The Hydrogen Atom in QED

The hydrogen atom is the canonical worked example of QED, and the most precisely tested system in atomic physics. This page walks through how the QED framework of QED and QED/historical.md actually produces the hydrogen spectrum, layer by layer.

The thread is: QED gives the full description of hydrogen as a bound state in Fock space, and the familiar Schrödinger / Dirac / Lamb-shift treatments are successive approximations in $\alpha =e^2/(4\pi \hslash c)\approx 1/137$.

1. Hydrogen as a Bound State in Fock Space

In QED, hydrogen is a bound state of the proton-electron-photon system — a non-perturbative state in ${\mathcal{F}}_{\text{QED}}\otimes {\mathcal{H}}_{\text{proton}}$ (the proton enters either as a fixed external source or as an additional fermion species). Schematically,

$$|H,n⟩=\int d^{3}p{d}^{3}q{\Phi }_n(\mathbf{p},\mathbf{q})b_{\mathbf{p}}^{†}{B}_{\mathbf{q}}^{†}|0⟩+(\text{photon-dressed admixtures}),$$

where:

• $b_{\mathbf{p}}^{†}$ creates an electron of momentum $\mathbf{p}$.
• $B_{\mathbf{q}}^{†}$ creates a proton of momentum $\mathbf{q}$.
• ${\Phi }_n(\mathbf{p},\mathbf{q})$ is the two-body relativistic wavefunction, satisfying the Bethe–Salpeter equation (the relativistic generalization of the Schrödinger equation for bound states).
• The "(photon-dressed admixtures)" indicate that the true bound state has nonzero overlap with sectors containing extra photons, virtual $e^{+}{e}^{-}$ pairs, and so on — these are the loop corrections.

This is the row labelled "Bound state" in fock-space-inventory.md §3, with $d^{†}$ (positron) replaced by $B^{†}$ (proton). Bound states are flagged there as non-perturbative because ${\Phi }_n$ cannot be obtained by ordinary perturbation theory in $\alpha$ — the Coulomb interaction must be summed to all orders to produce a bound state at all.

The full QED hydrogen state is therefore an object of considerable complexity. In practice one almost never works with $|H,n⟩$ directly; instead, one uses an expansion in two small parameters:

• $\alpha \approx 1/137$ — fine-structure constant.
• $m_e/m_p\approx 1/1836$ — electron-to-proton mass ratio.

Each level of the expansion adds physics. The next sections describe them in turn.

2. Hierarchy of Approximations

Level 0 — Schrödinger Hydrogen $(\sim m_e{c}^2{\alpha }^2)$

Treat the proton as an infinitely heavy classical Coulomb source $V(\mathbf{r})=-e^2/(4\pi r)$, and use the non-relativistic Schrödinger equation:

$$\left[-\frac{{\hslash }^2}{2m_e}{\nabla }^2-\frac{e^2}{4\pi r}\right]\psi (\mathbf{r})=E_n\psi (\mathbf{r}),E_n=-\frac{m_e{c}^2{\alpha }^2}{2n^2}.$$

This gives the gross structure of hydrogen: the Bohr energies $E_n=-13.6\mathrm{eV}/n^2$, the principal quantum number $n$, the orbital quantum numbers $\ell$, $m$, and (added by hand) the spin quantum number $m_s$. The eigenfunctions ${\psi }_{n\ell m}(\mathbf{r})$ are the familiar hydrogen wavefunctions involving Laguerre polynomials and spherical harmonics.

Position-space wavefunctions (closed form). Separation in spherical coordinates gives

$${\psi }_{n\ell m}(r,\theta ,\phi )=R_{n\ell }(r)Y_{\ell }^m(\theta ,\phi ),$$

with the radial functions

$$R_{n\ell }(r)=\sqrt{{\left(\frac{2}{n{a}_0}\right)}^3\frac{(n-\ell -1)!}{2n(n+\ell )!}}{e}^{-r/(n{a}_0)}{\left(\frac{2r}{n{a}_0}\right)}^{\ell }{L}_{n-\ell -1}^{2\ell +1}\left(\frac{2r}{n{a}_0}\right),$$

where $a_0=\hslash /(m_{e}c\alpha )\approx 5.29\times 10^{-11}\mathrm{m}$ is the Bohr radius and $L_k^{(\alpha )}$ are the associated Laguerre polynomials. The first few:

$${\psi }_{100}(\mathbf{r})=\frac{1}{\sqrt{\pi a_0^3}}{e}^{-r/a_0},{\psi }_{200}(\mathbf{r})=\frac{1}{\sqrt{32\pi a_0^3}}\left(2-\frac{r}{a_0}\right)e^{-r/(2a_0)},$$

$${\psi }_{210}(\mathbf{r})=\frac{1}{\sqrt{32\pi a_0^3}}\frac{r}{a_0}{e}^{-r/(2a_0)}\cos \theta .$$

In QED language, this is the leading-order term in a non-relativistic, no-radiation, classical-source expansion — zero of the QED machinery (no Fock space, no virtual photons, no antiparticles, no loops) is actually used. The proton is classical; the electron is first-quantized.

Level 1 — Fine Structure $(\sim m_e{c}^2{\alpha }^4)$

Replace the Schrödinger equation by the Dirac equation in an external Coulomb field:

$$(i{\gamma }^{\mu }({\partial }_{\mu }+ie{A}_{\mu }^{\text{ext}})-m_e)\psi =0,A_0^{\text{ext}}(\mathbf{r})=-\frac{e}{4\pi r},A_i^{\text{ext}}=0.$$

This is exactly the minimally coupled Dirac equation of QED/historical.md §2.1, with $A_{\mu }$ as a fixed external classical field rather than a quantized one. Expanding the Dirac-Coulomb solutions in powers of $(v/c{)}^2=O((Z\alpha {)}^2)$ around the Schrödinger result adds three corrections:

• Relativistic kinetic energy: ${\hat{T}}_{\text{rel}}=-\frac{{\hat{p}}^4}{8m_e^3{c}^2}$, from $\sqrt{{\hat{p}}^2{c}^2+m_e^2{c}^4}-m_e{c}^2\approx {\hat{p}}^2/(2m_e)-{\hat{p}}^4/(8m_e^3{c}^2)+\cdots$.
• Spin-orbit coupling: ${\hat{H}}_{\text{SO}}=\frac{1}{2m_e^2{c}^2}\frac{1}{r}\frac{dV}{dr}\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$, from the Dirac magnetic-moment interaction in the non-relativistic reduction. This couples orbital and spin angular momenta into the total $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$.
• Darwin term: ${\hat{H}}_{\text{D}}=\frac{\pi e^2}{2m_e^2{c}^2}{\delta }^3(\mathbf{r})$, a purely relativistic effect from the Zitterbewegung smearing (see fock-space-inventory.md §0.8.7) — the electron's effective position is averaged over a Compton wavelength.

The exact Dirac-Coulomb spectrum is

$$E_{n,j}=m_e{c}^2\left[{\left(1+\frac{(Z\alpha {)}^2}{(n-j-\frac{1}{2}+\sqrt{(j+\frac{1}{2}{)}^2-(Z\alpha {)}^2}{)}^2}\right)}^{-1/2}\right]-m_e{c}^2,$$

which expanded to leading correction in $(Z\alpha {)}^2$ gives the standard fine-structure formula:

$$E_{n,j}\approx -\frac{m_e{c}^2(Z\alpha {)}^2}{2n^2}\left[1+\frac{(Z\alpha {)}^2}{n}\left(\frac{1}{j+\frac{1}{2}}-\frac{3}{4n}\right)+O({\alpha }^4)\right].$$

Note that fine-structure energies depend only on $n$ and the total angular momentum $j$, not on $\ell$. So states like $2S_{1/2}$ and $2P_{1/2}$ (different $\ell$, same $j=\frac{1}{2}$) are degenerate at this level. This degeneracy is an "accidental" symmetry of the Coulomb potential — it doesn't survive once QED loop corrections (Level 2) are turned on.

Position-space wavefunctions (closed form — Darwin–Gordon, 1928). With orbital angular momentum no longer commuting with ${\hat{H}}_{\text{Dirac}}$ (spin–orbit mixes $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$), the angular dependence is carried by the spinor spherical harmonics

$${\Omega }_{j\ell m_j}(\theta ,\phi )=\sum _{m,m_s}⟨\ell ,m;\frac{1}{2},m_s|j,m_j⟩Y_{\ell }^m(\theta ,\phi )|m_s⟩,$$

and the exact 4-spinor wavefunction is

$${\psi }_{nj\kappa m_j}(\mathbf{r})=\left(\begin{array}{c}{g}_{n\kappa }(r){\Omega }_{j\ell m_j}(\theta ,\phi )\\ i{f}_{n\kappa }(r){\Omega }_{j{\ell }^{\prime }{m}_j}(\theta ,\phi )\end{array}\right),$$

where $\kappa =\mp (j+\frac{1}{2})$ for $j=\ell \pm \frac{1}{2}$, ${\ell }^{\prime }=2j-\ell$, and the upper / lower 2-component blocks have opposite parity (so $\ell$ and ${\ell }^{\prime }$ differ by 1). The radial functions are

$$\left(\begin{array}{c}{g}_{n\kappa }(r)\\ f_{n\kappa }(r)\end{array}\right)=N{r}^{\gamma -1}{e}^{-\lambda r}\left(\begin{array}{c}(m{c}^2+E)F_1\pm (m{c}^2-E)F_2\\ (m{c}^2-E)F_1\mp (m{c}^2+E)F_2\end{array}\right),$$

with parameters

$$\gamma =\sqrt{{\kappa }^2-(Z\alpha {)}^2},\lambda =\sqrt{m^2{c}^4-E^2}/(\hslash c),$$

and $F_{1,2}$ confluent hypergeometric functions ${}_1{F}_1$ in the variable $2\lambda r$. $N$ is a normalization constant. Explicit formulas are given in any relativistic-QM textbook (Bjorken–Drell, Greiner, Strange).

In the $Z\alpha \to 0$ limit these reduce to the Schrödinger wavefunctions of Level 0 (with the lower component vanishing as $v/c\to 0$).

This is still first-quantized electron physics in a classical Coulomb field. The full QED machinery — quantized $A_{\mu }$, Fock space, loops — has not been used.

Level 2 — Lamb Shift and True QED $(\sim m_e{c}^2{\alpha }^5)$

Now promote $A_{\mu }$ to a quantized field. New effects appear at order ${\alpha }^5\log \alpha$:

• Electron self-energy. The bound electron emits and reabsorbs virtual photons; the corresponding one-loop diagram shifts its energy. This is the dominant contribution to the Lamb shift. Conceptually: the electron is "dressed" by a cloud of virtual photons whose energy depends on the binding to the nucleus.

• Vacuum polarization (Uehling potential). Virtual $e^{+}{e}^{-}$ pairs in the photon propagator modify the Coulomb potential at short distances:

  $$V_{\text{eff}}(r)=-\frac{Z\alpha }{r}\left[1+\frac{2\alpha }{3\pi }{\int }_1^{\infty }dt\frac{\sqrt{t^2-1}}{t^2}\left(1+\frac{1}{2t^2}\right)e^{-2m_{e}rt/\hslash }+\cdots \right].$$

  The correction is exponentially suppressed beyond the Compton wavelength $\hslash /(m_{e}c)\approx 386\mathrm{fm}$, so it shifts only $S$-states (which have non-zero density at the origin).

• Anomalous magnetic moment. The electron has $g_e=2(1+a_e)$ with $a_e=\alpha /(2\pi )+O({\alpha }^2)$ — Schwinger's celebrated 1948 result. This modifies the spin-orbit and hyperfine couplings.

The combined effect lifts the Dirac-Coulomb degeneracy between $2S_{1/2}$ and $2P_{1/2}$ by the Lamb shift:

$$\Delta E_{2S_{1/2}-2P_{1/2}}\approx 1057\mathrm{MHz}\cdot h\approx 4.4\times 10^{-6}\mathrm{eV}.$$

Measured by Willis Lamb in 1947 and explained by Bethe within weeks (with the first one-loop QED computation), this was the founding triumph of QED. It is the first level at which QED's full Fock-space machinery — quantized photons, virtual particle loops, renormalization — is genuinely required.

Position-space wavefunctions: no closed form. There is no closed-form expression for the bound-state wavefunction once one-loop QED corrections are included. In practice the Dirac–Coulomb wavefunctions of Level 1 are used as a basis, and the QED corrections are computed perturbatively as matrix elements:

$$\Delta E_n^{\text{Lamb}}=⟨{\psi }_n^{\text{Dirac}}|{\hat{H}}_{\text{QED-correction}}|{\psi }_n^{\text{Dirac}}⟩+(\text{Bethe sum over intermediate states}),$$

where ${\hat{H}}_{\text{QED-correction}}$ packages the self-energy, vacuum polarization, and anomalous-moment contributions as effective operators (see §6 on NRQED). The one analytic piece available in closed form is the Uehling potential itself (the integral above), but the eigenvalue problem for the Coulomb + Uehling potential is not analytically solvable.

Level 3 — Hyperfine Structure $(\sim m_e{c}^2{\alpha }^4(m_e/m_p))$

Treat the proton as having spin ${\mathbf{S}}_p$ and a magnetic moment

$${\mathbf{\mu }}_p=g_p\frac{e}{2m_p}{\mathbf{S}}_p,g_p\approx \mathrm{5.586.}$$

The interaction $-{\mathbf{\mu }}_p\cdot {\mathbf{B}}_e$ between proton spin and the electron's magnetic field at the proton splits states with different total angular momentum $\mathbf{F}={\mathbf{J}}_e+{\mathbf{S}}_p$. For an $S$-state (no orbital field), the dominant contribution comes from the contact (Fermi) interaction at the origin.

For the $1S_{1/2}$ ground state, this gives the 21 cm line:

$$\Delta E_{F=1-F=0}\approx 5.9\times 10^{-6}\mathrm{eV},\nu \approx 1420\mathrm{MHz},\lambda \approx 21\mathrm{cm}.$$

This is the spectral line astronomers use to map neutral hydrogen across the Milky Way and beyond — visible to radio telescopes.

Note: $g_p\approx 5.586$ is anomalous (the proton is a composite QCD bound state; its $g$-factor is not 2). The numerical value of the hyperfine splitting therefore requires QCD input, not just QED. The QED part is the form of the interaction; the coefficient requires nuclear physics.

Position-space wavefunctions. The angular structure is now the doubly-coupled ${\Omega }_{j\ell m_j}(\theta ,\phi )\otimes |S_p,m_p⟩$ recombined into total $|F,m_F⟩$ via Clebsch–Gordan. The spatial / radial dependence is unchanged from Level 1 (Dirac–Coulomb radial functions); the hyperfine splitting is a small energy correction whose proportionality to $|\psi (0){|}^2$ at the origin makes the radial wavefunction enter only as an evaluated number, not as a modified function.

Level 4 — Higher-Order Corrections

Modern hydrogen spectroscopy probes effects at order ${\alpha }^6$ and beyond, requiring:

• Two-loop and higher QED corrections to self-energy and vacuum polarization.

• Nuclear recoil corrections. At leading order, replace $m_e$ everywhere by the reduced mass $\mu =m_e{m}_p/(m_e+m_p)$. Beyond that, relativistic recoil corrections of order ${\alpha }^4(m_e/m_p)$ enter.

• Finite proton size. The proton has an RMS charge radius $⟨r_p^2{⟩}^{1/2}\approx 0.84\mathrm{fm}$, modifying the Coulomb potential at very short distances:

  $$\Delta E_{\text{size}}\approx \frac{2\pi }{3}Z\alpha ⟨r_p^2⟩|\psi (0){|}^2.$$

  This affects only $S$-states and is the source of the proton radius puzzle (a long-standing $\sim 5\sigma$ discrepancy between hydrogen-spectroscopy and muonic-hydrogen / electron-scattering determinations of $r_p$, partially resolved in recent years).

• Weak-interaction parity violation. Tiny mixing of $S$ and $P$ states from $Z$-boson exchange — beyond QED proper, in the electroweak sector.

3. Numerical Hierarchy

For the $1S$–$2S$ transition in hydrogen ($\sim 10\mathrm{eV}$), the relative size of each contribution is roughly:

Modern spectroscopy (Garching, NIST, MPQ) measures the $1S$–$2S$ transition to 15 significant figures. Agreement with QED holds at this level, currently limited by uncertainty in the proton charge radius rather than by QED itself.

4. Closed-Form Solutions: What's Available

Position-space wavefunctions of hydrogen exist in closed form only at the first two levels of the hierarchy. Beyond Dirac–Coulomb, the bound state can only be characterized via systematic expansions or numerical methods.

4.1 What Is Closed-Form

The Dirac–Coulomb spectrum is exact to all orders in $Z\alpha$:

$$E_{nj}=m_e{c}^2{\left[1+\frac{(Z\alpha {)}^2}{(n-j-\frac{1}{2}+\sqrt{(j+\frac{1}{2}{)}^2-(Z\alpha {)}^2}{)}^2}\right]}^{-1/2}.$$

4.2 What Is Not Closed-Form

Every additional ingredient breaks closed-form solvability:

• Quantized photons (Lamb shift). Self-energy diagrams involve a logarithm $\log (m_e/⟨\Delta E⟩)$ — Bethe's logarithm — that depends on the entire bound-state spectrum. No closed form for the resulting energy shift.
• Quantum proton (recoil). Even the classical two-body relativistic problem has no separable closed form. Bethe–Salpeter restores Lorentz covariance at the cost of an integral equation in 4D.
• Vacuum polarization. The Uehling potential is closed-form (an integral representation), but the eigenvalue problem for Coulomb + Uehling potential is not.
• Asymptoticity of the QED series. At sufficiently high orders, the perturbative QED series for hydrogen is asymptotic, not convergent. An "all-orders sum" is not well-defined as a function of $\alpha$.
• Haag's theorem. The interacting QED bound state $|H,n⟩$ does not live in the same Hilbert space as the free Fock space (see QFT/remarks.md). There is no rigorous sense in which an "exact wavefunction" exists as a function of $\mathbf{x}$ in any rigorously constructed Hilbert space.

4.3 What Is Used in Practice

For state-of-the-art atomic-physics calculations:

• Dirac–Coulomb wavefunctions are the workhorse. They serve as the basis on which QED corrections are computed perturbatively.
• NRQED (Caswell–Lepage 1986) is an effective field theory in which $v/c\sim Z\alpha$ is the small parameter. The hydrogen Hamiltonian becomes a power series in $\alpha$ with coefficients computed once and reused; effective-Hamiltonian eigenstates are exact eigenfunctions at given order in $\alpha$.
• Bethe–Salpeter integral equations for the two-body wavefunction are solved numerically.
• Dimensional regularization gives analytical expressions for individual loop contributions, often involving zeta values and polylogarithms.

In a useful but loose sense, the "exact QED wavefunction" used in practice is the Dirac–Coulomb spinor, with QED corrections folded in via perturbation theory and effective Hamiltonians. This is a productive fiction, not a true exact QED state.

4.4 The Bethe–Salpeter Equation Explicitly

Although it has no closed-form solutions, the Bethe–Salpeter (BS) equation is the formally correct relativistic two-body bound-state equation in QFT, and it is fully explicit. For a two-fermion bound state with total 4-momentum $P^{\mu }$ and quantum numbers $n$, the Bethe–Salpeter amplitude is the matrix element

$${\chi }_P(x_1,x_2)\equiv ⟨0|T\hat{\psi }(x_1){\hat{\psi }}^{\prime }(x_2)|P,n⟩,$$

where $\hat{\psi }$ is the electron field and ${\hat{\psi }}^{\prime }$ is the second fermion's field (proton for hydrogen, positron for positronium). The homogeneous Bethe–Salpeter equation is

$$\boxed{{\chi }_P(x_1,x_2)=\int d^4{y}_1{d}^4{y}_2{d}^4{z}_1{d}^4{z}_2{S}_F(x_1-y_1)S_F^{\prime }(x_2-y_2)K(y_1,y_2;z_1,z_2){\chi }_P(z_1,z_2),}$$

or, schematically, ${\chi }_P=(S_F\otimes S_F^{\prime })K{\chi }_P$. Here:

• $S_F$, $S_F^{\prime }$ are the full (renormalized) one-particle propagators of the two fermions.
• $K$ is the two-particle irreducible (2PI) kernel — the sum of all Feynman diagrams that cannot be cut into two pieces by removing one electron line and one ${\hat{\psi }}^{\prime }$ line.

In momentum space, with relative momentum $q$ and total $P$, this becomes the integral eigenvalue equation

$${\chi }_P(q)=S_F(p_1)S_F^{\prime }(p_2)\int \frac{d^4{q}^{\prime }}{(2\pi {)}^4}K(q,q^{\prime };P){\chi }_P(q^{\prime }),p_{1,2}=\frac{P}{2}\pm q,$$

with bound-state masses appearing as the discrete values of $P^2=M^2$ for which non-trivial solutions exist.

The kernel order by order. $K$ is a perturbative sum, not closed-form. The leading non-trivial contribution is the single-photon-exchange ("ladder") kernel:

$$K^{(1)}(q,q^{\prime };P)=(-ie{\gamma }^{\mu }{)}_{\text{electron}}\otimes (+ie{\gamma }^{\nu }{)}_{\text{proton}}\frac{-i{\eta }_{\mu \nu }}{(q-q^{\prime }{)}^2+iϵ}.$$

Diagrammatically the resulting iteration (the ladder approximation) sums all "rung diagrams":

  e ─→─┬─→─┬─→─┬─→─ ...
       γ   γ   γ
  p ─→─┴─→─┴─→─┴─→─ ...

Higher orders add crossed-ladder diagrams, vertex corrections, self-energy insertions (which dress the propagators via Schwinger–Dyson), and vacuum-polarization insertions on the exchanged photon. Each is a higher power of $\alpha$.

Practical difficulties. Several issues make the raw 4D BS equation unwieldy:

• It is genuinely 4-dimensional — the relative momentum has a relative-energy component $q^0$ as well as $\mathbf{q}$, and there is no obvious reduction to a 3D Schrödinger-style problem.
• It has spurious solutions corresponding to anomalous dependence on relative time, which must be projected out.
• The off-shell kernel $K$ is gauge-dependent; physical bound-state energies are gauge-invariant only after summing the kernel to all orders.
• Even the toy ladder approximation has no closed-form solutions for full QED. The Wick–Cutkosky model (scalar electrodynamics with massless exchange) is a rare solvable case, via Wick rotation to a 4D harmonic-oscillator-like equation.

3D reductions. Because the 4D BS equation is hard to use directly, practical calculations use 3D reductions:

• Salpeter equation — instantaneous approximation $K(q,q^{\prime };P)\to K(\mathbf{q},{\mathbf{q}}^{\prime };P)$, integrating out $q^0$.
• Brezin–Itzykson–Zinn-Justin reduction — a more covariant 3D reduction that retains more relativistic structure than Salpeter.
• NRQED — integrate out high-energy modes ($|k|\gg m_{e}v$) once and for all, leaving an effective non-relativistic Hamiltonian with QED-computed matching coefficients. This is the modern workhorse.

Concrete example: positronium. The cleanest BS / NRQED application is positronium, where there is no QCD complication. Leading-order energies

$$E_n=-\frac{m_e{\alpha }^2}{4n^2}$$

(half the hydrogen Bohr energy because the reduced mass is $m_e/2$), and the leading hyperfine splitting between para- and ortho-positronium

$$\Delta E_{\text{hfs}}^{1S}=\frac{7}{12}{m}_e{\alpha }^4\approx 8.4\times 10^{-4}\mathrm{eV},$$

are now known through $O(m_e{\alpha }^7)$, with computed and measured values agreeing to $\sim 10$-digit precision.

In summary: the BS equation is essential conceptually — it defines what a relativistic bound state in QFT means — but is rarely used in its raw 4D form for production calculations. NRQED has replaced it as the practical tool.

4.5 Where the BS Equation Comes From

The BS equation is not postulated — it is a structural consequence of three ingredients that are already present in any QFT: the four-point Green's function, the 2PR/2PI organization of Feynman diagrams, and the spectral representation of correlation functions. Here is the standard derivation.

Step 1 — The four-point Green's function

For two fermion species $\hat{\psi }$, ${\hat{\psi }}^{\prime }$, define the four-point Green's function

$$G(x_1,x_2;y_1,y_2)\equiv ⟨0|T\hat{\psi }(x_1){\hat{\psi }}^{\prime }(x_2){\bar{\hat{\psi }}}^{\prime }(y_2)\bar{\hat{\psi }}(y_1)|0⟩.$$

This is the propagation amplitude for two fermions from $(y_1,y_2)$ to $(x_1,x_2)$, summing all interactions to all orders. It is what perturbation theory directly computes (a sum of all Feynman diagrams with two incoming and two outgoing fermion lines).

Step 2 — Two-particle reducibility

Organize diagrams in $G$ by two-particle reducibility:

• A diagram is two-particle reducible (2PR) if it can be cut into two pieces by removing one $\hat{\psi }$ line and one ${\hat{\psi }}^{\prime }$ line.
• It is two-particle irreducible (2PI) otherwise.

Define the 2PI kernel $K$ as the sum of all 2PI diagrams. Combinatorially, every diagram in $G$ is either 2PI or decomposes uniquely as a chain

$$\text{2PI block}\to \text{full propagators}\to \text{2PI block}\to \cdots$$

— i.e., $G$ is a sum of "chains" of 2PI kernels glued by full propagators. (This is just the definition of 2PI: cutting at any 2PR location separates the diagram into a smaller chain.)

Step 3 — Dyson's equation for $G$

Translating the chain decomposition into an equation gives Dyson's equation for the four-point function:

$$\boxed{G=G_0+G_{0}KG,}$$

where:

• $G_0(x_1,x_2;y_1,y_2)=S_F(x_1-y_1)S_F^{\prime }(x_2-y_2)$ is the disconnected propagator product (full one-particle propagators, no interaction between the two particles).
• The product is a 4-fold spacetime convolution.

This is exactly analogous to the Dyson equation $S_F=S_F^{(0)}+S_F^{(0)}\Sigma S_F$ for the one-particle propagator, but at the two-particle level. Iterating recovers the geometric series

$$G=G_0+G_{0}K{G}_0+G_{0}K{G}_{0}K{G}_0+\cdots ,$$

which is the chain decomposition above.

Step 4 — Bound states as poles in $G$

So far this is just a resummation; bound states have not yet appeared. They enter through the spectral content of $G$.

The four-point function, viewed as a function of the total momentum $P^{\mu }$ (Fourier-conjugate to the center-of-mass position), has discrete poles at $P^2=M_n^2$ for each bound-state mass $M_n$. Inserting a complete set of states between $T\hat{\psi }(x_1){\hat{\psi }}^{\prime }(x_2)$ and ${\bar{\hat{\psi }}}^{\prime }(y_2)\bar{\hat{\psi }}(y_1)$,

$$G=\sum _n\frac{i{\chi }_{P_n}(x_1,x_2){\bar{\chi }}_{P_n}(y_1,y_2)}{P^2-M_n^2+iϵ}+(\text{continuum}),$$

the residue at $P^2=M_n^2$ defines the Bethe–Salpeter amplitude

$${\chi }_{P_n}(x_1,x_2)\equiv ⟨0|T\hat{\psi }(x_1){\hat{\psi }}^{\prime }(x_2)|P_n⟩.$$

Step 5 — The BS equation as the residue equation

Plug the pole structure of $G$ back into the Dyson equation $G=G_0+G_{0}KG$ and match residues at $P^2=M_n^2$:

• The left-hand side has a simple pole with residue $\propto {\chi }_{P_n}{\bar{\chi }}_{P_n}$.
• The first term $G_0$ on the right has no pole at $P^2=M_n^2$ (free propagators do not bind).
• The pole on the right comes entirely from the $G$ in $G_{0}KG$, with residue $\propto G_{0}K{\chi }_{P_n}{\bar{\chi }}_{P_n}$.

Matching residues and dropping the common factor ${\bar{\chi }}_{P_n}$,

$$\boxed{{\chi }_{P_n}=G_{0}K{\chi }_{P_n}=(S_F\otimes S_F^{\prime })K{\chi }_{P_n}.}$$

This is the homogeneous Bethe–Salpeter equation stated in §4.4. Non-trivial solutions exist only for those values $P^2=M_n^2$ that are bound-state masses — the eigenvalue condition.

Status: postulated vs. derived

So the BS equation is not an independent postulate — it is a structural consequence of having any QFT with bound states.

Sanity check: non-relativistic limit

In the non-relativistic limit, the BS equation should reduce to the Schrödinger equation. Indeed:

• Take the kernel $K$ to be single-photon exchange (ladder).
• Take the instantaneous approximation ($K$ depends only on $\mathbf{q}-{\mathbf{q}}^{\prime }$, not $q^0-q^{\prime 0}$).
• Take both fermions to be non-relativistic ($p^0\approx m+{\mathbf{p}}^2/2m$, lower spinor components small).
• Define $\Psi (\mathbf{r})\equiv \int d{q}^0{\chi }_P(q^0,\mathbf{r})$ (integrate out relative time).

The result is

$$\left[-\frac{{\nabla }^2}{2\mu }-\frac{Z\alpha }{r}\right]\Psi (\mathbf{r})=E\Psi (\mathbf{r}),$$

the Schrödinger–Coulomb equation with reduced mass $\mu =m_e{m}_p/(m_e+m_p)$ and $E=M-m_e-m_p$ the binding energy. So the BS equation contains the Schrödinger hydrogen equation (Level 0) as its leading non-relativistic instantaneous-ladder limit, and Dirac–Coulomb (Level 1) as the next-order correction.

Historical note

The BS equation was derived by Bethe and Salpeter in 1951, motivated by the need to do bound-state calculations in QED beyond the Dirac–Coulomb level. Gell-Mann and Low gave a more rigorous derivation a few months later, formalizing the residue argument used above. It was the first relativistic two-body wave equation in QFT, and remains the canonical definition of a relativistic bound state — even though, as discussed, it is rarely used directly in modern computations (NRQED having largely replaced it).

5. The Spectrum, Schematically

Stacking all the corrections on the $n=2$ manifold gives the famous picture:

                                            
            n = 2
                            ┌── 2P₃/₂  ──── ── ── ── ── ── ── ── F=2,1
                            │        ↑
                            │   fine structure (~α⁴)
                            │        ↓
        Schrödinger ─────── ┤  2S₁/₂ ──── ──── ── ── ── ── ── ── F=1,0
        (Bohr energy)       │  2P₁/₂   } degenerate in Dirac    F=1,0
                            │              ↑
                            │         Lamb shift (~α⁵)
                            │              ↓
                            └── 2P₁/₂ ──── ── ── ── ── ── ── ── F=1,0
                                         splits 2S₁/₂ from 2P₁/₂

       Bohr            Dirac fine        QED Lamb        Hyperfine
       ~ α²            ~ α⁴              ~ α⁵            ~ α⁴ m_e/m_p

The same hierarchy applies to every level $n$, with the magnitude of each splitting falling off as a power of $n$.

6. Where This Sits in the QED Framework

The bound state is a non-perturbative object; ordinary scattering perturbation theory in $\alpha$ misses it entirely (a bound state appears as a pole in correlation functions, not as a finite-order Feynman diagram). The standard tools for it are NRQED (non-relativistic QED, an effective theory in which $v/c$ is small) and the Bethe–Salpeter equation, both of which sum the Coulomb interaction to all orders before perturbing in the residual radiative corrections.

7. Take-Aways

• Hydrogen is the cleanest, most precisely tested QED system — agreement at the part-in-$10^{15}$ level for the $1S$–$2S$ transition.
• The familiar Schrödinger / Dirac / Lamb / hyperfine pictures are successive approximations in the small parameters $\alpha$ and $m_e/m_p$, all derivable from the QED postulates.
• Levels 0 and 1 use no Fock-space machinery at all (classical or external Coulomb field); Level 2 (Lamb shift) is the first point where quantized photons and loop corrections are essential.
• The bound state itself is a non-perturbative vector in Fock space; perturbative Feynman diagrams compute corrections on top of an already-bound state, not the bound state itself.
• Hyperfine structure already takes us outside pure QED: the proton $g$-factor is set by QCD, and the proton charge radius enters as a hadronic input.