Trace relations and finite generation in quantum field theory

Establish whether finite-N trace relations are modified by renormalization in quantum field theory and determine whether a finite set of primary and secondary invariants can generate the full algebra of local operators.

Background

The paper identifies two unresolved obstacles to extending the invariant-theoretic framework from finite-dimensional matrix quantum mechanics to quantum field theory. Composite operators in quantum field theory require renormalization and may mix in the presence of ultraviolet divergences, potentially altering the trace relations. In addition, the finite-generation assumptions used in the Hilbert–Serre and Hironaka frameworks are not automatic for algebras of local operators, so it is unknown whether finite primary and secondary generating sets exist in general.

References

In quantum field theory there are two highly nontrivial generalizations. First, trace relations may themselves be modified by renormalization, since composite operators must be defined in the presence of UV divergences and can mix under renormalization. Second, the finite-generation assumptions underlying the usual Hilbert–Serre and Hironaka framework are no longer automatic: there need not be a finite set of primaries and secondaries generating the full algebra of local operators. We leave these fascinating questions for the future.

— A pedagogical introduction to invariant theory and finite-$N$ holography  (2608.22952 - Koch et al., 24 Aug 2026) in Discussion, bullet point 'Quantum Field Theory'