The unitary Master Ward Identity: Time slice axiom, Noether's Theorem and Anomalies

Abstract

The C*-algebraic formulation of generic interacting quantum field theories, recently presented by Detlev Buchholz and one of the authors (KF), is enriched by a unitary version of the Master Ward Identity, which was postulated some time ago by Franz Marc Boas, Ferdinand Brennecke and two of us (MD,KF). It is shown that the corresponding axiom implies the validity of the time slice axiom. Moreover, it opens the way for a new approach to Noether's Theorem where it yields directly the unitaries implementing the symmetries. It also unravels interesting aspects of the role of anomalies in quantum field theory. Contents 1. Introduction 1 2. Lagrangians, interactions and dynamical spacetimes 3 2.1. Local functionals and observables 3 2.2. Dynamical algebras 5 3. Algebraic adiabatic limit 6 4. Renormalization group 9 5. Master Ward Identity 15 6. Time slice Axiom and the relative Cauchy evolution 21 7. Symmetries of the Lagrangian and the anomalous Noether theorem 24 8. Renormalization group flow 28 9. Covariance 30 10. The unitary AMWI in perturbation theory 33 10.1. Perturbation theory 33 10.2. Scaling anomaly 37 10.3. Axial anomaly 40 11. Conclusions and outlook 42 Appendix A. Functionals and generalized fields 42 Appendix B. Interpolating metrics 44 Appendix C. Proof of ∆X ∈ Lie R c 45 References 46