Quantum Field Theory

Notes on the formal structure of relativistic quantum field theory, in the Wightman axiomatic style.

General formalism

1. Definitions and Preliminaries — Minkowski spacetime, the Lorentz/Poincaré groups, Wigner's classification, classical fields and Lagrangians, canonical structure, operator-valued distributions, Fock space, vacuum, correlation functions, the S-matrix, symmetries and Noether currents, gauge fields, regularization and renormalization.
2. Postulates of Quantum Field Theory — the ten Wightman-style postulates: relativistic state space, spectrum condition, unique vacuum, field operators, Poincaré covariance, microcausality, spin–statistics, vacuum cyclicity, dynamics from a local action, asymptotic completeness / S-matrix.
3. Modern Foundations — Wigner–Weinberg Derivation — the modern derivation of the QFT postulates from three primitive inputs (special relativity + quantum mechanics + cluster decomposition); fields, microcausality, spin–statistics, antiparticles, $CPT$, and gauge invariance for massless spin-1 all emerge as theorems.
4. Fock Space Inventory — what spaces, states, and operators exist after second quantization; clarifies the distinct roles of field operators, ladder operators, mode coefficients, and state vectors.
5. Particles as Excitations of Quantum Fields — concrete unpacking of the slogan "the electron is an excitation of the electron field", in terms of specific Fock-space vectors and operators.
6. Observables of QFT — the map of experimentally measurable quantities (cross sections, decay rates, branching ratios, asymmetries, bound-state energies, $g-2$ and form factors, IR-safe QCD observables, masses, couplings, etc.) and which part of the QFT machinery produces each. Concrete master formulas live in the observables/ subfolder:
   • Cross Sections — the master formula $d\sigma =(1/F)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$; flux factor, Lorentz-invariant phase space, Mandelstam variables, optical theorem, units (barns).
   • Decay Rates — the parallel master formula $d\Gamma =(1/2M)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$; lifetimes, branching ratios, the muon-lifetime worked example.
7. Remarks and Open Issues — the Wightman reconstruction theorem, Haag's theorem, gauge theories, the status of rigorous construction, the algebraic (Haag–Kastler) reformulation, and the status of measurement and collapse in QFT.

Specific theories

See theories/ for concrete QFTs:

• Quantum Electrodynamics (QED)