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The algebra of grand unified theories


Authors: John Baez and John Huerta
Journal: Bull. Amer. Math. Soc. 47 (2010), 483-552
MSC (2000): Primary 20C35, 81R05; Secondary 81-02
DOI: https://doi.org/10.1090/S0273-0979-10-01294-2
Published electronically: March 11, 2010
MathSciNet review: 2651086
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Abstract: The Standard Model is the best tested and most widely accepted theory of elementary particles we have today. It may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three ``grand unified theories'': theories that unify forces and particles by extending the Standard Model symmetry group $ \mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ to a larger group. These three are Georgi and Glashow's $ \mathrm{SU}(5)$ theory, Georgi's theory based on the group $ \Spin(10)$, and the Pati-Salam model based on the group $ \mathrm{SU}(2) \times \mathrm{SU}(2) \times \mathrm{SU}(4)$. In this expository account for mathematicians, we explain only the portion of these theories that involves finite-dimensional group representations. This allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve.


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  • 1. J. F. Adams, Lectures on Exceptional Lie Groups, Zafer Mahmoud and Mamoru Mimura, eds., University of Chicago Press, Chicago, 1996. MR 1428422 (98b:22001)
  • 2. M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), 3-38. MR 0167985 (29:5250)
  • 3. John C. Baez and Javier P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, Singapore, 1994. MR 1313910 (95k:83001)
  • 4. Stefano Bertolini, Luca Di Luzio and Michal Malinsky, Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal, available at http://arxiv.org/abs/hep-ph/0903.4049 arXiv:0903.4049.
  • 5. Lowell Brown, Quantum Field Theory, Cambridge Univ. Press, Cambridge, 1994. MR 1231339 (94h:81002)
  • 6. Claude Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, Springer, Berlin, 1996. MR 1636473 (99f:01028)
  • 7. Benedict Cassen and Edward U. Condon, On Nuclear Forces, Phys. Rev. 50 (1936), 846, reprinted in D. M. Brink, Nuclear Forces, Pergamon, Oxford, 1965, pp. 193-201.
  • 8. Robert P. Crease and Charles C. Mann, The Second Creation: Makers of the Revolution in Twentieth-Century Physics, Rutgers University Press, New Brunswick, New Jersey, 1996. MR 1677561 (2000f:81002)
  • 9. Andrzej Derdzinski, Geometry of the Standard Model of Elementary Particles, Springer, Berlin, 1992. MR 1219312 (95f:81101)
  • 10. Howard Georgi, The state of the art--gauge theories, in Particles and Fields--1974, ed. Carl E. Carlson, AIP Conference Proceedings 23, 1975, pp. 575-582.
  • 11. Howard Georgi, Lie Algebras in Particle Physics: From Isospin to Unified Theories, Westview Press, Boulder, Colorado, 1999. MR 644800 (83e:81089)
  • 12. Howard Georgi and Sheldon Glashow, Unity of all elementary-particle forces, Phys. Rev. Lett. 32(8) Feb 1974, 438-441.
  • 13. David Griffiths, Introduction to Elementary Particles, Wiley, New York 1987.
  • 14. Brian Hall, Lie Groups, Lie Algebras, and Representations, Springer, Berlin, 2003. MR 1997306 (2004i:22001)
  • 15. Werner Heisenberg, Zeitschr. f. Phys. 77 (1932), 1; English translation in D. M. Brink, Nuclear Forces, Pergamon, Oxford, 1965, pp. 144-154.
  • 16. Laurie Brown, Max Dresden, Lillian Hoddeson and Michael Riordan, eds., The Rise of the Standard Model, Cambridge Univ. Press, Cambridge, 1997. MR 1601138 (98h:81002)
  • 17. Kerson Huang, Quarks, Leptons & Gauge Fields, World Scientific, Singapore, 1992. MR 704019 (84k:81138)
  • 18. Chris Isham, Modern Differential Geometry for Physicists, World Scientific, Singapore, 1999. MR 1698234 (2000e:53001)
  • 19. T. D. Lee, Particle Physics and Introduction to Field Theory, Harwood, 1981. MR 633154 (83d:81001)
  • 20. Harry J. Lipkin, Lie Groups for Pedestrians, Dover, Mineola, New York, 2002. MR 1918338
  • 21. R. N. Mohapatra, Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics, Springer, 1992. MR 854671 (87j:81276)
  • 22. Gregory L. Naber, Topology, Geometry and Gauge Fields: Foundations, Springer, Berlin, 1997. MR 1444352 (99b:58001)
  • 23. Gregory L. Naber, Topology, Geometry and Gauge Fields: Interactions, Springer, Berlin, 2000. MR 1744816 (2001g:53058)
  • 24. Mikio Nakahara, Geometry, Topology, and Physics, Academic Press, 1983.
  • 25. Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World, Oxford University Press, 1988.
  • 26. Jogesh C. Pati, Proton decay: A must for theory, a challenge for experiment, available at http://arxiv.org/abs/hep-ph/0005095arXiv:hep-ph/0005095.
  • 27. Jogesh C. Pati, Probing grand unification through neutrino oscillations, leptogenesis, and proton decay, Int. J. Mod. Phys. A 18 (2003), 4135-4156. Also available at http://arxiv.org/abs/hep-ph/0305221arXiv:hep-ph/0305221.
  • 28. Jogesh C. Pati and Abdus Salam, Lepton number as the fourth ``color'', Phys. Rev. D 10 (1974), 275-289.
  • 29. Michael E. Peskin, Beyond the Standard Model, available at http://arxiv.org/abs/hep-ph/970549arXiv:hep-ph/970549.
  • 30. Michael E. Peskin and Dan V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995. MR 1402248 (97j:81001)
  • 31. Graham G. Ross, Grand Unified Theories, Benjamin/Cummings, 1985. MR 783830 (86c:81097)
  • 32. Lewis H. Ryder, Quantum Field Theory, Cambridge Univ. Press, Cambridge, 1996. MR 1404519 (97h:81001)
  • 33. Emilio Segrè, From X-Rays to Quarks: Modern Physicists and Their Discoveries, W.H. Freeman, San Francisco, 1980.
  • 34. Shlomo Sternberg, Group Theory and Physics, Cambridge Univ. Press, Cambridge, 1995. MR 1287387 (95i:20001)
  • 35. Mark Srednicki, Quantum Field Theory, Cambridge Univ. Press, 2007. Also available at http://www.physics.ucsb.edu/$ \sim$mark/qft.html http://www.physics.ucsb.edu/$ \sim$mark/qft.html.
  • 36. Anthony Sudbery, Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians, Cambridge Univ. Press, Cambridge, 1986. MR 2245407
  • 37. Robin Ticciati, Quantum Field Theory for Mathematicians, Cambridge Univ. Press, 1999. MR 1699269 (2000h:81002)
  • 38. Michael Tinkham, Group Theory and Quantum Mechanics, Dover, Mineola, New York, 2003.
  • 39. Edward Witten, Grand unification with and without supersymmetry, in Introduction to supersymmetry in particle and nuclear physics, O. Castanos, A. Frank, L. Urrutia, eds., Plenum Press, 1984, pp. 53-76. MR 758626
  • 40. Anthony Zee, Quantum Field Theory in a Nutshell, Princeton Univ. Press, Princeton, 2003. MR 1978227 (2004i:81003)

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Additional Information

John Baez
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: baez@math.ucr.edu

John Huerta
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: huerta@math.ucr.edu

DOI: https://doi.org/10.1090/S0273-0979-10-01294-2
Keywords: Grand unified theory, standard model, representation theory
Received by editor(s): May 8, 2009
Received by editor(s) in revised form: October 16, 2009
Published electronically: March 11, 2010
Additional Notes: This research was supported by a grant from the Foundational Questions Institute.
Article copyright: © Copyright 2010 John C. Baez and John Huerta

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