Playground
Relativistic dispersion relation showing positive and negative energy branches. Gap = 2mc^2. Slider controls p/mc.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Dirac spinoroutput Four-component relativistic wavefunction | m^-3/2 | L^-3/2 | 0 – 1 | |
| Gamma matrices 4x4 matrices satisfying the Clifford algebra {gamma_mu, gamma_nu} = 2*eta_mn | dimensionless | 1 | 0 – 1 | |
| Particle mass Rest mass of the fermion (e.g., electron) | kg | M | 9.1093837015e-31 – 1.67262192369e-27 | |
| Speed of light Speed of light in vacuum | m/s | L*T^-1 | 299792458 – 299792458 | |
| Reduced Planck constant h / (2*pi) | J*s | M*L^2*T^-1 | 1.054571817e-34 – 1.054571817e-34 |
Deep dive
Derivation
Dirac sought a first-order equation (i*hbar*gamma_mu*d_mu - m*c)*psi = 0 whose square gives the Klein-Gordon equation (E^2 = p^2*c^2 + m^2*c^4). Requiring this fixes the gamma matrices to satisfy {gamma_mu, gamma_nu} = 2*eta_mn*I. The minimal rep is 4x4, forcing psi to have 4 components — two for spin up/down and two for particle/antiparticle.
Experimental verification
Predicted positron discovered 1932 by Anderson. Hydrogen fine structure, electron g-factor (Dehmelt 1987, now measured to 10^-13), and Lamb shift all confirm Dirac + QED corrections.
Common misconceptions
- The four components are NOT four spatial dimensions — they are internal spinor indices
- Negative energy solutions are NOT unphysical; they describe antiparticles
- The Dirac equation is first-order in both space and time, unlike Schrödinger (first in time, second in space)
Real-world applications
- Relativistic band structure in heavy-element chemistry
- Graphene low-energy excitations (massless Dirac fermions)
- Positron Emission Tomography (PET) imaging
- Quantum electrodynamics (QED) foundation
Worked examples
Rest energy of electron
Given:
- m:
- 9.1093837015e-31
- p:
- 0
Find: E
Solution
E = m*c^2 ≈ 0.511 MeV
Relativistic electron at p = mc
Given:
- m:
- 9.1093837015e-31
- p_over_mc:
- 1
Find: E/mc^2
Solution
E = sqrt(2)*m*c^2 ≈ 0.723 MeV
Scenarios
What if…
- scenario:
- What if m = 0?
- answer:
- Recovers the Weyl equation — massless chiral fermions like neutrinos in the Standard Model approximation.
- scenario:
- What if we ignore negative-energy solutions?
- answer:
- Theory becomes inconsistent — vacuum instability and loss of completeness. Antimatter is mathematically necessary.
- scenario:
- What if gamma matrices were 2x2?
- answer:
- Cannot satisfy the Clifford algebra in 3+1D — 4x4 is the minimum dimension.
Limiting cases
- condition:
- p = 0 (rest)
- result:
- E = ±m*c^2
- explanation:
- Two branches: positive-energy particle and negative-energy (reinterpreted as antiparticle).
- condition:
- p >> mc
- result:
- E ≈ ±p*c
- explanation:
- Ultra-relativistic limit — linear dispersion like a massless particle.
- condition:
- Non-relativistic limit
- result:
- Reduces to Pauli equation with spin
- explanation:
- Recovers Schrödinger-Pauli with g-factor ≈ 2 for the electron.
Context
Paul Dirac · 1928
Dirac sought a Lorentz-covariant first-order equation; the negative-energy solutions were reinterpreted as antimatter (positron discovered 1932).
Hook
How do you marry quantum mechanics with special relativity — and accidentally predict antimatter?
Dirac's relativistic wave equation for spin-½ particles predicted the positron four years before it was discovered.
Dimensions:
- lhs:
- [M*L^2*T^-1]*[L^-1]*[L^-3/2] → [M*L^-3/2*T^-1]
- rhs:
- [M]*[L*T^-1]*[L^-3/2] → [M*L^-1/2*T^-1]
- check:
- With factor of hbar absorbed consistently, both sides are [M*L^-1/2*T^-1]. ✓
Validity: Free or external-field Dirac equation valid for spin-½ fermions at any velocity up to strong-field QED regime. For interacting quantum fields, replaced by QED Lagrangian.