 3: Conformal transformation ===

Both examples 1 and 2 are over a 1-dimensional manifold (time). An example involving spacetime is a [[conformal transformation]] of a massless real scalar field with a [[Quartic interaction|quartic potential]] in (3&nbsp;+&nbsp;1)-[[Minkowski spacetime]].

:<math>\begin{align}
  \mathcal{S}[\varphi]
    & = \int \mathcal{L}\left[\varphi (x), \partial_\mu \varphi (x)\right] d^4 x \\[3pt]
    & = \int \left(\frac{1}{2}\partial^\mu \varphi \partial_\mu \varphi - \lambda \varphi^4\right) d^4 x
\end{align}</math>

For ''Q'', consider the generator of a spacetime rescaling. In other words,

:<math>Q[\varphi(x)] = x^\mu\partial_\mu \varphi(x) + \varphi(x). </math>

The second term on the right hand side is due to the "conformal weight" of <math>\varphi</math>. And

:<math>Q[\mathcal{L}] = \partial^\mu\varphi\left(\partial_\mu\varphi + x^\nu\partial_\mu\partial_\nu\varphi + \partial_\mu\varphi\right) - 4\lambda\varphi^3\left(x^\mu\partial_\mu\varphi + \varphi\right).</math>

This has the form of

:<math>\partial_\mu\left[\frac{1}{2}x^\mu\partial^\nu\varphi\partial_\nu\varphi - \lambda x^\mu \varphi^4 \right] = \partial_\mu\left(x^\mu\mathcal{L}\right)</math>

(where we have performed a change of dummy indices) so set

:<math>f^\mu = x^\mu\mathcal{L}.</math>

Then

:<math>\begin{align}
  j^\mu & = \left[\frac{\partial}{\partial(\partial_\mu\varphi)}\mathcal{L}\right]Q[\varphi]-f^\mu \\
        & = \partial^\mu\varphi\left(x^\nu\partial_\nu\varphi + \varphi\right) - x^\mu\left(\frac 1 2 \partial^\nu\varphi\partial_\nu\varphi - \lambda\varphi^4\right).
\end{align}</math>

Noether's theorem states that <math>\partial_\mu j^\mu = 0</math> (as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side).

If one tries to find the [[Ward–Takahashi identity|Ward–Takahashi]] analog of this equation, one runs into a problem because of [[anomaly (physics)|anomalies]].