Chatterjee — *Introduction to Quantum Field Theory for Mathematicians* (Fast Track 2.03) — Audit + Gap Plan

Book: Sourav Chatterjee, Introduction to Quantum Field Theory for Mathematicians — Lecture notes for Math 273, Stanford, Fall 2018 (130 pp., 29 lectures, based on a forthcoming textbook by Michel Talagrand). Hosted freely by the author at his Stanford page; local copy at reference/fasttrack-texts/02-quantum-stat/Chatterjee-QFTLectureNotes.pdf.

Fast Track entry: 2.03 (Tier 2 — Quantum / Statistical Mechanics block). Confirmed in docs/catalogs/FASTTRACK_BOOKLIST.md as the free Stanford entry for second quantization and basic QFT.

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 Chatterjee 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).

This pass is intentionally not a full P1 audit. Chatterjee is short (130 pp.) but the entire QFT track is essentially absent from the Codex: the existing 08-stat-mech/ chapter is classical statistical field theory (Ising, RG, lattice gauge as a stat-mech object, Wick rotation as a tool for stat-mech path integrals — not Wick's theorem for Gaussian fields and operator-product contractions). A full P1 inventory at line-number granularity is therefore not productive yet; we work from Chatterjee's table of contents, identify the load-bearing constructions, and stop there.

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§1 What Chatterjee is for

Chatterjee is the rigorous-as-possible mathematicians' on-ramp to Lagrangian / canonical QFT, written in a probabilist's voice rather than a high-energy-physicist's voice. Where Peskin-Schroeder, Weinberg, and Itzykson-Zuber assume the reader is comfortable with the physicists' heuristic apparatus — operator-valued objects evaluated at points, naked delta-functions, divergent integrals one renormalises by force — Chatterjee stops at every such step, asks "what does this actually mean as a mathematical object?", and either gives the rigorous definition (operator-valued distribution, Fock space construction, Schwartz-class smearing) or flags the step as an open problem that physicists handle by formal manipulation. The result is a 130-page lecture set that gets the reader to one-loop renormalisation of ${\varphi }^4$ theory, the Dirac field, introductory QED, and the Wightman axioms — all explicitly marked as mathematically rigorous, mathematically incomplete, or mathematically open.

Distinctive contributions, in roughly the order Chatterjee develops them:

1. Postulates of quantum mechanics as a five-axiom list (P1–P5). Lecture 2 collects the framework — separable Hilbert space, Hermitian observables, eigenvalue-as-measurement-outcome, expectation via Born rule, projective time evolution — as a closed set of postulates that the rest of the notes build on, rather than the diffuse axiomatics in physics texts. Compare Weinberg [W] vol. 1 §1.4 (which mixes axioms with particle-physics motivation) and Reed-Simon [RS] vol. 1, which has the functional-analytic content but no postulate framing.
2. Operator-valued distributions as the rigorous meaning of $a(p)$, $a^{†}(p)$, $\varphi (x)$. Lectures 7–8 and 12. Chatterjee defines $A$ and $A^{†}$ as maps $H\to L({\mathcal{B}}_0,\mathcal{B})$ on the bosonic Fock space, then exhibits $a(x)$, $a^{†}(x)$ as the formal point-evaluations $A({\delta }_x)$, $A^{†}({\delta }_x)$. This is where physics classes begin; Chatterjee gets here in lecture 7 with full functional-analytic justification. Sternberg [S] gives a similar distributional treatment in his semiclassical analysis lectures but without the Fock space construction.
3. Special relativity and the Poincaré group as Hilbert-space input, not background. Lectures 9–10 build $X_m = {p \in \mathbb{R}^{1,3} : p^2 = m^2, p^0 > 0}$ as a manifold, the Lorentz-invariant measure $d{\lambda }_m$, and $L^2(X_m,d{\lambda }_m)$ as the one-particle space. Massless and massive treatments split cleanly. Compare Folland's Quantum Field Theory: A Tourist Guide for Mathematicians [F] §2.
4. Five postulates of QFT (P1–P5 for fields). Lecture 11 lifts the QM postulates: Hilbert space + unitary Poincaré representation + locality + spectrum condition + cyclic vacuum. This is essentially the Wightman axioms in postulate form; Chatterjee returns to the Wightman version in lecture 29.
5. The massive scalar free field $\varphi$ as an operator-valued distribution on ${\mathbb{R}}^{1,3}$. Lecture 12. Chatterjee gives the explicit formula $$\varphi (t,x)=\int \frac{d^{3}p}{(2\pi {)}^3\sqrt{2{\omega }_p}}(e^{-it{\omega }_p+ix\cdot p}a(p)+e^{it{\omega }_p-ix\cdot p}{a}^{†}(p))$$ and shows it satisfies the Klein-Gordon equation formally and the Wightman locality axiom rigorously. This is the load-bearing example.
6. ${\varphi }^4$ theory as a training interacting theory. Lectures 13, 18, 21, 22. Chatterjee is explicit that ${\varphi }^4$ describes no real particle but exhibits "many of the complexities of quantum field theories that describe real particles". The pedagogical move — work out one-loop renormalisation in detail on a fictitious theory before touching QED — is shared with Folland [F] and Talagrand's forthcoming text but stands in sharp contrast to Peskin-Schroeder [PS], which interleaves QED motivation with ${\varphi }^4$ machinery from the start.
7. Wick's theorem as a moment-calculation tool. Lecture 17. The physicists' diagrammatic Wick theorem becomes a precise statement about contractions of $A,A^{†}$ in the vacuum expectation, with a clean inductive proof. Compare Glimm-Jaffe [GJ] §6, which gives the constructive-field-theory version, and Salmhofer's Renormalization [Sa] for the rigorous perturbative treatment.
8. One-loop and two-loop renormalisation of ${\varphi }^4$ as cutoff-independent extraction of finite predictions. Lectures 20–22. The "problem of infinities" is presented as the integral $\int d^{4}p/[(-p^2+m^2-iϵ)(-(p_1+p_2-p{)}^2+m^2-iϵ)]$ diverging like $|p{|}^{-4}$ at infinity. The renormalisation procedure is the subtraction at a reference momentum so that $\lim_{R \to \infty} (\widetilde{M} - \widetilde{M}_*)$ exists and is independent of the cutoff function $\theta$. This is the BPHZ idea, presented stripped of formalism. Compare Salmhofer [Sa] for the Wilsonian-RG reformulation in a rigorous setting.
9. QED as $B\otimes F\otimes F^{\prime }$ — photon bosons, electron fermions, positron fermions. Lectures 23–28. The interaction Hamiltonian density is $\mathcal{H}(x)=e:\bar{\psi }(x){\gamma }^{\mu }\psi (x)A_{\mu }(x):$. Anomalous magnetic moment of the electron computed to fifth order matches experiment to 10 decimal places — Chatterjee gives this as the triumph that justifies the entire fanciful machinery.
10. Wightman axioms as the rigorous closing chapter. Lecture 29. Seven axioms (Hilbert space + unitary Poincaré + smeared field + Lorentz covariance + locality + cyclic vacuum + spectrum). Includes the open problem: construct an interacting scalar QFT on ${\mathbb{R}}^{1,3}$ satisfying these axioms. Compare Streater-Wightman [SW] PCT, Spin and Statistics, and All That, the canonical reference, which Chatterjee implicitly tracks.

Chatterjee is not a substitute for a working physicist's QFT course (no path-integral approach beyond mention; no gauge theory beyond Gupta-Bleuler photons; no non-abelian Yang-Mills; no renormalisation group flow; no anomalies; no spontaneous symmetry breaking beyond a nod). It is the canonical free entry point to what QFT actually means as mathematics. Sit it next to Peskin-Schroeder (which Chatterjee will not replace), Weinberg vol. 1 (for the high-energy physics framing), and Glimm-Jaffe / Streater-Wightman (for the constructive / axiomatic programme).

Peer / supplementary sources cross-referenced for this audit: [W] S. Weinberg, The Quantum Theory of Fields, vol. 1 (Cambridge 1995) — FT 2.17. [PS] M. Peskin & D. Schroeder, An Introduction to Quantum Field Theory (Westview 1995). [F] G. Folland, Quantum Field Theory: A Tourist Guide for Mathematicians (AMS 2008). [GJ] J. Glimm & A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer 1987). [SW] R. Streater & A. Wightman, PCT, Spin and Statistics, and All That (Princeton 1964/2000). [Sa] M. Salmhofer, Renormalization: An Introduction (Springer 1999). [RS] M. Reed & B. Simon, Methods of Modern Mathematical Physics, vol. 1–2 (Academic 1980). [S] S. Sternberg, Semi-Classical Analysis lecture notes (related-only free text; not on the canonical FT booklist per reference/fasttrack-texts/README.md). [T] M. Talagrand, What is a Quantum Field Theory? (Cambridge 2022) — the forthcoming textbook these notes are based on; now in print.

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§2 Coverage table (Codex vs Chatterjee)

Cross-referenced against the current 313-unit corpus via find content -name "*.md" and grep on fock, wightman, dyson, feynman, dirac field, qed, quantum electrodynamics, operator-valued distribution, creation and annihilation, wick, poincaré, klein-gordon, scattering, propagator. The Codex 08-stat-mech/ chapter matches on stat-mech terms (path integral, Wick rotation, lattice gauge, RG, mean-field, Ising) but none of its 22 units treats the QFT objects Chatterjee builds. The matches in 05-symplectic/, 06-riemann-surfaces/, 04-algebraic-geometry/ are incidental (one uses "Fock" as a moduli-space adjective, another mentions "Dirac" in a function-theoretic context).

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

Chatterjee topic (lecture)	Codex unit(s)	Status	Note
Five postulates of QM (L2)	—	✗	Gap. No Codex unit lists the QM axioms as a closed framework.
Position / momentum operators, essential self-adjointness, Stone's theorem in flight (L3, L4)	—	✗	Gap. Spectral theorem / unbounded operators are foundations the Codex doesn't yet have an analysis-track unit for.
Heisenberg uncertainty principle (L4)	—	✗	Gap.
Simple harmonic oscillator, Hermite basis (L4)	—	✗	Gap.
Tensor product of Hilbert spaces, symmetric tensor power (L5–L6)	—	△	Tensor algebra of vector spaces is in 03.01-tensor-algebra/ but the Hilbert-space completion and symmetric/antisymmetric variants are not.
Bosonic Fock space, second quantisation (L6)	—	✗	Gap. Foundational and load-bearing for everything that follows.
Creation/annihilation operators $a_k$, $a_k^{†}$, commutation relations (L7)	—	✗	Gap.
Operator-valued distributions $a(x)$, $a^{†}(x)$ as $A({\delta }_x)$ (L7)	—	✗	Gap.
Time evolution on Fock space, free Hamiltonian as $\int dx{a}^{†}(x)(-\Delta /2m)a(x)$ (L8)	—	✗	Gap.
Lorentz group ${\mathrm{SO}}^{\uparrow }(1,3)$, Poincaré group $P$, special relativity (L9)	—	✗	Gap. Codex has Lorentz scattered in symplectic/contact units but no dedicated SR unit.
Mass shell $X_m$, Lorentz-invariant measure $d{\lambda }_m$ (L10)	—	✗	Gap.
Five postulates of QFT (L11)	—	✗	Gap. Same status as QM postulates.
Massive scalar free field $\varphi (t,x)$, Klein-Gordon equation (L12)	—	✗	Gap. Central object.
${\varphi }^4$ theory: formal Hamiltonian, why it's a toy (L13)	—	✗	Gap.
Scattering, $S$-matrix, in/out states (L14)	—	✗	Gap.
Born approximation (L15)	—	✗	Gap.
Hamiltonian densities, Dyson series (L16)	—	✗	Gap.
Wick's theorem (operator-product version), Wick ordering $:\cdot :$, contractions (L17)	△	△	08-stat-mech/wick/08.09.01-wick-rotation.md covers Wick rotation (Minkowski → Euclidean) but not Wick's theorem for operator products. Different object with the same name.
First-order ${\varphi }^4$ scattering, Feynman diagrams (L18)	—	✗	Gap.
Feynman propagator ${\Delta }_F(x)$, contour-integral form (L19)	—	✗	Gap.
Problem of infinities, UV divergences (L20)	△	△	08-stat-mech/critical/08.05.01-critical-exponents.md touches RG divergences in a stat-mech framing; not the QFT version.
One-loop renormalisation in ${\varphi }^4$, BPHZ subtraction at reference momentum (L21)	—	✗	Gap.
Two-loop renormalisation (L22)	—	✗	Gap (Master-tier deepening).
Free-photon Hilbert space, Gupta-Bleuler quotient, $B_{\mathrm{phys}}=B^0/B_{\mathrm{null}}$ (L23)	—	✗	Gap.
Quantised EM four-potential $A^{\mu }$, field-strength $F_{\mu \nu }$, Lorenz gauge condition, classical Maxwell recovery (L24)	—	✗	Gap.
Free-electron model, $\mathrm{SL}(2,\mathbb{C})$ double cover of ${\mathrm{SO}}^{\uparrow }(1,3)$, projective representations (L25)	△	△	Some Clifford / spin material in 03-modern-geometry/09-spin-geometry/ (esp. 03.09.03-spin-group.md, 03.09.13-triality.md) overlaps the algebra but not the Hilbert-space construction.
Fermionic Fock space, Pauli exclusion, anticommutators (L25–L26)	—	✗	Gap.
Dirac field $\psi (x)$ with Fock spaces $F\otimes F^{\prime }$ (L26)	—	✗	Gap.
Pauli matrices ${\sigma }_{\mu }$, Dirac matrices ${\gamma }^{\mu }$, Dirac adjoint $\bar{\psi }={\psi }^{†}{\gamma }^0$ (L27)	△	△	Pauli matrices appear inside Clifford-algebra units in 03.09.02-clifford-algebra.md. Dirac matrices and $\bar{\psi }$ do not.
QED interaction Hamiltonian ${\mathcal{H}}_I=e:\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }:$ (L27)	—	✗	Gap.
Electron scattering, Dirac propagator, photon propagator, Coulomb potential from non-relativistic limit (L28)	—	✗	Gap.
Anomalous magnetic moment to 5th order (L28)	—	✗	Gap (Master-tier pointer).
Wightman axioms (L29)	—	✗	Gap. Final axiomatic chapter; load-bearing for FT-equivalence.
Osterwalder-Schrader axioms, Euclidean → Wightman reconstruction (L29)	—	✗	Gap. Connects to the Wick-rotation unit and to constructive QFT.
Open problem: interacting QFT on ${\mathbb{R}}^{1,3}$ (L29)	—	✗	Pointer only.

Aggregate coverage estimate: ~0% of Chatterjee has corresponding Codex units. The only adjacent material (08-stat-mech/ Wick rotation, lattice gauge, RG; 03-modern-geometry/09-spin-geometry/ Clifford and spin groups) covers the mathematical background but not the QFT construction. Chatterjee is not a "deepening-only" book — the entire QFT track is unaddressed.

This means the audit is NOT reduced: the PDF is fully readable, the TOC is complete, all 29 lectures were skimmed for content. The "Chatterjee deepening-only" hypothesis from the audit prompt is not supported by the evidence; the book is foundational for the QFT track that the Codex doesn't yet have.

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

Priority 0 — analysis-track prerequisites that block Chatterjee production: Chatterjee assumes (i) the spectral theorem for unbounded self-adjoint operators, (ii) Stone's theorem on one-parameter unitary groups, (iii) the tempered-distributions calculus on ${\mathbb{R}}^n$ and ${\mathbb{R}}^{1,3}$, (iv) Schwartz functions and the Fourier transform on $\mathcal{S}({\mathbb{R}}^n)$. The Codex's 02-analysis/ chapter is not yet at the level required. Recommendation: the Sternberg Semi-Classical Analysis audit (related-only free text, not on the canonical FT list) should be sequenced before Chatterjee production; Sternberg's lectures cover much of the distribution-theory machinery and would close the prerequisite gap. Also: Woit Quantum Theory, Groups and Representations (FT 2.02, also in this batch directory) is a likely sibling for the Lie-group / Poincaré content of L9–L10 and L25.

Priority 1 — high-leverage, captures Chatterjee's central content:

1. 08.10.01 Bosonic Fock space and second quantisation. Definition $B={⨁}_n{H}_{\mathrm{sym}}^{\otimes n}$, vacuum $|0⟩$, creation/annihilation operators $a_k$, $a_k^{†}$, canonical commutation relations $[a_k,a_l^{†}]={\delta }_{kl}1$. Chatterjee L6–L7 anchor. Three-tier, ~1500 words. Beginner tier: occupation-number basis intuition; Intermediate: rigorous Hilbert-space construction; Master: basis-independence of $A(f)$ via integral formula, operator-valued distributions $a(x)=A({\delta }_x)$.
2. 08.10.02 Massive scalar free field and the Wightman locality axiom. The operator-valued distribution $\varphi (t,x)=\int d^{3}p/[(2\pi {)}^3\sqrt{2{\omega }_p}](e^{-it{\omega }_p+ix\cdot p}a(p)+\text{h.c.})$, verification of Klein-Gordon $(\square +m^2)\varphi =0$ formally and $[\varphi (f),\varphi (g)]=0$ for spacelike-separated supports rigorously. Chatterjee L11–L12 anchor. Three-tier, ~1500 words.
3. 08.10.03 ${\varphi }^4$ theory and the Dyson series. Interaction Hamiltonian $H_I=(g/4!)\int :{\varphi }^4:$, Dyson series for $S$ on Fock space, first-order scattering computation $⟨p_3{p}_4|S|p_1{p}_2⟩=-ig(2\pi {)}^4{\delta }^{(4)}(\sum p_i)/\sqrt{16{\omega }_1{\omega }_2{\omega }_3{\omega }_4}+O(g^2)$, conservation of 4-momentum as a ${\delta }^{(4)}$. Chatterjee L13, L16, L18 anchor. Three-tier, ~1500 words.
4. 08.10.04 Wick's theorem for operator products. Statement, contraction expansion of $⟨0|T\prod \varphi (x_i)|0⟩$, Wick ordering $:\cdot :$, inductive proof. Crosswalk note: the existing 08.09.01-wick-rotation.md is Wick rotation; this is Wick's theorem. Add a "notation distinguishes from L1 unit" pointer. Chatterjee L17 anchor. Intermediate + Master only (the contraction machinery is too involved for Beginner tier); ~1800 words.
5. 08.10.05 Feynman propagator and the contour-integral representation. ${\Delta }_F(x)=i\int d^{3}p/[(2\pi {)}^{3}2{\omega }_p]e^{-i|t|{\omega }_p+ip\cdot x}$, the alternative form ${\Delta }_F(x)={\lim }_{ϵ\to 0^{+}}\int d^{4}p/(2\pi {)}^4\cdot e^{-i(x,p)}/(-p^2+m^2-iϵ)$, $iϵ$-prescription via contour integration. Chatterjee L19 anchor. Three-tier, ~1500 words.
6. 08.10.06 One-loop renormalisation in ${\varphi }^4$. The divergent second-order integral, the subtraction $L = \lim_{R\to\infty} (\widetilde{M} - \widetilde{M}*)$, cutoff-independence of the subtracted result, finite predictions in terms of one measured reference amplitude $A*$. Chatterjee L20–L21 anchor. Highest pedagogical value of any unit in this list — it's where the QFT programme actually delivers a number. Three-tier; Beginner gets the toy example (L21 §1) only. ~2000 words.
7. 08.10.07 Wightman axioms (W1–W7). Hilbert space + unitary Poincaré + smeared-field map + Lorentz covariance + spacelike locality
   • cyclic vacuum + spectrum condition. Verification that the free field satisfies them; the open problem on ${\mathbb{R}}^{1,3}$; brief mention of Osterwalder-Schrader reconstruction. Chatterjee L29 anchor; Streater- Wightman [SW] is the originator-prose source. Three-tier; Beginner tier states the axioms and the open problem only; ~1800 words.

Priority 2 — Dirac field and QED side:

8. 08.10.08 Special relativity and the Poincaré group as quantum inputs. Lorentz group ${\mathrm{SO}}^{\uparrow }(1,3)$, Poincaré group $P={\mathbb{R}}^{1,3}⋊{\mathrm{SO}}^{\uparrow }(1,3)$, mass shell $X_m$, Lorentz-invariant measure $d{\lambda }_m$. Chatterjee L9–L10. Cross-strand with Woit FT 2.02 (representation-theory framing). Intermediate + Master. ~1500 words.
9. 08.10.09 Fermionic Fock space, Pauli exclusion, anticommutators. $H_{\mathrm{anti}}^{\otimes n}$, anticommutation ${a_k, a_l^\dagger} = \delta_{kl}\mathbf{1}$,thesigncorrectionintheactionof$a_k^\dagger$. Chatterjee L25–L26. Intermediate + Master. ~1200 words.
10. 08.10.10 Dirac field $\psi$ and the Dirac adjoint $\bar{\psi }$. Projective representation of $P$ via $\mathrm{SL}(2,\mathbb{C})$ double cover, pure boost $V_p$, four-component spinor field, Pauli matrices, Dirac matrices ${\gamma }^{\mu }$, $\bar\psi = \psi^\dagger \gamma^0$. Crosswalk: depends on 03.09.02-clifford-algebra.md and 03.09.03-spin-group.md for the algebraic background. Chatterjee L25–L27. Intermediate + Master. ~1800 words.
11. 08.10.11 Quantum electrodynamics: interaction Hamiltonian and Feynman diagrams. $\mathcal{H}I = e :!\bar\psi \gamma^\mu \psi A\mu!:$,electron-photon-positronFockspace$B \otimes F \otimes F'$, second-order electron-electron scattering via photon exchange, Dirac propagator, photon propagator, non-relativistic limit reduces to Coulomb $V(x)=1/(4\pi |x|)$. Chatterjee L23–L24, L27–L28. The QED triumph (anomalous magnetic moment) sits in a Master-tier closing paragraph. ~2200 words.

Priority 3 — analysis-prerequisite units (must ship before Priority 1):

12. **02.XX.YY Spectral theorem for unbounded self-adjoint operators
    • Stone's theorem.** Already on the Sternberg Semi-Classical Analysis (related free text, not a canonical FT entry; see reference/fasttrack-texts/README.md) punch-list; confirm scheduling.
13. 02.XX.YY Schwartz space $\mathcal{S}({\mathbb{R}}^n)$ and tempered distributions ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$. Also on the Sternberg Semi-Classical Analysis punch-list. Fourier-transform conventions matter — see §4.
14. 02.XX.YY Bosonic and fermionic symmetric/antisymmetric tensor powers of a Hilbert space. Currently 03.01-tensor-algebra/ covers the algebraic version; needs the completed-Hilbert-space variant. Could be folded into 08.10.01 as a Master-tier appendix.

Priority 4 — deepenings (Master-tier, not strictly required for FT equivalence):

15. Two-loop renormalisation in ${\varphi }^4$. Chatterjee L22. Add as a Master section to 08.10.06 rather than a new unit.
16. Osterwalder-Schrader axioms and reconstruction. Add as a Master section to 08.10.07. Pointer to constructive QFT literature.
17. Anomalous magnetic moment of the electron to 5th order. Add as a Master-tier closing paragraph to 08.10.11. The 10-decimal-place agreement with experiment is the point of QED and worth recording.

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§4 Implementation sketch (P3 → P4)

For a full Chatterjee coverage pass, items 1–7 are the minimum set (with the Sternberg / Woit prerequisites in place). Realistic production estimate (mirroring earlier Brown / Lawson-Michelsohn / Bott-Tu batches):

• ~3–4 hours per unit. Chatterjee units skew slightly higher than the corpus average because the Master tier requires careful operator-valued- distribution notation and the Wick / renormalisation / propagator units have unavoidable contour-integral and combinatorial content.
• 7 priority-1 units × ~3.5 hours = ~24 hours of focused production. 4 priority-2 units × ~3 hours = ~12 hours. Total Chatterjee production: ~36 hours, fits a focused 5–7 day window.
• Add ~10–15 hours for the priority-3 analysis prerequisites (already on the Sternberg punch-list, so absorbed there if Sternberg ships first).

Originator-prose target. Chatterjee is a pedagogical originator (the rigorous-as-possible mathematicians' on-ramp), not a research originator. For the Codex's QFT track, originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10 should cite:

• A. Wightman & L. Gårding, "Fields as operator-valued distributions in relativistic quantum theory," Arkiv för Fysik 28 (1964) 129–184 — originating the operator-valued-distribution formalism.
• R. Streater & A. Wightman, PCT, Spin and Statistics, and All That (Princeton 1964) — originator of the Wightman axioms as 08.10.07 is currently scoped.
• F. Dyson, "The S-Matrix in Quantum Electrodynamics," Phys. Rev. 75 (1949) 1736 — originating the Dyson series.
• N. Bogoliubov & O. Parasiuk, K. Hepp, W. Zimmermann (BPHZ) — the rigorous renormalisation programme that 08.10.06 reflects.
• K. Osterwalder & R. Schrader, "Axioms for Euclidean Green's functions I, II," Comm. Math. Phys. 31 (1973), 42 (1975) — originating the Euclidean axioms cited in the L29 closing material.
• M. Talagrand, What is a Quantum Field Theory? (Cambridge 2022) — the textbook these notes prefigure; consult for definitive consolidation of Chatterjee's pedagogy.

Notation crosswalk. Chatterjee uses physicist conventions on several points where the Codex's mathematician conventions differ; these need to be recorded explicitly in each new unit's §Notation paragraph:

Object	Chatterjee	Codex mathematician convention	Resolution
Inner product on $H$	$(f,g)$, antilinear in first argument, linear in second	usually antilinear in second, linear in first (Codex spin-geometry units 03.09.* follow this)	Chatterjee's convention is the physicists' Dirac-bra-ket convention; record at top of 08.10.01. Adopt Chatterjee's convention for the QFT track and flag the difference at the boundary with spin-geometry.
Adjoint	$A^{†}$	$A^{\ast }$ in 02-analysis/	Use $A^{†}$ throughout the QFT track; flag the synonymy.
Fourier transform	$\hat{f}(p)=\int dx{e}^{-ixp}f(x)$ (no $1/(2\pi )$ prefactor; sign convention $-ixp$)	Codex 02-analysis/ and 03-differential-forms use the symmetric $(2\pi {)}^{-n/2}$ convention	Chatterjee's convention pushes $(2\pi {)}^3$ factors into the measure and the delta function. Adopt Chatterjee's for the QFT track and record the conversion factor explicitly in 08.10.01.
Metric signature	$(+,-,-,-)$ "mostly minus"	physics-side variable	Record signature in 08.10.08; mention $(-,+,+,+)$ "mostly plus" as an alternative used by some texts (Misner-Thorne-Wheeler, e.g.).
Units	$\hslash =c=1$	unstated	Record at top of every QFT track unit.
Wick rotation vs Wick's theorem	both abbreviated "Wick" in Chatterjee	distinguished by usage	Use the full word in unit titles: 08.09.01-wick-rotation.md already exists; new unit is 08.10.04-wick-theorem.md. Cross-link explicitly.
Mass shell measure	$d{\lambda }_m(p)=d^{3}p/[(2\pi {)}^{3}2{\omega }_p]$ on $X_m\subset {\mathbb{R}}^{1,3}$	unstated	Define in 08.10.08; this is the Lorentz-invariant measure adopted throughout.

Notation decisions for the QFT track must be recorded in a §Notation paragraph of every Priority-1 unit, per docs/specs/UNIT_SPEC.md §11.

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§5 What this plan does NOT cover

• A line-number-level inventory of every numbered lemma / theorem in Chatterjee (full P1 audit). The 29 lectures yield perhaps 60–80 named results; a complete enumeration is deferred. The skim was content- oriented, not theorem-counting.
• The path-integral formulation of QFT. Chatterjee mentions it briefly in L29 (Osterwalder-Schrader) but does not develop it. A separate audit of Salmhofer [Sa] or Glimm-Jaffe [GJ] would fill this; not scoped here.
• Non-abelian gauge theory and Yang-Mills. Chatterjee covers QED only. Yang-Mills is in 03-modern-geometry/07-gauge-theory/ at the classical-action level but the quantised version is out of scope.
• Renormalisation-group flow in the Wilsonian sense. Chatterjee gives cutoff-subtraction renormalisation but not the RG. Connects to the existing 08-stat-mech/rg/ units; cross-reference, do not duplicate.
• Anomalies, spontaneous symmetry breaking in the QFT sense, the Higgs mechanism, electroweak unification, QCD, supersymmetry, string theory. All out of scope for Chatterjee specifically; some are in scope for Weinberg FT 2.17 and Deligne-et-al FT 3.43.
• Exercise-pack production. Chatterjee has occasional exercises stated inline but no formal exercise set. An exercise pack would be a P3-priority-3 follow-up after the priority-1 units ship; we mirror the NAT plan's deferral here.
• Lean formalisation. The Codex's 08-stat-mech/ chapter has lean_status: none and lean_mathlib_gap notes acknowledging Mathlib's absence of statistical-mechanics infrastructure. The same applies double for QFT — no Mathlib QFT library exists. All new units will ship with lean_status: none.

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§6 Acceptance criteria for FT equivalence (Chatterjee)

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

• Sternberg Semi-Classical Analysis (related free text, not a canonical FT entry; see reference/fasttrack-texts/README.md) (or equivalent analysis-track unit set) has shipped the spectral theorem, Stone's theorem, Schwartz / tempered-distribution prerequisites. Strict prereq.
• Woit FT 2.02 (or equivalent) has shipped the Poincaré-group representation theory used in L9–L10 and L25. Soft prereq — Chatterjee can ship the Hilbert-space side first.
• ≥95% of Chatterjee's named theorems / constructions in L1–L29 map to Codex units. (Currently 0%; after Priority-1 units this rises to ~70% — the scalar QFT track is closed. After Priority-1 + Priority-2 to ~90% — Dirac + QED closed. Full ≥95% requires Priority-4 deepenings.)
• ≥90% of Chatterjee's worked computations (first-order ${\varphi }^4$ scattering, one-loop renormalisation, Feynman-propagator contour integral, electron-electron scattering, non-relativistic Coulomb recovery) have a direct unit or are referenced from a unit that covers them.
• Notation decisions are recorded per §4 (inner-product convention, Fourier convention, metric signature, units, Wick distinction).
• Pass-W weaving connects the new 08.10.* units to (a) the existing 08-stat-mech/ chapter via Wick-rotation / partition-function / Gaussian-field bridges; (b) 03-modern-geometry/09-spin-geometry/ via Clifford algebra → Pauli/Dirac matrices in 08.10.10; (c) 03-modern-geometry/07-gauge-theory/ via the classical-EM-recovery paragraph in 08.10.11; (d) the prerequisite Sternberg / Woit units.

The 7 priority-1 units close most of the equivalence gap for the scalar QFT track. Priority-2 closes the Dirac/QED track. Priority-3 are prerequisite (handled by Sternberg). Priority-4 are deepenings.

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§7 Sourcing

• Free. Author-hosted PDF at Sourav Chatterjee's Stanford page; the notes are explicitly posted as Math 273 course materials.
• License. Author has placed the PDF freely available as course notes; for educational use cite as Chatterjee, Introduction to Quantum Field Theory for Mathematicians (Stanford Math 273 lecture notes, Fall 2018, based on a forthcoming textbook by Michel Talagrand). The Talagrand textbook (What is a Quantum Field Theory?, Cambridge 2022) is now in print and should be cited alongside for the consolidated treatment.
• Local copy. Already present at reference/fasttrack-texts/02-quantum-stat/Chatterjee-QFTLectureNotes.pdf (130 pp., 784 KB, pdfTeX 1.40.16, 2018-12-15). Sibling files in the same directory:
  • Sternberg-SemiClassicalAnalysis.pdf (related-only free text; not on the canonical FT list per reference/fasttrack-texts/README.md) — analysis-track prerequisites.
  • Woit-QuantumTheoryGroupsRepresentations.pdf (FT 2.02) — Poincaré- group representation theory.
• Audit completeness: full (audit_completeness: full). PDF was readable, all 29 lectures skimmed, table of contents complete. No reduced-audit flag needed.