A twelve-unit advanced QFT textbook spanning operator-valued distributions, microcausality, quantization, functional integrals, LSZ scattering, gauge fixing, renormalization and effective field theory. University videos are followed by original native explanatory chapters, six-step intermediate derivations, worked calculations, verified scientific illustrations where available and primary-source reproduction guidance.
Published syllabus
A relativistic theory with creation and annihilation of excitations cannot be formulated as a fixed-number nonrelativistic wavefunction alone. Quantum fields are operator-valued distributions whose excitations carry particle quantum numbers. Causality, Lorentz covariance, locality and the spectral condition constrain their commutators and correlations.
A classical action S=∫d⁴x L(φ,∂φ) determines Euler–Lagrange equations under variations that vanish suitably on boundaries. Spacetime symmetries lead through Noether's theorem to conserved currents when the action is invariant. Surface terms can affect charges and boundary conditions even when they do not change bulk field equations.
Canonical quantization promotes a scalar field and its conjugate momentum to operators satisfying equal-time commutation relations. Fourier mode expansion expresses a free field as a continuum of oscillators labeled by momentum. Their creation operators build multiparticle Fock states. The Klein–Gordon equation allows both positive- and negative-frequency modes, with the appropriate relativistic field interpretation.
A fermionic spinor field satisfies anticommutation relations rather than bosonic commutators. Dirac gamma matrices represent the Clifford algebra compatible with Minkowski spacetime, and the spin-statistics theorem connects spin to exchange behavior under specific locality and positivity assumptions. Antiparticles emerge naturally in relativistic field quantization.
The path-integral formulation describes transition amplitudes through an oscillatory functional integral weighted by e^{iS/ℏ}. Gaussian path integrals reproduce free-field propagators. Interactions can be treated perturbatively by expanding the exponential, but an infinite-dimensional oscillatory integral requires a regulator or a careful definition. Wick rotation to Euclidean signature improves many mathematical properties under appropriate conditions.
The time-ordered two-point function is the Feynman propagator, encoding amplitudes and causal boundary prescriptions rather than a classical deterministic trajectory. For a free scalar field, its momentum-space denominator contains the mass shell and an iε instruction. Retarded propagators instead express causal response to sources; interchanging their prescriptions changes the physics.
The Dyson expansion generates perturbative scattering amplitudes ordered by interaction insertions. Wick's theorem contracts free fields and organizes terms as Feynman diagrams. Each diagram corresponds to a specified integral and symmetry factor, not a miniature photograph of real particle trajectories. The S-matrix requires a model of asymptotic scattering states.
Noether currents follow from continuous global action symmetries, while gauge redundancies express local descriptive freedom. Spontaneously broken global symmetries can support Goldstone modes under suitable conditions. Quantum anomalies occur when regularization obstructs a classical symmetry. Gauge redundancy must not be confused with a physical global symmetry transformation.
A gauge field A_mu enters matter dynamics through a covariant derivative and transforms with the local gauge convention. Non-Abelian field strength includes commutators and gauge boson self-interactions. Fixing gauge removes redundancy from perturbative calculations; BRST or related methods organize consistency. Observables must not depend on arbitrary gauge-fixing choices.
Loop integrals often diverge before regularization. Renormalization introduces scale-dependent parameters and counterterms constrained by symmetries, relating bare model symbols to measured quantities. Running couplings encode dependence on the chosen scale and scheme. A renormalized prediction must be independent of the arbitrary intermediate scale to the stated perturbative order.
Effective field theory organizes all interactions compatible with symmetries in an expansion suppressed by a high-energy scale. Operators of higher mass dimension are weighted by powers of that scale, and low-energy predictions gain controlled uncertainty when a separation of scales exists. Matching connects coefficients to the UV model or experiment.
Field-theoretic methods also describe collective excitations, response functions and phase transitions in condensed matter. A quasiparticle field need not represent a fundamental particle of nature; it may emerge from a many-body effective theory. Renormalization-group fixed points explain universal critical scaling, while lattice and continuum descriptions can match at long wavelengths.