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Structure of Phenomenological Lagrangians. I

1969/01/25 by Sidney Coleman, S. Coleman, J. Wess +1 · 2,182 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Black Holes and Theoretical Physics #Field (mathematics) #Group (periodic table) #Lie group #Manifold (fluid mechanics) #Mathematical physics #Mathematics #Numerical methods for differential equations #Phenomenological model #Physics #Pure mathematics #Quantum mechanics #Realization (probability) #Theoretical physics

paper · doi:10.1103/physrev.177.2239

published in Physical Review 177(5), 2239-2247 (American Institute of Physics)

openalex publication_date 1969/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

The general structure of phenomenological Lagrangian theories is investigated, and the possible transformation laws of the phenomenological fields under a group are discussed. The manifold spanned by the phenomenological fields has a special point, called the origin. Allowed changes in the field variables, which do not change the on-shell S matrix, must leave the origin fixed. By a suitable choice of fields, the transformations induced by the group on the manifold of the phenomenological fields can be made to have standard forms, which are described in detail. The mathematical problem is equivalent to that of finding all (nonlinear) realizations of a (compact, connected, semisimple) Lie group which become linear when restricted to a given subgroup. The relation between linear representations and nonlinear realization is discussed. The important special case of the chiral groups SU(2)\ifmmode×\else\texttimes\fiSU(2) and SU(3)\ifmmode×\else\texttimes\fiSU(3) is considered in detail.

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