Communications in Mathematical Physics

, Volume 372, Issue 1, pp 281–341 | Cite as

Anomalies in Time-Ordered Products and Applications to the BV–BRST Formulation of Quantum Gauge Theories

  • Markus B. FröbEmail author


We show that every (graded) derivation on the algebra of free quantum fields and their Wick powers in curved spacetimes gives rise to a set of anomalous Ward identities for time-ordered products, with an explicit formula for their classical limit. We study these identities for the Koszul–Tate and the full BRST differential in the BV–BRST formulation of perturbatively interacting quantum gauge theories, and clarify the relation to previous results. In particular, we show that the quantum BRST differential, the quantum antibracket and the higher-order anomalies form an \(L_\infty \) algebra. The defining relations of this algebra ensure that the gauge structure is well-defined on cohomology classes of the quantum BRST operator, i.e., observables. Furthermore, we show that one can determine contact terms such that also the interacting time-ordered products of multiple interacting fields are well defined on cohomology classes. An important technical improvement over previous treatments is the fact that all our relations hold off-shell and are independent of the concrete form of the Lagrangian, including the case of open gauge algebras.



It is a pleasure to thank Chris Fewster, Atsushi Higuchi, Stefan Hollands, Kasia Rejzner, Mojtaba Taslimi Tehrani and Jochen Zahn for discussions on (algebraic) quantum field theory, Igor Khavkine for comments on \(L_\infty \) algebras and a critical reading of the manuscript, Paweł Duch for pointing out a mistake and a simplification in the proof of Theorem 10, and the anonymous referee for a careful reading of the manuscript and for pointing out various mistakes and typos. This work is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 702750 “QLO-QG”.


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Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

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