7. Relativistic QM & Dirac Equation

First attempt: relativistic energy-momentum relation $E^2 = p^2c^2 + m^2c^4$

Promote to operators $E \to i\hbar\partial_t$, $\vec{p} \to -i\hbar\nabla$:

Or in covariant form: $(\Box + m^2)\phi = 0$ where $\Box = \partial_\mu\partial^\mu$

Problems:

• Second-order in time - requires both $\phi$ and $\dot{\phi}$ for initial conditions
• Probability density $\rho = \frac{i\hbar}{2mc^2}(\phi^*\dot{\phi} - \phi\dot{\phi}^*)$ not positive definite!
• No clear single-particle interpretation

Resolution: Klein-Gordon describes spin-0 particles in QFT framework, not wave function of single particle

Dirac (1928) sought first-order equation in time and space:

Where $\vec{\alpha} = (\alpha_1, \alpha_2, \alpha_3)$ and $\beta$ are to be determined

Require consistency with $E^2 = p^2c^2 + m^2c^4$:

Smallest matrices satisfying these are 4×4! $\psi$ must be 4-component spinor:

Introduce gamma matrices $\gamma^\mu = (\gamma^0, \gamma^1, \gamma^2, \gamma^3)$:

Clifford algebra: $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}\mathbb{I}$

Dirac equation:

Or using Feynman slash notation: $(i\!\!\!/\partial - m)\psi = 0$

Standard representation (Dirac-Pauli):

1. Conserved probability current:

Satisfies $\partial_\mu j^\mu = 0$; density $\rho = \psi^\dagger\psi \geq 0$ (positive!)

2. Spin naturally emerges:

Dirac particle has intrinsic spin-1/2 (not put in by hand!)

3. Negative energy solutions:

Free particle solutions have $E = \pm\sqrt{p^2c^2 + m^2c^4}$

Initially problematic - led to Dirac sea interpretation

Correctly interpreted in QFT: negative energy = antiparticles!

Plane wave solutions:

Four independent spinors for given momentum:

• Two positive energy: $u^{(1)}, u^{(2)}$ (spin up/down)
• Two negative energy: $v^{(1)}, v^{(2)}$ (antiparticle spin up/down)

At rest ($\vec{p} = 0$):

Write spinor as $\psi = \begin{pmatrix} \phi \\ \chi \end{pmatrix}$ (2-component each)

In limit $v \ll c$, $\chi$ is small and Dirac equation reduces to:

This is the Pauli equation with spin!

Including electromagnetic field $\vec{p} \to \vec{p} - e\vec{A}$:

Expanding $(\vec{\sigma} \cdot \vec{\pi})^2 = \vec{\pi}^2 + \vec{\sigma} \cdot (\vec{\pi} \times \vec{\pi})$ gives magnetic moment:

Predicts $g = 2$ for electron (confirmed experimentally!)

Negative energy solutions require reinterpretation:

Dirac sea (historical):

All negative energy states filled (Pauli exclusion)

Hole in sea = positive energy, opposite charge = antiparticle

Modern interpretation (QFT):

Negative frequency mode $v e^{ip \cdot x}$ reinterpreted as positive energy antiparticle $e^{-ip \cdot x}$

Prediction confirmed:

Positron (anti-electron) discovered by Anderson in 1932, just 4 years after Dirac equation!

All fundamental fermions now known to have antiparticles

Exact solution for Coulomb potential $V = -e^2/(4\pi\epsilon_0 r)$:

Where $\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137$ is the fine structure constant

Fine structure:

Energy depends on total angular momentum $j = l \pm 1/2$, not just $n$

Splitting of order $\alpha^2 E_n \sim 10^{-5}$ eV (precisely measured!)

Corrections account for:

• Spin-orbit coupling
• Relativistic kinetic energy correction
• Darwin term (zitterbewegung effect)

Dirac equation limitations:

• Single-particle interpretation breaks down at high energies ($E > 2mc^2$ allows pair creation)
• Cannot describe particle creation/annihilation processes
• Vacuum polarization and other QED effects require field theory

Resolution: Quantum Field Theory

Dirac field $\hat{\psi}(x)$ becomes operator creating/annihilating particles

$b_s^\dagger, d_s^\dagger$ create particles and antiparticles

QED (Quantum Electrodynamics) combining Dirac and Maxwell fields:

• Most precisely tested theory in physics ($g-2$ to 12 decimal places)
• Template for Standard Model of particle physics
• Foundation for all modern particle theory