1. OVERVIEW 3 If we choose a section of each torsor T (n+1) (E , Scl) → T (n) (E , Scl) we find an isomorphism T (∞) (E , Scl) ∼ series Scl + S(1) + 2 S(2) + · · · where each S(i) is a local functional, that is, a functional which can be written as the integral of a Lagrangian. Thus, this theorem allows one to quantize the theory associated to any classical action functional Scl. How- ever, there is an ambiguity to quantization: at each term in , we are free to add an arbitrary local functional to our action. 1.3. The main results of this book are all stated in the context of this theorem. In Chapter 4, I give a definition of an action of the group R 0 on the space of theories on Rn. This action is called the local renormalization group flow, and is a fundamental part of the concept of renormalizability developed by Wilson and others. The action of group R 0 on the space of theories on Rn simply arises from the action of this group on Rn by rescaling. The coeﬃcients of the action of this local renormalization group flow on any particular theory are the β functions of that theory. I include explicit calculations of the β function of some simple theories, including the φ4 theory on R4. This local renormalization group flow leads to a concept of renormaliz- ability. Following Wilson and others, I say that a theory is perturbatively renormalizable if it has “critical” scaling behaviour under the renormaliza- tion group flow. This means that the theory is fixed under the renormal- ization group flow except for logarithmic corrections. I then classify all possible renormalizable scalar field theories, and find the expected answer. For example, the only renormalizable scalar field theory in four dimensions, invariant under isometries and under the transformation φ → −φ, is the φ4 theory. In Chapter 5, I show how to include gauge theories in my definition of quantum field theory, using a natural synthesis of the Wilsonian effective action picture and the Batalin-Vilkovisky formalism. Gauge symmetry, in our set up, is expressed by the requirement that the effective action Seff[Λ] at each energy Λ satisfies a certain scale Λ Batalin-Vilkovisky quantum master equation. The renormalization group flow is compatible with the Batalin-Vilkovisky quantum master equation: the flow from scale Λ to scale Λ takes a solution of the scale Λ master equation to a solution to the scale Λ equation. I develop a cohomological approach to constructing theories which are renormalizable and which satisfy the quantum master equation. Given any classical gauge theory, satisfying the classical analog of renormalizability, I prove a general theorem allowing one to construct a renormalizable quan- tization, providing a certain cohomology group vanishes. The dimension

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