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QED

Field: Quantum Field Theory

Quantum Electrodynamics, the relativistic quantum field theory of electrodynamics.

Sequence of Expressions

The scattering amplitude M\mathcal{M} for a process involving NN external legs is given by the sum over all connected Feynman diagrams D\mathcal{D}: M=∑D(−1)∣D∣×(Product of propagators)×(Product of vertices)\mathcal{M} = \sum_{\mathcal{D}} (-1)^{|\mathcal{D}|} \times (\text{Product of propagators}) \times (\text{Product of vertices}) where the propagators are defined in momentum space as the inverse of the relevant differential operator (e.g., the photon propagator Dμν(q)=−igμνq2+iϵD_{\mu\nu}(q) = \frac{-i g_{\mu\nu}}{q^2 + i\epsilon}), and the vertices are determined by the interaction Lagrangian.
For a free electron field ψ(x)\psi(x), the Dirac equation is expressed covariantly as: (iγμ∂μ−m)ψ(x)=0\left(i \gamma^{\mu} \partial_{\mu} - m\right) \psi(x) = 0 where γμ\gamma^{\mu} are the Dirac gamma matrices satisfying {γμ,γν}=2ημνI\left\{ \gamma^{\mu}, \gamma^{\nu} \right\} = 2 \eta^{\mu\nu} I, and mm is the electron mass.
The vacuum polarization correction Π(q2)\Pi(q^2) modifies the photon propagator Dμν(q)D_{\mu\nu}(q) by adding a self-energy term: Dμν(q)=−igμνq2+−igμαq2Π(q2)−igανq2+O(Π2)D_{\mu\nu}(q) = \frac{-i g_{\mu\nu}}{q^2} + \frac{-i g_{\mu\alpha}}{q^2} \Pi(q^2) \frac{-i g_{\alpha\nu}}{q^2} + O(\Pi^2) The polarization tensor Π(q2)\Pi(q^2) is calculated from the electron-positron loop integral: Π(q2)=−e2∫d4k(2π)4Tr[γμSF(k+q)γνSF(k)]\Pi(q^2) = -e^2 \int \frac{d^4 k}{(2\pi)^4} \text{Tr} \left[ \gamma^{\mu} S_F(k+q) \gamma^{\nu} S_F(k) \right] where SF(k)=i(γ⋅k+m)k2−m2+iϵS_F(k) = \frac{i(\gamma \cdot k + m)}{k^2 - m^2 + i\epsilon} is the fermion propagator.
Theorem

Lamb Shift

The Lamb shift ΔEL\Delta E_{L} is a radiative correction to the energy levels of the hydrogen atom, calculated via the electron self-energy Σ(p)\Sigma(p) in the momentum space: ΔEL=⟨ψn∣Σ(En)−Σ(m)∣ψn⟩\Delta E_{L} = \langle \psi_n | \Sigma(E_n) - \Sigma(m) | \psi_n \rangle where Σ(En)\Sigma(E_n) is the electron self-energy evaluated at the energy eigenvalue EnE_n. This correction accounts for the virtual emission and reabsorption of photons by the electron.
The interaction between charged fermions (ψ\psi) and photons (AμA_{\mu}) is governed by the interaction Lagrangian density Lint\mathcal{L}_{int}: Lint=−eψˉγμAμψ\mathcal{L}_{int} = -e \bar{\psi} \gamma^{\mu} \textbf{A}_{\mu} \psi The resulting scattering amplitude M\mathcal{M} for a process like electron-electron scattering is derived from the Dyson series expansion of the S-matrix, incorporating the exchange of the photon field AμA_{\mu}.
The energy-time uncertainty relation is derived from the commutator of the Hamiltonian H^\hat{H} and an observable A^\hat{A}: ΔEΔt≥12∣⟨[H^,A^]⟩∣\Delta E \Delta t \ge \frac{1}{2} | \langle [\hat{H}, \hat{A}] \rangle | In the context of virtual particles, this relation dictates the minimum time Δt\Delta t required for a process involving an energy uncertainty ΔE\Delta E, leading to the virtual particle propagator 1/(p2−m2+iϵ)1/(p^2 - m^2 + i\epsilon).
The scattering amplitude M\mathcal{M} is calculated by summing contributions from all connected diagrams D\mathcal{D}. The rules define the components: M=∑D(Vertex Factor)×(∏internal linesPropagator)×(∏external linesExternal Field Factors)\mathcal{M} = \sum_{\mathcal{D}} \left( \text{Vertex Factor} \right) \times \left( \prod_{\text{internal lines}} \text{Propagator} \right) \times \left( \prod_{\text{external lines}} \text{External Field Factors} \right) Specifically, the vertex factor is −ieγμ-i e \gamma^{\mu}, the fermion propagator is iγ⋅(p−m)\frac{i}{\gamma \cdot (p - m)}, and the photon propagator is −igμνq2+iϵ\frac{-i g_{\mu\nu}}{q^2 + i\epsilon}.
The conservation of electric charge is a consequence of the U(1)U(1) gauge symmetry of the Lagrangian. By Noether's theorem, this symmetry implies the existence of a conserved current JμJ^{\mu}: ∂μJμ=0\partial_{\mu} J^{\mu} = 0 where the current JμJ^{\mu} is defined by the matter field ψ\psi and the gauge potential AμA_{\mu}: Jμ=eψˉγμψJ^{\mu} = e \bar{\psi} \gamma^{\mu} \psi The conservation law ensures that the total charge Q=∫d3xJ0Q = \int d^3 x J^0 remains constant over time.
The bare Lagrangian Lbare\mathcal{L}_{bare} is related to the renormalized Lagrangian Lren\mathcal{L}_{ren} by introducing counterterms δZi\delta Z_i: Lbare=Lren+Lcounter=Lren+12δZ3Fμν□−1Fμν+δZmψˉψ+…\mathcal{L}_{bare} = \mathcal{L}_{ren} + \mathcal{L}_{counter} = \mathcal{L}_{ren} + \frac{1}{2} \delta Z_3 F_{\mu\nu} \Box^{-1} F^{\mu\nu} + \delta Z_m \bar{\psi} \psi + \dots The physical parameters (mphys,ephysm_{phys}, e_{phys}) are obtained by absorbing the infinities (∝Λ\propto \Lambda) into these counterterms, ensuring that the physical observables remain finite.
For any pair of non-commuting Hermitian operators A^\hat{A} and B^\hat{B} representing physical observables, the generalized uncertainty principle states: σAσB≥12∣⟨[A^,B^]⟩∣\sigma_{A} \sigma_{B} \ge \frac{1}{2} | \langle [\hat{A}, \hat{B}] \rangle | Specifically, for position x^\hat{x} and momentum p^\hat{p}: σxσp≥ℏ2\sigma_{x} \sigma_{p} \ge \frac{\hbar}{2} where σA=⟨A^2⟩−⟨A^⟩2\sigma_{A} = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2} is the standard deviation.