Parisi-Wu — *Stochastic Quantization* (Fast Track 2.16) — Audit + Gap Plan

Primary book (secondary text): Mikio Namiki, Stochastic Quantization, Springer Lecture Notes in Physics, Monographs m9 (Springer-Verlag, Berlin / Heidelberg, 1992, x + 217 pp.). ISBN 978-3-540-54724-2 (print); 978-3-540-47217-9 (eBook). DOI 10.1007/978-3-540-47217-9.

Canonical review (co-anchor): Poul H. Damgaard, Helmuth Hüffel, "Stochastic Quantization," Physics Reports 152 no. 5–6 (1987) 227–398. DOI 10.1016/0370-1573(87)90144-X.

Originator paper: G. Parisi, Y.-S. Wu, "Perturbation theory without gauge fixing," Scientia Sinica 24 (1981) 483–496.

Fast Track entry: 2.16. FT 2.16 was catalogued as "Stochastic Quantization — Parisi et al" with source marked ? in docs/catalogs/FASTTRACK_BOOKLIST.md — no definitive book was pinned. This plan adopts Namiki LNP m9 (1992) as the canonical secondary text and Damgaard-Hüffel (1987) as the canonical review co-anchor, since Parisi-Wu themselves published only the short Sci. Sinica paper, not a book. Recorded 2026-05-18.

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 so that the Parisi-Wu stochastic-quantization (SQ hereafter) framework is covered to the FT-equivalence threshold of docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4. The framework is a specific reformulation of Euclidean QFT — modest punch-list expected (~5–8 units), not a full chapter rebuild.

Audit mode: REDUCED. No local PDF of Namiki LNP m9 or of the Damgaard-Hüffel Phys. Rep. review is present under reference/fasttrack-texts/02-quantum-stat/ (only Chatterjee QFT, Sternberg Semi-Classical, Woit QFT-Groups-Reps are on disk). Springer SpringerLink and Elsevier ScienceDirect both gated WebFetch (403 / IdP redirect). Plan is written from the canonical published structure of Namiki LNP m9 and Damgaard-Hüffel 1987 as documented across the secondary literature (Zinn-Justin QFT and Critical Phenomena ch. 17; Rivers Path Integral Methods in QFT §16; Henneaux-Teitelboim Quantization of Gauge Systems §20). Local PDF acquisition is the first P2 follow-up — see §7.

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§1 What the Parisi-Wu framework is for

The Parisi-Wu stochastic-quantization (SQ) programme reformulates Euclidean quantum field theory as the equilibrium distribution of a classical stochastic process in an extra "fictitious" time variable. Given a Euclidean action $S[\varphi ]$ on fields $\varphi (x)$ over ${\mathbb{R}}^d$, introduce an auxiliary parameter $\tau$ ("stochastic time") and let $\varphi (x,\tau )$ evolve under the Langevin equation $${\partial }_{\tau }\varphi (x,\tau )=-\frac{\delta S[\varphi ]}{\delta \varphi (x,\tau )}+\eta (x,\tau ),$$ where $\eta$ is Gaussian white noise with covariance $⟨\eta (x,\tau )\eta (x^{\prime },{\tau }^{\prime })⟩=2{\delta }^d(x-x^{\prime })\delta (\tau -{\tau }^{\prime })$. The associated Fokker-Planck equation has the unique stationary distribution $P_{\mathrm{st}}[\varphi ]=Z^{-1}{e}^{-S[\varphi ]}$ — i.e., the Euclidean path-integral measure. Quantum expectation values are recovered as $\tau \to \infty$ equal-time stochastic averages.

Distinctive contributions, in roughly the order Namiki LNP m9 develops them:

1. Langevin reformulation of Euclidean QFT. Parisi-Wu 1981. The Markov process in fictitious time replaces the Feynman path integral as the definition of the theory. Namiki LNP m9 ch. 2; Damgaard-Hüffel 1987 §2. Distinctive consequences: stochastic perturbation theory has a different graphical structure (rooted trees with stochastic-time propagators) than ordinary Feynman diagrams, but agrees in the $\tau \to \infty$ limit.
2. Fokker-Planck equivalence and equilibrium distribution. The $e^{-S[\varphi ]}/Z$ measure arises as the unique fixed point of the Fokker-Planck operator ${\partial }_{\tau }P={\partial }_{\varphi }[(\delta S/\delta \varphi )P+{\partial }_{\varphi }P]$. Namiki ch. 3; Damgaard-Hüffel §3. Convergence rate controlled by the spectrum of the Fokker-Planck operator; gapped for confining actions, gapless / power-law for free massless theories.
3. Stochastic gauge fixing (Zwanziger). For gauge theories the classical drift $-\delta S/\delta \varphi$ has zero modes along gauge orbits, so the Langevin equation does not relax to a unique distribution. Zwanziger 1981 showed one can add a drift term along gauge orbits that vanishes on physical observables but restores ergodicity. No Faddeev-Popov ghosts or BRST machinery are required: gauge fixing is implemented by a stochastic kernel rather than by gauge slices. Namiki ch. 5; Damgaard-Hüffel §6. This is the single most-cited practical advantage of SQ.
4. Parisi-Sourlas dimensional reduction. Parisi-Sourlas 1979 Phys. Rev. Lett. 43 744. A theory in $d$ dimensions with a quenched random source coupled to $\varphi$ reduces, at the level of correlation functions, to a supersymmetric theory in $d-2$ dimensions. The reduction is the closest pre-SQ analogue of the Langevin-trick: the stochastic noise plays the role of the random source. Namiki ch. 4; Zinn-Justin QFT and Critical Phenomena ch. 17.
5. The Nicolai map. Nicolai 1980 Phys. Lett. B 89 341 / Nucl. Phys. B 176 419. In supersymmetric theories there is a nonlinear change of field variables under which (a) the bosonic action becomes Gaussian and (b) the fermion determinant becomes the Jacobian of the transformation. The Nicolai-map field is exactly the equilibrium variable of a Langevin process with drift given by the superpotential. Stochastic quantization thus gives the most direct route to lattice formulations of SUSY theories. Namiki ch. 6; Damgaard-Hüffel §7.
6. Stochastic regularization. SQ admits a natural UV regulator — smear the noise correlator in fictitious time, $\langle\eta\eta\rangle \to 2 R_\Lambda(\tau - \tau')$with$R_\Lambda$ supported on $|\tau -{\tau }^{\prime }|<{\Lambda }^{-2}$ — that respects gauge invariance without ghosts. Bern-Halpern-Sadun-Taubes 1987 Nucl. Phys. B 284
   1. Namiki ch. 7; Damgaard-Hüffel §5.
7. Numerical stochastic perturbation theory and lattice implementation. The Langevin equation is the foundation of lattice QCD's hybrid-Monte-Carlo / Langevin algorithms (Parisi-Wu Langevin updates, Batrouni-Katz-Kronfeld-Lepage-Svetitsky-Wilson 1985 Phys. Rev. D 32 2736). Namiki ch. 8; this is where SQ leaves the continuum reformulation and becomes a practical numerical tool.
8. Complex-action problem and the sign problem. Generalised Langevin / complex Langevin (Parisi 1983, Klauder 1984) attempts to define $\int e^{-S}$ for actions $S$ taking complex values, where the Euclidean measure $e^{-S}$ is not a probability. Convergence is fragile and still an open problem (Aarts-Seiler-Stamatescu work, 2010s). Namiki ch. 9; pointer level in the Codex.

SQ is not a first introduction to QFT. It assumes Euclidean QFT (Wick-rotated path integrals, scalar ${\varphi }^4$, gauge theories, Yang-Mills), Langevin / Fokker-Planck theory at the level of Risken The Fokker-Planck Equation, and supersymmetric quantum mechanics. It is the canonical entry point if one wants the stochastic / dimensional-reduction angle on QFT — distinct from canonical quantization, path-integral quantization, or BRST-cohomological quantization.

References for §1 (peer-reviewed):

• M. Namiki, Stochastic Quantization, Springer LNP m9 (1992).
• P. H. Damgaard, H. Hüffel, "Stochastic Quantization," Phys. Rep. 152 (1987) 227.
• G. Parisi, N. Sourlas, "Random magnetic fields, supersymmetry, and negative dimensions," Phys. Rev. Lett. 43 (1979) 744.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Clarendon / Oxford 2002), ch. 17 "Stochastic differential equations."

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§2 Coverage table (Codex vs Namiki / Damgaard-Hüffel)

Cross-referenced against the current 313-unit corpus, focusing on 08-stat-mech/ (22 units across partition, mean-field, Onsager, RG, critical, gaussian, path-integral, lattice-gauge, wick) and on the nascent 12-quantum/ chapters. ✓ = covered, △ = partial / different framing, ✗ = not covered.

Namiki / D-H topic	Codex unit(s)	Status	Note
Euclidean path integral $\int \mathcal{D}\varphi e^{-S}$	08.07.01-path-integral	✓	Covered as stat-mech path integral.
Wick rotation $t\to -i\tau$	08.09.01-wick-rotation	✓	Covered.
Gaussian field $\varphi$ and free measure	08.06.01-gaussian-field	✓	Covered.
Lattice gauge action $S_W[U]$	08.08.01-wilson-lattice-gauge, 08.08.02-wilson-action	✓	Covered (Wilson action for SU($N$)).
Renormalisation group flow	08.04.01–08.04.04	✓	Real-space + Wilson-Fisher + $\beta$-function + block-spin.
Langevin equation ${\partial }_{\tau }\varphi =-\delta S/\delta \varphi +\eta$	—	✗	Gap (foundational for SQ).
Gaussian white-noise process; Wiener measure	—	✗	Gap. Codex has no stochastic-process unit at present.
Fokker-Planck equation ${\partial }_{\tau }P={\partial }_{\varphi }[(\delta S/\delta \varphi )P+{\partial }_{\varphi }P]$	—	✗	Gap.
Fluctuation-dissipation relation; detailed balance in field theory	—	✗	Gap.
Parisi-Wu stochastic-quantization theorem (equilibrium $=e^{-S}/Z$)	—	✗	Gap (central theorem).
Stochastic perturbation theory; tree-diagram expansion in $\tau$	—	✗	Gap. Master-tier deepening.
Zwanziger stochastic gauge fixing	—	✗	Gap. Distinctive contribution; no FP-ghost analogue exists yet in Codex either.
Faddeev-Popov ghosts / BRST (for contrast)	—	✗	Gap (separate; relevant only as the thing-SQ-replaces).
Parisi-Sourlas dimensional reduction	—	✗	Gap.
Nicolai map for SUSY theories	—	✗	Gap. Master-tier; pointer to SUSY QM.
Stochastic regularisation (Bern-Halpern-Sadun-Taubes)	—	✗	Gap (low priority — specialist).
Complex Langevin / sign problem	—	✗	Gap (low priority — pointer only).
Numerical Langevin / lattice updates	—	△	Touched in 08.08-lattice-gauge/ (Wilson action) but not the Langevin update rule itself.

Aggregate coverage estimate: SQ-specific content is ~0% covered. The supporting Euclidean / lattice / RG infrastructure is well-covered (roughly the upper half of the table), but the stochastic-process reformulation itself — Langevin, Fokker-Planck, Parisi-Wu theorem, Zwanziger, Parisi-Sourlas, Nicolai — is entirely absent. This is unsurprising: SQ is a specific QFT reformulation, not a foundational chapter, and FT 2.16 was not previously sourced.

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

Priority 0 — prerequisite gap to flag: The Codex has no stochastic-process unit anywhere (Langevin / Wiener / Fokker-Planck / Ornstein-Uhlenbeck). Adding at least an Ornstein-Uhlenbeck + Fokker-Planck pair is a hard prerequisite for any SQ unit. These belong in 02-analysis/12-ode/ (extending the existing Lyapunov-stability unit 02.12.08) or as a new 02-analysis/13-stochastic/ mini-chapter. Flag for the analysis-chapter lead; not produced by this audit.

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

1. 08.10.01 Langevin equation for fields and Gaussian white noise. Definition of ${\partial }_{\tau }\varphi =-\delta S/\delta \varphi +\eta$ with $⟨\eta \eta ⟩=2{\delta }^d\delta$. Standard examples: free scalar, ${\varphi }^4$, harmonic-oscillator analogue (finite-dim). Anchor: Namiki ch. 2; Damgaard-Hüffel §2; Zinn-Justin QFT and Critical Phenomena §17.1. Three-tier, ~1500 words. Foundational for the next four units.
2. 08.10.02 Fokker-Planck equation and equilibrium distribution. Derivation of ${\partial }_{\tau }P={\partial }_{\varphi }[(\delta S/\delta \varphi )P+{\partial }_{\varphi }P]$ from the Langevin equation; spectrum and gap; the $P_{\mathrm{st}}[\varphi ]=Z^{-1}{e}^{-S[\varphi ]}$ identification. Anchor: Namiki ch. 3; Damgaard-Hüffel §3. Three-tier; Master section covers the spectral-gap argument for convergence rate.
3. 08.10.03 Parisi-Wu stochastic-quantization theorem. Statement: Euclidean correlators $⟨\varphi (x_1)\cdots \varphi (x_n)⟩$ equal $\tau \to \infty$ equal-time stochastic averages of the Langevin process. Proof sketch via Fokker-Planck. Stochastic perturbation theory and rooted-tree expansion at Master tier. Originator citation: Parisi-Wu 1981 Sci. Sinica 24 483. Anchor: Namiki ch. 2–3; Damgaard-Hüffel §2–4. Three-tier; Beginner tier states only the equivalence, no proof.
4. 08.10.04 Stochastic gauge fixing (Zwanziger). The drift kernel that restores ergodicity for Yang-Mills without Faddeev-Popov ghosts. Originator: D. Zwanziger, "Covariant quantization of gauge theories in the function-space formulation," Nucl. Phys. B 192 (1981) 259. Anchor: Namiki ch. 5; Damgaard-Hüffel §6. Intermediate
   • Master; Beginner tier deferred (gauge-theory prerequisites too heavy). Note: depends on a future gauge-theory unit covering Yang-Mills classically; flag prereq.
5. 08.10.05 Parisi-Sourlas dimensional reduction. Statement of the theorem ($d$-dim theory with random source $\equiv$ SUSY theory in $d-2$ dim at the level of correlation functions); the canonical ${\varphi }^3$ example. Originator: Parisi-Sourlas 1979 Phys. Rev. Lett. 43 744. Anchor: Namiki ch. 4; Zinn-Justin §17.5. Three-tier; Beginner tier gives only the statement and the random-field-Ising motivation.

Priority 2 — SUSY-connection deepening:

6. 08.10.06 Nicolai map and SUSY field theories. Statement: nonlinear field redefinition trivialising the bosonic action in SUSY theories; identification with the Langevin equilibrium variable. Originator: H. Nicolai, "On a new characterization of scalar supersymmetric theories," Phys. Lett. B 89 (1980) 341, and "Supersymmetry and functional integration measures," Nucl. Phys. B 176 (1980) 419. Anchor: Namiki ch. 6; Damgaard-Hüffel §7. Master-only, ~1500 words. Pointer to SUSY-QM unit (which itself doesn't yet exist; flag prereq).

Priority 3 — applications / specialist (Master-tier, not strictly required for FT equivalence):

7. 08.10.07 Stochastic regularisation in gauge theories. Bern-Halpern-Sadun-Taubes regulator; UV cutoff in fictitious time preserving gauge invariance. Anchor: Namiki ch. 7; Damgaard-Hüffel §5. Master-only.
8. 08.10.08 Langevin updates and lattice numerics. The Parisi-Wu Langevin update as a Monte-Carlo algorithm; relation to hybrid-Monte-Carlo. Anchor: Namiki ch. 8; Batrouni-et-al 1985 Phys. Rev. D 32 2736. Could be added as a Master section to 08.08.01-wilson-lattice-gauge rather than as a new unit.

Priority 4 — pointer (optional, Master-only):

9. 08.10.09 Complex Langevin and the sign problem. Pointer unit; statement of the complex-Langevin proposal and the convergence-failure problem; reference to Aarts-Seiler-Stamatescu work as the modern follow-up. Anchor: Namiki ch. 9.

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§4 Implementation sketch (P3 → P4)

For a full SQ coverage pass, items 1–5 are the minimum set, with an Ornstein-Uhlenbeck / Fokker-Planck prerequisite in 02-analysis/. Realistic production estimate (mirroring earlier Brown / Lawson- Michelsohn batches):

• ~3–4 hours per unit. SQ units skew slightly higher than the corpus average because the master tier requires careful stochastic-calculus notation (Itô vs Stratonovich), and several units cite multiple originator papers.
• 5 priority-1 units × ~3.5 hours = ~17–18 hours of focused production.
• 1 priority-2 unit (Nicolai map) × ~4 hours = ~4 hours.
• 2 priority-3 units × ~2 hours each (Master-only, shorter) = ~4 hours.
• 1 priority-4 pointer × ~1.5 hours = ~1.5 hours.
• Total: ~27 hours focused production for full FT-equivalence coverage. Fits a focused 4–5 day window.

Prereq dependencies that must be flagged separately to the analysis-chapter and quantum-chapter leads:

• 02-analysis/13-stochastic/ (or expansion of 02.12-ode/): Brownian motion / Wiener measure; Itô calculus; Fokker-Planck for finite-dim systems. ~3 units, ~10 hours. Hard prereq for priority-1.
• Yang-Mills classical-field unit (currently absent from the Codex). Hard prereq for priority-1 item 4 (Zwanziger).
• SUSY quantum mechanics unit (currently absent). Hard prereq for priority-2 item 6 (Nicolai map).

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, originator-prose treatment should cite:

• G. Parisi, Y.-S. Wu, "Perturbation theory without gauge fixing," Sci. Sinica 24 (1981) 483 — originating the Langevin reformulation of QFT.
• D. Zwanziger, "Covariant quantization of gauge theories in the function-space formulation," Nucl. Phys. B 192 (1981) 259 — originating stochastic gauge fixing.
• G. Parisi, N. Sourlas, "Random magnetic fields, supersymmetry, and negative dimensions," Phys. Rev. Lett. 43 (1979) 744 — originating dimensional reduction; parallel framework that the SQ literature folds in.
• H. Nicolai, Phys. Lett. B 89 (1980) 341 — originating the Nicolai map.
• P. H. Damgaard, H. Hüffel, "Stochastic Quantization," Phys. Rep. 152 (1987) 227 — canonical review consolidating the framework.

Notation crosswalk. Namiki uses $t$ for fictitious / stochastic time; Damgaard-Hüffel and Zinn-Justin use $\tau$ (or $s$). The Codex already uses $\tau$ for Euclidean Wick-rotated time in 08.09.01-wick-rotation, which is a different variable than SQ's fictitious time. Recommend: use $\tau$ for Wick-rotated Euclidean time (unchanged) and reserve $s$ (or ${\tau }_{\mathrm{SQ}}$, on first use) for the SQ fictitious-time variable to prevent collision. Record this in a §Notation paragraph of 08.10.01 and 08.10.03. The white-noise covariance convention is $⟨\eta \eta ⟩=2\delta \delta$ (Damgaard-Hüffel); some sources (Namiki) drop the factor of 2 and rescale the drift. The Codex should adopt the Damgaard-Hüffel convention and note the rescaling at first use.

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§5 What this plan does NOT cover

• Connection between SQ / Nicolai map and spontaneous SUSY breaking. Active research area but tangential to FT 2.16 coverage. Defer.
• Witten 1982 J. Diff. Geom. connection ("Supersymmetry and Morse theory"). Same Langevin-like structure appears in Witten's Morse theory; that connection belongs in a Morse-theory unit (milnor-morse-theory.md punch-list), not here.
• Stochastic quantization in curved spacetime / cosmology. Pointer level only; defer.
• A line-number-level inventory of every named theorem in Namiki LNP m9 and in Damgaard-Hüffel. Full P1 audit deferred; this pass works from the canonical chapter structure.
• Exercise-pack production. Neither Namiki nor Damgaard-Hüffel includes formal exercise sets; a Codex exercise pack would have to be constructed from scratch and is a P3-priority-3 follow-up.
• Complex Langevin convergence theory beyond a pointer.
• Lattice numerical implementation beyond a Master-section in 08.08.01-wilson-lattice-gauge.

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§6 Acceptance criteria for FT equivalence (Parisi-Wu / SQ)

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

• The analysis-chapter stochastic-process prereqs have shipped (strict prereq): Wiener measure + Fokker-Planck at finite dim.
• ≥95% of Namiki LNP m9's named theorems in chapters 2–6 map to Codex units (currently 0%; after priority-1 units this rises to ~80%; after priority-1+2 to ~95%).
• ≥90% of Damgaard-Hüffel §2–7 results (the foundational core of the review) have a direct unit or are referenced from a unit that covers them.
• Notation decisions are recorded (see §4).
• Pass-W weaving connects the new 08.10.* units to 08.07.01-path-integral, 08.09.01-wick-rotation, 08.08.01-wilson-lattice-gauge, and forward to the SUSY-QM unit once it exists.

The 5 priority-1 units close most of the equivalence gap given the analysis-chapter stochastic prereqs are in place. Priority-2 closes the SUSY-connection (Nicolai) gap. Priority-3+4 are deepenings.

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§7 Sourcing

• Namiki LNP m9 (1992). Not free. Published by Springer; available through SpringerLink institutional access (https://link.springer.com/book/10.1007/978-3-540-47217-9; DOI 10.1007/978-3-540-47217-9) and via Library Genesis as scanned PDF. Not currently on disk under reference/fasttrack-texts/02-quantum-stat/. P2 acquisition follow-up: purchase or institutional download; add to reference/fasttrack-texts/02-quantum-stat/Namiki-StochasticQuantization-LNPm9.pdf to mirror the existing Chatterjee / Sternberg / Woit layout.
• Damgaard-Hüffel (1987) Phys. Rep. 152. Not free. Elsevier Physics Reports; DOI 10.1016/0370-1573(87)90144-X. Some early-1990s preprints of substantively the same content circulate on author pages; the published review itself is paywalled. P2 acquisition follow-up: institutional download; add to reference/fasttrack-texts/02-quantum-stat/Damgaard-Huffel-StochasticQuantization-PhysRep1987.pdf.
• Parisi-Wu 1981 Sci. Sinica. Original paper; scanned PDFs circulate. Cite by reference but no need for a local copy beyond the originator-prose snippet.
• Parisi-Sourlas 1979 Phys. Rev. Lett. 43. APS-paywalled original; an arXiv-equivalent does not exist (pre-arXiv). Citation only.
• Nicolai 1980 Phys. Lett. B 89 / Nucl. Phys. B 176. Elsevier- paywalled originals. Citation only.
• Zinn-Justin, QFT and Critical Phenomena 4th ed. Already on the Codex's "BUY" list for QFT coverage generally; ch. 17 is a strong free-standing secondary anchor for the Langevin / Fokker-Planck side and would let SQ priority-1 units be drafted without waiting for Namiki acquisition. Recommend prioritising Zinn-Justin acquisition first if budget forces a choice.
• License note. All anchor sources are commercial / paywalled. No author-hosted free PDF exists for any of them, in contrast to e.g. Brown's NAT. The Damgaard-Hüffel review is the most-quoted source in the SQ literature and is the most important acquisition.

Audit-mode note. This plan is marked REDUCED. The chapter-by- chapter punch-list maps onto Namiki LNP m9's standard structure (eight chapters: Langevin / Fokker-Planck / perturbation theory / Parisi-Sourlas / gauge fixing / Nicolai / regularisation / numerics / complex Langevin) and onto Damgaard-Hüffel §2–7. Once local PDFs are acquired, a follow-up P1-full audit should verify section numbers and named-theorem counts. Adjustments expected to be cosmetic (section-number corrections, possible reordering of priority-3 vs priority-4 items).