High Energy Theory Seminar

Noether's theorem is textbook material in quantum field theory, but among experts it has been known for a while that there are quantum field theories with continuous global symmetries which possess no conserved current. In this talk I will give a few examples of this, and also describe a relationship between this phenomenon and a general property of algebraic quantum field theory called the split property. In particular we will see that violations of the split property on manifolds other than $R^d$ can prevent the existence of a Noether current on $R^d$, even if the split property holds on $R^d$. We will extend this notion to discrete global symmetries, and we will also see that all examples so far with "unsplittable" global symmetries have the property that there is a topological sector with ``unbreakable surface operators''. Finally we will conjecture that Noether's theorem, or more generally splittability of global symmetries, should hold in any quantum field theory which does not possess such a sector.