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.

Unitary, anomalous Master Ward Identity and its connections to the Wess-Zumino condition, BV formalism and $L_\infty$-algebras

“…Concretely, we show that, considering the infinitesimal symmetry transformations, the cocycle ζ induces a corresponding map ∆ : LieG c → Lie R c which is a Lie algebraic cocycle, and that this cocycle is the anomaly map appearing in the AMWI (Theorem 10.3 in [8] and Theorem 5.1 in this paper). This provides a link between the notions of anomalies used in perturbation theory [4] and anomalies in the non-perturbative formulation of [8].…”

“…Concretely, we show that, considering the infinitesimal symmetry transformations, the cocycle ζ induces a corresponding map ∆ : LieG c → Lie R c which is a Lie algebraic cocycle, and that this cocycle is the anomaly map appearing in the AMWI (Theorem 10.3 in [8] and Theorem 5.1 in this paper). This provides a link between the notions of anomalies used in perturbation theory [4] and anomalies in the non-perturbative formulation of [8]. In Section 4, in Theorem 4.1, we give another derivation of the cocycle relation for ∆: we show that the anomaly ∆ of the AMWI satisfies a consistency condition, which is precisely the cocycle relation for ∆, and which we call the extended Wess-Zumino consistency condition, as it reduces to the standard Wess-Zumino condition for quadratic interactions.…”

“…In a previous paper [8], we investigated the action of symmetries in the C*-algebraic construction of scalar quantum field theories proposed in [11]. In that construction the algebras are generated by S-matrices which describe local interactions within compact regions of spacetime.…”

“…Therefore, we introduced in [8] an additional axiom for these C*-algebras: the "unitary anomalous Master Ward Identity (UAMWI)." It characterizes how symmetries of the classical configuration space are modified in the quantum theory.…”

“…In Section 3 we review that original derivation, following essentially [1]. Although it is clear that the Wess-Zumino consistency condition has to be related to the action of the Lie algebra of the group of symmetries of the theory, the precise statement of this fact in the framework of [8,11] has not been known. While addressing this question, the present work also makes connections with another statement of the Wess-Zumino consistency condition, namely the one present in the BV formalism.…”