EigenChris: Spinors for Beginners
Visual, intuitive introduction to spinor mathematics essential for quantum field theory
About This Series
EigenChris's "Spinors for Beginners" series provides an exceptionally clear, visual introduction to spinor theory. Understanding spinors is essential for the Dirac field and fermionic quantum field theory. These 25 lectures build from basic concepts through to the Dirac equation and quantum field theory.
π¨ Visual Approach
Geometric intuition with excellent diagrams and animations
π Progressive Build
From Jones vectors through Clifford algebras to QFT
π QFT Connection
Direct path to understanding Dirac and Weyl spinors
Why Study Spinors?
For Quantum Field Theory
- β’ Dirac Field: Electrons, quarks - all fermions are spinor fields
- β’ Weyl Spinors: Massless fermions and chiral theories
- β’ Lorentz Group: Spinors transform under SL(2,β)
- β’ Spin-Statistics: Understanding why fermions anticommute
- β’ Links to Dirac Field (Part I)
Mathematical Beauty
- β’ Double Covers: SU(2) β SO(3), SL(2,β) β SOβΊ(1,3)
- β’ Clifford Algebras: Unify rotations and reflections
- β’ Geometric Algebra: Powerful framework beyond matrices
- β’ Representation Theory: Irreps of Lie groups
- β’ Foundation for understanding gauge theory structure
Course Structure
Introduction (1-7)
Jones vectors, polarization, Pauli matrices, geometric picture
Clifford Algebras (11-15)
Geometric algebra, spin groups, ideals, fermions
Lie Theory (16-20)
Lie groups/algebras, SU(2) reps, Lorentz group
QFT Applications (21-25)
Klein-Gordon, Dirac, Proca, Maxwell equations
Part I: Introduction & Pauli Spinors (Lectures 1-10)
Introduction (Overview + Table of Contents)
Course roadmap: from Jones vectors to quantum field theory, why spinors matter
Video Lecture
Spinors for Beginners 1: Introduction
Complete overview of the spinor series and learning path
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Jones Vectors and Light Polarization
Physical introduction: polarized light as the first example of spinors
Video Lecture
Spinors for Beginners 2: Jones Vectors and Light Polarization
Classical optics provides intuition for quantum spin
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Polarizations and SU(2) Matrices
Connection to rotation groups: SU(2), U(2), SO(3), O(3) relationships
Video Lecture
Spinors for Beginners 3: Polarizations and SU(2) Matrices
How spinors transform under rotations
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Quantum Spin States (Stern-Gerlach Experiment)
Physical realization: measuring electron spin, quantum measurement
Video Lecture
Spinors for Beginners 4: Quantum Spin States
Experimental foundation for spinor theory
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
The Flagpole and Complex Projective Line (CPΒΉ)
Geometric visualization: Riemann sphere, Bloch sphere, projective geometry
Video Lecture
Spinors for Beginners 5: The Flagpole and Complex Projective Line
Beautiful geometric picture of spinor space
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Pauli Vectors and Pauli Matrices
The fundamental Ο matrices: Οβ, Οβ, Οβ and their geometric meaning
Video Lecture
Spinors for Beginners 6: Pauli Vectors and Pauli Matrices
Foundation of spinor algebra
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Equivalence of Quaternions, Sigma Matrices, and SU(2)
Deep connection: how quaternions, Pauli matrices, and SU(2) are all the same structure
Video Lecture
Spinors for Beginners 6.1: Equivalence of Quaternions, Sigma Matrices, and SU(2)
Unifying different mathematical structures
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Square Root of a Vector (Factoring Vectors into Spinors)
Key insight: spinors are "square roots" of vectors - fundamental to QFT
Video Lecture
Spinors for Beginners 7: Square Root of a Vector
Why Dirac needed spinors for his equation
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Are the Pauli Matrices also Vectors? (Intro to Spinor Spaces)
Distinguishing vector spaces from spinor spaces, index notation
Video Lecture
Spinors for Beginners 8: Are the Pauli Matrices also Vectors?
Understanding the structure of spinor spaces
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Pauli Spinors vs Weyl Spinors vs Dirac Spinors
Different types of spinors in QFT: when and why to use each
Video Lecture
Spinors for Beginners 9: Pauli Spinors vs Weyl Spinors vs Dirac Spinors
Taxonomy of spinors in physics
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
SU(2) double covers SO(3), SL(2,β) double covers SOβΊ(1,3)
The famous 2:1 covering: why spinors pick up minus signs under 2Ο rotation
Video Lecture
Spinors for Beginners 10: SU(2) double covers SO(3)
Understanding the spin-statistics connection
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Part II: Clifford Algebras & Spin Groups (Lectures 11-15)
What is a Clifford Algebra?
Geometric algebra, Grassmann algebra, exterior algebra - the grand unification
Video Lecture
Spinors for Beginners 11: What is a Clifford Algebra?
Foundation of modern geometric algebra
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
How the Spin Group Generalizes Quaternions to any Dimension
Spin(n) groups: rotations via Clifford algebra in arbitrary dimensions
Video Lecture
Spinors for Beginners 12: How the Spin Group Generalizes Quaternions
Extending spinors beyond 3D
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Ideals and Projectors (Idempotents)
Algebraic structure: how to extract spinor spaces from Clifford algebras
Video Lecture
Spinors for Beginners 13: Ideals and Projectors
Mathematical machinery for spinors
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Minimal Left Ideals (and Pacwoman Property)
Constructing spinor representations systematically
Video Lecture
Spinors for Beginners 14: Minimal Left Ideals
Algebraic construction of spinors
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Nilpotents, Fermions, and Maximally Isotropic Subspaces
Deep connection: nilpotent elements and fermionic creation operators
Video Lecture
Spinors for Beginners 15: Nilpotents, Fermions, and Maximally Isotropic Subspaces
Why fermions anticommute - geometric perspective
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Part III: Lie Groups & Representation Theory (Lectures 16-20)
Lie Groups and Lie Algebras
Systematic treatment: generators, structure constants, exponential map
Video Lecture
Spinors for Beginners 16: Lie Groups and Lie Algebras
Essential for understanding gauge theories
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
The spin 1/2 representations of SU(2) and SL(2,β)
Fundamental spinor representations: rotation group and Lorentz group
Video Lecture
Spinors for Beginners 17: The spin 1/2 representations
The defining representations of spin
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Irreducible Representations of SU(2) (Ladder Operators)
Higher spin representations: j = 0, 1/2, 1, 3/2, ... using raising/lowering operators
Video Lecture
Spinors for Beginners 18: Irreducible Representations of SU(2)
Complete classification of angular momentum
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Tensor Product Representations of su(2) [Clebsch-Gordan coefficients]
Adding angular momentum: β decomposition and CG coefficients
Video Lecture
Spinors for Beginners 19: Tensor Product Representations
Combining spin states in quantum mechanics
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lorentz Group / Algebra Representation Theory
Relativistic spinors: (1/2,0) and (0,1/2) representations, left/right Weyl spinors
Video Lecture
Spinors for Beginners 20: Lorentz Group Representation Theory
Foundation for relativistic quantum mechanics
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Part IV: Quantum Field Theory Applications (Lectures 21-25)
Introduction to Quantum Field Theory from the ground up
QFT foundations: why fields, Lagrangian formalism, relativistic wave equations
Video Lecture
Spinors for Beginners 21: Introduction to Quantum Field Theory
Bridging spinor theory to QFT
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Dirac Equation and Gamma Matrices Deep Dive (+ chirality)
The famous equation: Ξ³α΅βα΅€Ο + mΟ = 0, gamma matrix algebra, chiral projections
Video Lecture
Spinors for Beginners 22: Dirac Equation and Gamma Matrices
Complete treatment of the Dirac equation
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Klein-Gordon Equation (derivation + solutions)
Scalar field equation: (β‘ + mΒ²)Ο = 0, plane wave solutions, issues with probability
Video Lecture
Spinors for Beginners 23: Klein-Gordon Equation
Simplest relativistic wave equation
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Proca and Maxwell Equations (derivation + solutions)
Vector field equations: massive (Proca) and massless (Maxwell) spin-1 fields
Video Lecture
Spinors for Beginners 24: Proca and Maxwell Equations
Gauge bosons and photons
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Dirac Equation (derivation + solutions) - Chiral, Weyl, Mass, Dirac Basis
Complete Dirac field: all representations, Weyl vs Dirac basis, massless limit
Video Lecture
Spinors for Beginners 25: Dirac Equation (complete treatment)
Bringing everything together: spinors in QFT
π‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
How to Use This Series
π Before Studying QFT
Watch lectures 1-20 before starting MIT Part I: Dirac Field
- β’ Build geometric intuition for spinors
- β’ Understand SU(2) and Lorentz group reps
- β’ Master Clifford algebra basics
- β’ Be ready for Dirac equation derivation
π Alongside QFT Course
Use lectures 21-25 as visual companions to MIT lectures
- β’ Lecture 22-25 complement Dirac Field
- β’ Lecture 21 pairs with Classical Field Theory
- β’ Lectures 11-15 deepen understanding of spinor quantization
- β’ Return to lectures 16-20 for gauge theory prep
Integrated Study Path
EigenChris 1-10: Build spinor intuition (Jones vectors β double covering)
MIT Part I: Lagrangian formalism, scalar fields
EigenChris 11-20: Clifford algebras and Lorentz group rep theory
MIT + EigenChris 21-25: Dirac field with visual supplements
Continue MIT Parts II-V: With solid spinor foundation
Additional Resources
- β’ Full YouTube Playlist:Spinors for Beginners by EigenChris
- β’ Creator:@eigenchris on YouTube
- β’ Other Series:Tensor Calculus, General Relativity, Differential Geometry
- β’ Prerequisites:Linear algebra, basic complex numbers, some quantum mechanics helpful