Skip to Main Content

Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The algebra of grand unified theories
HTML articles powered by AMS MathViewer

by John Baez and John Huerta PDF
Bull. Amer. Math. Soc. 47 (2010), 483-552


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 $\textrm {Spin}(10)$, 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.
  • J. F. Adams, Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. With a foreword by J. Peter May; Edited by Zafer Mahmud and Mamoru Mimura. MR 1428422
  • M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl, suppl. 1, 3–38. MR 167985, DOI 10.1016/0040-9383(64)90003-5
  • John Baez and Javier P. Muniain, Gauge fields, knots and gravity, Series on Knots and Everything, vol. 4, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1313910, DOI 10.1142/2324
  • Stefano Bertolini, Luca Di Luzio and Michal Malinsky, Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal, available at arXiv:0903.4049.
  • Lowell S. Brown, Quantum field theory, Cambridge University Press, Cambridge, 1992. MR 1231339, DOI 10.1017/CBO9780511622649
  • Claude Chevalley, The algebraic theory of spinors and Clifford algebras, Springer-Verlag, Berlin, 1997. Collected works. Vol. 2; Edited and with a foreword by Pierre Cartier and Catherine Chevalley; With a postface by J.-P. Bourguignon. MR 1636473
  • 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.
  • Robert P. Crease and Charles C. Mann, The second creation, Rutgers University Press, New Brunswick, NJ, 1996. Makers of the revolution in twentieth-century physics; With a foreword by Timothy Ferris; Revised reprint of the 1986 original. MR 1677561
  • Andrzej Derdziński, Geometry of the standard model of elementary particles, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. MR 1219312, DOI 10.1007/978-3-642-50310-8
  • 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.
  • Howard Georgi, Lie algebras in particle physics, Frontiers in Physics, vol. 54, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1982. From isospin to unified theories; With an introduction by Sheldon L. Glashow. MR 644800
  • Howard Georgi and Sheldon Glashow, Unity of all elementary-particle forces, Phys. Rev. Lett. 32(8) Feb 1974, 438–441.
  • David Griffiths, Introduction to Elementary Particles, Wiley, New York 1987.
  • Brian C. Hall, Lie groups, Lie algebras, and representations, Graduate Texts in Mathematics, vol. 222, Springer-Verlag, New York, 2003. An elementary introduction. MR 1997306, DOI 10.1007/978-0-387-21554-9
  • Werner Heisenberg, Zeitschr. f. Phys. 77 (1932), 1; English translation in D. M. Brink, Nuclear Forces, Pergamon, Oxford, 1965, pp. 144–154.
  • Lillian Hoddeson, Laurie Brown, Michael Riordan, and Max Dresden (eds.), The rise of the standard model, Cambridge University Press, Cambridge, 1997. Particle physics in the 1960s and 1970s; Papers from the Third International Symposium on the History of Particle Physics held at Stanford University, Stanford, CA. MR 1601138, DOI 10.1017/CBO9780511471094
  • Kerson Huang, Quarks, leptons & gauge fields, World Scientific Publishing Co., Singapore, 1982. MR 704019
  • Chris J. Isham, Modern differential geometry for physicists, 2nd ed., World Scientific Lecture Notes in Physics, vol. 61, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1698234, DOI 10.1142/3867
  • T. D. Lee, Particle physics and introduction to field theory, Contemporary Concepts in Physics, vol. 1, Harwood Academic Publishers, Chur, 1981. Translated from the Chinese. MR 633154
  • Harry J. Lipkin, Lie groups for pedestrians, Dover Publications, Inc., Mineola, NY, 2002. Reprint of the second edition [North-Holland, Amsterdam, 1966; MR0210386 (35 #1279)]. MR 1918338
  • Rabindra N. Mohapatra, Unification and supersymmetry, Contemporary Physics, Springer-Verlag, New York, 1986. The frontiers of quark-lepton physics. MR 854671, DOI 10.1007/978-1-4757-1928-4
  • Gregory L. Naber, Topology, geometry, and gauge fields, Texts in Applied Mathematics, vol. 25, Springer-Verlag, New York, 1997. Foundations. MR 1444352, DOI 10.1007/978-1-4757-2742-5
  • Gregory L. Naber, Topology, geometry, and gauge fields, Applied Mathematical Sciences, vol. 141, Springer-Verlag, New York, 2000. Interactions. MR 1744816, DOI 10.1007/978-1-4757-6850-3
  • Mikio Nakahara, Geometry, Topology, and Physics, Academic Press, 1983.
  • Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World, Oxford University Press, 1988.
  • Jogesh C. Pati, Proton decay: A must for theory, a challenge for experiment, available at arXiv:hep-ph/0005095.
  • 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 arXiv:hep-ph/0305221.
  • Jogesh C. Pati and Abdus Salam, Lepton number as the fourth “color”, Phys. Rev. D 10 (1974), 275–289.
  • Michael E. Peskin, Beyond the Standard Model, available at arXiv:hep-ph/970549.
  • Michael E. Peskin and Daniel V. Schroeder, An introduction to quantum field theory, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1995. Edited and with a foreword by David Pines. MR 1402248
  • Graham G. Ross, Grand unified theories, Frontiers in Physics, vol. 60, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, MA, 1984. MR 783830
  • Lewis H. Ryder, Quantum field theory, 2nd ed., Cambridge University Press, Cambridge, 1996. MR 1404519, DOI 10.1017/CBO9780511813900
  • Emilio Segrè, From X-Rays to Quarks: Modern Physicists and Their Discoveries, W.H. Freeman, San Francisco, 1980.
  • S. Sternberg, Group theory and physics, Cambridge University Press, Cambridge, 1994. MR 1287387
  • Mark Srednicki, Quantum Field Theory, Cambridge Univ. Press, 2007. Also available at$\sim$mark/qft.html.
  • Anthony Sudbery, Quantum mechanics and the particles of nature, Cambridge University Press, Cambridge, 1986. An outline for mathematicians. MR 2245407
  • R. Ticciati, Quantum field theory for mathematicians, Encyclopedia of Mathematics and its Applications, vol. 72, Cambridge University Press, Cambridge, 1999. MR 1699269, DOI 10.1017/CBO9780511526428
  • Michael Tinkham, Group Theory and Quantum Mechanics, Dover, Mineola, New York, 2003.
  • Edward Witten, Grand unification with and without supersymmetry, Introduction to supersymmetry in particle and nuclear physics (Mexico City, 1981) Plenum, New York, 1984, pp. 53–76. MR 758626
  • A. Zee, Quantum field theory in a nutshell, Princeton University Press, Princeton, NJ, 2003. MR 1978227
Similar Articles
  • Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 20C35, 81R05, 81-02
  • Retrieve articles in all journals with MSC (2000): 20C35, 81R05, 81-02
Additional Information
  • John Baez
  • Affiliation: Department of Mathematics, University of California, Riverside, California 92521
  • Email:
  • John Huerta
  • Affiliation: Department of Mathematics, University of California, Riverside, California 92521
  • Email:
  • 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.
  • © Copyright 2010 John C. Baez and John Huerta
  • Journal: Bull. Amer. Math. Soc. 47 (2010), 483-552
  • MSC (2000): Primary 20C35, 81R05; Secondary 81-02
  • DOI:
  • MathSciNet review: 2651086