Table of Contents
- Introduction
- What Are Spinors?
- Motivation from Lorentz Group Representations
- Spinor Transformation Properties
- Types of Spinors
- Dirac Spinors and 4-Component Representation
- Gamma Matrices and Clifford Algebra
- Representations of Gamma Matrices
- Dirac Basis
- Weyl (Chiral) Basis
- Majorana Basis
- Properties of Gamma Matrices
- The Fifth Gamma Matrix: \(\gamma^5\)
- Spinor Bilinears
- Lorentz Covariance and Spinors
- Projection Operators
- Charge Conjugation and Antiparticles
- Fierz Identities
- Applications in Field Theory
- Conclusion
1. Introduction
Spinors and gamma matrices are fundamental in describing fermions (particles with half-integer spin) in relativistic quantum field theory. They form the algebraic backbone of the Dirac equation and the Standard Model. Understanding their transformation properties and internal structure is crucial for mastering particle physics and quantum field theory.
2. What Are Spinors?
Spinors are mathematical objects that transform under the spinor representation of the Lorentz group. Unlike vectors or tensors, spinors pick up a sign change under a full \( 2\pi \) rotation:
\[
\psi \rightarrow -\psi
\]
This property distinguishes them from classical objects and makes them suitable for describing fermions like electrons and quarks.
3. Motivation from Lorentz Group Representations
The Lorentz group \( SO(1,3) \) has representations built from the covering group \( SL(2, \mathbb{C}) \). Under this group:
- Left-handed Weyl spinors transform as \( (1/2, 0) \)
- Right-handed Weyl spinors transform as \( (0, 1/2) \)
- Dirac spinors combine both
4. Spinor Transformation Properties
Under Lorentz transformations:
\[
\psi(x) \rightarrow S(\Lambda)\psi(\Lambda^{-1}x)
\]
where \( S(\Lambda) \) is a matrix in the spinor representation satisfying:
\[
S^{-1}(\Lambda)\gamma^\mu S(\Lambda) = \Lambda^\mu{}_\nu \gamma^\nu
\]
5. Types of Spinors
- Weyl spinors: 2-component, massless fermions, chiral basis
- Dirac spinors: 4-component, massive fermions, combine two Weyl spinors
- Majorana spinors: real spinors, \( \psi = \psi^C \)
6. Dirac Spinors and 4-Component Representation
Dirac spinors \( \psi \) are four-component complex objects. A general solution to the Dirac equation involves:
\[
\psi = \begin{pmatrix}
\chi \
\eta
\end{pmatrix}
\]
where \( \chi \) and \( \eta \) are 2-component spinors.
7. Gamma Matrices and Clifford Algebra
The gamma matrices \( \gamma^\mu \) satisfy:
\[
\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu} I
\]
This is the Clifford algebra associated with the Minkowski metric \( \eta^{\mu\nu} = \text{diag}(1, -1, -1, -1) \).
8. Representations of Gamma Matrices
The algebra permits many equivalent representations:
- Dirac basis: standard for massive fermions
- Weyl (chiral) basis: separates left/right spinors
- Majorana basis: used for real-valued spinors
9. Dirac Basis
In the Dirac basis:
\[
\gamma^0 =
\begin{pmatrix}
I & 0 \
0 & -I
\end{pmatrix},
\quad
\gamma^i =
\begin{pmatrix}
0 & \sigma^i \
-\sigma^i & 0
\end{pmatrix}
\]
10. Weyl (Chiral) Basis
In the chiral basis:
\[
\gamma^0 =
\begin{pmatrix}
0 & I \
I & 0
\end{pmatrix},
\quad
\gamma^i =
\begin{pmatrix}
0 & \sigma^i \
-\sigma^i & 0
\end{pmatrix}
\]
\[
\gamma^5 =
\begin{pmatrix}
-I & 0 \
0 & I
\end{pmatrix}
\]
11. Majorana Basis
Used in theories involving real spinors where \( \psi = \psi^C \). All gamma matrices are imaginary, making \( \psi \) real-valued.
12. Properties of Gamma Matrices
- Traces:
\[
\text{Tr}(\gamma^\mu) = 0, \quad \text{Tr}(\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu}
\] - Antisymmetric products:
\[
\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu, \gamma^\nu]
\]
13. The Fifth Gamma Matrix: \(\gamma^5\)
Defined as:
\[
\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3
\]
Properties:
- Anticommuting: \( \{\gamma^5, \gamma^\mu\} = 0 \)
- \( (\gamma^5)^2 = I \)
- Used in defining chirality
14. Spinor Bilinears
Useful for constructing Lorentz-invariant or covariant quantities:
- Scalar: \( \bar{\psi}\psi \)
- Vector: \( \bar{\psi}\gamma^\mu\psi \)
- Pseudoscalar: \( \bar{\psi}\gamma^5\psi \)
- Axial vector: \( \bar{\psi}\gamma^\mu\gamma^5\psi \)
- Tensor: \( \bar{\psi}\sigma^{\mu\nu}\psi \)
15. Lorentz Covariance and Spinors
The transformation properties of spinors ensure that bilinear combinations transform as scalars, vectors, or tensors under Lorentz transformations.
16. Projection Operators
To isolate chiral components:
\[
P_L = \frac{1 – \gamma^5}{2}, \quad P_R = \frac{1 + \gamma^5}{2}
\]
These satisfy \( P_L + P_R = 1 \), \( P_L^2 = P_L \), and \( P_R^2 = P_R \).
17. Charge Conjugation and Antiparticles
The charge conjugate spinor:
\[
\psi^C = C\bar{\psi}^T
\]
where \( C \) is the charge conjugation matrix satisfying:
\[
C\gamma^\mu C^{-1} = -(\gamma^\mu)^T
\]
Used in defining Majorana fermions.
18. Fierz Identities
These are algebraic identities that relate different spinor bilinears and are important in simplifying expressions in fermionic interaction terms.
19. Applications in Field Theory
- Dirac equation and spin-½ dynamics
- Anomalies and symmetry breaking
- Neutrino mass models (Majorana vs Dirac)
- Spinor field quantization in QED and the Standard Model
20. Conclusion
Spinors and gamma matrices are essential for describing fermions in quantum field theory. Their algebraic structure and transformation properties provide the tools to handle spin-½ particles consistently with relativity and quantum mechanics. Mastery of spinors is foundational for further study in particle physics, gauge theory, and string theory.
