Dirac Equation, Antiparticles and the g=2 Prediction

Derivation

Dirac Equation, Antiparticles and the g=2 Prediction

D-278 Home PU-304 Threads matter · symmetry · waves Depends on Klein-Gordon Equation and the Yukawa Potential, pauli-spin-matrices

Statement

Demanding a wave equation first order in time whose plane-wave solutions still satisfy the relativistic dispersion \( E^2 = \mathbf{p}^2 c^2 + m^2 c^4 \) forces the Hamiltonian \( \hat H = c\,\vec{\alpha}\cdot\hat{\mathbf p} + \beta m c^2 \) with four anticommuting matrices \( \alpha_i,\beta \). The smallest realisation is \(4\times4\), giving the Dirac equation \( i\hbar\,\gamma^\mu\partial_\mu\psi - mc\,\psi = 0 \). Its four components describe a spin-\(\tfrac12\) particle and its antiparticle, and minimal coupling in the non-relativistic limit yields the Pauli equation with gyromagnetic ratio \( g = 2 \).

Why it matters

The Dirac equation is the first equation in physics from which spin, the electron magnetic moment, and antimatter emerge as unavoidable consequences of Lorentz invariance plus a single linear-in-time postulate, rather than being inserted by hand as in the Pauli theory. The prediction \( g = 2 \) — confirmed to the level of the QED anomaly \( g/2 = 1.00116\ldots \) — was the decisive early triumph of relativistic quantum mechanics.

It also resolves the pathology of the Klein-Gordon equation (a second-order-in-time equation with an indefinite conserved density) by supplying a positive-definite probability current, at the price of introducing negative-energy solutions that Dirac reinterpreted as antiparticles. This launched quantum field theory and the modern understanding of the vacuum.

Assumptions

Linearity in \( \partial_t \).If we keep second-order time derivatives we recover Klein-Gordon; the probability density is no longer positive-definite and single-particle interpretation fails.

Linearity in \( \nabla \) with Lorentz covariance.Special relativity treats space and time symmetrically, so first order in time forces first order in space; drop this and the equation is not a Lorentz scalar/spinor equation.

Each component satisfies the relativistic dispersion \( E^2=\mathbf p^2c^2+m^2c^4 \).This is the physical input fixing the mass shell; without it the coefficients \( \alpha_i,\beta \) are unconstrained and no particle mass is defined.

The coefficients are constant, dimensionless, position-independent objects.Allowing them to be functions of \(x\) spoils translation invariance and the clean anticommutator algebra; the minimal-matrix argument collapses.

Hermiticity of \( \hat H \), i.e. \( \alpha_i^\dagger=\alpha_i,\ \beta^\dagger=\beta \).Required for real energies and unitary evolution; if dropped the spectrum can become complex and probability is not conserved.

Derivation

\[ \hat H \psi = \left( c\,\vec{\alpha}\cdot\hat{\mathbf p} + \beta m c^2 \right)\psi, \qquad i\hbar\,\frac{\partial\psi}{\partial t} = \hat H\psi \]

Most general Hamiltonian linear in \( \hat{\mathbf p}=-i\hbar\nabla \) and in \( m \), with as-yet-unknown constant coefficients \( \alpha_x,\alpha_y,\alpha_z,\beta \). A

\[ \hat H^2\psi = -\hbar^2\frac{\partial^2\psi}{\partial t^2} \quad\Longrightarrow\quad \hat H^2 \overset{!}{=} c^2\hat{\mathbf p}^2 + m^2c^4 \]

Iterating the equation once and demanding every component obey the relativistic dispersion (the mass shell). This is the physical constraint that fixes the algebra. A

\[ \hat H^2 = c^2\sum_{i,j}\alpha_i\alpha_j\,\hat p_i\hat p_j + mc^3\sum_i(\alpha_i\beta+\beta\alpha_i)\hat p_i + \beta^2 m^2c^4 \]

Expand the square. The \( \hat p_i \) commute with each other and with the constant matrices, so the ordering issue is entirely in the matrices. B

\[ c^2\sum_{i,j}\alpha_i\alpha_j\,\hat p_i\hat p_j = c^2\sum_{i\le j}\tfrac12\{\alpha_i,\alpha_j\}\,\hat p_i\hat p_j \]

Because \( \hat p_i\hat p_j=\hat p_j\hat p_i \) is symmetric, only the symmetric part \( \tfrac12\{\alpha_i,\alpha_j\} \) of the matrix product survives the contraction. B

\[ \{\alpha_i,\alpha_j\}=2\delta_{ij}\mathbf 1,\qquad \{\alpha_i,\beta\}=0,\qquad \beta^2=\mathbf 1 \]

Matching term by term to \( c^2\hat{\mathbf p}^2+m^2c^4 \): the cross terms in \( \hat p_i \) must vanish and each square must give unity. These are the defining (Clifford) relations. B

\[ \alpha_i = -\beta\alpha_i\beta \;\Rightarrow\; \mathrm{Tr}\,\alpha_i = -\mathrm{Tr}\big(\beta\alpha_i\beta\big) = -\mathrm{Tr}\,\alpha_i = 0 \]

From \( \{\alpha_i,\beta\}=0 \) and \( \beta^2=\mathbf1 \), using cyclicity of the trace. The same argument gives \( \mathrm{Tr}\,\beta=0 \). All four matrices are traceless. C

\[ \alpha_i^2=\beta^2=\mathbf1 \;\Rightarrow\; \text{eigenvalues } \pm1;\quad \mathrm{Tr}=0 \;\Rightarrow\; \#(+1)=\#(-1) \;\Rightarrow\; N \text{ even} \]

Eigenvalues square to 1 so are \( \pm1 \); tracelessness forces equal numbers of each, so the dimension \(N\) is even. \(N=2\) is exhausted by \( \mathbf1,\sigma_x,\sigma_y,\sigma_z \) which give only three mutually anticommuting traceless matrices, not four. Hence \( N=4 \) is minimal. C

\[ \alpha_i=\begin{pmatrix}0&\sigma_i\\\sigma_i&0\end{pmatrix},\qquad \beta=\begin{pmatrix}\mathbf1&0\\0&-\mathbf1\end{pmatrix} \]

A concrete \(4\times4\) (Dirac) representation in \(2\times2\) blocks built from the Pauli matrices, which satisfy \( \{\sigma_i,\sigma_j\}=2\delta_{ij} \). One checks all relations of Step 5 hold. B

\[ \gamma^0=\beta,\quad \gamma^i=\beta\alpha_i,\qquad \{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\mathbf1 \]

Multiply the wave equation \( i\hbar\partial_t\psi=\hat H\psi \) on the left by \( \beta/c \) and define \( \gamma^\mu \). The Clifford relations of Step 5 become the covariant algebra with \( \eta=\mathrm{diag}(+,-,-,-) \). B

\[ \boxed{\; i\hbar\,\gamma^\mu\partial_\mu\psi - mc\,\psi = 0\;},\qquad \partial_\mu=\left(\tfrac1c\partial_t,\ \nabla\right) \]

Manifestly covariant Dirac equation. Its plane waves \( \psi\propto e^{-ip\cdot x/\hbar} \) satisfy \( (\gamma^\mu p_\mu-mc)u=0 \), whose solvability condition \( \det(\gamma^\mu p_\mu-mc)=0 \) returns \( E=\pm\sqrt{\mathbf p^2c^2+m^2c^4} \): both signs of energy appear. B

\[ \hat H=c\,\vec\alpha\cdot\hat{\mathbf p}+\beta mc^2,\qquad [\hat{\mathbf L},\hat H]\neq0,\quad [\hat{\mathbf L}+\tfrac{\hbar}{2}\vec\Sigma,\ \hat H]=0,\quad \vec\Sigma=\begin{pmatrix}\vec\sigma&0\\0&\vec\sigma\end{pmatrix} \]

Orbital angular momentum alone is not conserved; conservation is restored only by adding an intrinsic \( \tfrac{\hbar}{2}\vec\Sigma \). The eigenvalues \( \pm\tfrac{\hbar}{2} \) show the particle carries spin \( \tfrac12 \) — an output, not an input. C

\[ \hat{\mathbf p}\to\vec\pi=\hat{\mathbf p}-q\mathbf A,\qquad i\hbar\partial_t\psi=\big(c\,\vec\alpha\cdot\vec\pi+\beta mc^2+q\Phi\big)\psi \]

Minimal coupling to an electromagnetic field \( (\Phi,\mathbf A) \) with charge \(q\), the unique gauge-covariant substitution. A

\[ \psi=\begin{pmatrix}\varphi\\\chi\end{pmatrix}e^{-imc^2t/\hbar},\qquad \chi\approx\frac{\vec\sigma\cdot\vec\pi}{2mc}\,\varphi \]

Split into upper/lower two-spinors, remove the rest-energy phase, and take the non-relativistic limit \( |\varepsilon|,\,|q\Phi|\ll mc^2 \). The lower ("small") component is suppressed by \( \sim v/c \). B

\[ (\vec\sigma\cdot\vec\pi)^2=\vec\pi^2+i\vec\sigma\cdot(\vec\pi\times\vec\pi),\qquad \vec\pi\times\vec\pi=-q(\hat{\mathbf p}\times\mathbf A+\mathbf A\times\hat{\mathbf p})=i\hbar q\,\mathbf B \]

Use the Pauli identity \( (\vec\sigma\cdot\mathbf a)(\vec\sigma\cdot\mathbf b)=\mathbf a\cdot\mathbf b+i\vec\sigma\cdot(\mathbf a\times\mathbf b) \). Here \( \mathbf a=\mathbf b=\vec\pi \) do not commute, and \( \hat{\mathbf p}\times\mathbf A+\mathbf A\times\hat{\mathbf p}=-i\hbar\,\nabla\times\mathbf A=-i\hbar\mathbf B \). B

\[ \left[\frac{\vec\pi^2}{2m}-\frac{q\hbar}{2m}\,\vec\sigma\cdot\mathbf B+q\Phi\right]\varphi=\varepsilon\,\varphi \]

Substitute Step 14 into the upper-component equation. This is the Pauli equation; the spin-magnetic term appears automatically. B

\[ -\frac{q\hbar}{2m}\vec\sigma\cdot\mathbf B \equiv -\vec\mu\cdot\mathbf B,\qquad \vec\mu=\frac{q}{m}\,\mathbf S=g\,\frac{q}{2m}\,\mathbf S,\quad \mathbf S=\tfrac{\hbar}{2}\vec\sigma \;\Rightarrow\; \boxed{g=2} \]

Identify the coefficient of \( \mathbf B \) as a magnetic moment. Comparing with the definition \( \vec\mu=g\,(q/2m)\mathbf S \) gives \( g=2 \) with no free parameter. B

Result

\[ i\hbar\,\gamma^\mu\partial_\mu\psi - mc\,\psi = 0,\qquad \{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\mathbf1,\qquad \vec\mu=g\,\frac{q}{2m}\mathbf S,\quad g=2 \]

Reading. Linearising the relativistic energy relation cannot be done with numbers; it requires four anticommuting \(4\times4\) matrices. The resulting four-component wavefunction automatically carries spin \( \tfrac12 \) (two states) and comes in two energy signs (particle and antiparticle, two more). Coupling to a magnetic field produces a magnetic moment exactly twice the naive classical value, \( g=2 \) — a parameter-free prediction. The negative-energy branch, reinterpreted, is the positron.

Units check. In \( i\hbar\gamma^\mu\partial_\mu\psi \), \( [\hbar\partial_\mu]=\mathrm{J\,s\cdot m^{-1}}=\mathrm{kg\,m\,s^{-1}} \) (momentum); \( [mc]=\mathrm{kg\,m\,s^{-1}} \) matches. For the moment, \( [\,q\hbar/2m\,]=\mathrm{C\cdot J\,s/kg}=\mathrm{A\,m^2}=\mathrm{J\,T^{-1}} \), the units of magnetic moment, and \( -\vec\mu\cdot\mathbf B \) has units \( \mathrm{J\,T^{-1}\cdot T=J} \), an energy. \(g\) is dimensionless.

Limiting cases

\( m\to0 \): the equation decouples into two independent 2-component Weyl equations \( i\hbar(\partial_t\mp c\,\vec\sigma\cdot\nabla)\psi_\pm=0 \) of definite chirality — massless spin-\(\tfrac12\) fermions.
\( \mathbf p\to0 \) (rest frame): \( \hat H\to\beta mc^2 \), eigenvalues \( \pm mc^2 \); the upper two components are the particle at rest, the lower two the antiparticle.
Non-relativistic \( v\ll c \): reduces to the Pauli equation (Step 15); the small component scales as \( \chi/\varphi\sim v/c \).
\( \mathbf A=0,\ \Phi\) central: exact solution gives the relativistic hydrogen spectrum with fine structure and spin-orbit coupling emerging together, without ad hoc addition.
\( c\to\infty \): the \( mc^2 \) rest energy dominates; the two energy branches separate infinitely and antiparticles decouple, recovering single-branch Schrödinger-Pauli dynamics.

Breaks when

Strong fields, \( q\Phi \gtrsim 2mc^2 \) (e.g. \(Z\gtrsim137\), or \(E\gtrsim1.3\times10^{18}\,\mathrm{V/m}\)): the positive- and negative-energy branches overlap and single-particle interpretation collapses (Klein paradox, spontaneous pair creation). Only quantum field theory is consistent here.
Localisation below the Compton wavelength, \( \Delta x\lesssim\hbar/mc \): position measurements inject enough energy \( \sim mc^2 \) to create pairs; the one-particle position operator suffers Zitterbewegung at frequency \( 2mc^2/\hbar \) and loses its naive meaning.
Precision magnetic moment, \( g=2 \) exactly: fails at the \( 10^{-3} \) level — the measured \( g/2=1.001159652\ldots \) includes radiative (QED) corrections \( \alpha/2\pi+\ldots \) that this tree-level equation cannot capture.
Composite or strongly-interacting particles (proton, neutron): the \( g=2 \) prediction applies only to structureless point Dirac particles; the proton has \( g\approx5.59 \) from quark substructure.

Failure modes

Treating \( \alpha_i,\beta \) as numbers. Students try to solve \( \alpha_i^2=1,\ \alpha_i\alpha_j=-\alpha_j\alpha_i \) with scalars and conclude "no solution" — missing that matrices are required.
Forgetting \( \vec\pi\times\vec\pi\neq0 \). In Step 14, \( \vec\pi \) components do not commute; writing \( (\vec\sigma\cdot\vec\pi)^2=\vec\pi^2 \) loses the entire \( \vec\sigma\cdot\mathbf B \) term and hence \( g=2 \).
Confusing the four components with spin-up/down alone. Two of the four are the antiparticle degrees of freedom, not extra spin states.
Sign/units slips in minimal coupling. Using \( \hat{\mathbf p}+q\mathbf A \) instead of \( \hat{\mathbf p}-q\mathbf A \), or mixing Gaussian and SI, flips the moment or mis-scales \(g\).
Claiming Dirac "explains" \(g=2\) exactly. It predicts the tree-level value; the observed anomaly is a QED effect and its omission is a physics error, not rounding.
Assuming positive-definite single-particle probability solves everything. The current \( j^\mu=c\bar\psi\gamma^\mu\psi \) is positive-definite, but negative-energy states still force the field-theoretic (many-body) reinterpretation.

Discussion

The deepest lesson of the derivation is that spin is not an optional add-on but a consequence of demanding a Lorentz-covariant, first-order, positive-probability wave equation. The Clifford algebra \( \{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu} \) is the arithmetic square root of the Minkowski metric; its irreducible representation in four dimensions is precisely a spinor. The two extra components beyond the naive one enforce a doubling that we read physically as spin, and the two energy signs as particle/antiparticle. Nothing was assumed about intrinsic angular momentum — it fell out of Step 11.

The \( g=2 \) result is a structural prediction: the same matrix \( \vec\sigma \) that generates spin also multiplies \( \mathbf B \), so the ratio of spin magnetic moment to spin angular momentum is fixed to twice the orbital value. Historically this settled a puzzle from atomic spectra where a factor of 2 (and a compensating relativistic factor \( \tfrac12 \) in spin-orbit coupling, the Thomas precession) had resisted explanation. The Dirac equation produces both the factor of 2 and the Thomas \( \tfrac12 \) automatically and consistently.

Antiparticles are the most radical output. The negative-energy solutions cannot be discarded (they are needed for completeness), so Dirac proposed the "sea": a filled negative-energy vacuum whose holes are positive-energy, positive-charge particles — the positron, found by Anderson in 1932. The modern view replaces the sea with field quantisation, where \( \psi \) is an operator that annihilates particles and creates antiparticles, but the physical content — every fermion has an antiparticle of opposite charge and equal mass — survives intact.

At the field-theoretic level the "breaks when" regimes are not failures but signposts: the Klein paradox, Zitterbewegung, and vacuum instability all signal that particle number is not conserved once energies reach \( mc^2 \). The Foldy-Wouthuysen transformation makes this precise by block-diagonalising \( \hat H \) order by order in \( 1/mc^2 \), yielding the Pauli term, spin-orbit coupling, and the Darwin term as a controlled expansion — and showing explicitly that a clean single-particle position operator exists only to each finite order, never exactly. The residual anomaly \( a_e=(g-2)/2 \) is then computed in QED from virtual photon loops, and its agreement with experiment to twelve significant figures is among the most precise confirmations in all of physics.

Common misconceptions. (i) "Dirac derived spin \( \tfrac12 \) from spin \( \tfrac12 \)" — no; spin appears only via Step 11 as the operator needed to conserve total angular momentum. (ii) "The four components are the four states of a spin-\(\tfrac32\) particle" — no; they are (spin up/down) \(\times\) (particle/antiparticle). (iii) "\( g=2 \) is exact" — it is the tree-level value; the measured anomaly is a real, calculable correction.

Worked examples

Electron spin magnetic moment and Zeeman splitting in \( B=1.00\,\mathrm{T} \).

Predict the moment magnitude and the energy gap between \( m_s=\pm\tfrac12 \) states using \( g=2 \). Symbols first, then numbers. A

\[ \mu_z = g\,\frac{q}{2m}\,S_z = g\,\frac{(-e)}{2m}\left(\pm\frac{\hbar}{2}\right)=\mp\, g\,\frac{e\hbar}{4m}=\mp\, g\,\frac{\mu_B}{2},\qquad \mu_B\equiv\frac{e\hbar}{2m} \]

Insert charge \( q=-e \) and \( S_z=\pm\hbar/2 \); define the Bohr magneton. A

\[ \mu_B=\frac{(1.602\times10^{-19})(1.055\times10^{-34})}{2(9.109\times10^{-31})}=9.274\times10^{-24}\ \mathrm{J\,T^{-1}} \]

Evaluate the Bohr magneton in SI. With \( g=2 \), \( |\mu_z|=\mu_B \). A

\[ \Delta E = g\,\mu_B B = 2(9.274\times10^{-24})(1.00)=1.855\times10^{-23}\ \mathrm{J}=1.16\times10^{-4}\ \mathrm{eV} \]

Energy gap \( E=-\vec\mu\cdot\mathbf B \) between the two spin projections, difference \( \Delta m_s=1 \). B

\[ |\mu_z|=\mu_B=9.27\times10^{-24}\ \mathrm{J\,T^{-1}},\qquad \Delta E=1.86\times10^{-23}\ \mathrm{J}=116\ \mu\mathrm{eV} \]

Reading. An electron's spin moment is one Bohr magneton, and in a 1 T field the spin sublevels split by \(116\ \mu\mathrm{eV}\), corresponding to an ESR frequency \( \Delta E/h\approx28\ \mathrm{GHz} \). Halving \(g\) to the classical value 1 would halve both — the factor of 2 is directly observable.

Units check. \( \mathrm{J\,T^{-1}\cdot T=J} \), an energy; dividing by \( h \) (\(\mathrm{J\,s}\)) gives \( \mathrm{s^{-1}} \), a frequency.

Pair-production threshold and Zitterbewegung frequency for an electron.

Use the two energy branches \( E=\pm\sqrt{\mathbf p^2c^2+m^2c^4} \) to find the minimum energy gap and the interference frequency between them. A

\[ E_{\rm gap}=E_+ - E_-\big|_{\mathbf p=0}=2mc^2,\qquad \omega_Z=\frac{E_+-E_-}{\hbar}\bigg|_{\mathbf p=0}=\frac{2mc^2}{\hbar} \]

The smallest separation of the branches is at rest; a superposition of positive- and negative-energy states beats at this frequency (Zitterbewegung). B

\[ 2mc^2=2(9.109\times10^{-31})(2.998\times10^{8})^2=1.637\times10^{-13}\ \mathrm{J}=1.022\ \mathrm{MeV} \]

Evaluate the rest-energy gap. This is the threshold to create an electron-positron pair. A

\[ \omega_Z=\frac{1.637\times10^{-13}}{1.055\times10^{-34}}=1.55\times10^{21}\ \mathrm{rad\,s^{-1}} \]

Divide the gap by \( \hbar \). The associated length scale is the reduced Compton wavelength \( \lambda\!\!\!\!-\,=\hbar/mc=3.86\times10^{-13}\,\mathrm{m} \). B

\[ E_{\rm gap}=2m_ec^2=1.022\ \mathrm{MeV},\qquad \omega_Z=1.55\times10^{21}\ \mathrm{rad\,s^{-1}} \]

Reading. Creating a positron alongside an electron costs at least 1.022 MeV, exactly the negative-energy gap the Dirac equation predicted. The enormous Zitterbewegung frequency and sub-picometre length scale explain why single-particle localisation fails below the Compton wavelength.

Units check. \( \mathrm{kg\,(m/s)^2=J} \); \( \mathrm{J/(J\,s)=s^{-1}} \), an angular frequency.

Problems

Show directly that no set of ordinary (commuting) numbers \( \alpha_1,\alpha_2,\alpha_3,\beta \) can satisfy \( \{\alpha_i,\alpha_j\}=2\delta_{ij} \) and \( \{\alpha_i,\beta\}=0 \), and explain what property of matrices rescues the construction.
Solution
For numbers, \( \{\alpha_i,\alpha_j\}=2\alpha_i\alpha_j \), so \( i\neq j \) demands \( \alpha_i\alpha_j=0 \), forcing at least one factor to vanish, contradicting \( \alpha_i^2=1 \). Equivalently \( \alpha_1\alpha_2=-\alpha_2\alpha_1 \) is impossible for nonzero commuting numbers. Matrices need not commute, so \( \alpha_i\alpha_j=-\alpha_j\alpha_i \) is realisable; the objects must be non-commuting (matrix) quantities of even dimension \( \ge4 \).
Using cyclicity of the trace and \( \beta^2=\mathbf1 \), prove \( \mathrm{Tr}\,\alpha_i=0 \). Then, given the eigenvalues of \( \alpha_i \) are \( \pm1 \), deduce that the matrix dimension is even.
Solution
From \( \{\alpha_i,\beta\}=0 \): \( \alpha_i=-\beta\alpha_i\beta^{-1}=-\beta\alpha_i\beta \). Taking the trace and using \( \mathrm{Tr}(ABC)=\mathrm{Tr}(BCA) \): \( \mathrm{Tr}\,\alpha_i=-\mathrm{Tr}(\beta\alpha_i\beta)=-\mathrm{Tr}(\alpha_i\beta\beta)=-\mathrm{Tr}\,\alpha_i \), hence \( \mathrm{Tr}\,\alpha_i=0 \). Since eigenvalues are \( \pm1 \) and the trace (their sum) is zero, the number of \(+1\)s equals the number of \(-1\)s, so the total dimension \( N \) is even.
Verify the Pauli identity \( (\vec\sigma\cdot\mathbf a)(\vec\sigma\cdot\mathbf b)=\mathbf a\cdot\mathbf b\,\mathbf1+i\,\vec\sigma\cdot(\mathbf a\times\mathbf b) \) for commuting vectors \( \mathbf a,\mathbf b \), and state precisely where the derivation of \( g=2 \) uses the case \( \mathbf a=\mathbf b=\vec\pi \) with non-commuting components.
Solution
Using \( \sigma_i\sigma_j=\delta_{ij}\mathbf1+i\epsilon_{ijk}\sigma_k \): \( (\vec\sigma\cdot\mathbf a)(\vec\sigma\cdot\mathbf b)=a_ib_j\sigma_i\sigma_j=a_ib_j(\delta_{ij}\mathbf1+i\epsilon_{ijk}\sigma_k)=\mathbf a\cdot\mathbf b\,\mathbf1+i\vec\sigma\cdot(\mathbf a\times\mathbf b) \). With \( \mathbf a=\mathbf b=\vec\pi \) the cross term would vanish for commuting components, but \( [\pi_i,\pi_j]=iq\hbar\,\epsilon_{ijk}B_k\neq0 \), so \( \vec\pi\times\vec\pi=i\hbar q\,\mathbf B\neq0 \). This nonzero commutator is exactly the source of the \( -\tfrac{q\hbar}{2m}\vec\sigma\cdot\mathbf B \) term, hence of \( g=2 \).
In the non-relativistic limit the lower spinor is \( \chi\approx\dfrac{\vec\sigma\cdot\vec\pi}{2mc}\varphi \). Estimate the ratio \( |\chi|/|\varphi| \) for a hydrogen-atom electron (take \( v\approx\alpha c \), \( \alpha\approx1/137 \)) and comment on when the approximation fails.
Solution
With \( \pi\sim p\sim mv \), \( |\chi|/|\varphi|\sim p/(2mc)=v/(2c)\approx\alpha/2\approx1/274\approx3.6\times10^{-3} \). The small component is a few parts in a thousand, justifying the Pauli reduction for light atoms. It fails for high \(Z\) where \( v\sim Z\alpha\,c \) approaches \(c\) (relativistic corrections and eventually the \( Z\gtrsim137 \) instability), or in strong fields where \( q\Phi \) is no longer small compared with \( mc^2 \).
An electron sits in a uniform field \( B=2.50\,\mathrm{T} \). Using the Dirac prediction \( g=2 \), compute the spin Zeeman splitting \( \Delta E \) in eV and the corresponding ESR frequency. Then compute the fractional shift if the true anomalous value \( g=2.00232 \) is used.
Solution
\( \Delta E=g\mu_B B=2(9.274\times10^{-24})(2.50)=4.637\times10^{-23}\,\mathrm{J}=2.90\times10^{-4}\,\mathrm{eV} \). Frequency \( \nu=\Delta E/h=4.637\times10^{-23}/6.626\times10^{-34}=7.00\times10^{10}\,\mathrm{Hz}=70.0\,\mathrm{GHz} \). Using \( g=2.00232 \) scales everything by \( 2.00232/2=1.00116 \), a fractional increase of \( 1.16\times10^{-3} \): \( \Delta E\to2.903\times10^{-4}\,\mathrm{eV} \), \( \nu\to70.08\,\mathrm{GHz} \). The \( \sim0.1\% \) shift is the measurable QED anomaly beyond the tree-level Dirac value.