Haag–Łopuszański–Sohnius

Haag–Łopuszański–Sohnius
Name	Haag–Łopuszański–Sohnius
Type	Theorem
Field	Theoretical Physics; Mathematical Physics
Contributors	Rudolf Haag; Jan Łopuszański; Martin Sohnius
Year	1975
Related	Coleman–Mandula theorem; supersymmetry; superalgebra; Wess–Zumino model

Haag–Łopuszański–Sohnius is a theorem that extends the constraints of the Coleman–Mandula theorem by classifying allowable symmetry extensions of relativistic quantum field theories to include graded structures, thereby providing a rigorous foundation for supersymmetry in four-dimensional Minkowski space. It was formulated by Rudolf Haag, Jan Łopuszański, and Martin Sohnius in 1975 and addresses the compatibility of internal symmetries such as isospin, flavor SU(3), and gauge symmetry with spacetime symmetries represented by the Poincaré group and its algebra. The result identifies supersymmetry algebras, including central charges, as the only nontrivial extensions consistent with the assumptions that underlie relativistic scattering theory used by S-matrix approaches in the era of the S-matrix theory program.

Introduction

The Haag–Łopuszański–Sohnius theorem refines conclusions of the Coleman–Mandula theorem by allowing graded (fermionic) generators to mix internal symmetries like SU(2), SU(3), U(1), and SU(N) with spacetime symmetries of the Poincaré group and its universal cover Spin(1,3). It shows that the only nontrivial extension of the direct product of internal symmetry algebras and the Poincaré algebra under the standard axioms of relativistic quantum field theory are Lie superalgebras that contain supercharges transforming as spinors under Lorentz group representations such as the chiral Weyl spinor or Majorana spinor. This conclusion paved the way for model building exemplified by the Wess–Zumino model, Minimal Supersymmetric Standard Model, and extended constructions like N=2 supersymmetry and N=4 supersymmetry.

Historical Background and Development

The theorem emerged in the mid-1970s amid efforts by researchers at institutions including CERN, Institut des Hautes Études Scientifiques, and universities across Europe and North America to reconcile particle spectra with spacetime symmetries following the success of the Standard Model. Prior work by Gerard 't Hooft and Steven Weinberg on renormalization, as well as the S-matrix constraints articulated by Murray Gell-Mann and Maurice Lévy, framed the context. The original Coleman–Mandula theorem by Sidney Coleman and Jeffrey Mandula precluded nontrivial mixing of internal and spacetime symmetries in nontrivial scattering theories; Haag, Łopuszański, and Sohnius relaxed the assumption of Lie algebraic symmetry to include graded algebras, influenced by earlier algebraic results from Victor Kac and others in the theory of superalgebras. The theorem influenced subsequent discoveries such as supergravity by Daniel Z. Freedman, Sergio Ferrara, and Peter van Nieuwenhuizen and string-theoretic developments by Michael Green, John Schwarz, and Edward Witten.

Statement of the Theorem

Under assumptions common to axiomatic formulations of scattering theory—existence of a nontrivial S-matrix as in work by Heisenberg and others, a mass gap as in models studied by Raymond Stora, analyticity properties of amplitudes used by Gerard 't Hooft, and finite particle types below any mass—any Lie algebra of symmetries that extends the direct sum of the Poincaré algebra and a compact internal symmetry algebra must be a direct sum of the Poincaré algebra, a compact internal algebra, and a graded extension generated by spinor charges. Concretely, allowed extensions are Lie superalgebras whose odd part transforms in spinor representations of SO(1,3), possibly including central charges that commute with all generators, as later exploited in BPS states and Seiberg–Witten theory.

Supersymmetry and Graded Lie Algebras

The theorem legitimizes the use of Lie superalgebra frameworks like 4), N), and 4) in particle physics and AdS/CFT correspondence contexts. Supercharges Q carry spinor indices and satisfy anticommutation relations that close onto the energy–momentum operators P and possible central elements Z, mirroring structures used in the Wess–Zumino model and super Yang–Mills theory. The graded structure distinguishes even generators (forming a Lie algebra like su(N)) from odd generators, enabling fermionic transformations that mix multiplets such as chiral multiplets and vector multiplets. Central extensions recognized here became central to understanding solitonic objects in string theory and M-theory studied by Joseph Polchinski and Edward Witten.

Proof Outline and Key Techniques

The proof strategy adapts methods from the proof of the Coleman–Mandula theorem and incorporates representation theory of the Lorentz group and analytic properties of scattering amplitudes as developed by Enrico Fermi-era axiomatic programs. Haag, Łopuszański, and Sohnius analyze possible additional symmetry generators and classify their commutators and anticommutators with momentum and Lorentz generators, using constraints from the spin-statistics theorem and positivity of energy emphasized by Eugene Wigner and Arthur Wightman. They show that bosonic generators beyond the Poincaré and compact internal algebras must be trivial, whereas fermionic generators can produce nontrivial graded closures consistent with locality and causality, yielding superalgebra structures with allowed central charges characterized by invariant tensors of the internal algebra.

Implications and Applications in Physics

The theorem underpins theoretical frameworks including the Minimal Supersymmetric Standard Model, supergravity, and supersymmetric quantum field theorys used in renormalization studies by Pauli-era successors and in duality analyses by Nathan Seiberg and Edward Witten. It explains why supersymmetry offers cancellations of ultraviolet divergences leveraged by Gerard 't Hooft and Martinus Veltman techniques and motivates searches for superpartners at colliders like the Large Hadron Collider where experiments by ATLAS and CMS probe signatures predicted by supersymmetric scenarios. The classification of central charges connects to Bogomol'nyi–Prasad–Sommerfield bounds and topological charges studied in soliton literature and monopole dynamics investigated by Goddard and Olive.

Extensions, Generalizations, and Limitations

Generalizations relax some assumptions to explore non‑Lorentzian settings, non‑local theories, and higher-dimensional spacetimes considered in string theory and M-theory, where algebras like M-algebra and extended supersymmetry algebras appear. Limitations include reliance on analyticity and asymptotic completeness assumptions challenged in conformal theories such as N=4 supersymmetric Yang–Mills theory and in quantum gravity contexts explored by Juan Maldacena and Steven Weinberg. Alternative symmetry structures like quantum groups studied by Vladimir Drinfeld and noncommutative deformations explored by Seiberg lie outside the theorem's original scope and motivate ongoing research linking algebraic classification to physical model building.

Category:Theorems in physics