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For $U\in SU(2)$ with $U\mapsto AU{B}^{-1}$, the chiral group acts transitively. A derivative-free invariant is therefore constant. At $U=I$, its diagonal stabilizer acts on tangent vectors by the irreducible three-dimensional rotation representation; the only invariant symmetric metric is a multiple of ${\delta }_{ab}$. Transporting this metric gives the displayed unique Lorentz-invariant two-derivative action up to normalization and total derivatives. Equivalently, $U^{†}{\partial }_{\mu }U$ is traceless, eliminating a separate double-trace invariant. Writing $U=\exp (i{\pi }_a{\tau }_a/F)$ and using $\mathrm{tr}({\tau }_a{\tau }_b)=2{\delta }_{ab}$ gives canonically normalized pion kinetic terms.

For a complex scalar field multiplet with canonical kinetic term and scalar potential invariant under $\Phi \mapsto e^{i\epsilon T}\Phi$, a constant Hermitian matrix $T$ gives the displayed Noether current. Vary the field and its conjugate independently in the Noether current for a first-derivative scalar field. A common alternative convention reverses both the parameter and current sign. Its divergence vanishes on shell.

Wave-function renormalization rescales a field so that its kinetic term or propagator residue has the chosen normalization. Its scale dependence gives the field's anomalous dimension.

Localizing the rigid supersymmetry of a free Wess–Zumino model produces a derivative of the supersymmetry parameter multiplying the supercurrent. The Noether gauging procedure introduces a gravitino coupled to this current. Closure also requires a vierbein, the Einstein-Hilbert action, covariant matter kinetic terms and higher-order interactions. The leading gravitino-matter coupling contains the derivative of the scalar, not its undifferentiated value.

After shell integration and $x^{\prime }=bx$, restore the kinetic term by ${\varphi }^{\prime }=b^{-(d-2)/2}\sqrt{1+\Delta Z}\varphi$. An operator with $n$ scalar fields and $m$ derivatives scales by $b^{m+n(d-2)/2-d}$. In four dimensions this gives the displayed factor with $m+n-4$; the general exponent is $m+n(d-2)/2-d$. The coupling shift and wavefunction renormalization arise before the rescaling.

The trace-free metric variation is $(P_{1}c{)}_{ab}={\nabla }_a{c}_b+{\nabla }_b{c}_a-{\gamma }_{ab}{\nabla }_d{c}^d$. A symmetric trace-free antighost exponentiates the Faddeev-Popov determinant. In light-cone worldsheet coordinates its two chiral kinetic terms are $b_{++}{\partial }_{-}{c}^{+}$ and $b_{--}{\partial }_{+}{c}^{-}$.

For the left-parameter convention and ghost kinetic term $(\partial b)Dc$, localization of the BRST symmetry yields this conserved Noether current. Subtract the total-derivative contribution $K^{\mu }=h^a(D^{\mu }c{)}^a$ in the Lagrangian variation. The BRST charge is the spatial integral of $j^0$. The antighost term has the displayed sign because the parameter and antighost field derivative anticommute.

In a Minkowski-space convention with metric $(+---)$ and canonically normalized gauge fields, the displayed Lagrangian density supplies their kinetic term. The Yang-Mills field strength contains a quadratic gauge-potential term, which generates cubic and quartic Yang-Mills self-interactions when the field-strength square is expanded. The coupling can instead be placed as an overall prefactor by rescaling the gauge potential; the normalization convention must be fixed before comparing coefficients.