24 1. INTRODUCTION In more pedestrian terms, the extended classical action functional en- codes the following data: (1) the original action functional on the original space of fields (2) the Lie bracket on the space of infinitesimal gauge symmetries, (3) the way this Lie algebra acts on the original space of fields. In order to construct a quantum theory, one asks that the extended action satisfies the quantum master equation. This is a succinct way of encoding the following conditions: (1) The Lie bracket on the space of infinitesimal gauge symmetries satisfies the Jacobi identity. (2) This Lie algebra acts in a way preserving the action functional on the space of fields. (3) The Lie algebra of infinitesimal gauge symmetries preserves the “Lebesgue measure” on the original space of fields. That is, the vector field on the original space of fields associated to every infin- itesimal gauge symmetry is divergence free. (4) The adjoint action of the Lie algebra on itself also preserves the “Lebesgue measure”. Again, this says that a vector field associated to every infinitesimal gauge symmetry is divergence free. Unfortunately, the quantum master equation is an ill-defined expression. The 3rd and 4th conditions above are the source of the problem: the diver- gence of a vector field on the space of fields is a singular expression, involving the same kind of singularities as those appearing in one-loop Feynman dia- grams. 9.4. This form of the quantum master equation violates our philosophy: we should always express things in terms of the effective actions. The quan- tum master equation above is about the original “infinite energy” action, so we should not be surprised that it doesn’t make sense. The solution to this problem is to combine the BV formalism with the effective action philosophy. To give an effective action in the BV formalism is to give a functional Seff[Λ] on the energy ≤ Λ part of the extended space of fields (i.e., the space of ghosts, fields, anti-fields and anti-ghosts). This energy Λ effective action must satisfy a certain energy Λ quantum master equation. The reason that the effective action philosophy and the BV formalism work well together is the following. Lemma. The renormalization group flow from scale Λ to scale Λ carries solutions of the energy Λ quantum master equation into solutions of the energy Λ quantum master equation. Thus, to give a gauge theory in the effective BV formalism is to give a collection of effective actions Seff[Λ] for each Λ, such that Seff[Λ] satis- fies the scale Λ QME, and such that Seff[Λ ] is obtained from Seff[Λ] by

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