6 · relativistic quantum mechanics and qed

relativistic wave equations

A free particle of mass $m$ and spin $s$ transforms according to an irreducible unitary representation of the Poincare group. In a momentum-space basis its wave function carries a finite-dimensional representation of the little group and satisfies the mass-shell condition $p^2=m^2$. For the most important low-spin cases the momentum-space description can be rewritten as a local differential equation on configuration space.

spin 0. the Klein-Gordon equation $$ (\square+m^2)\phi=0 $$ follows from $p^2=m^2$ upon the substitution $p_\mu\to i\partial_\mu$. the conserved Klein-Gordon current is not positive-definite, signalling that a single-particle probability interpretation is incomplete.

spin $\tfrac12$. the Dirac equation $$ (i\gamma^\mu\partial_\mu-m)\psi=0 $$ realises the massive spin-$\tfrac12$ representation. the four-component spinor decomposes, in the rest frame, into two-component pieces associated with positive and negative energy. the bilinear $\bar\psi\psi$ is a Lorentz scalar and the current $\bar\psi\gamma^\mu\psi$ is positive-definite on solutions of definite charge sign, permitting a consistent single-particle reading at the level of the free theory.

spin 1 (massless). the free Maxwell equations in the Lorentz gauge, $$ \square A^\mu=0,\qquad\partial_\mu A^\mu=0, $$ together with residual gauge freedom, realise the massless helicity-$\pm 1$ representation. the field strength $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is gauge-invariant and satisfies the source-free Maxwell system.

example (dirac algebra). the matrices $\gamma^\mu$ satisfy $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$. plane-wave solutions $u(p,s)e^{-ip\cdot x}$ and $v(p,s)e^{ip\cdot x}$ furnish positive- and negative-frequency bases used throughout QED calculations.

These equations describe free particles. Interactions are introduced either by minimal coupling (for the electromagnetic field) or by the systematic construction of an interacting field theory (Chapter 7).

electrons and photons

The electron is identified with the positive-energy subspace of the Dirac representation; its antiparticle, the positron, corresponds to the negative-energy subspace, reinterpreted via hole theory or, more cleanly, via the charge-conjugation automorphism of the Dirac algebra. The photon is the massless helicity-$\pm 1$ particle associated with the Maxwell field.

The interaction between them is fixed by the requirement of local $U(1)$ gauge invariance: the Dirac operator is replaced by the covariant derivative $i\partial_\mu\to i\partial_\mu-eA_\mu$, yielding $$ \bigl(i\gamma^\mu(\partial_\mu+ieA_\mu)-m\bigr)\psi=0. $$ The electromagnetic field itself acquires a source term given by the Dirac current, recovering the classical Maxwell-Dirac system as the formal limit $\hbar\to 0$.

remark (coupling convention). the sign in $\partial_\mu+ieA_\mu$ matches the mostly plus or mostly minus metric convention used for the Dirac operator; what matters physically is that the covariant derivative realises a connection on the $U(1)$ charge bundle and that $e$ is the elementary charge appearing in Coulomb's law.

fock space and variable particle number

A single-particle Hilbert space $\mathcal{H}$ is inadequate once interactions can change particle number. The appropriate state space is the Fock space $$ \mathcal{F}(\mathcal{H})=\bigoplus_{n=0}^\infty\mathcal{H}^{\otimes n}_{\mathrm{sym/antisym}}, $$ where the symmetric (antisymmetric) product is taken for bosons (fermions). The vacuum $|0\rangle$ spans the zero-particle sector. Creation and annihilation operators $a^\dagger(\mathbf{p},s)$ and $a(\mathbf{p},s)$ implement transitions between sectors and satisfy the CCR or CAR appropriate to the statistics. Any state of definite particle content is obtained by acting on the vacuum with a finite number of creation operators.

For photons the one-particle space is the massless helicity-$\pm 1$ representation; for electrons and positrons it is the positive-energy Dirac representation, with charge conjugation relating the two. The full Fock space of quantum electrodynamics is the graded tensor product of the photonic and electronic Fock spaces.

example (one-photon state). a normalized one-photon wave packet is $\int d^3k\,f_\lambda(k)\,a_\lambda^\dagger(k)|0\rangle$ with $\sum_\lambda\int|f_\lambda|^2=1$. multi-photon and multi-electron states are built analogously, with antisymmetrization automatic for fermionic creators.

quantization of the electromagnetic field

The free electromagnetic field is expanded in normal modes, $$ A_\mu(x)=\sum_{\lambda=\pm 1}\int\frac{d^3k}{(2\pi)^{3/2}\sqrt{2\omega_k}} \Bigl(\varepsilon_\mu^{(\lambda)}(k)a_\lambda(k)e^{-ik\cdot x}+\mathrm{h.c.}\Bigr), $$ where $\varepsilon^{(\lambda)}$ are transverse polarization vectors and $\omega_k=|\mathbf{k}|$. The mode operators obey $$ [a_\lambda(k),a_{\lambda'}^\dagger(k')]=\delta_{\lambda\lambda'}\delta^3(\mathbf{k}-\mathbf{k}'). $$ The physical Hilbert space is the subspace of the (possibly indefinite-metric) photon Fock space that satisfies the Gupta-Bleuler condition, or an equivalent BRST-cohomology condition introduced later. The resulting theory reproduces the Planck spectrum, the Einstein $A$ and $B$ coefficients, and the selection rules for multipole radiation.

example (transversality). in Coulomb gauge one imposes $\nabla\cdot\mathbf{A}=0$ and retains only the two physical helicities. covariant gauges keep all four components of $A^\mu$ and remove unphysical modes by a constraint on physical states.

radiation processes

Elementary radiative processes are computed in first-order perturbation theory. The interaction Hamiltonian density is $$ \mathcal{H}_{\mathrm{int}}=e\bar\psi\gamma^\mu\psi A_\mu. $$ Matrix elements between Fock states yield spontaneous emission and absorption of photons by bound electrons, Thomson and Compton scattering, bremsstrahlung, and pair production and annihilation.

Phase-space integration and the use of free Dirac and photon wave functions produce differential and total cross-sections that can be compared directly with experiment. Polarization sums and spin averages are performed with the standard trace techniques over Dirac matrices and photon polarization tensors.

example (thomson limit). for photon energy $\omega\ll m$, Compton scattering reduces to the classical Thomson cross-section $\sigma_T=8\pi\alpha^2/(3m^2)$ after spin and polarization averages, providing a low-energy check on the QED Feynman rules.

scattering theory and the $S$-matrix

In the interaction picture the time-evolution operator from $t=-\infty$ to $t=+\infty$ defines the scattering operator $S$. Asymptotic free states $|i\rangle$ and $|f\rangle$ (elements of Fock space) are related by $|f\rangle=S|i\rangle$. The transition amplitude is $$ S_{fi}=\langle f|S|i\rangle. $$ Unitarity of $S$ encodes probability conservation and implies the optical theorem. For short-range or suitably screened interactions the $S$-matrix elements between wave-packet states yield measurable cross-sections after the usual flux and density-of-states factors are extracted.

remark (asymptotic states). the identification of in and out states with free Fock vectors is an idealization valid for theories with a mass gap or with careful infrared dressing. soft photons require additional care already visible in inclusive cross-sections.

propagators and perturbation theory

Higher-order processes are organised by the Dyson expansion of the $S$-operator. Each term is a time-ordered product of interaction Hamiltonians. Wick's theorem converts time-ordered products into normal-ordered products and contractions. The contractions are the free propagators.

The electron propagator is $$ S_F(x-y)=\langle 0|T\psi(x)\bar\psi(y)|0\rangle =\int\frac{d^4p}{(2\pi)^4}\frac{i(\not p+m)}{p^2-m^2+i\varepsilon}e^{-ip\cdot(x-y)}, $$ and the photon propagator in Feynman gauge is $$ D_F^{\mu\nu}(x-y)=\langle 0|TA^\mu(x)A^\nu(y)|0\rangle =\int\frac{d^4k}{(2\pi)^4}\frac{-ig^{\mu\nu}}{k^2+i\varepsilon}e^{-ik\cdot(x-y)}. $$

Feynman diagrams provide a graphical calculus for the resulting expressions. External lines correspond to free particle wave functions, internal lines to propagators, and vertices to factors of the coupling $e\gamma^\mu$. The rules automatically incorporate Fermi statistics through the anticommutativity of the electron operators.

example (tree-level vertex). the elementary QED vertex is $-ie\gamma^\mu$ in momentum space, with momentum conservation at the vertex. attaching free spinors and polarization vectors builds the tree amplitudes of Compton scattering and $e^+e^-$ annihilation.

concrete calculations

Standard textbook processes illustrate the method: Moller scattering ($e^-e^-\to e^-e^-$) and Bhabha scattering ($e^+e^-\to e^+e^-$) at tree level; Compton scattering to order $e^2$; pair annihilation $e^+e^-\to\gamma\gamma$; and, at one loop, the Lamb shift and the anomalous magnetic moment of the electron (requiring the first indications of renormalization).

In each case the amplitude is reduced to a Dirac trace, the trace is evaluated with standard identities, and the resulting Lorentz-invariant functions of the Mandelstam variables are integrated against the appropriate phase-space measure to produce cross-sections or decay rates.

example (anomalous moment). the one-loop vertex correction yields $a_e=(g-2)/2=\alpha/(2\pi)$ for the electron, historically the first precision QED success and a prototype for renormalized loop computations.

limitations of the single-particle and fixed-order pictures

The Dirac hole theory and the first-quantized Klein-Gordon theory encounter difficulties with causality, pair production, and vacuum polarization once the electromagnetic coupling is turned on. Already at second order, the vacuum polarization amplitude involves the loop integral $$ \Pi^{\mu\nu}(q)\propto e^2\int\frac{d^4k}{(2\pi)^4} \operatorname{Tr}\bigl[\gamma^\mu S_F(k)\gamma^\nu S_F(k+q)\bigr], $$ which is ultraviolet divergent and cannot be interpreted inside a fixed-particle-number Hilbert space. These difficulties are resolved only by passing to a genuine multi-particle formulation in which electrons, positrons, and photons are quanta of underlying fields $\psi$ and $A_\mu$.

Likewise, a generic $L$-loop amplitude scales as an integral $\int^{\Lambda} d^{4L}k\,/\,k^{2L}$ cut off at momentum $\Lambda$, producing poles in the dimensional regulator $\varepsilon=4-d$ that must be absorbed by local counterterms. Both issues motivate the structural reconstruction of quantum field theory undertaken in the following chapters.

summary

Relativistic wave equations realise the Poincare representations of Chapter 5 as local differential equations. Interactions that change particle number force the transition from a single-particle Hilbert space to Fock space. Quantization of the electromagnetic field together with the Dirac field produces quantum electrodynamics.

Transition amplitudes are matrix elements of the $S$-operator; they are computed perturbatively by means of Feynman diagrams whose building blocks are free propagators and interaction vertices. The resulting cross-sections and decay rates constitute the concrete phenomenological content of the theory. The computational success of the framework simultaneously reveals its structural limitations (causality, vacuum polarization, and ultraviolet divergences), thereby preparing the ground for the axiomatic and renormalization-group analyses that follow.