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arxiv: 2603.10761 · v2 · submitted 2026-03-11 · 🧮 math-ph · hep-th· math.MP· math.PR

Recognition: 2 theorem links

· Lean Theorem

From path integral quantization to stochastic quantization: a pedestrian's journey

Dario Benedetti , Ilya Chevyrev , Razvan Gurau
Authors on Pith no claims yet

Pith reviewed 2026-05-15 12:48 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MPmath.PR
keywords path integral quantizationstochastic quantizationEuclidean quantum field theoryconstructive field theoryFeynman expansionTaylor interpolationspanning forests
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0 comments X

The pith

Path integral and stochastic quantizations are equivalent for generic scalar Euclidean quantum field theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives two new proofs that path integral quantization and stochastic quantization produce the same results for generic scalar Euclidean quantum field theories. Both proofs use Taylor interpolations indexed by forests, following the style of constructive field theory. One proof checks the equivalence term by term inside the Feynman expansion, where the forests act as spanning forests on the graphs. The other proof works directly with the path integral measure and does not require expanding the full perturbation series. A sympathetic reader cares because the result unifies two standard but technically distinct ways of defining the same quantum field theories.

Core claim

The central claim is that path integral quantization and stochastic quantization of generic scalar Euclidean quantum field theories are equivalent. The proofs are constructed via Taylor interpolations indexed by forests: the first at the level of individual terms in the Feynman series, with forests appearing as spanning forests in the graphs, and the second directly at the level of the path integral without performing the full perturbative expansion.

What carries the argument

Taylor interpolations indexed by forests, used to interpolate between the two quantizations while controlling the difference term by term or at the integral level.

If this is right

  • Equivalence holds separately for every term in the Feynman expansion.
  • Equivalence can be established without expanding the perturbation series at all.
  • The same forest interpolation technique applies uniformly to generic scalar theories.
  • Results proven in one quantization scheme transfer immediately to the other.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The forest method might extend to show equivalence with other quantization procedures such as canonical or lattice approaches.
  • Practical computations could choose whichever quantization is easier for a given observable and import the result to the other framework.
  • Similar interpolations could be used to relate different regularization schemes inside a single quantization method.

Load-bearing premise

The Taylor interpolations indexed by forests from constructive field theory apply to generic scalar Euclidean quantum field theories without extra regularity conditions.

What would settle it

Compute a low-order correlation function in a specific scalar theory such as phi^4 in four dimensions and check whether the path-integral and stochastic versions differ beyond the order allowed by the forest interpolation.

Figures

Figures reproduced from arXiv: 2603.10761 by Dario Benedetti, Ilya Chevyrev, Razvan Gurau.

Figure 1
Figure 1. Figure 1: A combinatorial map with half-edges D = {r, v1 , v2 , v3 , v4 , w1 , w2 , w3 , u1 , u2}, root r, and permutations, written in cycle notation, σ = (r)(v 1 v 2 v 3 v 4 )(w 1w 2w 3 )(u 1u 2 ) and α = (rv1 )(v 2w 1 )(v 3u 1 )(v 4w 3 )(u 2w 2 ). The root r is a fixed point (r) of the permutation σ. The cycles of the permutation σ, denoted (h, σ(h), σ2 (h), . . .), are the vertices of the combinatorial map G, th… view at source ↗
Figure 2
Figure 2. Figure 2: Unlabeled rooted map corresponding to the map in Fig. 1. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of unlabeled abstract graphs rooted at the external vertex on the left. For [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Unlabeled rooted maps with up to three 3-valent vertices and the distinguished “keep [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Recursive trees with up to four non-root vertices, where we have added a root [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Unlabeled rooted combinatoiral trees with up to four non-root vertices. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Rooted plane trees with up to four non-root vertices. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Ternary trees. We have grouped together the ternary trees that differ only by the [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A recursive tree T and its set of branches. The branches B13 and B25 are isomorphic as rooted trees, but they are distinct as generalized recursive trees and as elements of Br(T). 1 3 4 2 5 1 2 3 4 α1 B13 B4 B25 α1(B13) = 1, α1(B4) = 2, α1(B25) = 4 1 3 4 2 5 1 2 3 4 α2 B13 B4 B25 α2(B13) = 4, α2(B4) = 3, α2(B25) = 1 [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Two example injections α1, α2 ∈ I4(T). There are in total 4 · 3 · 2 = 4! 1! such injections. We claim that Φ(t0,x1) · · · Φ(t0,xn) = X q≥0 X T ∈T |V (T)|=q+1 dT (0)≤n X α∈In(T) An(T, α) Yn ν=1 ϕ 0 ν xν  . (6.8) Before proving (6.8), we note that it implies Eq. (6.6). Indeed, for a fixed tree T with dT (0) = m, summing over α ∈ In(T) corresponds to choosing which of the m root derivatives acts on which f… view at source ↗
Figure 11
Figure 11. Figure 11: The top diagram shows an example of (T, α, S) for n = 4 with (T, α) taken from the left diagram in [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Feynman graphs at zero, first and second order contributing to the two point [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Triple tadpoles. At next order we have triple tadpoles (represented in [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Tadpole-sunset graphs and a new primitive graph. [PITH_FULL_IMAGE:figures/full_fig_p037_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Zero order contribution from the mating of the two trees with no internal vertex. [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The connected first order contributions. [PITH_FULL_IMAGE:figures/full_fig_p039_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Mated trees leading to sunset graphs. Sunset contributions to [PITH_FULL_IMAGE:figures/full_fig_p040_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The one-edge graph. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Taylor interpolation of tadpoles. This is the inverse of what depicted in Fig. 16. [PITH_FULL_IMAGE:figures/full_fig_p042_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Taylor interpolation for the sunset graph. [PITH_FULL_IMAGE:figures/full_fig_p042_20.png] view at source ↗
read the original abstract

We give two novel proofs that the path integral and stochastic quantizations of generic scalar Euclidean quantum field theories are equivalent. Our proofs rely on Taylor interpolations indexed by forests, in the fashion of constructive field theory. The first proof works at the level of individual terms in the Feynman expansion, with the forests appearing as spanning forests in Feynman graphs. The second one works at the level of the path integral and avoids the full expansion of the Feynman perturbation series.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to provide two novel proofs that the path integral and stochastic quantizations of generic scalar Euclidean quantum field theories are equivalent. The proofs rely on Taylor interpolations indexed by forests in the style of constructive field theory. The first proof equates the quantizations term-by-term in the Feynman expansion by identifying forests as spanning forests in graphs; the second works directly at the level of the path integral without expanding the full perturbative series.

Significance. If the proofs are valid for the stated class of theories, the work would offer a rigorous bridge between two standard quantization methods, with the direct path-integral proof providing a potentially useful non-perturbative route. The explicit use of forest-indexed expansions from constructive QFT is a methodological strength that could aid future equivalence checks or calculations.

major comments (2)
  1. [Abstract] Abstract: the claim of equivalence for 'generic scalar Euclidean quantum field theories' is not supported by stated regularity conditions; forest Taylor interpolations are known to require dimension-specific bounds and analyticity or small-coupling assumptions that are controlled only for polynomial potentials in d=2,3, yet no such restrictions or convergence radii are provided.
  2. [First proof section] First proof (Feynman-graph level): the identification of forests as spanning forests in graphs equates the measures only if the Taylor remainder is controllable term-by-term, but the manuscript supplies no explicit estimate showing that the remainder vanishes or is bounded for non-polynomial interactions outside the constructive regime.
minor comments (2)
  1. The informal title 'a pedestrian's journey' is atypical for a journal article; a more descriptive title would better reflect the technical content.
  2. Notation for the forest-indexed interpolations could be introduced with a short preliminary subsection to aid readers unfamiliar with constructive-field-theory conventions.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight important points about the scope of our claims and the rigor of the estimates, which we address point by point below. We will revise the manuscript to clarify assumptions and limitations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of equivalence for 'generic scalar Euclidean quantum field theories' is not supported by stated regularity conditions; forest Taylor interpolations are known to require dimension-specific bounds and analyticity or small-coupling assumptions that are controlled only for polynomial potentials in d=2,3, yet no such restrictions or convergence radii are provided.

    Authors: We agree that the abstract's reference to 'generic' scalar Euclidean QFTs is too broad without explicit regularity conditions. The forest-indexed Taylor interpolations follow the constructive field theory approach and are rigorously controlled for polynomial potentials in d=2,3 under standard bounds. In the revised manuscript we will update the abstract, introduction, and a new dedicated subsection to state the precise assumptions (polynomial interactions, d=2,3, small-coupling or analyticity conditions as needed) and briefly discuss convergence radii. This will align the stated scope with the proofs. revision: yes

  2. Referee: [First proof section] First proof (Feynman-graph level): the identification of forests as spanning forests in graphs equates the measures only if the Taylor remainder is controllable term-by-term, but the manuscript supplies no explicit estimate showing that the remainder vanishes or is bounded for non-polynomial interactions outside the constructive regime.

    Authors: The first proof identifies the forest structures arising from the Taylor interpolation with spanning forests of the Feynman graphs, thereby equating the two sides term by term within the perturbative expansion. We acknowledge that the manuscript does not supply new explicit remainder estimates for interactions outside the polynomial constructive regime. We will add a clarifying remark after the proof stating that the term-by-term identification holds whenever the Taylor remainders are known to be controllable (as established in the constructive literature for polynomial cases), and that extension to general non-polynomial interactions would require additional estimates not derived here. revision: partial

standing simulated objections not resolved
  • Explicit estimates showing that the Taylor remainder vanishes or is bounded for non-polynomial interactions outside the constructive regime

Circularity Check

0 steps flagged

No significant circularity; proofs derive equivalence from interpolation methods.

full rationale

The manuscript presents two novel proofs that equate path-integral and stochastic quantizations of generic scalar Euclidean QFTs by applying forest-indexed Taylor interpolations in the style of constructive field theory. The first proof operates term-by-term on Feynman graphs with spanning forests; the second works directly at the path-integral level without full perturbative expansion. These steps import standard interpolation techniques as external tools rather than defining the target equivalence into the inputs or fitting parameters to the result. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain appears; the central claim remains independent of the paper's own fitted values or prior results by the same authors. The derivation is therefore self-contained against external benchmarks in constructive QFT.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on techniques from constructive field theory without introducing new free parameters or entities in the abstract.

axioms (1)
  • domain assumption Taylor interpolations indexed by forests apply to generic scalar Euclidean QFTs as in constructive field theory
    Invoked to structure the proofs at Feynman and path integral levels.

pith-pipeline@v0.9.0 · 5370 in / 1122 out tokens · 36945 ms · 2026-05-15T12:48:34.346096+00:00 · methodology

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