Group-Algebraic Characterization of Spin Particles by Francis Howard
Channel: Math 4 Wisdom
Date: 2023-05-30

"Group-Algebraic Characterization of Spin Particles: Semi-Simplicity, SO(2N) Structure and Iwasawa Decomposition"

Francis Howard gives this inaugural Math 4 Wisdom Distinguished Lecture.

Slides: http://www.math4wisdom.com/files/FrancisHoward_Group_Algebraic_Math4Wisdom_20230529.pdf

This is joint work with his advisors Mahouton Norbert Hounkonnou and Kinvi Kangni.  All three are affiliated with International Chair of Mathematical Physics and Applications, (ICMPA-UNESCO Chair), University of Abomey-Calavi, 072 B.P. 50 Cotonou, Republic of Benin.

Francis Howard is awaiting his PhD defense this year and applying for PostDoc positions.

Francis Howard presented his lecture to Thomas Gajdosik of Vilnius University and Andrius Kulikauskas of Math 4 Wisdom.

1:50 Motivation
5:25 Para-fermionic algebra
6:50 Question by Thomas Gajdosik whether there should be both creation and annihilation operators
16:00 Para-fermionic algebra - continued
17:15 Semi-simple Lie algebra generated by creation and annihilation operators
18:35 Lie algebra of spin group
23:00 Spin Lie group and its Lie algebra
26:50 Spin Lie Group
30:30 SU(2) and Wigner coefficients
32:30 Spin Semi-simplicity
45:00 Francis answers questions from Andrius about almost complex manifolds, and also about twofold and fourfold covers, and the meaning of fermions and bosons as mathematical structures
53:15 Real Lie algebra of Spin particle
1:00:40 The difference between mathematical spin and physical spin.
1:01:40 Iwasawa Decomposition of Spin(1/2) particle
1:05:45 Iwasawa decomposition of Lie algebra and Lie group Levels
1:14:00 Andrius checks his understanding regarding the decomposition
1:17:50 Theorem V: Main Result: Particle Decomposition 
1:28:30 Questions by Andrius on parameters for the torus, and on the decomposition and its relation to "division of everything", and on decomposition of SU(2) into rotations into real, complex and quaternionic matrices. Francis relates the latter to the polar decomposition, which is particular to SU(2). Andrius associates that to the learning three-cycle.
1:39:50 Is Spin(n) in Mathematics the same as the Spin(J) in particle Physics?
1:43:00 SU(5) understood as the standard model SU(3) X SU(2) X U(1) shows that mathematical spin and physical spin cannot be the same.
1:46:20 Concluding Remarks
1:47:45 Encouraging and supportive remarks by Thomas
1:51: 00 Question on behalf of John Harland, how could the fine structure constant have evolved? Andrius: Could they likewise evolve the transcendental numbers pi and e which are involved in the decomposition?
1:55:20 Andrius: What better word could there be for physical spin?
1:58:20 Andrius: How to fit gravity and the standard model? Francis explains that SU(5) can be thought in two ways as the standard model and as gravity, which suggests that particles can be defined in terms of gravitational wave equations
2:02:40 Invitation to viewers to study with us
2:04:25 Andrius: What could be the role of the combinatorics of the Jacobi polynomials?
2:09:30 Andrius asks: What is a particle? Francis: A particle is a matter or anti-matter or dark matter but it must be either a fermion (half integral spin), a boson (integral spin) or a Higgs.
2:10:40 Andrius explains why there are four classical families of Lie groups/algebras based on their root systems. What could that say about bosons and fermions? Francis relates bosons and fermions.
2:16:30 Thomas says a concluding prayer.

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