Reflections on Noether's second theorem and the energy-momentum tensor

Abstract

Through symmetry of the action under global spacetime translations, Noether's first theorem infamously entails an energy-momentum tensor (EMT) that is neither symmetric nor gauge-invariant. In a prior work [Phys. Rev. D 106, 125012 (2022)], I had obtained a symmetric and gauge-invariant EMT by using Noether's second theorem instead, with local spacetime translations as the symmetry group. However, the derivation therein was flawed, containing a faulty assumption about the transformation rule for spinor fields. In this work, I revisit the derivation of [Phys. Rev. D 106, 125012 (2022)], both correcting the faulty step and simplifying the derivation for broader accessibility. The end result is an EMT for quantum chromodynamics that is gauge-invariant, but not symmetric.

Figures (2)

• Figure 1: Depiction of a local translation. Left panel: a scalar function $\phi$ of one spatial variable, $x$. Middle panel: $x$ is transformed by moving every spatial point, and $\phi$ is reparametrized to take the same values at the moved points. Right panel: $\delta_{\xi}\phi$ is evaluated by taking the difference between the transformed and original curve, per $x$ value.
• Figure 2: Depiction of a Lie group homomorphism $\rho : \mathrm{ML}(4,\mathbb{R}) \rightarrow \mathrm{GL}(4,\mathbb{R})$, which maps the double cover of $\mathrm{GL}(4,\mathbb{R})$ (here marked $\mathrm{ML}(4,\mathbb{R})$) onto $\mathrm{GL}(4,\mathbb{R})$. Since $\mathrm{SL}(2,\mathbb{R})$ is a subgroup of $\mathrm{GL}(4,\mathbb{R})$, its double cover (here marked $\mathrm{Mp}(2,\mathbb{R})$) must be a subgroup of $\mathrm{ML}(4,\mathbb{R})$---and $\rho$ must likewise map $\mathrm{Mp}(2,\mathbb{R})$ onto $\mathrm{SL}(2,\mathbb{R})$. A faithful matrix representation of $\mathrm{ML}(4,\mathbb{R})$ can only exist if a there is a faithful matrix representation of its subgroup $\mathrm{Mp}(2,\mathbb{R})$.