Quantum Field Theory

Notes on relativistic quantum field theory: the formal structure (Wightman axiomatic style), the computational pipeline that turns a Lagrangian into a measured rate (canonical quantization → perturbation theory → Feynman diagrams → cross sections), and the specific theories of the Standard Model.

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.

From fields to amplitudes — the computational core

The formalism above is top-down and axiomatic; the observables at the end are master formulas. The pages below build the constructive machinery in between — the route a working field theorist takes from a Lagrangian to a number:

$$\mathcal{L}[\varphi ]\to \text{canonical quantization}\to \text{perturbation theory}\to \overline{|\mathcal{M}{|}^2}\to \sigma ,\Gamma .$$

Classical field theory (the groundwork for quantization):

• Lagrangian and Hamiltonian Field Theory — the action principle for fields, Euler–Lagrange equations, conjugate momenta, the Hamiltonian and Poisson brackets, counting degrees of freedom.
• Symmetries, Noether's Theorem and Currents — the proof of Noether's theorem; the $U(1)$ current, the energy–momentum tensor (canonical and Belinfante), and angular momentum.

Free fields and canonical quantization (the concrete realization of Fock space):

• The Free Scalar Field — equal-time commutators, mode expansion, vacuum energy and normal ordering, microcausality, the Feynman propagator, the complex scalar and antiparticles.
• The Dirac Field — why spin-$\frac{1}{2}$ forces anticommutators, Pauli exclusion, the fermion propagator, charge conjugation.
• The Free Vector Field — Proca vs. Maxwell, gauge redundancy, $R_{\xi }$ gauges and the photon propagator, polarization sums.
• Discrete Symmetries C, P, T on Fields — the action of $C$, $P$, $T$ on each field, bilinear table, individual violation, and the $CPT$ theorem.

Interactions and perturbation theory (the calculational pipeline):

• The Interaction Picture and the Dyson Series — $H=H_0+H_{\text{int}}$, the time-ordered exponential $S=T\exp (i\int {\mathcal{L}}_{\text{int}})$.
• Wick's Theorem and Contractions — reducing time-ordered products to normal-ordered terms and propagators.
• Feynman Diagrams and Feynman Rules — the diagram dictionary, QED rules, connected/amputated/1PI classes, symmetry factors.
• The LSZ Reduction Formula — from correlators to S-matrix elements: leg amputation, on-shell projection, wavefunction renormalization $Z$.
• The S-Matrix, Cross Sections and Decay Rates — assembling $\mathcal{M}$ into the observable master formulas; spin sums, phase space, the optical theorem.
• Tree-Level Worked Examples — ${\varphi }^4$, Yukawa, and $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ walked end-to-end; Mandelstam variables and crossing.

Path-integral quantization (the functional route to the same physics):

• The Functional Integral and Generating Functional — $Z[J]=\int \mathcal{D}\varphi e^{iS+i\int J\varphi }$; correlators by source differentiation; Feynman rules with automatic symmetry factors; the Euclidean/statistical-mechanics analogy.
• Fermions and Grassmann Integration — anticommuting variables, Berezin integration, the fermionic determinant and loop sign.
• The Effective Action and 1PI Generating Functional — $W[J]$, the Legendre transform $\Gamma [\varphi ]$, 1PI vertices, the effective potential, the loop = $\hslash$ expansion.
• Ward–Takahashi Identities and Schwinger–Dyson Equations — symmetries of the measure become exact relations among correlators; the QED Ward identity ($Z_1=Z_2$); anomalies as the failure case.

Renormalization and the renormalization group (taming quantum corrections):

• Loop Integrals and Divergences — the superficial degree of divergence, power counting and renormalizability, Feynman parameters, Wick rotation.
• Regularization — cutoff, Pauli–Villars, dimensional regularization ($\overline{\text{MS}}$), lattice; regulator-independence of physics.
• Renormalization and Counterterms — bare vs. renormalized quantities, counterterms, renormalizability classes, BPHZ.
• The Renormalization Group — the Callan–Symanzik equation, $\beta$-functions and anomalous dimensions, running couplings, fixed points.
• Wilsonian RG and Effective Field Theory — integrating out shells, relevant/marginal/irrelevant operators, "every QFT is an EFT", matching and decoupling.

Gauge theories (quantizing the self-interacting force carriers):

• Non-Abelian Gauge Theory (Yang–Mills) — the gauge principle for $SU(N)$, matrix-valued $A_{\mu }$, the self-interacting field strength, the connection/curvature picture.
• Gauge Fixing and Faddeev–Popov Ghosts — the path-integral overcounting problem, the Faddeev–Popov determinant, anticommuting ghosts and unitarity.
• BRST Symmetry and Unitarity — the residual fermionic symmetry, nilpotency, the physical Hilbert space as BRST cohomology, Slavnov–Taylor identities.
• Asymptotic Freedom — the negative Yang–Mills $\beta$-function, antiscreening vs. screening, confinement and ${\Lambda }_{\text{QCD}}$.

Spontaneous symmetry breaking (mass generation):

• Spontaneous Symmetry Breaking and Goldstone's Theorem — symmetry of the vacuum vs. the dynamics, the Mexican-hat potential, one massless boson per broken generator, pseudo-Goldstones.
• The Higgs Mechanism — gauging a broken symmetry, the eaten Goldstone, massive gauge bosons, $R_{\xi }$ gauges and renormalizability, the electroweak Higgs.
• The Effective Potential and Coleman–Weinberg — radiative symmetry breaking, the one-loop potential, vacuum stability.

Anomalies (classical symmetries broken by quantization):

• The Chiral (ABJ) Anomaly — the triangle diagram and the Fujikawa measure, ${\partial }_{\mu }{j}^{\mu 5}\sim ϵFF$, ${\pi }^0\to \gamma \gamma$.
• Anomaly Cancellation and Consistency — why gauge anomalies are fatal, the Standard-Model cancellation, 't Hooft matching, Witten's $SU(2)$ anomaly.

Advanced and nonperturbative topics:

• Effective Field Theory — top-down/bottom-up EFT, power counting, matching and running; Fermi theory, chiral perturbation theory, SMEFT.
• Solitons, Instantons and Topology — topological charge, kinks/vortices/monopoles, instantons, the $\theta$-vacuum and strong-CP.
• Conformal Field Theory — conformal symmetry at RG fixed points, primary operators and the OPE, the bootstrap, 2D CFT and holography.
• Lattice Field Theory — Euclidean discretization, Wilson gauge action, confinement from the area law, Monte Carlo, fermion doubling.
• Finite-Temperature and Finite-Density QFT — the Matsubara formalism, thermal propagators, symmetry restoration and phase transitions.
• Supersymmetry — the unique boson–fermion extension of Poincaré, superpartners, the hierarchy problem, non-renormalization theorems.
• QFT in Curved Spacetime — the observer-dependent vacuum, the Unruh effect, Hawking radiation, gravity as an EFT.

Specific theories

See theories/ for concrete QFTs:

• Quantum Electrodynamics (QED)