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| Coleman–Mandula theorem | |
|---|---|
| Name | Coleman–Mandula theorem |
| Field | Theoretical physics |
| Published | 1967 |
| Authors | Sidney Coleman; Jeffrey Mandula |
| Keywords | Quantum field theory, S-matrix, Symmetry, Lie algebra |
Coleman–Mandula theorem The Coleman–Mandula theorem is a result in theoretical physics concerning the possible symmetries of the S-matrix in relativistic quantum field theories. It constrains how spacetime symmetries like those of Special relativity and internal symmetries such as those associated with Noether's theorem can combine, and it motivated developments including Supersymmetry, Supergravity, and aspects of String theory. The theorem was proven by Sidney Coleman and Jeffrey Mandula in 1967 and has influenced research at institutions like Princeton University, Harvard University, and CERN.
The theorem addresses the structure of continuous symmetries in interacting relativistic quantum theories characterized by an analytic, nontrivial S-matrix. It arose amid efforts by researchers at places such as Harvard University, Massachusetts Institute of Technology, Caltech, Institute for Advanced Study, Stanford University, and Brookhaven National Laboratory to reconcile internal symmetries seen in models like Quantum electrodynamics and Quantum chromodynamics with spacetime symmetries from Poincaré group and Lorentz group. Early twentieth-century developments by figures including Albert Einstein, Hendrik Lorentz, Hermann Minkowski, and later quantum pioneers such as Paul Dirac, Enrico Fermi, and Richard Feynman provided the backdrop for the theorem’s formalization.
Roughly stated, for a nontrivial, analytic S-matrix in four-dimensional Minkowski spacetime under reasonable physical conditions, any symmetry of the S-matrix that includes the Poincaré group must be a direct product of the Poincaré group and an internal symmetry group; no nontrivial mixing of spacetime and internal symmetries is allowed except via conserved charges that commute with translations. The theorem’s conclusion informed later work by researchers at CERN, DESY, SLAC National Accelerator Laboratory, and universities including Columbia University and Yale University.
Key assumptions include: the existence of a nontrivial, analytic S-matrix describing scattering; a finite number of particle types below any mass threshold; mass gaps between stable particle states; and generators of symmetries that are represented by Lie algebras with well-behaved action on one-particle states. These hypotheses relate to frameworks developed by scientists at Bell Labs, Bell Laboratories, Los Alamos National Laboratory, and theorists such as Wolfgang Pauli, Julian Schwinger, and Murray Gell-Mann whose work on symmetries and conserved currents influenced the formal assumptions. The theorem presumes locality in the sense used in axiomatic approaches championed at Bôiteux, Princeton, and the Institute for Advanced Study.
The proof proceeds by analyzing how symmetry generators act on asymptotic one-particle states and on the analytic structure of the scattering amplitude, using techniques from S-matrix theory familiar to researchers at Imperial College London, University of Cambridge, University of Oxford, and University of Chicago. Coleman and Mandula show that any symmetry generator that does not commute with translations would create an infinite tower of particle states unless it reduces to the known spacetime symmetries; by invoking analyticity and cluster decomposition—concepts used by theorists at Rutgers University, University of California, Berkeley, and University of California, Santa Barbara—they rule out nontrivial extensions mixing internal and spacetime symmetries. The argument leverages results in representation theory studied by mathematicians at Princeton University, University of Cambridge, and École Normale Supérieure.
Subsequent work relaxed or altered assumptions to permit richer symmetry structures. The Haag–Łopuszański–Sohnius extension, introduced in the early 1970s by researchers connected to CERN and University of Warsaw, showed that inclusion of graded Lie algebras allows supersymmetry, leading to constructions by Peter Freund, Julius Wess, Bruno Zumino, and others that underpin Supersymmetry and Supergravity models developed at Caltech, Massachusetts Institute of Technology, University of Texas at Austin, and UCLA. In lower-dimensional systems and in theories without a mass gap—contexts investigated at Princeton University, Perimeter Institute, and Simons Center for Geometry and Physics—conformal and infinite-dimensional algebras such as the Virasoro algebra and affine Kac–Moody algebras provide further generalizations, influencing research at Yale University, Cambridge, and Oxford.
The theorem explains why internal symmetries like SU(3), SU(2), and U(1) in the Standard Model coexist with Poincaré invariance without nontrivial mixing, a structure codified by laboratories such as Fermilab, CERN, SLAC, and DESY. Its constraints shaped model-building approaches in particle physics pursued by groups at Brookhaven National Laboratory, Los Alamos National Laboratory, Argonne National Laboratory, and university departments at Columbia University and University of Chicago. By motivating supersymmetry, the theorem indirectly influenced searches at colliders including the Large Hadron Collider and experiments at Tevatron and informed theoretical programs in String theory research centers like Institute for Advanced Study and Perimeter Institute.
Known loopholes arise when one or more assumptions are violated: allowing infinite particle types below a mass threshold (as in some String theory spectra), relaxing analyticity or locality (as considered in proposals related to Noncommutative geometry and models explored at University of Toronto and Imperial College London), working in lower spacetime dimensions, or admitting graded Lie algebras leading to supersymmetry per the Haag–Łopuszański–Sohnius result. Such exceptions have been central to developments by researchers at Princeton University, Harvard University, Stanford University, and Cambridge.