John Baez is a mathematical physicist, professor of mathematics at UC Riverside, a researcher at the Centre for Quantum Technologies in Singapore, and a researcher at the Topos Institute in Berkeley, CA. John has worked on an impressively wide range of topics, pure and applied, ranging from loop quantum gravity, applications of higher categories to physics, applied category theory, environmental issues and math related to engineering and biology, and most recently on applying network theory to scientific software.

Additionally, John is a prolific writer and blogger. This first began with John’s column This Week's Finds in Mathematical Physics, which ran 300 issues between 1993 and 2010, which then continued in the form of his ongoing blog Azimuth. Last but not least, John is also a host and contributor of the popular blog The ncategory Cafe.

In this episode, we dive into John Baez and John Huerta’s paper “The Algebra of Grand Unified Theories” which was awarded the Levi Conant Prize in 2013. The paper gives a crash course in the representation theory underlying the Standard Model of particle physics and its three most well known Grand Unified Theories (GUTs): the SU(5), SO(10) (aka Spin(10)), and PatiSalam theories. The main result of BaezHuerta is that the particle representations underlying the three GUTs can in fact be unified via a commutative diagram. We dive deep into the numerology of the standard model to see how the SU(5) theory naturally arises. We then make brief remarks about the SO(10) and PatiSalam theories in order to state the BaezHuerta theorem about their organization into a commutative square: a unification among grand unifications!

#math #physics #standardmodel #quantum #theoryofeverything #grouptheory #particles #ericweinstein #briankeating

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Correction:
1;29;01: The formula for hypercharge in the bottom right note should be Y = 2(QI_3) instead of Y = (QI_3)/2.

Notes:
1) While we do provide a crash course on SU(2) and spin, some representation theory jargon is used at times in our discussion. Those unfamiliar should just forge ahead!
2) We work in Euclidean signature instead of Lorentzian signature. Other than keeping track of minus signs, no essential details are changed.

Part I. Introduction
00:00 : Introduction
05:50 : Climate change
09:40 : Crackpot index
14:50 : Eric Weinstein, Brian Keating, Geometric Unity
18:13 : Overview of “The Algebra of Grand Unified Theories” paper
25:40 : Overview of Standard Model and GUTs
34:25 : SU(2), spin, isospin of nucleons
40:22 : SO(4), Spin(4), double cover
44:24 : three kinds of spin

Part II. Zoology of Standard Model
49:35 : Electron and neutrino
58:40 : Quarks
1:04:51 : The three generations of the Standard Model
1:08:25 : Isospin quantum numbers
1:17:11 : U(1) representations (“charge”)
1:29:01 : Hypercharge.
1:34:00 : Strong force and color
1:36:50 : SU(3)
1:40:45 : Antiparticles

Part III. SU(5) numerology
1:41:16 : 32 = 2^5 particles
1:45:05 : Mapping SU(3) x SU(2) x U(1) to SU(5) and hypercharge matching
2:05:17 : Exterior algebra of C^5 and more hypercharge matching
2:37:32 : SU(5) rep extends Standard Model rep

Part IV. How the GUTs fit together
2:41:42 : SO(10) rep: brief remarks
2:46:28 : PatiSalam rep: brief remarks
2:47:17 : Commutative diagram: main result
2:49:08 : What about the physics? Spontaneous symmetry breaking and the Higgs mechanism

References:
J. Baez & J. Huerta. "The Algebra of Grand Unified Theories". https://arxiv.org/abs/0904.1556
Baez's Crackpot Index: https://math.ucr.edu/home/baez/crackp...

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