Courses

These three graduate-level courses provide cumulative routes through quantum field theory. Use them when you want the pacing of a lecture sequence, where each calculation prepares the next one. The subject volumes and Reference area serve a different purpose: they organize definitions, results, and conventions for topic-by-topic lookup.

The courses are reconstructed from handwritten notes taken by Jie Ren in one-semester courses taught by Alexander M. Polyakov. They are independently edited and substantially expanded teaching documents—not verbatim transcripts or official course publications.

Choose a course

QFT I — Introduction to Quantum Field Theory

Preparation. Graduate quantum mechanics and special relativity; no prior path-integral course is required.

Progression. Fock space and free fields → propagators and path integrals → scattering and analytic structure → thermal fields → spinors, gauge fields, and renormalization.

QFT II — Advanced Quantum Field Theory

Preparation. A first graduate QFT course, including path integrals, Feynman rules, one-loop renormalization, spinors, and basic gauge theory.

Progression. Renormalization and Wilsonian RG → OPE and gauge dynamics → Fermi surfaces and two-dimensional fields → sigma models, topology, confinement, real-time methods, and horizons.

QFT III — Selected Topics in High-Energy Physics

Preparation. QFT I plus familiarity with spontaneous symmetry breaking; RG and statistical mechanics are helpful, while selected QFT II topics can be read in parallel.

Progression. Ising duality and critical phenomena → conformal field theory → gauge-invariant observables and random surfaces → compact phases and topological defects → confinement, strings, and geometry.

QFT I builds the perturbative and conceptual foundation. QFT II develops a scale-dependent, semiclassical, topological, and real-time view of field theory. QFT III connects lattice models and critical points to operator algebras, defects, extended objects, and geometry.

How the sequence fits together

Each course is ordered internally and contains forty numbered lessons. Across the three courses, however, the structure is not a single 120-lesson chain. QFT I is the common foundation; QFT II and QFT III are complementary advanced routes.

There are certain overlaps. Renormalization, two-dimensional field theory, topological sectors, and confinement first appear as calculational methods in QFT II and return as parts of a wider lattice, conformal, or geometric picture in QFT III. Reading the overlapping lessons side by side is often more useful than forcing a strict order between the two advanced courses.

Choose by preparation and goal

• This is your first graduate QFT course. Begin with QFT I and read it linearly. The normalization checks and short exercises in the early lessons build the habits needed for the later scattering and loop calculations.
• You completed QFT before but need a diagnostic. Try the QFT I warm-ups. If they are routine, compare the QFT II and QFT III diagnostics before choosing where to enter.
• You want renormalization, many-body scaling, topology, or real-time QFT. Choose QFT II after securing the QFT I material on Green functions, perturbation theory, spinors, and gauge fields.
• You want critical phenomena, CFT, defects, or the path toward strings and geometry. Choose QFT III after QFT I. Consult the overlapping QFT II lessons on Wilsonian RG, two-dimensional fields, sigma models, and confinement as needed.

Work through a lesson

A productive pass has four stages:

1. Identify the assumptions before using a formula. Signature, normalization, boundary conditions, and expansion parameters are part of the result.
2. Reproduce at least one consequential derivation with the page closed. Recognition is not yet command of the method.
3. Test the answer with dimensions, symmetries, free or special limits, and the relevant sign conventions. Then do the short exercises before opening their solutions.
4. Use the linked topic and Reference pages after the calculation. They place the lecture result in a broader, topic-centered account.

The courses inherit the site-wide global conventions. A lesson states an explicit translation whenever the handwritten source or a standard reference uses a different metric, Fourier phase, field normalization, or gauge convention.

Provenance and corrections

The numbered lessons are editorial teaching units, not the number of class meetings. Added explanations, intermediate derivations, figures, exercises, and cross-references are the work of the site editor. Any remaining transcription, interpretation, or scientific errors are likewise the editor’s responsibility.

Exact identities, controlled approximations, semiclassical mechanisms, and structural hints should be read at the level stated on the relevant page. If a formula appears to disagree with another source, first compare conventions and domains of validity. Please report a surviving scientific error, a broken link, or an accessibility problem through Send feedback.