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Quantum Mechanics

Minimal Coupling of Scalar and Dirac Equations

Minimal coupling replaces the ordinary derivative by a gauge-covariant one. For a particle with signed charge qq in a prescribed classical four-potential,

Aμ=(Φc,A),Dμ=∂μ+iqℏAμ.A^\mu=\left(\frac{\Phi}{c},\mathbf A\right), \qquad D_\mu=\partial_\mu+\frac{iq}{\hbar}A_\mu.

This single substitution gives the minimally coupled Klein–Gordon and Dirac equations and reduces to the familiar kinetic momentum

π=−iℏ∇−qA.\boldsymbol\pi=-i\hbar\nabla-q\mathbf A.

The electron is obtained by setting q=−eq=-e, with e>0e>0. No sign is added by hand afterward. The construction below applies to scalar and spinor wave equations in classical electromagnetic backgrounds. For the nonrelativistic Schrödinger derivation, see Minimal Coupling in Wave Mechanics.

Required background. Metric and Units, Four-Vectors, the Klein–Gordon Equation, and the Covariant Dirac Equation supply the sign and unit conventions and the free operators to be coupled. Classical electromagnetic potentials and fields are assumed.

Electromagnetic fields and gauge transformations

Section titled “Electromagnetic fields and gauge transformations”

The electromagnetic fields are

E=−∇Φ−∂tA,B=∇×A.\mathbf E=-\nabla\Phi-\partial_t\mathbf A, \qquad \mathbf B=\nabla\times\mathbf A.

With the mostly-minus metric,

Aμ=(Φc,−A).A_\mu=\left(\frac{\Phi}{c},-\mathbf A\right).

Define the field-strength tensor by

Fμν=∂μAν−∂νAμ.F_{\mu\nu} = \partial_\mu A_\nu-\partial_\nu A_\mu.

Its components include

F0i=Eic,Fij=−ϵijkBk.F_{0i}=\frac{E_i}{c}, \qquad F_{ij}=-\epsilon_{ijk}B_k.

For a real scalar function χ(x)\chi(x), the gauge transformation is

Aμ′=Aμ−∂μχ,ψ′=eiqχ/ℏψ,A_\mu'=A_\mu-\partial_\mu\chi, \qquad \psi'=e^{iq\chi/\hbar}\psi,

or, equivalently,

A′=A+∇χ,Φ′=Φ−∂tχ.\mathbf A'=\mathbf A+\nabla\chi, \qquad \Phi'=\Phi-\partial_t\chi.

The signs of DμD_\mu, the potential transformation, and the matter phase are one linked choice. Changing only one of them destroys gauge covariance.

Directly,

Dμ′ψ′=(∂μ+iqℏAμ′)eiqχ/ℏψ=eiqχ/ℏ(∂μ+iqℏAμ)ψ=eiqχ/ℏDμψ.\begin{aligned} D_\mu'\psi' &= \left( \partial_\mu+\frac{iq}{\hbar}A_\mu' \right)e^{iq\chi/\hbar}\psi \\ &= e^{iq\chi/\hbar} \left( \partial_\mu+\frac{iq}{\hbar}A_\mu \right)\psi \\ &= e^{iq\chi/\hbar}D_\mu\psi. \end{aligned}

Thus DμψD_\mu\psi transforms with the same local phase as ψ\psi.

Curvature of the covariant derivative

Section titled “Curvature of the covariant derivative”

Ordinary partial derivatives commute on the flat coordinate patch, but covariant derivatives do not:

[Dμ,Dν]ψ=iqℏFμνψ.[D_\mu,D_\nu]\psi = \frac{iq}{\hbar}F_{\mu\nu}\psi.

The electromagnetic field strength is therefore the local obstruction to commuting two gauge-covariant derivatives. In spatial language, with πi=−iℏ∂i−qAi\pi_i=-i\hbar\partial_i-qA_i,

[πi,πj]=iqℏ(∂iAj−∂jAi)=iqℏϵijkBk.\begin{aligned} [\pi_i,\pi_j] &= iq\hbar (\partial_iA_j-\partial_jA_i) \\ &= iq\hbar\epsilon_{ijk}B_k. \end{aligned}

Here AiA_i in the three-vector expression is the Euclidean component of A\mathbf A, not the lowered spatial component of the four-potential. The kinetic momenta commute only where the relevant magnetic field vanishes.

Three minimally coupled wave equations

Section titled “Three minimally coupled wave equations”

The nonrelativistic Schrödinger Hamiltonian is

HS=π22m+qΦ.H_S = \frac{\boldsymbol\pi^2}{2m}+q\Phi.

The wave-mechanics derivation explains its probability current and the ordering of the momentum operators.

For a charged complex scalar amplitude, the minimally coupled equation is

(DμDμ+m2c2ℏ2)ϕ=0.\left( D_\mu D^\mu +\frac{m^2c^2}{\hbar^2} \right)\phi=0.

The order matters when the potential varies: DμDμD_\mu D^\mu is an operator composition, not an algebraic square of commuting symbols. A real scalar cannot carry this nontrivial continuous U(1)U(1) charge representation by itself; a charged scalar field is complex.

For a spinor,

(iℏc γμDμ−mc2)ψ=0.\left( i\hbar c\,\gamma^\mu D_\mu-mc^2 \right)\psi=0.

All three formulas use the same signed qq, potentials, and gauge phase.

Dirac Hamiltonian in an electromagnetic background

Section titled “Dirac Hamiltonian in an electromagnetic background”

Use

D0=1c∂t+iqℏcΦ,D_0 = \frac{1}{c}\partial_t +\frac{iq}{\hbar c}\Phi,

and the lowered spatial potential Ai(4)=−Ai(3)A_i^{(4)}=-A_i^{(3)}. Separating the time derivative in the covariant Dirac equation gives

iℏ∂tψ=HDψ,i\hbar\partial_t\psi = H_D\psi,

where

HD=cα⋅π+βmc2+qΦ.\boxed{ H_D = c\boldsymbol\alpha\cdot\boldsymbol\pi +\beta mc^2 +q\Phi }.

This formula makes the canonical/kinetic distinction explicit:

p=−iℏ∇,π=p−qA.\mathbf p=-i\hbar\nabla, \qquad \boldsymbol\pi=\mathbf p-q\mathbf A.

The canonical momentum depends on the representation and gauge choice; the kinetic momentum is gauge covariant and is tied to mechanical velocity. When A(x,t)\mathbf A(\mathbf x,t) varies in space, derivatives act on the potential as well as on the spinor, so operator order cannot be inferred from classical commuting symbols.

For an electron, q=−eq=-e gives

π=p+eA,qΦ=−eΦ.\boldsymbol\pi=\mathbf p+e\mathbf A, \qquad q\Phi=-e\Phi.

Both signs follow from the same substitution.

Squaring exposes spin–field coupling

Section titled “Squaring exposes spin–field coupling”

Minimal coupling makes the Dirac operator sensitive to the noncommutativity of DμD_\mu. With σμν=i[γμ,γν]/2\sigma^{\mu\nu}=i[\gamma^\mu,\gamma^\nu]/2,

(γμDμ)2=DμDμ+q2ℏσμνFμν.(\gamma^\mu D_\mu)^2 = D_\mu D^\mu +\frac{q}{2\hbar}\sigma^{\mu\nu}F_{\mu\nu}.

Therefore every minimally coupled Dirac solution obeys the second-order equation

(DμDμ+m2c2ℏ2+q2ℏσμνFμν)ψ=0.\left( D_\mu D^\mu +\frac{m^2c^2}{\hbar^2} +\frac{q}{2\hbar}\sigma^{\mu\nu}F_{\mu\nu} \right)\psi=0.

The extra field-strength term is absent for a minimally coupled scalar. Its magnetic part is the algebraic seed of the Pauli spin coupling and the tree-level Dirac value g=2g=2.

SI and Gaussian-unit translations

Section titled “SI and Gaussian-unit translations”

This page uses SI-normalized potentials, for which

iℏDμ=iℏ∂μ−qAμ.i\hbar D_\mu=i\hbar\partial_\mu-qA_\mu.

Some Gaussian-unit texts write a substitution of the form

pμ⟶pμ−qcAμG.p_\mu\longrightarrow p_\mu-\frac{q}{c}A_\mu^{\mathrm G}.

The displayed 1/c1/c is tied to how the Gaussian four-potential and charge are defined. It must not be inserted into this page’s SI formula while retaining the SI Aμ=(Φ/c,A)A^\mu=(\Phi/c,\mathbf A). Translate the entire electromagnetic unit system, including field definitions and charge units, rather than moving a single factor of cc.

Minimal does not mean most general

Section titled “Minimal does not mean most general”

Minimal coupling is the interaction generated by replacing ordinary derivatives with DμD_\mu. It fixes the tree-level magnetic moment of an elementary Dirac particle to g=2g=2. Symmetry also permits nonminimal effective operators. The leading spinor example has the tensor structure

ψˉσμνFμνψ,\bar\psi\sigma^{\mu\nu}F_{\mu\nu}\psi,

with a coefficient determined by microscopic physics or experiment. In QED, radiative corrections generate an anomalous magnetic moment. Writing that term is not a failure of gauge covariance; it is a statement that the low- energy theory contains more than the minimal operator.

The coefficient and its SI signs are matched explicitly on Magnetic Moment.

This page treats AμA_\mu as a prescribed classical background. When the electromagnetic field itself fluctuates quantum mechanically, photons carry energy and momentum and radiative corrections require QED. Pair creation already becomes possible when the matter field is quantized in a prescribed classical electric background; quantized photons are not a prerequisite. The first-quantized equation used here does not itself provide the vacuum and particle-number observables needed to calculate that process.

Common pitfalls

Section titled “Common pitfalls”

Replacing qq by positive ee everywhere. The electron has q=−eq=-e. Substitute once into the full Hamiltonian so scalar and vector terms retain consistent signs.

Mixing SI and Gaussian formulas. A lone 1/c1/c cannot be translated in isolation. Fix the four-potential and charge normalization first.

Expanding π2\boldsymbol\pi^2 as commuting algebra. Momentum derivatives act on A\mathbf A, and different kinetic-momentum components fail to commute in a magnetic field.

Changing the gauge phase but not DμD_\mu. Gauge covariance requires three consistent signs: the potential shift, matter phase, and covariant derivative.

Exercises

Section titled “Exercises”

1. Gauge covariance

Section titled “1. Gauge covariance”

Start with zero potentials and a plane wave ψ=uexp⁡[i(p⋅x−Et)/ℏ]\psi=u\exp[i(\mathbf p\cdot\mathbf x-Et)/\hbar], where uu is a constant scalar or spinor amplitude satisfying the appropriate free equation. Apply the time-independent gauge function χ=λxy\chi=\lambda xy, with real constant λ\lambda. Find the new potentials and show that the kinetic-momentum eigenvalue remains p\mathbf p, although the canonical momentum acting on the transformed wave changes.

Solution

The new potentials are Φ′=0\Phi'=0 and A′=(λy,λx,0)\mathbf A'=(\lambda y,\lambda x,0). Their curl vanishes: ∂xAy′−∂yAx′=λ−λ=0\partial_x A'_y-\partial_y A'_x=\lambda-\lambda=0. The transformed amplitude is ψ′=eiqλxy/ℏψ\psi'=e^{iq\lambda xy/\hbar}\psi. Therefore

−iℏ∇ψ′=[p+qλ(y,x,0)]ψ′,(−iℏ∇−qA′)ψ′=pψ′.-i\hbar\nabla\psi' =\bigl[\mathbf p+q\lambda(y,x,0)\bigr]\psi', \qquad (-i\hbar\nabla-q\mathbf A')\psi'=\mathbf p\psi'.

The extra spatially varying phase changes the canonical-momentum expression, but the potential subtracts precisely that change from the kinetic momentum. No electric or magnetic field has been introduced.

2. Kinetic-momentum commutator

Section titled “2. Kinetic-momentum commutator”

Derive [πi,πj][\pi_i,\pi_j] in position representation.

Solution

Using [pi,f]=−iℏ∂if[p_i,f]=-i\hbar\partial_i f,

[πi,πj]=−q[pi,Aj]−q[Ai,pj]=iqℏ(∂iAj−∂jAi)=iqℏϵijkBk.\begin{aligned} [\pi_i,\pi_j] &= -q[p_i,A_j]-q[A_i,p_j] \\ &= iq\hbar(\partial_iA_j-\partial_jA_i) \\ &= iq\hbar\epsilon_{ijk}B_k. \end{aligned}

3. Aharonov–Bohm phase

Section titled “3. Aharonov–Bohm phase”

Consider two coherent paths around a static, inaccessible magnetic-flux region. Both paths are field free and have the same scalar-potential contribution. Orient the closed curve CC along the first path and back along the second. Show that the magnetic contribution to their relative phase is determined by the enclosed flux.

Solution

The vector-potential contribution to the phase is

Δθ=qℏ∮CA⋅dℓ.\Delta\theta = \frac{q}{\hbar}\oint_C\mathbf A\cdot d\boldsymbol\ell.

Stokes’ theorem gives

Δθ=qℏ∫SB⋅dS=qΦBℏ.\Delta\theta = \frac{q}{\hbar}\int_S\mathbf B\cdot d\mathbf S = \frac{q\Phi_B}{\hbar}.

The orientation of the surface normal agrees with CC by the right-hand rule. The observable phase is defined modulo 2π2\pi. A single open-path phase is gauge dependent, while the closed-loop difference is unchanged by A↦A+∇χ\mathbf A\mapsto\mathbf A+\nabla\chi for single-valued χ\chi. For time-dependent fields, the phase generally also involves the scalar potential and must be evaluated along the two paths in spacetime.

4. Squared Dirac operator

Section titled “4. Squared Dirac operator”

Derive the coefficient of σμνFμν\sigma^{\mu\nu}F_{\mu\nu} in (γμDμ)2(\gamma^\mu D_\mu)^2.

Solution

Separate symmetric and antisymmetric products:

γμγνDμDν=DμDμ+14[γμ,γν][Dμ,Dν].\gamma^\mu\gamma^\nu D_\mu D_\nu = D_\mu D^\mu +\frac14[\gamma^\mu,\gamma^\nu][D_\mu,D_\nu].

Using [γμ,γν]=−2iσμν[\gamma^\mu,\gamma^\nu]=-2i\sigma^{\mu\nu} and [Dμ,Dν]=iqFμν/ℏ[D_\mu,D_\nu]=iqF_{\mu\nu}/\hbar gives

(γμDμ)2=DμDμ+q2ℏσμνFμν.(\gamma^\mu D_\mu)^2 = D_\mu D^\mu +\frac{q}{2\hbar}\sigma^{\mu\nu}F_{\mu\nu}.

References

Section titled “References”