22 1. INTRODUCTION Now let us state how one classifies scalar field theories, in general. Theorem 8.1.1. Let R(k)(Rn) denote the space of renormalizable scalar field theories on Rn, invariant under translation, defined modulo n+1 . Then, R(k+1)(Rn) R(k)(Rn) is a torsor for the vector space of local action functionals S(φ) which are a sum of terms of non-negative dimension. Further, R(0)(Rn) is canonically isomorphic to the space of local action functionals of the form S(φ) = 1 2 Rn φ D φ + cubic and higher terms, of non-negative dimension. As before, the choice of a renormalization scheme leads to a section of each of the torsors R(k+1)(Rn) R(k)(Rn), and so to a bijection between the space of renormalizable scalar field theories and the space of series 1 2 φ D φ + i Si where each Si is a translation invariant local action functional of non- negative dimension, and S0 is at least cubic. Applying this to R4, we find the following. Corollary 8.1.2. Renormalizable scalar field theories on R4, invariant under SO(4) R4 and under φ −φ, are in bijection with Lagrangians of the form L (φ) = D φ + bφ4 + cφ2 for a, b, c R[[ ]], where a = 1 2 modulo and b = 0 modulo . More generally, there is a finite dimensional space of non-free renormal- izable theories in dimensions n = 3, 4, 5, 6, an infinite dimensional space in dimensions n = 1, 2, and none in dimensions n 6. ( “Finite dimensional” means as a formal scheme over Spec R[[ ]]: there are only finitely many R[[ ]]-valued parameters). Thus we find that the scalar field theories traditionally considered to be “renormalizable” are precisely the ones selected by the Wilsonian definition advocated here. However, in this approach, one has a conceptual reason for why these particular scalar field theories, and no others, are renormalizable. 9. Gauge theories We would like to apply the Wilsonian philosophy to understand gauge theories. In Chapter 5, we will explain how to do this using a synthesis of Wilsonian ideas and the Batalin-Vilkovisky formalism.
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