Woit — Quantum Theory, Groups and Representations (Fast Track 2.02) — Audit + Gap Plan

Book: Peter Woit, Quantum Theory, Groups and Representations: An Introduction (Springer 2017; revised-and-expanded "under construction" draft dated 20 October 2025, 49 chapters + exercise appendix, ~580 pp.). Hosted free by the author at Columbia (https://www.math.columbia.edu/~woit/QM/qmbook.pdf).

Fast Track entry: 2.02 (the QM-via-representation-theory slot of the Quantum-Theory-and-Statistical-Physics strand; paired with Sternberg Group Theory and Physics (1.15) and Sourav Chatterjee QFT Lecture Notes (2.03) as the free open-access bridge from classical mechanics into non-relativistic and relativistic free-field 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 — and, more critically here, an entire new chapter directory — so that Quantum Theory, Groups and Representations (WQGR 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).

This pass is not a full P1 audit. WQGR is a 49-chapter living draft; a complete line-number inventory is a multi-week job and is deferred. This plan works from the published TOC, the preface's explicit list of "differences from standard physics presentations," and the first three chapters (sampled directly), produces the gap punch-list, and stops there.

The audit surface here is larger than any prior Fast Track audit so far. The Codex has substantial machinery for the pure-math half of WQGR (Lie groups, Lie algebras, representation theory, symplectic geometry, Clifford algebras, spin geometry) — that infrastructure is at ~50% of what Woit assumes. But the Codex has no chapter for the quantum-mechanics half: no Hilbert-space axioms, no Schrödinger equation, no Heisenberg algebra, no Stone–von Neumann, no metaplectic representation, no harmonic-oscillator quantization, no canonical/anticanonical commutation relations, no Klein-Gordon, no Dirac equation, no Wigner classification of Poincaré representations, no field quantization. The QM-side gap is total.

§1 What WQGR is for

WQGR is the canonical mathematician-facing introduction to quantum theory built around unitary representations of Lie groups from page one. Where Griffiths or Sakurai (and every standard physics text) introduces QM by motivating the Schrödinger equation from classical mechanics and treats group theory as a later "applications" chapter, Woit inverts the order: the Lie group acting on phase space is the primary object, its unitary representation on $H$ is the quantum theory, and "quantization" is the Stone–von Neumann passage from the classical Heisenberg Lie algebra to the unique (up to unitary equivalence) irreducible representation realising the canonical commutation relations. This is Hermann Weyl's 1928 Gruppentheorie und Quantenmechanik programme [1] — the "Gruppenpest" that physicists revolted against — updated with ninety years of post-Wigner, post-Mackey, post-Kirillov hindsight.

Distinctive contributions, in roughly the order WQGR develops them:

Quantization as the search for a unitary representation, not as a substitution rule. The classical phase space carries a Heisenberg Lie algebra of linear functions under Poisson bracket; quantization is the passage to its unique irreducible unitary representation (Stone–von Neumann theorem). The Schrödinger and Bargmann–Fock pictures are then two coordinatisations of the same representation related by an intertwiner (the Bargmann transform).
Spin from $S U (2)$ representation theory, before the Dirac equation. Chapters 3–8 develop the entire $S U (2)$ / $S O (3)$ rep-theoretic edifice (Pauli matrices, two-state systems, weight decomposition, raising and lowering operators, spherical harmonics, Casimir operator) before any wave-equation appears. The spin-1/2 particle in a magnetic field is computed as a representation-theoretic problem (chapter 7), not as a Pauli-equation problem.
Symplectic geometry first, Lagrangian formalism last. The Poisson bracket and moment map are introduced in chapters 14–15 as the natural language for classical mechanics; the Lagrangian formalism and Noether's theorem only appear in chapter 35, after most of the non-relativistic theory has been built. Woit's preface is explicit: "operators generating symmetry transformations are derived using the moment map… not by invoking Noether's theorem." This is the editorial move Sternberg [2] also makes; WQGR pushes it further.
The Groenewold–van Hove no-go theorem, foregrounded. Chapter 17 states and discusses the obstruction to extending quantization beyond the Heisenberg subalgebra of polynomial observables — a result almost universally hidden in physics texts. This frames why geometric quantization (Woodhouse [3], Bates–Weinstein [4]) is needed at all.
Semidirect products and Mackey-style induced representations. Chapters 18–20 build the Euclidean group $E (n)$ , its representations via Mackey's induction, and treats the non-relativistic free particle as a representation of $E (3)$ . Chapter 42 then applies the identical machinery to the Poincaré group to recover Wigner's 1939 classification of relativistic particles by mass and spin. This is the cleanest parallel-construction of non-relativistic and relativistic QM in print.
Metaplectic representation and Bogoliubov transformations. Chapters 20, 23–26 develop the metaplectic / oscillator representation of $S p (2 d, R)$ as the projective unitary representation arising from the symplectic action on the Heisenberg group, with the projective factor (the Maslov line bundle) tracked explicitly. Chapter 24 then identifies Bogoliubov transformations with $S U (1, 1)$ symmetries of the oscillator phase space — making "squeezed states" a representation- theoretic identity rather than an ad hoc construction.
Bosonic / fermionic parallelism via Clifford algebras and Lie superalgebras. Chapters 27–32 develop the fermionic oscillator, anticommuting variables (Grassmann calculus), and the spinor representation as the fermionic analogue of Heisenberg/metaplectic, with Lie superalgebras as the unifying structure. This is Berezin's programme [5]; Woit's chapter 32 is an explicit "Parallels Between Bosonic and Fermionic Quantization" summary table.
Lagrangian QFT only at the end, and only as a complement. Chapters 35–48 develop multi-particle quantum systems, field quantization, gauge symmetry, Klein-Gordon, Dirac, and the Standard Model in 13 chapters — but the entire path uses the Hamiltonian/representation- theoretic framework already built, with the Lagrangian/path-integral picture introduced in chapter 35 as one viewpoint among others, not the foundational one.

WQGR is not a first introduction to QM. It assumes linear algebra and multivariable calculus only (the preface is emphatic about this), but it moves at graduate-mathematician pace. It is the canonical entry point to QM-as-representation-theory if one wants the Hamiltonian / Stone–von-Neumann / Mackey programme rather than the Lagrangian / path-integral / functional-integral programme of Peskin– Schroeder or Weinberg [6]. The two programmes are complementary; the Fast Track explicitly chooses Woit as the entry, with Chatterjee 2.03 and Weinberg 2.17 as the path-integral / interacting-QFT complement.

Peer sources triangulating Woit's distinctive editorial move:

[1] H. Weyl, Gruppentheorie und Quantenmechanik (1928); English The Theory of Groups and Quantum Mechanics (Dover reprint 1950) — the originating text of QM-via-representation-theory.
[2] S. Sternberg, Group Theory and Physics (Cambridge 1994; FT 1.15) — the closest contemporary peer, sharing Woit's moment-map-first and Lie-superalgebra emphasis; differs in skipping the Stone–von Neumann theorem and going harder on crystallographic groups.
[3] N. M. J. Woodhouse, Geometric Quantization (Oxford 1991) — the canonical reference for the polarization/half-density/Maslov refinements WQGR alludes to in chapter 17.5.
[4] S. Bates and A. Weinstein, Lectures on the Geometry of Quantization (Berkeley Math Lecture Notes 1997) — the Weinstein-school geometric-quantization companion; freely hosted.
[5] F. A. Berezin, The Method of Second Quantization (Academic Press 1966) — originating the Grassmann-calculus / fermionic-oscillator treatment Woit follows in chapters 27–30.
[6] B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Springer GTM 222, 2nd ed. 2015) — the closest pure-math companion for the Lie-theoretic prerequisites; Hall and Woit cross-reference each other.

§2 Coverage table (Codex vs WQGR)

Cross-referenced against the current 313-unit corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered.

WQGR chapter / topic	Codex unit(s)	Status	Note
Ch. 1 — Axioms of QM (state, observable, dynamics, Born rule)	—	✗	Gap. No Codex unit on Hilbert-space axioms.
Ch. 2 — $U (1)$ and its representations; charge operator	`07.01.01-group-representation`	△	Group-rep definition present; $U (1)$ classification by integer weight + physical interpretation as charge — not covered.
Ch. 3 — Two-state systems; Pauli matrices; commutation relations	—	✗	Gap. Pauli matrices nowhere in Codex.
Ch. 4 — Linear algebra review (vector spaces, duals, inner products, orthogonal/unitary groups)	`01.01.*`, `03.03.03-orthogonal-group`	✓	Codex linear algebra and $O (n)$ coverage adequate.
Ch. 5 — Lie algebras and Lie algebra representations	`03.04.01-lie-algebra`, `07.06.01-lie-algebra-representation`, `07.06.02-universal-enveloping-algebra`, `05.01.01-symplectic-vector-space` (adjacent)	△	Abstract Lie algebra + rep present; Woit's explicit complexification / skew-vs-self-adjoint dictionary not covered.
Ch. 6 — Rotation and spin groups in 3 and 4D; quaternions; $Spin (3) ≅ S U (2)$	`03.09.03-spin-group`, `03.03.03-orthogonal-group`	△	Spin group present at the abstract Clifford level; the explicit quaternion / $S U (2)$ double cover / 4D version not unit-level.
Ch. 7 — Spinor representation; spin-1/2 in a magnetic field; Bloch sphere; complex projective space	—	✗	Gap. Spinor rep at the rep-theoretic level (vs. spinor bundle in `03.09.05`) not covered; Bloch sphere absent.
Ch. 8 — Representations of $S U (2)$ and $S O (3)$ ; weight decomposition; raising/lowering; spherical harmonics; Casimir	`07.06.03-root-system`, `07.06.06-verma-module`, `07.06.07-weyl-character-formula` (general); — (specifically $S U (2)$ )	△	The general highest-weight theory is shipped; the explicit $S U (2)$ worked example with raising/lowering and spherical-harmonic decomposition of $L^{2} (S^{2})$ is not.
Ch. 9 — Tensor products, entanglement, addition of spin; characters	`07.01.06-tensor-product-of-representations`, `07.01.03-character`	△	Tensor-product-of-reps present; the QM-flavored entanglement / Clebsch-Gordan / addition-of-angular-momentum framing missing.
Ch. 10 — Momentum and the free particle; representations of $R$ and $R^{3}$	—	✗	Gap.
Ch. 11 — Fourier analysis; distributions; Schwartz space	`02.*` partial	△	Schwartz/tempered distributions not yet a Codex unit; Fourier transform on $R^{n}$ partially covered in analysis chapter (verify).
Ch. 12 — Position operator; Dirac notation; Heisenberg uncertainty; propagators; Green's functions	—	✗	Gap.
Ch. 13 — Heisenberg Lie algebra; Heisenberg group; Schrödinger representation	—	✗	Gap (CRITICAL — load-bearing for the entire Woit edifice).
Ch. 14 — Poisson bracket and symplectic geometry; classical mechanics on phase space	`05.02.02-poisson-bracket`, `05.01.02-symplectic-manifold`	✓	Codex symplectic chapter covers this well.
Ch. 15 — Hamiltonian vector fields, moment map	`05.02.01-hamiltonian-vector-field`, `05.04.01-moment-map`	✓	Codex coverage adequate.
Ch. 16 — Quadratic polynomials and the symplectic group; $S p (2 d, R)$	`05.01.03-symplectic-group`	△	Symplectic group present; the explicit identification of quadratic Hamiltonians with $sp (2 d, R)$ via the moment map is not.
Ch. 17 — Canonical quantization; Groenewold–van Hove no-go theorem; quantization and symmetries	—	✗	Gap (CRITICAL — Woit's pedagogical pivot point). No-go theorem entirely absent.
Ch. 18 — Semidirect products; Euclidean group $E (n)$ ; semidirect-product Lie algebras	—	✗	Gap. Codex covers product Lie groups but not semidirect products as a unit.
Ch. 19 — Quantum free particle as a representation of $E (2), E (3)$	—	✗	Gap.
Ch. 20 — Representations of semidirect products; intertwining operators; metaplectic representation introduced	—	✗	Gap (CRITICAL — first appearance of the metaplectic rep).
Ch. 21 — Central potentials; $so (4)$ symmetry of the Coulomb potential; hydrogen atom	—	✗	Gap. The Pauli–Fock $so (4)$ derivation of the hydrogen spectrum is iconic; absent from Codex.
Ch. 22 — Harmonic oscillator; creation / annihilation operators; Bargmann–Fock representation	—	✗	Gap (CRITICAL).
Ch. 23 — Coherent states; harmonic-oscillator propagator; Bargmann transform	—	✗	Gap.
Ch. 24–25 — Metaplectic representation in detail; $S U (1, 1)$ and Bogoliubov transformations; arbitrary $d$	—	✗	Gap. No metaplectic-rep machinery anywhere.
Ch. 26 — Complex structures and quantization; compatible triples	`05.08.01-compatible-triple` (verify), `05.07-almost-complex` partial	△	Compatible-triple language exists in symplectic chapter; the quantization-from-complex-structure perspective is not.
Ch. 27 — Fermionic oscillator; CAR	—	✗	Gap.
Ch. 28 — Complex Weyl and Clifford algebras	`03.09.02-clifford-algebra`	△	Clifford algebra present at the geometric level; Woit's "Weyl-and-Clifford as bosonic-vs-fermionic" parallel framing is not.
Ch. 29 — Clifford algebras and geometry; rotations as iterated reflections	`03.09.02-clifford-algebra`, `03.09.03-spin-group`, `03.09.11-clifford-chessboard`	✓	Adequately covered.
Ch. 30 — Anticommuting variables; Grassmann algebra; pseudo-classical mechanics	—	✗	Gap. Grassmann calculus / Berezin integration absent.
Ch. 31 — Fermionic quantization; spinors via Bargmann–Fock	—	✗	Gap.
Ch. 32 — Bosonic/fermionic parallels (summary chapter)	—	✗	Gap (low priority — synthetic summary unit).
Ch. 33 — Supersymmetric oscillator; SUSY QM and differential forms	—	✗	Gap.
Ch. 34 — Pauli equation and the Dirac operator (non-relativistic)	`03.09.08-dirac-operator`	△	Geometric Dirac operator present; Pauli equation as a rep-theoretic object on $E (3)$ not covered.
Ch. 35 — Lagrangian methods; Noether's theorem; path integral	`05.00.04-noether-theorem`, `08.07.01-path-integral`	△	Both shipped, but in classical / stat-mech context, not the QM context Woit uses.
Ch. 36–37 — Multi-particle systems; field quantization; second quantization	—	✗	Gap.
Ch. 38 — Symmetries and non-relativistic quantum fields; internal symmetries; spatial symmetries	—	✗	Gap.
Ch. 39 — Quantization of infinite-dimensional phase spaces; inequivalent representations; Schwinger term; spontaneous symmetry breaking; renormalization	—	✗	Gap.
Ch. 40 — Minkowski space; Lorentz group and its Lie algebra	—	✗	Gap.
Ch. 41 — Representations of the Lorentz group; Dirac $γ$ -matrices in $Cl (3, 1)$	`03.09.02-clifford-algebra`, `03.09.11-clifford-chessboard`	△	Clifford signatures classified; the specific Lorentz-rep / Dirac- $γ$ presentation is not a unit.
Ch. 42 — Poincaré group and its representations; Wigner classification by orbits	—	✗	Gap (CRITICAL — Wigner 1939 is foundational).
Ch. 43 — Klein-Gordon equation; scalar quantum fields	—	✗	Gap.
Ch. 44 — Relativistic scalar quantum fields with internal/external symmetries	—	✗	Gap.
Ch. 45 — $U (1)$ gauge symmetry and EM fields; curvature; non-Abelian gauge	`03.07.05-yang-mills-action`	△	Yang-Mills action shipped; the elementary $U (1)$ -gauge-from-rep-theory derivation is not a separate unit.
Ch. 46 — Quantization of the EM field; the photon; Coulomb / covariant gauge	—	✗	Gap.
Ch. 47 — Dirac equation and spin-1/2 fields; Majorana, Weyl, Dirac spinors	`03.09.05-spinor-bundle`, `03.09.13-triality`	△	Spinor bundle at the geometric level; the spinor fields of QFT not a unit.
Ch. 48 — Introduction to the Standard Model	—	✗	Gap (low priority — survey chapter).
Appendix A — Conventions	—	n/a	Notation crosswalk; see §4.
Appendix B — Exercise pack (≈300 exercises)	—	✗	Gap. Exercise production deferred.

Aggregate coverage estimate: ~12% of WQGR has corresponding Codex units (mostly the pure-math prerequisites in chapters 4, 5, 9, 14, 15, 29, and partial credit on chapters 8 and 35). The QM half (chapters 1–3, 10–13, 17, 19–28, 30–48) is ~0% covered. Coverage at the chapter-level is below the 80% "already shipped" threshold — this is not a flag-as-mostly-done book.

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

Priority 0 — structural prerequisite: Open a new top-level chapter directory content/09-quantum-mechanics/ (or expand 08-stat-mech/ to a combined 08-quantum-and-stat-mech/). Woit's content does not fit naturally into any of the existing eight chapters; about 35 of the units below are QM/QFT proper and need their own home. This is a structural decision that should be made before any unit-level work.

Priority 1 — high-leverage, captures Woit's central content (15 units):

09.01.01 Hilbert-space axioms of QM (state vector, self-adjoint observables, Schrödinger dynamics, Born rule). WQGR §1.2 anchor.
09.01.02 $U (1)$ representations and the charge operator. WQGR §2. Builds on 07.01.01.
09.01.03 Two-state systems and Pauli matrices. WQGR §3. Worked example: spin-1/2 in a magnetic field via $S U (2)$ rep theory. Worked example: Bloch sphere as $C P^{1}$ .
09.01.04 $S U (2)$ representations: weight decomposition, raising and lowering, spherical harmonics, Casimir. WQGR §8. Specialisation of 07.06.06 to $S U (2)$ ; deserves its own unit because it is load-bearing for every later QM calculation.
09.02.01 The Heisenberg Lie algebra and Heisenberg group. WQGR §13. CRITICAL — load-bearing.
09.02.02 Schrödinger representation and the Stone–von Neumann uniqueness theorem. WQGR §13.3, with proof via Mackey / von Neumann
1. CRITICAL.
09.02.03 Canonical quantization and the Groenewold–van Hove no-go theorem. WQGR §17. CRITICAL — frames why higher-degree quantization needs more machinery.
09.03.01 Semidirect product Lie groups and Lie algebras. WQGR §18. Euclidean group $E (n)$ as the canonical example.
09.03.02 Mackey induction and representations of semidirect products with abelian normal factor. WQGR §20.
09.03.03 Quantum free particle as a representation of $E (3)$ . WQGR §19.
09.04.01 Quantum harmonic oscillator: creation and annihilation operators; Bargmann–Fock representation; Bargmann transform as intertwiner. WQGR §22–§23. CRITICAL.
09.04.02 Coherent states. WQGR §23.
09.04.03 Metaplectic representation of $S p (2 d, R)$ and the projective factor. WQGR §20, §24–§25. CRITICAL.
09.05.01 Poincaré group and its Lie algebra. WQGR §42.1.
09.05.02 Wigner classification of irreducible representations of the Poincaré group by orbits in momentum space. WQGR §42.2–§42.3. CRITICAL.

Priority 2 — fermionic side, field quantization, Dirac (10 units):

09.04.04 Fermionic oscillator; canonical anticommutation relations. WQGR §27.
09.04.05 Grassmann algebra and pseudo-classical mechanics; Berezin integration. WQGR §30.
09.04.06 Spinor representation via fermionic Bargmann–Fock; bosonic-fermionic parallels table. WQGR §31–§32.
09.05.03 Lorentz group representations and Dirac $γ$ -matrices in $Cl (3, 1)$ . WQGR §41.
09.06.01 Klein-Gordon equation and free scalar quantum field. WQGR §43.
09.06.02 Dirac equation and free spin-1/2 quantum field; Majorana, Weyl, Dirac spinor fields. WQGR §47.
09.06.03 Multi-particle systems and field operators; second quantization. WQGR §36–§37.
09.06.04 Internal and spatial symmetries of non-relativistic quantum fields. WQGR §38.
09.06.05 $U (1)$ gauge symmetry and EM fields; quantization of the photon in Coulomb / covariant gauge. WQGR §45–§46.
09.06.06 Inequivalent irreducible representations in infinite dimensions; Schwinger term; spontaneous symmetry breaking. WQGR §39. Pointer unit.

Priority 3 — depth-completions and connecting tissue (6 units):

09.01.05 Quantum dynamics in the Heisenberg picture; propagators in position and momentum space; Green's functions. WQGR §12, §23.
09.02.04 Tensor product of representations and addition of spin; Clebsch–Gordan. WQGR §9. Specialisation of 07.01.06.
09.04.07 Complex structures and quantization; squeezed states. WQGR §26.
09.04.08 Supersymmetric quantum mechanics and the de Rham complex. WQGR §33.
09.05.04 Hydrogen atom via $so (4)$ symmetry of the Coulomb potential. WQGR §21.
Extend 05.04.01-moment-map (existing) with a Master section on the WQGR §15.3 perspective: moment map as the bridge from Lie-group actions to self-adjoint quantum operators.

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

09.07.01 Path integral as a complement to Hamiltonian quantization. WQGR §35.3–§35.4. Connects to 08.07.01-path-integral.
09.07.02 Introduction to the Standard Model: non-Abelian gauge fields, fundamental fermions, Yukawa couplings, Higgs. WQGR §48. Pointer unit only.
09.07.03 Pointer to geometric quantization (Woodhouse, Bates–Weinstein) and to deformation quantization (Kontsevich formality). WQGR §17.5.

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

WQGR uses $∣ ψ ⟩$ (Dirac ket) for state vectors and $\langle \cdot, \cdot \rangle$ (mathematician convention, linear in the second slot) for Hermitian inner products. Codex 01.01.* uses standard mathematician notation with no kets. Adopt physics ket notation in the 09-quantum-mechanics/ chapter only, with a parenthetical $∣ ψ ⟩ = ψ$ on first use.
WQGR uses $†$ for adjoint; Codex symplectic / Lie chapters use $*$ . Adopt $†$ within 09-* and cross-reference to $*$ at chapter boundary.
WQGR uses $ℏ$ explicitly and reserves the right to set $ℏ = 1$ context-dependently. Codex convention should mirror.
WQGR notation for the Heisenberg algebra: $h_{2 d + 1}$ with basis ${q_{j}, p_{j}, c}$ and central element $c$ . Adopt this; record cross-reference to the (rare) Codex notation in 03.11.01-central- extension.

§4 Implementation sketch (P3 → P4)

For a full WQGR coverage pass, items 1–15 are the minimum set. Realistic production estimate (mirroring earlier batches):

~4–5 hours per unit for the QM-side units; these are higher than the corpus average because (a) each carries an explicit worked computation (hydrogen spectrum, harmonic-oscillator spectrum, Wigner orbits), (b) the notation/style is a hybrid of physics ket-notation and rigorous mathematician prose, and (c) the master tier needs the careful projective-representation / Maslov-line-bundle discussion.
15 priority-1 units × ~4.5 hours = ~70 hours of focused production. Plus the priority-2 fermionic/field-quantization batch (~45 hours). Plus the priority-3 depth completions (~25 hours). Total ~140 hours for full FT equivalence. Fits a 3–4-week focused window.
Lean formalisation is deferred for this strand — none of the QM units have current Mathlib analogues at the unit-spec depth.

Originator-prose target. WQGR's editorial choices (representation- first, moment-map-first, Stone–von Neumann at the centre) trace back to:

Weyl (1928), Gruppentheorie und Quantenmechanik — originator of the QM-via-rep-theory programme. Cite in 09.01.01.
von Neumann (1931), "Die Eindeutigkeit der Schrödingerschen Operatoren," Math. Ann. 104, 570–578 — original Stone–von Neumann uniqueness proof. Cite in 09.02.02.
Wigner (1939), "On unitary representations of the inhomogeneous Lorentz group," Ann. Math. 40, 149–204 — original classification. Cite in 09.05.02.
Mackey (1968), Induced Representations of Groups and Quantum Mechanics — induction machinery used in WQGR ch. 20.
Groenewold (1946) / van Hove (1951) — no-go theorem originators. Cite in 09.02.03.
Bargmann (1961), "On a Hilbert space of analytic functions and an associated integral transform," Comm. Pure Appl. Math. 14, 187–214 — Bargmann–Fock space and the Bargmann transform.
Berezin (1966), The Method of Second Quantization — Grassmann calculus origin. Cite in 09.04.05.

Woit himself is not the originator of these results — he is a synthesiser in the Weyl tradition. Cite Woit (2017) as the modern consolidating reference and as the FT canonical anchor. Supplement liberally with Hall (2015) for Lie-theoretic prerequisites and Sternberg (1994; FT 1.15) for the moment-map / Lie-superalgebra material — Sternberg has overlapping coverage on roughly half the priority-1 units and should be cross-cited where his treatment is cleaner (notably on the Heisenberg algebra and on the spin-1/2 magnetic-field computation).

Recommended sequencing. Priority-1 units should ship in the order listed; in particular 09.02.01 → 09.02.02 → 09.02.03 is a hard chain (no skipping), and 09.03.01 → 09.03.02 → 09.03.03 likewise. 09.04.01 (harmonic oscillator) is the pedagogical climax of the non-relativistic half and should be staged as a wave terminal.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in WQGR (full P1 audit; deferred — WQGR is ~580 pp. + 49 chapters).
Exercise-pack production. WQGR appendix B carries ~300 exercises across 21 sub-appendices, many of them substantial calculations (e.g., derive the hydrogen spectrum from $so (4)$ ). Exercise pack is a P3-priority-4 follow-up after the priority-1 units ship.
Decision on whether to open content/09-quantum-mechanics/ as a new top-level chapter or expand the existing 08-stat-mech/. Flagged in §3 Priority 0; not decided here.
Geometric quantization in the Woodhouse / Bates–Weinstein sense — only pointed at in 09.07.03. A full geometric-quantization audit belongs to a separate plan (Woodhouse / Bates–Weinstein as their own Fast Track entry, currently uncatalogued).
Interacting QFT, renormalisation, the BPHZ programme. WQGR explicitly punts to "for further reading" (chapter 49); the Fast Track defers this to Weinberg 2.17–2.19 / Chatterjee 2.03 / Costello.
Path-integral formulation as a primary track. Covered as a complement pointer in 09.07.01 only.
Crystallographic point groups and the finite-group rep-theory side of QM (Wigner's Group Theory and Quantum Mechanics 1931 / Tinkham). WQGR explicitly omits this; Sternberg 1.15 is the FT anchor for that material.

§6 Acceptance criteria for FT equivalence (Woit)

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

The Priority 0 structural decision (new chapter directory) is recorded.
Priority-1 units 1–15 have all shipped.
≥95% of WQGR's named theorems in chapters 1–34 (non-relativistic half) map to Codex units (currently ~12%; after priority-1 units this rises to ~65%; after priority-1+2 to ~85%; full ≥95% requires priority-3 + selective priority-4).
≥80% of WQGR's chapters 35–48 (Lagrangian + relativistic + Standard Model survey) map to Codex units or are referenced from a pointer unit.
≥90% of WQGR's worked computations in chapters 1–34 have a direct unit or are exercised in a referenced unit. The signature computations — $S U (2)$ spectrum / spherical harmonics, harmonic-oscillator spectrum via raising/lowering, hydrogen via $so (4)$ , Wigner's classification — all need explicit worked-example status.
Notation decisions are recorded (see §3 crosswalk).
Pass-W weaving connects the new 09-quantum-mechanics/ units laterally to 03.09-spin-geometry/, 05-symplectic/, 07-representation-theory/, and to the new 09.05.* Poincaré/Wigner units from the Klein-Gordon / Dirac directions.

The 15 priority-1 units close most of the equivalence gap. Priority-2 closes the fermionic/QFT half. Priority-3 fills the depth pieces. Priority-4 is survey-pointer deepening only.

§7 Sourcing

Free. Author-hosted PDF at https://www.math.columbia.edu/~woit/QM/qmbook.pdf (and at https://www.math.columbia.edu/~woit/QMbook/). The "revised and expanded, under construction" draft (dated 20 October 2025) is the most recent version and supersedes the 2017 Springer edition for citation purposes. Page numbering used in §2 and §3 of this plan is drawn from the 20 October 2025 draft.
License. Author retains copyright (©2021 Peter Woit, all rights reserved) but hosts the draft freely for educational use. Springer publishes the 2017 edition under standard textbook terms (ISBN 978-3-319-64610-7). For Codex citation use the author's hosted PDF as the primary anchor and cite "Woit, Quantum Theory, Groups and Representations: An Introduction, Springer 2017 (revised draft, Columbia University, hosted free)."
Local copy. Present at reference/fasttrack-texts/02-quantum-stat/Woit-QuantumTheoryGroupsRepresentations.pdf.
Audit completeness. Full. TOC read, preface read, chapters 1–2 sampled directly; remaining chapters audited at TOC granularity. No reduced-flag.

Woit — *Quantum Theory, Groups and Representations* (Fast Track 2.02) — Audit + Gap Plan