Meeting Notes 2025-03-04

This Week’s Finds

No finds this week.

 

Focus Session

Speaker: Joshua
Title: Spinors in 3+1D: Chirality and the Riesz Definition

 

1. Overview

• This is the second talk on spinors, focusing on new features in 3+1 dimensions.
• Two goals:
  1. Identify complications that arise in 3+1D, notably chirality absent in Pauli spinors.
  2. Present Riesz’s definition: “spinors are members of minimal left ideals of a Clifford algebra.”

 

2. Main Content

1. 3+1D Complications & Chirality
   • Pauli spinors live in \(\mathbb{R}^3\); no intrinsic notion of handedness.
   • In \(\mathbb{R}^{3,1}\), the gamma matrices \(\gamma^\mu\) generate \(\mathrm{Cl}(3,1)\), and one defines \(\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3\).
   • Chiral (Weyl) projectors \(P_{L,R} = \frac{1}{2}(1 \mp \gamma^5)\) split a Dirac spinor \(\psi\) into left- and right-handed components:
     • \(\psi_L = P_L \psi\)
     • \(\psi_R = P_R \psi\)
   • Physical significance: chirality in 3+1D underlies parity violation in weak interactions.
2. Riesz Definition of Spinors
   • Clifford algebra \(\mathrm{Cl}(3,1)\) generated by basis vectors \(e_\mu\) with \(e_\mu e_\nu + e_\nu e_\mu = 2\eta_{\mu\nu}\).
   • A minimal left ideal is of the form \(\mathrm{Cl}(3,1)\,f\) where \(f\) is a primitive idempotent, e.g.
     \(P = \frac{1}{2}(1 + e_0e_1e_2e_3)\), satisfying \(P^2 = P\).
   • Elements of \(\mathrm{Cl}(3,1)\,P\) transform as spinors under the spin group \({\rm Spin}(3,1)\).
   • Equivalence to matrix representations: one recovers 4-component Dirac spinors by choosing a matrix realization of \(\mathrm{Cl}(3,1)\).

 

3. Discussion Points

• Geometric interpretation of chirality in terms of volume elements in the algebra.
• How different choices of idempotent affect the explicit form of spinor ideals.
• Relation between Riesz’s abstract definition and concrete Weyl/Dirac representations.

 

Up Next

Next Talk:

• Alice on Characteristic Classes and Their Applications

 

References

1. Riesz, “Clifford Numbers and Spinors,” Arkiv för Matematik, 1958.
2. Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus.
3. Spinor (Wikipedia)
4. Clifford Algebra (Wikipedia)