Table of Contents
- Introduction
- Motivation and Historical Context
- The Need for a Relativistic Wave Equation
- Inadequacy of the Klein-Gordon Equation
- Dirac’s Insight: First-Order Equation
- Mathematical Form of the Dirac Equation
- Gamma Matrices and Clifford Algebra
- Covariant Form and Notation
- Plane Wave Solutions and Spinors
- Interpretation of Solutions
- Probability Current and Density
- Negative Energy States and Antiparticles
- Dirac Sea and Hole Theory
- Spin and Magnetic Moment of the Electron
- Chirality and Helicity
- Lorentz Covariance of the Dirac Equation
- Dirac Field Quantization
- Applications in Particle Physics
- Experimental Validations
- Conclusion
1. Introduction
The Dirac equation is a cornerstone of modern physics, providing the first successful description of spin-½ particles that is consistent with both quantum mechanics and special relativity. It predicts antimatter and lays the foundation for quantum electrodynamics (QED) and modern quantum field theory.
2. Motivation and Historical Context
Formulated by Paul Dirac in 1928, the equation addressed the need for a relativistic theory of the electron that preserved the probabilistic interpretation of quantum mechanics and incorporated intrinsic spin.
3. The Need for a Relativistic Wave Equation
The non-relativistic Schrödinger equation:
\[
i \frac{\partial \psi}{\partial t} = -\frac{1}{2m} \nabla^2 \psi + V\psi
\]
is not Lorentz invariant. A relativistic theory must adhere to the relation:
\[
E^2 = p^2 + m^2
\]
4. Inadequacy of the Klein-Gordon Equation
While the Klein-Gordon equation is Lorentz invariant:
\[
(\Box + m^2) \phi = 0
\]
it fails to:
- Properly describe spin-½ particles.
- Provide a positive-definite probability density.
- Avoid negative energy complications in a satisfactory manner.
5. Dirac’s Insight: First-Order Equation
Dirac proposed a first-order differential equation in both time and space:
\[
(i \gamma^\mu \partial_\mu – m)\psi = 0
\]
This allowed the theory to remain Lorentz invariant and consistent with quantum mechanical interpretations.
6. Mathematical Form of the Dirac Equation
In natural units \( (\hbar = c = 1) \):
\[
(i \gamma^\mu \partial_\mu – m)\psi(x) = 0
\]
where:
- \( \psi(x) \): 4-component Dirac spinor
- \( \gamma^\mu \): gamma matrices
- \( m \): mass of the particle
7. Gamma Matrices and Clifford Algebra
The gamma matrices \( \gamma^\mu \) satisfy the Clifford algebra:
\[
\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu} I_4
\]
where:
- \( \eta^{\mu\nu} \): Minkowski metric (mostly minus signature)
- \( I_4 \): \( 4 \times 4 \) identity matrix
Common representations: Dirac, Weyl, and Majorana.
8. Covariant Form and Notation
The equation can be rewritten:
\[
\gamma^\mu \partial_\mu \psi = m \psi
\quad \Rightarrow \quad
\not{\partial} \psi = m \psi
\]
where \( \not{\partial} = \gamma^\mu \partial_\mu \) is the Feynman slash notation.
9. Plane Wave Solutions and Spinors
Solutions are of the form:
\[
\psi(x) = u(p) e^{-ipx}, \quad \bar{\psi}(x) = \bar{u}(p) e^{ipx}
\]
and satisfy:
\[
(\not{p} – m)u(p) = 0, \quad \bar{u}(p)(\not{p} – m) = 0
\]
These spinors encode spin degrees of freedom.
10. Interpretation of Solutions
- \( u(p) \): Positive-energy spinors
- \( v(p) \): Negative-energy spinors (later interpreted as antiparticles)
- Spinors transform under Lorentz transformations and encode spin-½ structure.
11. Probability Current and Density
Define:
\[
j^\mu = \bar{\psi} \gamma^\mu \psi
\]
This satisfies the continuity equation:
\[
\partial_\mu j^\mu = 0
\]
and gives a positive-definite time component \( j^0 \), unlike the Klein-Gordon theory.
12. Negative Energy States and Antiparticles
The equation admits negative-energy solutions. Dirac interpreted this as implying the existence of antiparticles, with the positron being the antiparticle of the electron — later experimentally discovered.
13. Dirac Sea and Hole Theory
Dirac proposed that all negative energy states are filled (Dirac sea). A hole in the sea appears as a positron — a particle with positive energy and opposite charge.
14. Spin and Magnetic Moment of the Electron
The Dirac equation naturally incorporates spin and predicts the g-factor of the electron as:
\[
g = 2
\]
Corrections arise from quantum electrodynamics and match experimental precision.
15. Chirality and Helicity
Define projection operators:
\[
P_L = \frac{1}{2}(1 – \gamma^5), \quad P_R = \frac{1}{2}(1 + \gamma^5)
\]
These project the spinor into left- and right-chiral components. In the massless limit, chirality equals helicity.
16. Lorentz Covariance of the Dirac Equation
The Dirac equation transforms covariantly under Lorentz transformations. The spinor field transforms via the spinor representation of the Lorentz group.
17. Dirac Field Quantization
The field operator is expanded as:
\[
\psi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \sum_s \left[ a_{p,s} u_s(p) e^{-ipx} + b^\dagger_{p,s} v_s(p) e^{ipx} \right]
\]
Anticommutation relations:
\[
\{a_{p,s}, a^\dagger_{p’,s’}\} = (2\pi)^3 \delta^3(p – p’)\delta_{ss’}
\]
18. Applications in Particle Physics
- Predicts antimatter
- Basis for quantum electrodynamics (QED)
- Describes fermions in the Standard Model
- Used in relativistic corrections in atomic physics
19. Experimental Validations
- Prediction and discovery of the positron
- Fine structure of hydrogen
- Electron’s magnetic moment
- Pair production and annihilation
20. Conclusion
The Dirac equation revolutionized physics by combining relativity and quantum theory in a consistent framework for spin-½ particles. It predicted antimatter and laid the foundation for modern particle physics. Its conceptual and mathematical structure continues to influence theoretical developments to this day.
