2009/01/01 by Yuji Igarashi, Katsumi Itoh, Hidenori Sonoda · 4 citations
Mathematics · Physics and Astronomy · #Abelian group #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Field (mathematics) #Formalism (music) #Functional renormalization group #Quantum Chromodynamics and Particle Interactions #Quantum field theory #Realization (probability) #Renormalization #Renormalization group #Rotation formalisms in three dimensions #Symmetry (geometry) #hep-th
paper · pdf · doi:10.1143/ptps.181.1
166 pages, 27 figures
openalex publication_date 2009/01/01 · arxiv created 2009/09/02 · arxiv updated 2015/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We review the use of the exact renormalization group for realization of symmetry in renormalizable field theories. The review consists of three parts. In part I (§§2–4), we start with the perturbative construction of a renormalizable field theory as a solution of the exact renormalization group (ERG) differential equation. We show how to characterize renormalizability by an appropriate asymptotic behavior of the solution for a large momentum cutoff. Renormalized parameters are introduced to control the asymptotic behavior. In part II (§§5–9), we introduce two formalisms to incorporate symmetry: one by imposing the Ward-Takahashi identity, and the other by imposing the generalized Ward-Takahashi identity via sources that generate symmetry transformations. We apply the two formalisms to concrete models such as QED, YM theories, and the Wess-Zumino model in four dimensions, and the O(N) non-linear sigma model in two dimensions. We end this part with calculations of the abelian axial and chiral anomalies. In part III (§§10 and 11), we overview the Batalin-Vilkovisky formalism adapted to the Wilson action of a bare theory with a UV cutoff. We provide a few appendices to give details and extensions that can be omitted for the understanding of the main text. The last appendix is a quick summary for the reader's convenience.