Dirac Equation — Physica

Equivalent forms

i\hbar \gamma^\mu \partial_\mu \psi = mc\psi

Spin, antimatter, and the g-factor of ~2 all fall out of one equation.

Symbol	Name	SI unit	Dimension	Range
$psi$	Dirac spinoroutput Four-component relativistic wavefunction	m^-3/2	L^-3/2	0 – 1
$gamma_mu$	Gamma matrices 4x4 matrices satisfying the Clifford algebra {gamma_mu, gamma_nu} = 2*eta_mn	dimensionless	1	0 – 1
$m$	Particle mass Rest mass of the fermion (e.g., electron)	kg	M	9.1093837015e-31 – 1.67262192369e-27
$c$	Speed of light Speed of light in vacuum	m/s	L*T^-1	299792458 – 299792458
$hbar$	Reduced Planck constant h / (2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34

Unit systems

Symbol	SI	CGS	Imperial
$psi$	m^-3/2	cm^-3/2	ft^-3/2
$m$	kg	g	lb
$c$	m/s	cm/s	ft/s
$hbar$	J*s	erg*s	ftlbfs

Where it holds

Free or external-field Dirac equation valid for spin-

\tfrac{1}{2}

fermions at any velocity up to strong-field QED regime. For interacting quantum fields, replaced by QED Lagrangian.

Dimensional analysis

[M*L^2*T^-1]*[L^-1]*[L^-3/2] \to [M*L^-3/2*T^-1]

[M]*[L*T^-1]*[L^-3/2] \to [M*L^-1/2*T^-1]

With factor of hbar absorbed consistently, both sides are [M*L^-1/2*T^-1].

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Discovery

Paul Dirac · 1928

Dirac sought a Lorentz-covariant first-order equation; the negative-energy solutions were reinterpreted as antimatter (positron discovered 1932).

Try this

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.

Research status: stable

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

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)

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.

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.

Limiting cases

p = 0

(rest)⟶

E = \pm m*c^2

Two branches: positive-energy particle and negative-energy (reinterpreted as antiparticle).

p >> mc⟶

E \approx \pm p*c

Ultra-relativistic limit — linear dispersion like a massless particle.

Non-relativistic limit⟶ Reduces to Pauli equation with spinRecovers Schrödinger-Pauli with g-factor

\approx 2

for the electron.

What if…

What if

m = 0

?

Recovers the Weyl equation — massless chiral fermions like neutrinos in the Standard Model approximation.

What if we ignore negative-energy solutions?

Theory becomes inconsistent — vacuum instability and loss of completeness. Antimatter is mathematically necessary.

What if gamma matrices were 2x2?

Cannot satisfy the Clifford algebra in 3+1D — 4x4 is the minimum dimension.

1

Rest energy of electron

Given ·

m:: 9.1093837015e-31
p:: 0

Find · E

Steps

At $p = 0$ : $E = m*c^2$
$= 9.1093837015e-31 * (299792458)^2$
$\approx 8.187e-14\,\mathrm{J} \approx 0.511\,\mathrm{MeV}$

Result ·

E = m*c^2 \approx 0.511\,\mathrm{MeV}

2

Relativistic electron at p = mc

Given ·

m:: 9.1093837015e-31
p over mc:: 1

Find · E/mc^2

Steps

$E^2 = (pc)^2 + (mc^2)^2 = (mc^2)^2 + (mc^2)^2 = 2(mc^2)^2$
$E = \sqrt{2} * 0.511\,\mathrm{MeV}$
$E \approx 0.723\,\mathrm{MeV}$

Result ·

E = \sqrt{2}*m*c^2 \approx 0.723\,\mathrm{MeV}

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