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Haag–Łopuszański–Sohnius theorem

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Super-Algebra and Super-Geometry

Contents

Idea

The Coleman theorem says that the only possible symmetry Lie algebra of the S-matrix of a 4-dimensional quantum field theory with mass gap is a direct sum of the Poincare Lie algebra and some other Lie algebra of internal gauge symmetries, hence that there cannot be a non-trivial mixing between the spacetime symmetry and the internal symmetry at the level of Lie algebras and in the presence of a mass gap.

(This applies to fundamental local symmetry of the S-matrix, not to global symmetry after spontaneous symmetry breaking.)

The theorem of Haag–Łopuszański–Sohnius 75 makes the analogous statement for super Lie algebras: the only possible super Lie algebra symmetry of an S-matrix is a direct sum of the super Poincare Lie algebra (“supersymmetry”) and another super Lie algebra.

Notice that the super Poincare Lie algebra itself does mix the plain bosonic Poincare Lie algebra symmetry with a kind of “internal” symmetry, see also at extended supersymmetry.

References

Last revised on April 29, 2019 at 12:31:37. See the history of this page for a list of all contributions to it.