2010/03/11 by John C. Baez, John Huerta · 139 citations
Physics and Astronomy · Mathematics · #Black Holes and Theoretical Physics #Noncommutative and Quantum Gravity Theories #Particle physics theoretical and experimental studies #Group (periodic table) #Physics #Mathematics #Spin (aerodynamics) #Standard Model (mathematical formulation) #Theoretical physics #Group theory #Symmetry (geometry) #Symmetry group #Algebra over a field #Pure mathematics #Quantum mechanics #Geometry
paper · pdf · doi:10.1090/s0273-0979-10-01294-2
published in Bulletin of the American Mathematical Society 47(3), 483-552 (American Mathematical Society)
openalex publication_date 2010/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper U left-parenthesis 1 right-parenthesis times normal upper S normal upper U left-parenthesis 2 right-parenthesis times normal upper S normal upper U left-parenthesis 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm U(1) × \mathrm SU(2) × \mathrm SU(3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a larger group. These three are Georgi and Glashow’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal upper U left-parenthesis 5 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm SU(5)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> theory, Georgi’s theory based on the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Spin left-parenthesis 10 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Spin</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>10</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\textrm Spin(10)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the Pati–Salam model based on the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal upper U left-parenthesis 2 right-parenthesis times normal upper S normal upper U left-parenthesis 2 right-parenthesis times normal upper S normal upper U left-parenthesis 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm SU(2) × \mathrm SU(2) × \mathrm SU(4)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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.