Generalized symmetries and Noether's theorem in QFT

TL;DR

The paper analyzes when continuous global symmetries in QFT can be generated by local Noether currents in the presence of generalized (non-local) symmetries. Using a refined twist classification (additive vs complete) within a local QFT framework and the split property, it proves that generalized symmetries cannot be charged under Noether charges, and that charged generalized symmetries must be non-compact, yielding a continuum of non-local sectors. This topological obstruction provides a new proof and generalizations of the Weinberg–Witten theorem to arbitrary dimensions and representations, and clarifies when Noether's strong version can hold. The results have implications for scale versus conformal invariance, Coleman–Mandula, and global symmetries in quantum gravity, distinguishing weak/noether twisting from the strong version.

Abstract

We show that generalized symmetries cannot be charged under a continuous global symmetry having a Noether current. Further, only non-compact generalized symmetries can be charged under a continuous global symmetry. These results follow from a finer classification of twist operators, which naturally extends to finite group global symmetries. They unravel topological obstructions to the strong version of Noether's theorem in QFT, even if under general conditions a global symmetry can be implemented locally by twist operators (weak version). We use these results to rederive Weinberg-Witten's theorem within local QFT, generalizing it to massless particles in arbitrary dimensions and representations of the Lorentz group. Several examples with local twists but without Noether currents are described. We end up discussing the conditions for the strong version to hold, dynamical aspects of QFT's with non-compact generalized symmetries, scale vs conformal invariance in QFT, connections with the Coleman-Mandula theorem and aspects of global symmetries in quantum gravity.

Figures (6)

• Figure 1: Geometric configuration characterizing the definition of a twist operator $\tau_g(R,Z)$. The topology of the region $R$ is not required to be trivial for the twist to be defined. The twist is constructed so as to effect the symmetry transformation in ${\cal A}(R)$ and leave the operators in ${\cal A}(\bar{R})$ invariant. The buffer zone $Z$ with non-zero size $\epsilon$ is required for the operator to exist in the QFT.
• Figure 2: Geometric configuration used for the definition of twist operators out of Noether currents. The smearing function $\alpha(\vec{x})$ is equal to one for the future and past of $R$ in the time slice $(-\delta,\delta)$ and is equal to zero in the future ans past of $\bar{R}$ in the same time slice.
• Figure 3: Geometric configurations appropriate for the definition of concatenability of twist operators. The left hand side represents the existence of twist operators $\tau_g(A, Z_A)$ and $\tau_g(B, Z_B)$. The right hand side represents the existence of a twist operator $\tau_g(C,Z_C)$ for $C=A\cup B\cup Z$ and $Z=(Z_A\cup Z_B)\cap Z'$. We say $\tau_g(A, Z_A)$ and $\tau_g(B, Z_B)$ are "concatenable" if $\tau_g(AZB,Z_A\cup Z_B)$ is the product of $\tau_g(A, Z_A)$ and $\tau_g(B, Z_B)$ .
• Figure 4: A ball $B$ with its buffer zone $Z_B$. We have a region $R$ with buffer zone $Z$ inside it. Region $R$ might display generalized symmetries. When the classes associated with the generalized symmetry are charged under the global symmetry group additive twists in $R$ and $B-R$ cannot concatenate to twists in $B$.
• Figure 5: Geometric configuration that facilitates solving for $\nabla \times A=B$ inside the ring. The surfaces of constant $r$ foliate the ring with topological torus $T_r$.