Spin from the Dirac Equation

Before 1928, spin was a fact without a reason. Experiments — the fine structure of spectral lines, the Stern-Gerlach beam splitting in two, the anomalous Zeeman effect — demanded that the electron carry an intrinsic angular momentum of $\tfrac12\hbar$ and a magnetic moment twice as large as a classical spinning charge would give. Pauli encoded this by hand: he bolted a two-component spinor and the gyromagnetic factor $g\approx 2$ onto the Schrödinger equation as empirical inputs. Nothing explained why. Then Dirac, asking only for a wave equation linear in time that squares to the relativistic energy relation, found spin and $g=2$ falling out as unavoidable consequences of Lorentz covariance. The mechanism is sharp and provable. Requiring the equation to be first order forces four-component spinors and a set of anticommuting matrices; two of those components are nothing but spin up and spin down. The diagnostic is conservation: the free Dirac Hamiltonian does not commute with the orbital angular momentum $\mathbf L=\mathbf r\times\mathbf p$ , so $\mathbf L$ alone is not conserved in a rotationally invariant theory. The deficit is repaired exactly by an operator $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ built from the same matrices, so that only the total $\mathbf J=\mathbf L+\mathbf S$ is conserved. The electron carries angular momentum even at rest. This article owns two machine-proved results. First, the conservation identity: $[H,\mathbf L]\ne 0$ and $[H,\mathbf S]\ne 0$ separately, but $[H,\mathbf L+\mathbf S]=0$ — total angular momentum is the conserved rotation generator, and its half-integer spin eigenvalues $\pm\tfrac{\hbar}{2}$ mark a spin- $\tfrac12$ particle. Second, the magnetic moment: coupling to an electromagnetic field and reducing to the non-relativistic limit produces a term $-\tfrac{q\hbar}{2mc}\boldsymbol\sigma\cdot\mathbf B$ , from which $g=2$ drops out with no adjustable knob. That $g=2$ is Dirac's prediction; the tiny excess measured today ( $g=2.0023193$ ) is the QED anomaly, a story that lies beyond this equation.

Key takeaways

The Dirac equation, built only to be first-order and Lorentz-covariant, forces four-component spinors carrying an internal angular momentum $\tfrac12\hbar$ — spin is not a postulate but a theorem.
Conservation (owned): the free Dirac Hamiltonian gives $[H,\mathbf L]=-i\hbar c\,\boldsymbol\alpha\times\mathbf p\ne 0$ and $[H,\mathbf S]=+i\hbar c\,\boldsymbol\alpha\times\mathbf p$ , so only $\mathbf J=\mathbf L+\mathbf S$ commutes with $H$ — total angular momentum is conserved, orbital and spin separately are not.
Spin operator $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ has eigenvalues $\pm\tfrac{\hbar}{2}$ and $\mathbf S^2=\tfrac34\hbar^2=\tfrac12(\tfrac12+1)\hbar^2$ : an intrinsic spin- $\tfrac12$ .
g = 2 (owned): minimal coupling plus the identity $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2-\tfrac{q\hbar}{c}\boldsymbol\sigma\cdot\mathbf B$ yields the Pauli term $-\tfrac{q\hbar}{2mc}\boldsymbol\sigma\cdot\mathbf B$ , fixing the gyromagnetic ratio at exactly $2$ — no free parameter.
Scope: $g=2$ is the Dirac prediction; the measured $g=2.0023193$ (anomaly $a_e\approx 0.00116$ ) is a QED radiative correction that lies beyond this article.
Historically this converted the empirical Pauli spin and the anomalous $g$ -factor into derived quantities — a landmark triumph of relativistic quantum mechanics.
Key idea: writing a Lorentz-covariant first-order wave equation forces the four-component spinor and the Clifford algebra that automatically supply spin $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ and the gyromagnetic ratio $g=2$ .

The core insight

Spin- $\tfrac12$ and $g=2$ are forced outputs of a Lorentz-covariant first-order wave equation: the four-component spinor and the gamma-matrix Clifford algebra automatically supply an internal angular momentum $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ — the piece that must be added to $\mathbf L$ for total $\mathbf J=\mathbf L+\mathbf S$ to commute with $H$ — and a magnetic moment with gyromagnetic ratio exactly $2$ .

Where this lives today

The $g=2$ of the Dirac equation anchors one of physics' most precise dialogues between theory and experiment. Modern Penning-trap experiments determine the electron's spin and cyclotron frequencies with extraordinary precision, allowing the $g$ -factor to be extracted to better than one part in $10^{12}$ ; the current value $g=2.00231930436$ deviates from the Dirac $2$ by a measured, not assumed, amount. On the theory side, quantum electrodynamics computes the anomaly $a_e=(g-2)/2$ as a power series in $\alpha$ , now carried to five loops. The agreement is so sharp that the comparison is effectively reversed: the measured electron g-factor is used to determine the fine-structure constant $\alpha$ , providing one of its most precise values. The muon's analogous $g-2$ is a live frontier, where a persistent tension with the Standard-Model prediction has for years been a watched hint of new physics. All of these developments rest on the baseline established here: before radiative corrections are included, relativistic quantum mechanics predicts exactly $g=2$ — a consequence of the Dirac equation and Lorentz covariance, not an adjustable parameter.

Figure

Dirac g = 2 is the baseline; QED adds the small measured anomaly

The physical picture

To be relativistic and first order in time, a wave equation must square to the Klein-Gordon equation. That single requirement forces the coefficients multiplying the momentum components to satisfy anticommutation relations. Ordinary numbers cannot do this, so the coefficients must be matrices. The smallest matrices that work are $4\times4$ , so the wavefunction acquires four internal components. Two of them turn out to be spin up and spin down: spin is the price of relativistic linearity, extracted before any electromagnetic field is even mentioned. Angular momentum conservation is the diagnostic that reveals it. In a rotationally invariant theory the generator of rotations must commute with the Hamiltonian. Compute $[H,\mathbf L]$ for the Dirac Hamiltonian and it does not vanish — orbital angular momentum alone is not conserved. But the failure is precise: it is cancelled term-for-term by adding $\tfrac{\hbar}{2}\boldsymbol\Sigma$ , a matrix built from the same gamma matrices. The mathematics is telling us that the electron carries angular momentum even when sitting motionless at the origin, where $\mathbf L=\mathbf r\times\mathbf p$ can contribute nothing. That leftover, frame-independent angular momentum is intrinsic spin. The magnetic moment $g=2$ is a fingerprint of the very same algebra. When the electron's kinetic momentum couples to a magnetic field, the operator $\boldsymbol\sigma\cdot\boldsymbol\pi$ has to be squared, and the Pauli identity turns the cross term into $-\tfrac{q\hbar}{c}\boldsymbol\sigma\cdot\mathbf B$ . The coefficient is fixed by the Pauli algebra, not chosen to fit data — which is exactly why $g$ comes out to $2$ with no adjustable knob. The four-component Dirac spinor encodes both spin and particle-antiparticle degrees of freedom. In the nonrelativistic limit, the upper and lower components become weakly coupled, with relativistic corrections of order $v/c$ giving rise to effects such as spin-orbit coupling.

Figure

Orbital L is not conserved; adding spin S restores conservation of total J

From covariance to spin and g = 2

The two owned results share one root — the Clifford algebra hidden in the Dirac operator — but they answer different questions. The first asks what rotation generator is conserved and discovers spin; the second asks how the electron responds to a magnetic field and discovers $g=2$ . We prove each in turn. The free Dirac Hamiltonian $H=c\,\boldsymbol\alpha\cdot\mathbf p+\beta mc^2$ , the anticommutators $\{\alpha^i,\alpha^j\}=2\delta^{ij}$ , and the four-component structure are owned by The Dirac Equation and Dirac Gamma Matrices and Clifford Algebra, recalled here by name.

Derivation AOnly total angular momentum is conserved, forcing spin

We show that the free Dirac Hamiltonian fails to commute with the orbital angular momentum $\mathbf L$ , and that the failure is repaired exactly by a spin operator $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ built from the gamma matrices, so that $\mathbf J=\mathbf L+\mathbf S$ is the conserved rotation generator. The strategy is to compute $[H,\mathbf L]$ and $[H,\mathbf S]$ separately and watch them cancel; the physics is that a rotationally invariant Hamiltonian must commute with some angular momentum, and the mathematics dictates that it be $\mathbf L$ plus an intrinsic piece. Work with $H=c\,\boldsymbol\alpha\cdot\mathbf p+\beta mc^2$ , where the mass term commutes with everything angular and drops out.

Step 1. Commute $H$ with orbital $\mathbf L$ . Only the $c\,\boldsymbol\alpha\cdot\mathbf p$ part matters, and using $[L_i,p_j]=i\hbar\epsilon_{ijk}p_k$ ,

[H,\mathbf L]=c\,[\boldsymbol\alpha\cdot\mathbf p,\mathbf L]=-i\hbar c\,(\boldsymbol\alpha\times\mathbf p) \qquad (1)

where: $\boldsymbol\alpha\times\mathbf p$ is the vector with components $\epsilon_{ijk}\alpha_j p_k$ . Orbital angular momentum is not conserved: $[H,\mathbf L]\ne 0$ whenever the electron has momentum. Something is missing.

Step 2. Introduce the spin matrices. Define $\boldsymbol\Sigma=\mathrm{diag}(\boldsymbol\sigma,\boldsymbol\sigma)$ , equivalently $\boldsymbol\Sigma=-\tfrac{i}{2}\boldsymbol\alpha\times\boldsymbol\alpha$ , and set $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ . Their commutator with $H$ uses $[\alpha\cdot\mathbf p,\Sigma_z]=2i(\boldsymbol\alpha\times\mathbf p)_z$ :

[H,\mathbf S]=+i\hbar c\,(\boldsymbol\alpha\times\mathbf p) \qquad (2)

Spin alone is not conserved either — but its non-conservation is exactly the negative of the orbital one.

Step 3. Add them. Equations (1) and (2) are equal and opposite, so

[H,\mathbf J]=[H,\mathbf L+\mathbf S]=-i\hbar c\,(\boldsymbol\alpha\times\mathbf p)+i\hbar c\,(\boldsymbol\alpha\times\mathbf p)=0 \qquad (3)

where: $\mathbf J=\mathbf L+\mathbf S$ is the total angular momentum. This is the whole point: the Dirac Hamiltonian is rotationally invariant, but the conserved generator is $\mathbf J$ , not $\mathbf L$ — the electron must carry the intrinsic $\mathbf S$ for angular momentum to close.

Step 4. Read off the spin. The eigenvalues of each $\Sigma_i$ are $\pm 1$ , so $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ has projections $\pm\tfrac{\hbar}{2}$ , and $\mathbf S^2=\tfrac34\hbar^2=\tfrac12(\tfrac12+1)\hbar^2$ — the spectrum of a spin- $\tfrac12$ . Spin is a theorem of the equation, not an add-on. As a concrete check: with $\alpha^i=[[0,\sigma_i],[\sigma_i,0]]$ and $\boldsymbol\Sigma=\mathrm{diag}(\boldsymbol\sigma,\boldsymbol\sigma)$ , for a test momentum $\mathbf p=(0.7,-1.1,0.4)$ the matrices satisfy $[H,L_z]=-i(\boldsymbol\alpha\times\mathbf p)_z\ne 0$ , $[H,S_z]=+i(\boldsymbol\alpha\times\mathbf p)_z$ , and $[H,L_z+S_z]=0$ exactly; each $\Sigma_i$ has eigenvalues $\{-1,-1,+1,+1\}$ and $\mathbf S^2=\tfrac34\hbar^2\,\mathbb 1$ .

Go deeper · Derivation B · 2 minThe magnetic moment g = 2 drops out of the nonrelativistic reduction

Derivation BThe magnetic moment g = 2 drops out of the nonrelativistic reduction

We couple the electron to an electromagnetic field by minimal substitution and show that the nonrelativistic limit produces a magnetic-moment term with gyromagnetic ratio exactly $2$ . The key step is the Pauli identity: squaring the operator $\boldsymbol{\sigma}\!\cdot\!\boldsymbol{\pi}$ produces a term proportional to $\boldsymbol{\sigma}\!\cdot\!\mathbf{B}$ because the kinetic-momentum components do not commute in a magnetic field. That coefficient is $g=2$ ; there is no place to insert a free parameter. The full block-elimination reduction to the Pauli equation is owned downstream; here we isolate the algebraic core that fixes $g$ .

Step 1. Minimal coupling. Replace $\mathbf p\to\boldsymbol\pi=\mathbf p-\tfrac{q}{c}\mathbf A$ , the kinetic momentum. In a static field the two-component upper spinor obeys an equation containing $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2$ :

\boldsymbol\pi=\mathbf p-\tfrac{q}{c}\mathbf A,\qquad [\pi_i,\pi_j]=\tfrac{i\hbar q}{c}\,\epsilon_{ijk}B_k \qquad (4)

where: $\mathbf B=\nabla\times\mathbf A$ and $q$ is the electron charge ( $q=-e$ ). The components of $\boldsymbol\pi$ do not commute in a magnetic field — this is where the moment comes from.

Step 2. Apply the Pauli identity. For any operator vector, $(\boldsymbol\sigma\cdot\mathbf a)(\boldsymbol\sigma\cdot\mathbf b)=\mathbf a\cdot\mathbf b+i\boldsymbol\sigma\cdot(\mathbf a\times\mathbf b)$ . With $\mathbf a=\mathbf b=\boldsymbol\pi$ ,

(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2+i\boldsymbol\sigma\cdot(\boldsymbol\pi\times\boldsymbol\pi)=\boldsymbol\pi^2-\tfrac{q\hbar}{c}\,\boldsymbol\sigma\cdot\mathbf B \qquad (5)

where: $(\boldsymbol\pi\times\boldsymbol\pi)_k=\tfrac{i\hbar q}{c}B_k$ from (4) is nonzero only because the momenta fail to commute. The classical $\boldsymbol\pi^2$ is the orbital kinetic energy; the extra $-\tfrac{q\hbar}{c}\boldsymbol\sigma\cdot\mathbf B$ is a purely quantum, spin-dependent piece that no classical squaring would produce.

Step 3. Read the moment. Dividing by $2m$ , the spin-field term in the Hamiltonian is $-\tfrac{q\hbar}{2mc}\boldsymbol\sigma\cdot\mathbf B$ . Writing it as $-\boldsymbol\mu\cdot\mathbf B$ with $\boldsymbol\mu=g\,\tfrac{q}{2mc}\mathbf S$ and $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\sigma$ gives

-g\,\frac{q}{2mc}\cdot\frac{\hbar}{2}\,\boldsymbol\sigma\cdot\mathbf B=-\frac{q\hbar}{2mc}\,\boldsymbol\sigma\cdot\mathbf B\ \Rightarrow\ \frac{g}{2}=1\ \Rightarrow\ g=2 \qquad (6)

where: matching the two coefficients forces $g=2$ with no free parameter. This is the punchline: the same Clifford algebra that gave spin also fixes the gyromagnetic ratio, and it fixes it at $2$ — twice the classical value of a spinning charge.

Step 4. Bound the claim. $g=2$ is the Dirac prediction of this equation. Experiment measures $g=2.0023193\ldots$ , corresponding to the anomalous magnetic moment $a_e=(g-2)/2\approx0.00116$ , which arises entirely from QED radiative corrections (the electron interacting with its own photon field) and lies entirely beyond the single-particle Dirac equation. As a concrete check: the commutator $[\pi_x,\pi_y]=i\hbar q B_z/c$ holds symbolically; the identity $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2-\tfrac{q\hbar}{c}\boldsymbol\sigma\cdot\mathbf B$ gives cross-term coefficient $-qB$ for $\mathbf B=B\hat z$ ; matching coefficients yields $g=2$ exactly; and the measured $g=2.0023193$ gives $a_e=0.00115965$ (QED, out of scope).

Limiting cases

case	result	meaning
electron at rest ( $\mathbf p=0$ )	$[H,\mathbf L]=0$ , $[H,\mathbf S]=0$	at rest even $\mathbf L$ and $\mathbf S$ are separately conserved; spin still $\tfrac{\hbar}{2}$
no magnetic field ( $\mathbf B=0$ )	$(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2$	the spin–field interaction vanishes; the intrinsic spin remains, although it does not contribute to the energy
weak field, non-relativistic	$H_{\rm Pauli}=\tfrac{\boldsymbol\pi^2}{2m}-\tfrac{q\hbar}{2mc}\boldsymbol\sigma\cdot\mathbf B+q\phi$	the Pauli equation with $g=2$ (owned downstream)
classical spinning charge	$g=1$	half the Dirac value — the factor-of-2 "anomaly" the old quantum theory could not explain
with QED corrections	$g=2.0023193$	the measured value; the $0.00116$ excess is beyond this equation

Common trapg = 2 is the Dirac prediction, not the measured value 2.0023193

It is tempting to say "the Dirac equation gives the electron's magnetic moment," full stop — but that overstates the case in a way that matters. The single-particle Dirac equation predicts $g=2$ exactly: a clean, parameter-free result that was already a triumph, since the old quantum theory could not explain why the spin $g$ -factor was twice the orbital value. But the measured value is $g=2.0023193043$ , and the difference — the anomalous moment $a_e=(g-2)/2\approx 0.00116$ — is not in the Dirac equation. It arises from quantum electrodynamics: the electron emits and reabsorbs virtual photons, the leading correction being Schwinger's $a_e=\alpha/2\pi$ . So the honest statement is layered: single-particle Dirac theory predicts $g=2$ ; QED predicts the small excess; and the agreement of QED with experiment to twelve significant figures is among the most precise confirmations in all of physics. Do not claim to derive $2.0023193$ from the Dirac equation alone — you cannot, and the gap is the whole subject of the anomaly.

Worked examples

Example 1 — the angular-momentum commutators, explicitly (confidence builder). Take the free Dirac Hamiltonian and a concrete momentum, and verify that $\mathbf L$ and $\mathbf S$ do not commute separately with $H$ , whereas their sum $\mathbf J=\mathbf L+\mathbf S$ does.

Step 1. Set up the matrices. Use $\alpha^i=\begin{pmatrix}0&\sigma_i\\ \sigma_i&0\end{pmatrix}$ , $\beta=\begin{pmatrix}\mathbb 1&0\\ 0&-\mathbb 1\end{pmatrix}$ , and $\Sigma_i=\begin{pmatrix}\sigma_i&0\\ 0&\sigma_i\end{pmatrix}$ . Pick the test momentum $\mathbf p=(0.7,-1.1,0.4)$ (units $\hbar=c=1$ ).

Step 2. Compute the orbital piece. The $z$ -component of $\boldsymbol\alpha\times\mathbf p$ is $(\boldsymbol\alpha\times\mathbf p)_z=\alpha_x p_y-\alpha_y p_x=\alpha_x(-1.1)-\alpha_y(0.7)$ , a nonzero $4\times4$ matrix. Therefore

[H,L_z]=-i(\boldsymbol\alpha\times\mathbf p)_z\ \boxed{\ne 0}.

Since the commutator is nonzero, orbital angular momentum is not conserved.

Step 3. Compute the spin piece and add. The commutator $[\alpha\cdot\mathbf p,\Sigma_z]=2i(\boldsymbol\alpha\times\mathbf p)_z$ , so with $S_z=\tfrac12\Sigma_z$ ,

[H,S_z]=+i(\boldsymbol\alpha\times\mathbf p)_z,\qquad [H,L_z+S_z]=-i(\boldsymbol\alpha\times\mathbf p)_z+i(\boldsymbol\alpha\times\mathbf p)_z=\boxed{0}.

The two contributions cancel identically (machine-verified to zero). Total $J_z$ is conserved; the electron's spin is exactly the piece needed to close it.

Example 2 — the g = 2 Zeeman splitting (exam level). Use the Dirac magnetic moment to compute the Zeeman energy splitting of a free electron spin in a laboratory field, and contrast it with a classical charge.

Step 1. Write the interaction. The Pauli term is $H_Z=-\boldsymbol\mu\cdot\mathbf B$ with $\boldsymbol\mu=-g\,\mu_B\,\mathbf S/\hbar$ and $g=2$ , so for $\mathbf B=B\hat z$ the energy is $E=g\,\mu_B\,B\,m_s$ with $m_s=\pm\tfrac12$ .

Step 2. Insert numbers. For a magnetic field of $1\,\mathrm{T}$ , $\mu_B B = 5.788 \times 10^{-5}\,\mathrm{eV}$ . With $g=2$ the two spin states sit at

E_\pm=\pm\,g\,\mu_B B\,\tfrac12=\pm\,\mu_B B=\pm\,\mathbf{5.788\times10^{-5}\ \text{eV}}.

Step 3. Read the splitting. The gap between spin-up and spin-down is

\Delta E=g\,\mu_B B=2\times5.788\times10^{-5}=\mathbf{1.158\times10^{-4}\ \text{eV}}\quad(B=1\ \text{T}).

This corresponds to a resonance frequency $\Delta E/h\approx 28\ \text{GHz/T}$ — the electron-spin-resonance frequency. A classical spinning charge with $g=1$ would give exactly half this splitting; the doubled value, seen directly in the anomalous Zeeman effect and Stern-Gerlach-type experiments (recalled by name in Spin Angular Momentum), was the empirical fact that Dirac's $g=2$ finally explained.

Uhlenbeck's factor of two, and Dirac's derivation

Origins · 1 ming-factor and how this was discovered

g-factor

introduced 1925; Dirac's g=2 in 1928 · Leiden, Cambridge

The gyromagnetic ratio began as an embarrassment. In 1925 George Uhlenbeck and Samuel Goudsmit proposed that the electron spins, carrying angular momentum $\tfrac12\hbar$ — but to fit the observed spectra its magnetic moment had to be twice what a classical spinning charge would produce. This factor of two, the spin $g$ -factor $g=2$ , was pure phenomenology: nobody knew why the ratio should be doubled, and a naive classical model gave $g=1$ . Llewellyn Thomas partly rescued the situation in 1926, showing that relativistic kinematics (Thomas precession) supplied a compensating $\tfrac12$ in the spin-orbit energy, but the fundamental $g=2$ remained an input. Then in 1928 Paul Dirac wrote his linear relativistic equation and $g=2$ simply appeared — a forced consequence of the gamma-matrix algebra, with no assumption about the electron's internal structure. It was among the most persuasive early confirmations that the equation was genuinely explanatory. The gap between $2$ and the measured $2.0023193$ waited until Schwinger's 1948 QED calculation of the anomaly $a_e=\alpha/2\pi$ .

History

1925

— Uhlenbeck and Goudsmit propose electron spin with an unexplained g-factor of 2

1926

— Thomas precession supplies the factor-½ in spin-orbit coupling

1927

— Pauli builds two-component spinors and g=2 into wave mechanics by hand

1928

— Dirac's equation derives spin-½ and g=2 from Lorentz covariance

1947

— Kusch and Foley measure the anomalous moment, g slightly above 2

1948

— Schwinger computes the QED anomaly a_e = α/2π ≈ 0.00116

Going further

References. Bjorken & Drell, Relativistic Quantum Mechanics, §1.4, §4.3 · Sakurai, Advanced Quantum Mechanics, §3.2–3.3 · Griffiths, Introduction to Quantum Mechanics, 3rd ed., §12 · Dirac, Proc. R. Soc. A 117, 610 (1928) · Schwinger, Phys. Rev. 73, 416 (1948) (the anomaly) · Peskin & Schroeder, §3.3. Two results are derived here. (1) The conservation identity: $[H,\mathbf L]=-i\hbar c(\boldsymbol\alpha\times\mathbf p)\ne 0$ , $[H,\mathbf S]=+i\hbar c(\boldsymbol\alpha\times\mathbf p)$ , $[H,\mathbf L+\mathbf S]=0$ , confirmed for $\mathbf p=(0.7,-1.1,0.4)$ , forcing spin $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ with $\mathbf S^2=\tfrac34\hbar^2$ . (2) The gyromagnetic ratio $g=2$ from $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2-\tfrac{q\hbar}{c}\boldsymbol\sigma\cdot\mathbf B$ , with $[\pi_x,\pi_y]=i\hbar qB_z/c$ , cross-term coefficient $-qB$ , and $g/2=1$ . The measured $g=2.0023193$ ( $a_e\approx 0.00116$ ) is a QED radiative correction stated but not derived here — the anomaly and the Schwinger term belong to quantum electrodynamics, and the Zeeman scale $\mu_B B=5.788\times10^{-5}\ \text{eV/T}$ is noted for reference. The systematic order-by-order reduction to the Pauli equation and the higher spin-orbit and Darwin terms are developed in Nonrelativistic Reduction and the Pauli Equation and Foldy-Wouthuysen Transformation by name.

Builds on: The Dirac Equation · Dirac Gamma Matrices and Clifford Algebra · Dirac Spinors and Plane-Wave Solutions · The Klein-Gordon Equation

Leads to: Nonrelativistic Reduction and the Pauli Equation · Foldy-Wouthuysen Transformation · Charge Conjugation and the Positron

Does the Dirac equation really predict that the electron spins, or just that it has two extra components?

It predicts genuine spin, and the distinction is the point. The four components alone could be mere bookkeeping; what makes two of them spin is the conservation argument. In a rotationally invariant theory the conserved rotation generator must commute with $H$ , and for the Dirac Hamiltonian that generator is not $\mathbf L$ but $\mathbf L+\mathbf S$ . The operator $\mathbf S=\tfrac{\hbar}{2}\boldsymbol\Sigma$ has eigenvalues $\pm\tfrac{\hbar}{2}$ , obeys the angular-momentum algebra, and contributes to $\mathbf J$ even for an electron at rest — every property we demand of a spin. So it is not a formal relabelling: the equation forces a physical, intrinsic angular momentum of magnitude $\tfrac12\hbar$ .

Why is the classical g-factor 1 but the electron's is 2?

For a classical charge distribution rotating rigidly, the magnetic moment and the angular momentum come from the same circulating mass-and-charge, so their ratio (the $g$ -factor) is $1$ . Spin is different: it is intrinsic angular momentum, not literal circulation, and the relation between the electron's spin and its magnetic moment is set by the Dirac algebra rather than by a rotating-charge picture. The reduction shows that this intrinsic ratio is exactly $2$ — twice the classical orbital value. Historically this was baffling; a spinning ball of charge simply cannot give $g=2$ . The resolution is that the electron is not a spinning ball, and the Dirac equation, which knows nothing about internal structure, delivers $g=2$ from pure algebra.

If neither L nor S is conserved, why do we still use them as quantum numbers?

Because for many systems a different operator restores their usefulness. In a central potential the total $\mathbf J$ , its projection $J_z$ , and $\mathbf J^2$ are all conserved, so $j$ and $m_j$ are good quantum numbers — that is what the fine-structure levels are labelled by. Orbital $\ell$ and spin $s$ individually are conserved only in the non-relativistic limit, where the coupling between them (spin-orbit) is a small correction; there $\ell$ , $s$ , $m_\ell$ , $m_s$ are approximately good. So the honest statement is that $\mathbf J$ is the exactly conserved quantity, while $\mathbf L$ and $\mathbf S$ separately are good approximate labels only when relativistic spin-orbit effects are negligible.

Is g = 2 exact in the Dirac equation, or approximate?

It is exact — at the level of the single-particle Dirac equation. The reduction gives the spin term $-\tfrac{q\hbar}{2mc}\boldsymbol\sigma\cdot\mathbf B$ with a coefficient that is a pure number, and matching it to the definition of $g$ forces $g=2$ identically, with no expansion parameter and no higher-order corrections within this equation. The subtlety is that the real electron is not described by the single-particle equation alone but by quantum electrodynamics, where it continually emits and absorbs virtual photons. Those field-theoretic effects shift $g$ upward by the anomaly $a_e\approx 0.00116$ . So " $g=2$ " is exact for the Dirac equation and slightly wrong for the physical electron; the difference is precisely the domain of QED.

How does spin appearing here relate to the Pauli equation I already know?

The Pauli equation is the non-relativistic limit of exactly this calculation. In ordinary quantum mechanics you write down a two-component spinor and insert the term $-\tfrac{q\hbar}{2mc}\boldsymbol\sigma\cdot\mathbf B$ with $g=2$ as an empirical fact. The Dirac equation, reduced to leading order in $v/c$ , produces that same two-component equation — the Pauli equation — with the spin term and its $g=2$ coefficient derived rather than assumed. So this article explains where the Pauli equation comes from: it is the shadow the Dirac equation casts in the slow-electron limit. The systematic version of this reduction, keeping the next-order spin-orbit and Darwin terms, is the Foldy-Wouthuysen transformation.

Could a spinless relativistic particle exist, or does relativity always force spin?

Relativity does not force spin in general — it forces spin here because we insisted on a first-order equation. The Klein-Gordon equation is a perfectly good relativistic wave equation for a spin- $0$ particle (the pion, the Higgs boson): second order in time, single-component, no spinor structure and no intrinsic angular momentum. Spin- $\tfrac12$ emerges specifically from demanding linearity in $\partial_t$ , which pushes you into four-component spinors and the gamma algebra. Higher spins have their own equations (Rarita-Schwinger for spin- $\tfrac32$ ). The lesson is precise: the Dirac equation's first-order structure is what makes it the equation of a spin- $\tfrac12$ particle, and $g=2$ is bound up with that same choice.