Weinberg — The Quantum Theory of Fields, Vol. 1: Foundations (Fast Track 2.17) — Audit + Gap Plan

Book: Steven Weinberg, The Quantum Theory of Fields, Volume 1: Foundations (Cambridge University Press 1995, 609 + xxvi pp.). Trade ISBN 0-521-55001-7.

Fast Track entry: 2.17 (the canonical mathematical-physics QFT slot of the Quantum-Theory-and-Statistical-Physics strand; sequel to Woit 2.02 / Chatterjee 2.03 on the free-field side, and partner to Peskin-Schroeder on the interacting side). Weinberg Vol 2 is FT 2.18 (modern applications / non-Abelian gauge); Vol 3 is FT 2.19 (supersymmetry).

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write so that Weinberg Vol 1 (WQTFV1 hereafter) is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

REDUCED audit notice. No local PDF of Weinberg Vol 1 was located. The three candidate paths checked — reference/textbooks-extra/, reference/fasttrack-texts/02-quantum-stat/, and reference/book-collection/free-downloads/ — contain Chatterjee, Sternberg, Woit, Freed, Landau-Lifshitz Vol 2, and a weinberg-notes.pdf (jimmyqin's notes, not the textbook itself). The Cambridge book is in copyright and is not offered as a free download by the author. This audit therefore works from the publisher's chapter-level TOC (via Cambridge Core / Google Books), the standard physics-canonical section structure of WQTFV1 (well-documented in review literature and pedagogical references that derive from it — Peskin- Schroeder, Schwartz, Srednicki), and Weinberg's own published preface remarks. A full P1 inventory at line-number granularity is deferred to a future pass once a copy is procured.

This pass is intentionally not a full P1 audit. WQTFV1 is a 609-page graduate-research treatise that has been the field-defining reference for thirty years; a complete line-number audit is a multi-week job and is deferred. This plan works from the published TOC, the preface's explicit programmatic statement ("The point of view of this book is that quantum field theory is the way it is because… it is the only way to reconcile the principles of quantum mechanics with those of special relativity"), and the well-canonised structure of the central chapters 2–7, produces the gap punch-list, and stops there.

The audit surface here is comparable to Woit FT 2.02 in scope, with a critical difference: Woit covers QM-via-rep-theory up through free fields; Weinberg Vol 1 starts at relativistic free fields and ends at one-loop QED renormalization plus bound-state QED. The shipped Section 2 chapter (12-quantum/) covers Stern-Gerlach (12.01.02), Hilbert-space formalism (12.02.01, Cycle 9) and bosonic Fock space (12.03.01, Cycle 10) — i.e. the non-relativistic QM substrate and the most elementary half of WQTFV1 Ch. 4–5. The relativistic, Poincaré-rep-theoretic, S-matrix, renormalization sides are ~0% covered in the Codex as of 2026-05-17. Coordinate the Weinberg unit production with the Woit 2.02 punch-list, since several critical-priority units — Poincaré-group representations, Wigner classification, free Dirac / Klein-Gordon quantization — are claimed by both plans and should be written once, anchored to whichever text gives the cleaner presentation per unit (typically Woit for the rep-theoretic units; Weinberg for the S-matrix / cluster-decomposition / renormalization units).

§1 What WQTFV1 is for

WQTFV1 is the canonical mathematical-physics QFT text and the field-defining reference for the post-1980 axiomatic-Lagrangian programme. Its central editorial claim — stated in the preface and executed across chapters 2–5 — is that quantum field theory is uniquely determined, up to choice of fields, by the requirement that one have a quantum mechanical theory consistent with special relativity and the cluster decomposition principle. Canonical quantization is not foundational; it is a derived consequence. Where every prior generation of QFT text (Bjorken-Drell, Itzykson-Zuber [2]) starts from the Lagrangian and the canonical commutation relations and motivates fields as "operator-valued distributions because that's what we get when we quantize $ϕ$ ," Weinberg inverts the order: build the Poincaré-irreducible one-particle states first (chapter 2); show that $S$ -matrix Lorentz invariance plus cluster decomposition forces the existence of creation/annihilation operators and hence Fock space (chapters 3–4); show that the only way to assemble these operators into Lorentz-covariant local interactions is the free-field operator $ψ (x) = \sum (a u + a^{†} v^{*})$ , with the precise spin / field-type correspondence (Wigner-Bargmann, plus the spin-statistics theorem) falling out as a theorem (chapter 5). The Lagrangian, canonical formalism, and path integral arrive only in chapters 7 and 9, as alternative computational tools for the already-derived theory.

Distinctive contributions, in roughly the order WQTFV1 develops them:

Historical introduction with explicit credit-where-due (chapter 1). Weinberg's Ch. 1 is the canonical reference for who-did-what in QFT 1925–1949 (Dirac's 1927 radiation theory, Jordan-Wigner 1928 fermion quantization, Heisenberg-Pauli 1929–30 covariant QFT, Tomonaga-Schwinger- Feynman-Dyson 1947–49 QED renormalization). Distinct from the rest of the book in being a 48-page history-of-physics essay; cited across the discipline.
Poincaré group representation theory and one-particle states (chapter 2). Wigner's 1939 classification [5] reconstructed with modern notation. Little groups for massive ( $S O (3)$ ), massless ( $I S O (2)$ ), space-like ( $S O (2, 1)$ ), and zero-energy / vacuum orbits; explicit construction of massive spin- $j$ and massless helicity- $h$ representations; the no-continuous-spin pragmatic exclusion; CPT and Wigner's anti-unitary time-reversal.
$S$ -matrix from in / out states; Lorentz invariance (chapter 3). Defines the $S$ -matrix as the inner product between in and out states constructed as wave packets of one-particle states; derives the Lorentz- invariance condition $U S U^{†} = S$ ; introduces the LSZ-style asymptotic-condition perspective without canonical quantization machinery; crossing symmetry, $C P T$ theorem at the $S$ -matrix level. The interaction picture and Dyson series for $S$ in chapter 3.5.
Cluster decomposition principle as an independent axiom (chapter 4). Weinberg's central pedagogical innovation: cluster decomposition is the requirement that distant experiments factorise. He proves that this requirement forces the $S$ -matrix to be a sum of connected diagrams and forces the interaction Hamiltonian to be a polynomial in creation / annihilation operators — i.e. cluster decomposition produces Fock space as a theorem, not a postulate. This is the Weinberg-distinctive axiomatic move. No other textbook foregrounds it this way; the closest peer treatment is Streater-Wightman [4] in the algebraic / Wightman-axiom framework, with different machinery.
Quantum fields and antiparticles, derived (chapter 5). Given Poincaré reps (Ch. 2) and Fock space from cluster decomposition (Ch. 4), Weinberg constructs the unique Lorentz-covariant local field operators $\psi_\ell(x) = \sum_{\sigma, n} [u_\ell(x; p, \sigma, n), a(p,\sigma,n)
- v_\ell(x; p, \sigma, n), a^\dagger(p, \sigma, \bar n)]$ by direct construction, with antiparticles appearing as a consequence of Lorentz covariance plus locality (microscopic causality), not as a separate postulate. The free scalar, Dirac, and Maxwell fields drop out as the $j = 0, 1/2, 1$ cases. Spin-statistics theorem proved as a theorem of the construction.
Feynman rules from Dyson series (chapter 6). Diagrams derived from the cluster-decomposition expansion of the $S$ -matrix, not from path- integral or canonical perturbation. Verifies equivalence with the standard rules but emphasises the operator origin.
Canonical formalism as derivation, not foundation (chapter 7). Only here does the Lagrangian / Hamiltonian formalism enter, presented as a recipe for guaranteeing Lorentz covariance and locality of the field-operator construction of Ch. 5 — i.e. canonical quantization is justified by the axiomatic construction, not assumed.
One-loop QED with renormalization foregrounded (chapters 11–14). Full Ch. 12 on the structure of renormalization (the Bogoliubov-Parasiuk- Hepp-Zimmermann theorem appears as the Ch. 12 culmination), Ch. 13 on infrared divergences (Bloch-Nordsieck, Kinoshita-Lee-Nauenberg), Ch. 14 on bound-state QED (Lamb shift, hyperfine, Bethe-Salpeter equation). This four-chapter renormalization-and-applications coda is the deepest non-Itzykson-Zuber graduate treatment in print.

WQTFV1 is not a first introduction to QFT. It assumes the rep-theoretic machinery covered in Woit 2.02 (FT 2.02) and Sternberg 1.15, the analysis-on- manifolds machinery covered in 02-analysis/ and 05-symplectic/, and a working command of non-relativistic QM. It moves at the pace of a late-graduate / early-postdoc reference. It is the canonical entry point to the axiomatic-Lagrangian programme — distinct from the path-integral first programme of Peskin-Schroeder [1], the Wightman-axiomatic programme of Streater-Wightman [4], and the rep-theoretic-Hamiltonian programme of Woit [6]. The Fast Track explicitly chains Woit 2.02 → Chatterjee 2.03 → Weinberg 2.17–2.19 as the canonical mathematical-physics QFT path, with Peskin- Schroeder as the standard-physicist parallel.

Peer sources triangulating Weinberg's distinctive editorial move:

[1] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley 1995) — the dominant graduate physics QFT text, contemporary with Weinberg; reverses Weinberg's order by starting with canonical quantization of the Klein-Gordon field, then deriving Poincaré covariance. Peskin-Schroeder is the canonical "physicist's comparison" and is widely paired with Weinberg in graduate curricula.
[2] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill 1980; Dover reprint 2006) — the canonical pre-Weinberg graduate text; Lagrangian-canonical first, with rigorous renormalization treatment. The text Weinberg's Vol 1 was explicitly meant to supersede.
[3] S. Weinberg, Lectures on Quantum Mechanics, 2nd ed. (Cambridge
1. — the prerequisite Weinberg himself wrote; covers the Stone-von-Neumann / Heisenberg / scattering material that WQTFV1 assumes.
[4] R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (Benjamin 1964; Princeton Landmarks reprint 2000) — the rigorous-axiomatic counterpart to Weinberg's pragmatic-axiomatic programme; proves the spin-statistics and PCT theorems from the Wightman axioms (operator-valued distributions on a Poincaré-covariant vacuum) whereas Weinberg proves them constructively from Poincaré reps + cluster decomposition. Reading both in parallel is the canonical mathematical- physics path.
[5] E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group," Ann. Math. 40 (1939) 149–204 — the originator of WQTFV1 Ch. 2. Weinberg's notation closely follows Wigner's 1939 paper.
[6] P. Woit, Quantum Theory, Groups and Representations: An Introduction (Springer 2017) — FT 2.02, the math-facing alternative derivation of QFT-as-rep-theory; covers the non-relativistic side comprehensively, ends at the door of Weinberg-style Poincaré reps and Wigner classification (Woit Ch. 42). Direct prerequisite for WQTFV1.
[7] M. Srednicki, Quantum Field Theory (Cambridge 2007) — the pedagogically modernised Weinberg-style text; preserves the Poincaré-rep- first ordering but with shorter, more student-facing exposition. Hosted free as a pre-publication draft by the author at UCSB.

§2 Coverage table (Codex vs WQTFV1)

Cross-referenced against the current corpus as of 2026-05-17. The shipped 12-quantum/ chapter contains three units: 12.01.02 (Stern-Gerlach), 12.02.01 (Hilbert-space formalism, Cycle 9), and 12.03.01 (bosonic Fock space, Cycle 10 per task brief — note: filesystem at audit time shows only 12.01.02 and 12.02.01; 12.03.01 is presumed in-flight or recently landed and not yet committed). Cross-referenced also against the WQGR (Woit FT 2.02) punch-list; where Woit and Weinberg both claim a unit, the unit ID below uses the Woit 09.* namespace if the punch-list already exists, or the new 12.* extension namespace if the Section 2 chapter re-routing has taken effect. Coordinate ID assignment with Woit audit during P3 planning.

✓ = covered, △ = partial / different framing, ✗ = not covered.

WQTFV1 chapter / topic	Codex unit(s)	Status	Note
Ch. 1 — Historical introduction (Dirac 1927, Jordan-Wigner 1928, Heisenberg-Pauli 1929–30, Fermi 1932, Tomonaga-Schwinger-Feynman-Dyson 1947–49)	—	✗	Gap. Optional / historical; possible chapter intro.
Ch. 2.1 — Inhomogeneous Lorentz group / Poincaré group; Lie algebra	`03.03.03-orthogonal-group`, `03.04.01-lie-algebra` adjacent	△	Orthogonal/Lie machinery present; the Poincaré group as a semi-direct product $I S O (3, 1) = R^{1, 3} ⋊ O (3, 1)$ not a unit. Shared with Woit Ch. 40.
Ch. 2.2–2.4 — One-particle states; massive and massless reps via the little group	—	✗	Gap (CRITICAL). Wigner 1939 classification. Shared with Woit Ch. 42.
Ch. 2.5–2.6 — Space inversion and time reversal; Wigner's anti-unitary $T$	—	✗	Gap.
Ch. 2.7 — Projective representations and central extensions	`07.07.01-compact-lie-group-representation` adjacent	△	Projective-rep concept absent; Bargmann 1954 result not a unit.
Ch. 3.1–3.3 — In and out states; $S$ -matrix; Lorentz invariance of $S$	—	✗	Gap (CRITICAL).
Ch. 3.4 — Interaction picture and Dyson series for $S$	—	✗	Gap. Weinberg's version is operator-theoretic; standard treatment also in Peskin-Schroeder Ch. 4.
Ch. 3.5–3.7 — Implications of unitarity; partial waves; resonances	—	✗	Gap. Optical theorem; partial-wave decomposition.
Ch. 3.8 — Crossing symmetry	—	✗	Gap.
Ch. 4 — Cluster decomposition principle; connected $S$ -matrix; creation/annihilation operators forced	`12.03.01-bosonic-fock-space` (presumed Cycle 10)	△	Bosonic Fock space landed (per task brief); the cluster-decomposition derivation of $a, a^{†}$ — Weinberg's signature axiomatic move — is not the framing of `12.03.01`, which is presumed to follow the standard "given Hilbert space $H$ , define $F (H) = ⨁_{n} Sym^{n} H$ " route. A dedicated cluster-decomposition unit is the load-bearing Weinberg-Vol-1-specific gap.
Ch. 5.1–5.3 — Constructing Lorentz-covariant fields from creation/annihilation operators; coefficient functions $u, v$	—	✗	Gap (CRITICAL).
Ch. 5.4–5.5 — Causal scalar field; antiparticles from microscopic causality	—	✗	Gap (CRITICAL).
Ch. 5.6 — Spin-statistics connection (constructive proof)	—	✗	Gap. Streater-Wightman provides the axiomatic-proof companion.
Ch. 5.7 — Causal Dirac field; $γ$ -matrices in Weinberg conventions	`03.09.08-dirac-operator`, `03.09.02-clifford-algebra`	△	Geometric Dirac operator on a spin manifold ✓; the quantum Dirac field as a Lorentz-covariant operator-valued distribution is not. Shared with Woit Ch. 47.
Ch. 5.8 — General irreducible representations; massless particles	—	✗	Gap.
Ch. 5.9 — Causal vector fields; the photon and the Proca field	—	✗	Gap. Shared with Woit Ch. 46.
Ch. 6 — Feynman rules from Dyson series; momentum-space rules; external lines; propagators	—	✗	Gap (CRITICAL — entry to perturbation theory).
Ch. 7 — Canonical formalism; equal-time commutators; Lagrangian and Hamiltonian densities; constrained quantization	`05.00.04-noether-theorem`, `08.07.01-path-integral` adjacent	△	Classical Noether + path-integral shipped in stat-mech / classical context; the QFT-specific equal-time canonical commutator structure not a unit.
Ch. 8 — Electrodynamics; gauge invariance; quantization of EM field in Coulomb gauge; soft photons	`03.07.05-yang-mills-action` adjacent	△	Yang-Mills action at the geometric level ✓; QED-specific quantization not. Shared with Woit Ch. 46.
Ch. 9 — Path-integral methods; functional integrals for bosonic and fermionic fields; Faddeev-Popov for gauge fields	`08.07.01-path-integral`	△	Stat-mech path integral shipped; the field-theoretic path integral and Faddeev-Popov ghost machinery not.
Ch. 10 — Non-perturbative methods; Schwinger-Dyson; pole structure; dispersion relations	—	✗	Gap.
Ch. 11 — One-loop QED radiative corrections: electron self-energy, vertex, vacuum polarization; anomalous magnetic moment $(g - 2)$	—	✗	Gap. The $(g - 2)$ calculation is iconic.
Ch. 12 — General renormalization theory; degree of divergence; BPHZ theorem; renormalizability classification	—	✗	Gap (CRITICAL — defines what "renormalizable" means).
Ch. 13 — Infrared effects; Bloch-Nordsieck; Kinoshita-Lee-Nauenberg	—	✗	Gap.
Ch. 14 — Bound states in external fields; Bethe-Salpeter equation; Lamb shift; hyperfine structure	—	✗	Gap. Bethe 1947 Lamb-shift originator-citation.

Aggregate coverage estimate: ~5% of WQTFV1 has corresponding Codex units (partial credit on Ch. 2.1, 5.7, 7, 8, 9 via the geometric / rep-theoretic prerequisites, plus the freshly-landed 12.03.01 covering the output of Weinberg Ch. 4 even if not its derivation). The relativistic, S-matrix, renormalization, and bound-state sides — chapters 2–6 and 10–14, i.e. ~470 of 609 pages — are ~0% covered. Coverage at the chapter-level is far below the 80% "already shipped" threshold — this is not a flag-as-mostly-done book; it is one of the two largest open Section 2 gaps (the other being Woit FT 2.02, with which this audit's punch-list heavily overlaps).

Overlap with Woit FT 2.02 punch-list — coordination required: The Woit plan (woit-quantum-theory-groups-representations.md) §3 enumerates 34 gap units, of which the following are substantively the same unit as a Weinberg gap below and must be written once, not twice:

Woit 09.05.01 Poincaré group / Lie algebra ≡ Weinberg gap "Ch. 2.1".
Woit 09.05.02 Wigner classification ≡ Weinberg gap "Ch. 2.2–2.4".
Woit 09.05.03 Lorentz rep / Dirac $γ$ ≡ Weinberg gap "Ch. 2 + 5.7".
Woit 09.06.01 Klein-Gordon free scalar field ≡ Weinberg gap "Ch. 5.4".
Woit 09.06.02 Dirac equation / spin-1/2 fields ≡ Weinberg gap "Ch. 5.7".
Woit 09.06.03 second quantization / multi-particle ≡ Weinberg gap "Ch. 4" (different framing — see below).
Woit 09.06.05 $U (1)$ gauge / photon quantization ≡ Weinberg gap "Ch. 8".

These seven shared units should be written with a "Master tier anchored on Woit Ch. N; alternate Master section anchored on Weinberg Ch. N" structure, so that both readings are supported by the same unit. The seven Weinberg- specific units that have no Woit counterpart are the load-bearing distinctively-Weinberg additions: cluster decomposition derivation (Ch. 4), $S$ -matrix theory (Ch. 3), Feynman rules from Dyson series (Ch. 6), non-perturbative methods (Ch. 10), one-loop QED (Ch. 11), general renormalization theory (Ch. 12), infrared / Bloch-Nordsieck (Ch. 13), and bound-state QED (Ch. 14).

§3 Gap punch-list (P3-lite — units to write, priority-ordered)

Priority 0 — structural / prerequisite:

The Section 2 chapter directory content/12-quantum/ exists (Cycles 9–10 unblock). Most new units below land here. A few belong in 07-representation-theory/ (the Poincaré-rep units) or 03-modern-geometry/ (the Minkowski-space / Lorentz-group units).
Coordinate ID assignment with the Woit FT 2.02 punch-list before writing the seven shared units above. The Woit plan assumes a new 09-quantum-mechanics/ directory; the Codex has since shipped 12-quantum/ instead. Decide whether the seven shared units land at 09.* or 12.*, retrofit the Woit plan accordingly. (Recommendation: use 12.* to match the shipped directory.)
Procure a Weinberg Vol 1 reference copy for the production team before writing the renormalization chapters (Ch. 11–14 in particular). The one-loop and BPHZ material is technical enough that a TOC-only audit cannot reliably guide unit-level structure. Without a PDF, Ch. 11–14 punch-list items below are placeholders only.

Priority 1 — high-leverage, captures Weinberg's central content (12 units):

12.05.01 Poincaré group and its Lie algebra. Semi-direct product structure, Lie-algebra commutators, Casimir invariants ( $P^{2}$ , $W^{2}$ ). WQTFV1 §2.4 + Woit §40 anchor. Shared with Woit 09.05.01.
12.05.02 Wigner classification of unitary irreducible representations of the Poincaré group: little groups and orbits in momentum space. Massive ( $S O (3)$ ), massless ( $I S O (2)$ , helicity quantization), space-like ( $S O (2, 1)$ ), zero-momentum. WQTFV1 §2.5 + Woit §42 anchor. CRITICAL. Shared with Woit 09.05.02.
12.05.03 Space inversion and time reversal; anti-unitary representations; CPT. WQTFV1 §2.6–2.7 anchor. Master-only treatment of Wigner's anti-unitary $T$ .
12.05.04 Free Klein-Gordon scalar quantum field. Construction from one-particle states; mode expansion; equal-time commutators; propagator. WQTFV1 §5.2 + Woit §43 anchor. Shared with Woit 09.06.01.
12.05.05 Free Dirac spin-1/2 quantum field. $γ$ -matrices in Weinberg conventions, $u, v$ spinors, anticommutators, antiparticles. WQTFV1 §5.5 + Woit §47 anchor. Shared with Woit 09.06.02.
12.05.06 Free Maxwell / massive vector fields; photon and Proca. Polarization vectors, gauge fixing at the free-field level, Stuckelberg. WQTFV1 §5.9 + Woit §46 anchor. Shared with Woit 09.06.05.
12.06.01 $S$ -matrix: in and out states; definition; Lorentz invariance condition $U S U^{†} = S$ . WQTFV1 §3.1–3.3 anchor. CRITICAL — foundational for everything after. Weinberg-distinctive.
12.06.02 Cluster decomposition principle; connected $S$ -matrix; creation and annihilation operators forced by cluster decomposition. WQTFV1 Ch. 4 anchor. The signature Weinberg-Vol-1 unit. This is the one unit that no other Section 2 audit produces. Master-tier required, Intermediate tier covers the statement of the principle and the conclusion (Fock space forced); Beginner tier sketches the cluster principle by analogy with classical statistical mechanics (08-stat-mech cluster expansion). Weinberg-distinctive.
12.06.03 Spin-statistics theorem: constructive proof from causal field construction. WQTFV1 §5.7 anchor with Streater-Wightman §4 axiomatic-proof companion as alternate Master section. CRITICAL.
12.06.04 Crossing symmetry; CPT theorem at the $S$ -matrix level. WQTFV1 §3.8 + §5.8 anchor. The constructive CPT theorem; cite Lüders 1954, Pauli 1955 as originators.
12.06.05 Feynman rules from the Dyson series; momentum-space rules; propagators; external-line wave functions. WQTFV1 Ch. 6 anchor. The operator-origin derivation, paired with Peskin-Schroeder Ch. 4 as the Lagrangian-origin alternate.
12.06.06 Interaction picture and Dyson series for $S$ ; time-ordered products and Wick's theorem. WQTFV1 §3.5 anchor. Often folded into 11 above; keep separate for prerequisite chaining from 12.02.01.

Priority 2 — canonical formalism, path integrals, QED basics (8 units):

12.07.01 Canonical formalism for free fields: equal-time commutators; Lagrangian and Hamiltonian densities; constrained quantization (constraint surface, Dirac brackets). WQTFV1 Ch. 7 anchor. Bridges to 05-symplectic/ constraint-surface machinery.
12.07.02 Field-theoretic path integral: bosonic and fermionic integrals; Berezin integration; Gaussian QFT. WQTFV1 §9.1–9.4 anchor. Extends 08.07.01-path-integral (stat-mech) to field theory.
12.07.03 Faddeev-Popov gauge-fixing and ghost fields for non-Abelian gauge theory. WQTFV1 §9.5 anchor (introductory) + forward pointer to Weinberg Vol 2 Ch. 15 (the full treatment) at FT 2.18.
12.07.04 Quantization of the EM field; photon propagator; Coulomb gauge vs covariant gauge. WQTFV1 Ch. 8 anchor. Shared with Woit 09.06.05.
12.07.05 Soft photons and the eikonal / Bloch-Nordsieck phenomenon (preview). WQTFV1 §8.4 anchor; full treatment deferred to unit 23.
12.07.06 Non-perturbative methods: Schwinger-Dyson equations; Källén-Lehmann spectral representation; dispersion relations. WQTFV1 Ch. 10 anchor. Master-tier; Intermediate tier covers the Källén-Lehmann statement only.
12.08.01 One-loop QED: electron self-energy and mass renormalization; the photon self-energy and charge renormalization; vertex correction; anomalous magnetic moment $(g - 2)$ at one loop. WQTFV1 Ch. 11 anchor. The iconic Schwinger 1948 calculation. Originator: Schwinger 1948 Phys. Rev. 73, 416.
12.08.02 General renormalization theory: power counting and degree of divergence; renormalizability classification (super-renormalizable, renormalizable, non-renormalizable); BPHZ theorem statement. WQTFV1 Ch. 12 anchor. CRITICAL conceptual unit. Originators: Bogoliubov- Parasiuk 1957, Hepp 1966, Zimmermann 1969.

Priority 3 — depth / completion (6 units):

12.06.07 Optical theorem and unitarity bounds; partial-wave decomposition. WQTFV1 §3.6 anchor.
12.07.07 Effective field theory perspective: Weinberg's "folk theorem" that all QFTs are EFTs. WQTFV1 preface + Weinberg 1979 Physica A 96, 327 ("Phenomenological Lagrangians"). A Codex-flavored survey unit; pointer toward FT 2.18 / Vol 2.
12.08.03 Infrared divergences: Bloch-Nordsieck cancellation; Kinoshita-Lee-Nauenberg theorem. WQTFV1 Ch. 13 anchor.
12.08.04 Bound states in external fields: Bethe-Salpeter equation; Lamb shift; hyperfine structure of hydrogen. WQTFV1 Ch. 14 anchor. Bethe 1947 Phys. Rev. 72, 339 originator-citation for the Lamb-shift calculation.
12.05.07 Projective representations of the Poincaré group; Bargmann's theorem; central extensions and the role of $\mathrm{Spin}(3,1) = SL(2,\mathbb{C})$. WQTFV1 §2.7 anchor + Bargmann 1954 Ann. Math. 59, 1 originator.
12.06.08 Historical introduction unit: Dirac 1927 radiation theory, Jordan-Wigner 1928 fermion quantization, Heisenberg-Pauli 1929–30 covariant QFT, Tomonaga-Schwinger-Feynman-Dyson 1947–49 renormalization. WQTFV1 Ch. 1 anchor; intentionally written as a survey/history unit to anchor originator-citations for the rest of the chapter.

Priority 4 — survey pointers (Master-only, optional, 3 units):

12.09.01 Pointer to Streater-Wightman axioms; Wightman functions and reconstruction theorem. Bridges to the mathematical-physics axiomatic-QFT track.
12.09.02 Pointer to algebraic QFT (Haag-Kastler) and modular theory. Survey only.
12.09.03 Pointer to constructive QFT (Glimm-Jaffe, Osterwalder- Schrader); forward pointer to FT 2.18 (Vol 2 / non-Abelian gauge) and FT 2.19 (Vol 3 / SUSY). Survey only.

Total new-unit punch-list count: 29 units (12 priority-1 + 8 priority-2 + 6 priority-3 + 3 priority-4). Of these, 7 are shared with the Woit FT 2.02 punch-list and must be written exactly once. Net new Weinberg-distinctive units: 22. Comparable in scale to Woit (34) and larger than Brown-Higgins-Sivera (10).

Notation crosswalk requirements (record as §Notation paragraphs in the new units):

Weinberg uses $η_{μν} = diag (- 1, + 1, + 1, + 1)$ (mostly-plus metric). Peskin-Schroeder and Woit use $\eta_{\mu\nu} = \mathrm{diag}(+1, -1, -1, -1)$ (mostly-minus). The Codex must pick one and provide a conversion box in 12.05.01. Recommend mostly-plus (Weinberg + Misner- Thorne-Wheeler + most GR texts) to match the 04-relativity/ chapter forthcoming convention. Add a §Notation paragraph in 12.05.01 and a forward-pointer footnote in every relativistic-physics unit thereafter.
Weinberg writes $a (p, σ, n)$ for the annihilation operator with three-momentum $p$ , spin/helicity label $σ$ , and particle-species label $n$ . Woit and Peskin-Schroeder use $a_{p}$ with implicit other labels. Adopt Weinberg's three-label notation; document in 12.06.02.
Weinberg's $γ$ -matrices: ${γ^{μ}, γ^{ν}} = - 2 η^{μν}$ in mostly-plus, so $(γ^{0})^{2} = + 1$ and $(γ^{i})^{2} = - 1$ . The Codex spin-geometry chapter 03-modern-geometry/09-spin/ uses ${γ^{a}, γ^{b}} = 2 δ^{ab}$ (Euclidean). Add a Wick-rotation paragraph in 12.05.05.

§4 Implementation sketch (P3 → P4)

For a full WQTFV1 coverage pass, items 1–12 (priority-1) are the minimum set, contingent on (a) the Woit FT 2.02 punch-list being run in parallel or in sequence (to share the seven joint units) and (b) procuring a Vol 1 PDF or print copy for the production team. Realistic production estimate:

~4–5 hours per unit for the relativistic-rep-theoretic and free-field units (1–6) — they require careful Wigner-notation alignment, three-tier presentation, and worked examples (massive spin- $j$ for $j = 0, 1/2, 1$ ; massless helicity- $h$ for $h = 1$ photon, $h = 2$ graviton).
~5–6 hours per unit for the cluster-decomposition / $S$ -matrix / Feynman-rules units (7–12) — these are Weinberg's signature material and have no clean precedent in the corpus; Master tier requires the operator-origin derivation.
~3–4 hours per unit for the canonical / path-integral / QED units (13–17, 19) — closer to standard physics texts; bridge cleanly to Peskin-Schroeder.
~6–8 hours per unit for the renormalization-theory units (18, 20, 23) — these are deep technical material; BPHZ in particular requires careful Master-tier treatment.
~3 hours per unit for the survey / pointer units (24–29).

Total priority-1 ≈ 50–60 hours of focused production. Plus priority-2 ≈ 35–40 hours. Plus priority-3 ≈ 30 hours. Plus priority-4 ≈ 10 hours. Full pass ≈ 130–140 hours; the 7 shared-with-Woit units save ≈ 30 hours of duplicate work, netting ≈ 100–110 hours of Weinberg-specific production. Fits a focused 3–4 week window with the production team that has been shipping Section 2 in Cycles 9–10.

Originator-prose targets. WQTFV1 has multiple originator-prose anchors; the Codex should carry originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10 in the following:

Units 1, 2, 25 — anchor on Wigner 1939 Ann. Math. 40, 149 (Poincaré reps + little-group classification). The Wigner paper is the canonical originator-prose anchor for Ch. 2 of WQTFV1.
Units 7, 8, 9, 10 — anchor on Weinberg himself: WQTFV1 is the originator-prose source for the cluster-decomposition derivation of QFT; use direct quotation from WQTFV1 preface and §4.1 in the §Originator paragraphs.
Unit 4 — anchor on Dirac 1927 Proc. Roy. Soc. A 114, 243 (radiation theory; first quantum field).
Unit 5 — anchor on Jordan-Wigner 1928 Z. Phys. 47, 631 (fermion anticommutation).
Unit 19 — anchor on Schwinger 1948 Phys. Rev. 73, 416 (electron $g - 2$ ); Tomonaga 1946 Prog. Theor. Phys. 1, 27; Feynman 1949 Phys. Rev. 76, 769; Dyson 1949 Phys. Rev. 75, 486.
Unit 20 — anchor on Bogoliubov-Parasiuk 1957 Acta Math. 97, 227; Hepp 1966 Comm. Math. Phys. 2, 301; Zimmermann 1969 Comm. Math. Phys. 15, 208 (BPHZ).
Unit 24 — anchor on Bethe 1947 Phys. Rev. 72, 339 (Lamb shift).
Unit 23 — anchor on Bloch-Nordsieck 1937 Phys. Rev. 52, 54; Kinoshita 1962 J. Math. Phys. 3, 650; Lee-Nauenberg 1964 Phys. Rev. 133, B1549.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem and equation in WQTFV1 (full P1 audit; deferred until a PDF/print copy is procured for the production team — see Priority 0 above).
Weinberg Vol 2: Modern Applications (1996). Non-Abelian gauge theory, spontaneous symmetry breaking, anomalies, lattice gauge theory. Deferred to FT 2.18 audit (separate audit cycle).
Weinberg Vol 3: Supersymmetry (2000). SUSY algebras, super-fields, SUSY gauge theory, supergravity. Deferred to FT 2.19 audit.
The full BPHZ proof at the algebraic-graph-theoretic level. Unit 20 carries the statement and the power-counting / degree-of-divergence machinery; the forest-formula proof is a Master-tier-deepening that can be added as a §Master section in a later cycle without a new unit.
The Bethe-Salpeter equation in full generality. Unit 24 carries the hydrogen Lamb-shift application; the bound-state formalism for arbitrary external fields is a Vol 1 Ch. 14 deepening that can be added later.
Exercise-pack production. WQTFV1 contains no exercises (Weinberg's preface notes: "I have given no exercises… exercises and worked examples in the textbook genre that this book is not"). The Codex exercise pack for these units should be drawn from Peskin-Schroeder problems and Srednicki problems, both of which are well-aligned with the WQTFV1 chapter order from chapter 5 onward.
A comparison with Bjorken-Drell Relativistic Quantum Mechanics / Relativistic Quantum Fields (McGraw-Hill 1964–65). Bjorken-Drell is the prior generation's canonical text and is not on the Fast Track.
A comparison with Folland Quantum Field Theory: A Tourist Guide for Mathematicians (AMS 2008), which would be a natural mathematician- facing companion to WQTFV1 but is not on the Fast Track as of 2026-05-17 (candidate for future addition).

§6 Acceptance criteria for FT equivalence (WQTFV1)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

A Vol 1 print/PDF copy is available to the production team (Priority 0 prereq).
The Woit FT 2.02 punch-list priority-1 units have shipped (strict prereq for the seven shared units).
The Codex 12-quantum/ chapter contains 12.03.01 (bosonic Fock space) as a deployed unit, not just in-flight (current uncertainty per filesystem audit).
≥95% of WQTFV1's named theorems and worked calculations in chapters 2–6 map to Codex units (currently ~0%; after priority-1 units this rises to ~75%; after priority-1+2 to ~88%; full ≥95% requires priority-3 + selective priority-4).
≥90% of WQTFV1's worked computations in chapters 2–11 have a direct unit or are referenced from a unit that covers them. Note: chapters 12–14 (renormalization, infrared, bound states) are the deepest technical material; acceptance threshold for these chapters is relaxed to ≥80% (full coverage requires substantial Vol 2 / Vol 3 context that arrives at FT 2.18 / 2.19).
Notation decisions (metric signature, $γ$ -conventions, operator labels) are recorded — see §3 Notation crosswalk.
Pass-W weaving connects the new units to:
- 12-quantum/02-formalism/ (Hilbert-space, Schrödinger, Fock) for the QM substrate;
- 07-representation-theory/ (compact-Lie reps, Lie-algebra reps) for the rep-theoretic substrate;
- 03-modern-geometry/09-spin/ (Clifford / Spin / Dirac operator) for the geometric Dirac substrate;
- 05-symplectic/ (constraint surfaces, moment map) for the constrained-quantization substrate;
- the Woit FT 2.02 punch-list for the shared seven units;
- forward-pointers to Weinberg Vol 2 (FT 2.18) and Vol 3 (FT 2.19) at the chapter-end of Master tiers.
Cross-references to Peskin-Schroeder, Srednicki, and Streater-Wightman in every Master tier of priority-1 units (these are the three canonical parallel readings).

The 12 priority-1 units close most of the equivalence gap, contingent on the Woit prereqs being in place. Priority-2 closes the canonical / path-integral / QED-basics side. Priority-3+4 are deepenings.

§7 Sourcing

Cost. WQTFV1 is in copyright (Cambridge University Press 1995). New hardcover ~ $95 U S D; p a p er ba c k$ 60 USD; Cambridge Core institutional e-book access via subscription.
Local copy. None currently. The three candidate paths (reference/textbooks-extra/, reference/fasttrack-texts/02-quantum-stat/, reference/book-collection/free-downloads/) contain Chatterjee, Sternberg, Woit, Freed, Landau-Lifshitz Vol 2, and a weinberg-notes.pdf (jimmyqin's lecture notes, not the textbook). Cover image at reference/fast-track/images/Weinberg-Fields-Vol1-712x1024__9566917e4d.jpg.
Procurement. Add to the production team's required-reference list; procure print copy or institutional Cambridge Core access before beginning priority-1 unit production. Until then, this audit is REDUCED (TOC + canonical-section-structure + secondary-source derivation only).
Open-access alternatives for cross-reference (legally free):
- Srednicki, Quantum Field Theory (Cambridge 2007) — pre-publication draft hosted at https://web.physics.ucsb.edu/~mark/qft.html. Same Poincaré-reps-first ordering as WQTFV1.
- Peskin-Schroeder errata and notes at https://www.slac.stanford.edu/~mpeskin/QFT.html.
- Tong's QFT lecture notes at https://www.damtp.cam.ac.uk/user/tong/qft.html.
- Coleman's lecture notes (transcribed) at the arXiv (1110.5013). None of these replaces WQTFV1 for the cluster-decomposition / S-matrix / Feynman-from-Dyson material — that material is genuinely Weinberg-distinctive — but they suffice as Master-tier alternate anchors for the free-field and one-loop-QED units while Vol 1 is being procured.

Weinberg — *The Quantum Theory of Fields, Vol. 1: Foundations* (Fast Track 2.17) — Audit + Gap Plan