Pith. sign in

REVIEW 3 major objections 5 minor 45 references

Renormalising Feynman diagrams with multi-indices

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Multi-index BPHZ renormalisation produces an explicit formula for the renormalised measure's shifted couplings.

desk verdict A serious multi-index Hopf algebra for BPHZ renormalisation; the equivalence theorems are convincing, but the main theorem's proof has an unproved multiplicativity step that must be fixed. read the letter →

arxiv 2501.08151 v3 pith:27LYVEZ2 submitted 2025-01-14 math-ph math.MPmath.PRmath.RA

Yvain Bruned , Yingtong Hou This is my paper · ORCID
classification math-phmath.MPmath.PRmath.RA MSC 81T1581T1816T05
keywords BPHZrenormalisationmulti-indicesFeynmandiagramsHopfalgebraextraction-contractioncoproductrenormalisedmeasurePhi-4cumulantexpansion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the BPHZ renormalisation of a quantum field theory measure can be carried out without drawing Feynman diagrams: it is enough to renormalise a 'multi-index', a monomial that counts how many vertices of each arity appear in the diagram. The central result is that this multi-index renormalisation is equivalent to the usual diagrammatic BPHZ renormalisation, and that applying it to the cumulant expansion of a partition function reproduces the partition function of the same Hermite-polynomial measure with shifted couplings, with an explicit formula for the shifts. A sympathetic reader would care because finding the renormalised measure, the shifts in the Lagrangian that absorb divergences, is normally a hard combinatorial problem, and this paper reduces it to algebra on vertex counts. The construction is illustrated on the $\Phi^4$ measure, where the known mass shift and vacuum term come out of the formula.

What carries the argument

The machinery is the extraction-contraction coproduct $\Delta_m$ on the algebra of multi-indices, with each multi-index $z^\beta = \prod_k z_k^{\beta(k)}$ recording the multiplicities of vertices of each arity. It is the adjoint, under inner products weighted by the symmetry factors $S_m(z^\beta) = \prod_k \beta(k)!(k!)^{\beta(k)}$ and $S_f(\Gamma)=|\mathrm{Aut}(\Gamma)|$, of a simultaneous insertion product $\star_m$, built from an insertion product $z^\beta \blacktriangleright z^\alpha = \sum_k (D^k z^\beta)(\partial_{z_k} z^\alpha)$ with $D=\sum_k z_{k+1}\partial_{z_k}$. Two structural identities carry the argument: the counting map $\Phi$ from Feynman diagrams to multi-indices is a morphism for insertions, $\Phi(\Gamma_1 \star_f \Gamma_2) = \Phi(\Gamma_1)\star_m \Phi(\Gamma_2)$, and its adjoint $P$ lifts a multi-index to the sum of all diagrams formed by pairing its half-edges, with $S_m = N S_f$ counting the pairings.

What would settle it

Compute, for the rule $\{2,4\}$ and the Green's function of $\Phi^4_3$, the coefficient of a fixed multi-index, say $z_3^2 z_4$ or a four-loop diagram, on both sides of $(P\otimes P)\Delta_m z^\beta = \Delta_f P z^\beta$; any disagreement in the symmetry-factor-weighted sums would falsify Theorem 4.11, and hence the shift formula. A direct numerical check would compare the $\gamma_k$ from Theorem 4.13 with a forest-formula BPHZ computation of the renormalised cumulant at the same order.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 4.13: for a model with a rule $R$, the multi-index BPHZ renormalisation $\hat M_m = (\Pi_m A_m \otimes \mathrm{id})\Delta_m^-$ acts on the formal exponential $\exp(-\int \sum_{k\in K(R)} \alpha_k H_k(X(x))\,dx)$ as the identity on the polynomial form, producing $\exp(-\int \sum_{k\in K(R)\cup\{0\}} (\alpha_k+\gamma_k) H_k(X(x))\,dx)$ with explicit coefficients $\gamma_k = -\sum_{z^{\check\delta}\in \check M_{R,k}}\sum_{z^\delta\in M^-} \frac{\Pi_m A_m(z^\delta) \langle D^k z^\delta, z^{\check\delta}\rangle}{k!\, \hat S_m(z^{\check\delta}) S_m(z^\delta)} \Upsilon^\alpha_m[z^{\check\delta}]$. Theorems 4.10 and 4.11 establish the route to this formula: $\Pi_m \hat M_m = \Pi_f \hat M_f P$, i.e. renormalising the multi-index and then lifting to Feynman diagrams by pairing half-edges gives the same answer as BPHZ-renormalising the diagram itself.

Load-bearing premise

The equivalence and the explicit shift formula are proved only for renormalisation that subtracts the bare divergent value of each subgraph, not its derivatives or higher Taylor terms; if a model needs derivative counterterms, the main theorems do not directly apply.

Editorial extensions

If this is right

  • If Theorem 4.13 is correct, the renormalised $\Phi^4$ measure is obtained with no diagram-by-diagram bookkeeping: the only shift in the interaction is $\gamma_2 = 8\alpha^2 \Pi_m(z_2^3)$ for the mass term, with $\gamma_4=0$ and a vacuum contribution $\gamma_0 = \alpha^2 \Pi_m(z_2^4)/2 - \alpha^3 \Pi_m(z_3^4)/6$.
  • The identity $\Pi_m \hat M_m = \Pi_f \hat M_f P$ means any valuation on Feynman diagrams that factors through the pairing of half-edges can be renormalised by the same algebraic recipe, so the method extends beyond vacuum diagrams of a single kernel to decorated and oriented settings.
  • The group structure on the BPHZ characters of multi-indices gives a composition law for renormalisation schemes, so changing the rule or the degree cutoff corresponds to a convolution product on the vertex-counting algebra.
  • Because the formula for $\gamma_k$ is explicit and built from vertex multiplicities, it avoids solving the coupled system of equations that the diagrammatic cumulant expansion would impose on the shifted couplings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The pairing-counting identity $S_m(\Phi(\Gamma)) = N(\Gamma) S_f(\Gamma)$ is, in effect, a labelled-versus-unlabelled enumeration of pairings; it could be read as a generating-function statement for configuration models of random graphs, where the same 'pair half-edges of prescribed degrees' combinatorics appears.
  • The paper restricts to subtraction of the divergent value only. A natural testable extension is to decorate multi-indices with monomials and derivatives, as sketched in the introduction, and check whether the equivalence $\Pi_m \hat M_m = \Pi_f \hat M_f P$ survives with the same symmetry factors; the paper does not prove this.
  • The explicit $\gamma_k$ formula might be used as a computational shortcut in perturbative calculations: rather than enumerate forests of divergent subgraphs, one evaluates a fixed sum over multi-indices, which could be implemented symbolically for arbitrary order in $\alpha$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces a Hopf algebra on multi-indices for BPHZ renormalisation, with an extraction-contraction coproduct and a simultaneous insertion product, and a symmetry factor chosen so that the counting map and its adjoint are dual. It proves an equivalence between the multi-index renormalisation \hat M_m and the Feynman-diagram BPHZ renormalisation \hat M_f (Theorems 4.10 and 4.11), and then claims that the BPHZ-renormalised cumulant expansion is the cumulant expansion of a renormalised measure of the same polynomial form with couplings \alpha_k+\gamma_k, with an explicit formula for \gamma_k (Theorem 4.13). The construction is illustrated by the \Phi^4_3 measure.

Significance. The result is significant if correct: it provides a diagram-free algebraic route to the renormalised measure and fills a gap between pre-Feynman diagrams and Feynman-diagram renormalisation. The symmetry-factor setup is original and the main equivalence theorems, if fully proved, would be a useful bridge between multi-index renormalisation for SPDEs and the Connes\textendash Kreimer programme for QFT. The paper is commendably explicit about its scope restriction to renormalisation without higher-order Taylor terms. The main theorems are not empirically fitted; they are derived from definitional adjoint relations, so there is no circularity. However, the proof of the central Theorem 4.13 contains a serious gap, and Theorem 4.7 relies on an unproved adaptation of literature results. These need to be repaired before the claims can be accepted.

major comments (3)
  1. [Section 4.2, proof of Theorem 4.13] The assertion that \hat M_m^* is multiplicative, introduced with 'One can easily observe, due to the multiplicativity of \hat S_m(\cdot)/S_m(\cdot)', is not proved and is not an immediate consequence of the displayed formula. The formula for \hat M_m^* z^\theta is a double sum over divergent forests and target monomials; factorising it as \prod_k (\hat M_m^* z_k)^{\theta(k)} requires a nontrivial compatibility between the simultaneous insertion product \bar\star_{mR} and the monomial product z^\theta, together with the S-factor weights. This is load-bearing: without multiplicativity, the conclusion that the renormalised exponential is the exponential of a shifted polynomial measure collapses. Please provide a complete proof or a detailed combinatorial verification, or a counterexample if false.
  2. [Section 4.1, Theorem 4.7] The proof says 'the main idea is firstly combining and adapting the proofs of Lemma 12 and Lemma 14 in [42]' but does not reproduce the adaptation. The adjoint relation between \Delta_f and \star_f is the key step in Theorem 4.11 and hence in Theorem 4.10. In particular, the counting of automorphism classes for repeated \Gamma_i in a forest, and the passage from single insertion to simultaneous insertion via the Guin\textendash Oudom construction, need to be written out. As it stands, the proof is a sketch, not a verification.
  3. [Section 4.2, degree restrictions after the bullet list] The paragraph 'It can be verified that all the properties and theorems in the previous chapters are still valid as long as this degree restriction described above is put accordingly...' is an assertion without proof. Since the twisted antipode \mathcal A_m and the equivalence theorems rely on the coproduct and insertion product vanishing on the complement of the negative-degree spaces, this verification is necessary. Please provide it, or at least state the precise conditions under which the adjointness and morphism properties survive the restriction.
minor comments (5)
  1. [Abstract and Sections 1.2, 2.2] The abstract should state the restriction to counterterms without higher-order Taylor terms; as written, the abstract claims a general method for the renormalised measure in QFT, while Sections 1.2 and 2.2 explicitly limit the results to the no-higher-order case. This is a clarity issue, not a correctness issue.
  2. [Throughout] There are several typos and small infelicities: 'Hemite' in Section 3.1, 'digrams' in Section 1.2, 'Propisition' in the references, and a duplicated 'the' in the introduction. A careful proofreading pass is needed.
  3. [Section 5.3] In the example, expressions such as '21035' and '4 × 4!3' are ambiguous; they should be typeset with explicit multiplication signs (e.g., 210 \cdot 3^5 and 4 \cdot (4!)^3) to avoid confusion.
  4. [Theorem 4.13] The space \check M_{R,k} is defined tersely; the arity condition and the role of the rule R in the summation should be spelt out more fully for readability.
  5. [End of Section 4.2, after Remark 4.14] The paragraph claiming a Hopf algebra on \langle M^-\rangle and a coaction identity says 'One can see easily'; this is another sketch that should be expanded or replaced by a precise reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the multi-index renormalisation is built from definitions and adjointness rather than fitted to the target, and the equivalence theorems are proved internally; the only caveat is an unproved multiplicativity assertion inside Theorem 4.13, which is a proof gap rather than a circular reduction.

full rationale

The paper's construction is a definitional algebraic derivation, not a fit: the symmetry factors Sm and Sf, the counting map Phi and its adjoint P are set up so that P reproduces the half-edge pairing coefficients (Proposition 4.3), and Theorems 4.10 and 4.11 are proved from adjointness and the morphism property of the insertion products rather than assumed. Corollary 4.5 and Lemma 4.12 are internal rewritings of the cumulant expansion with the multi-index valuation defined to match the Feynman diagram valuation, so they do not smuggle the conclusion into the input. The one load-bearing unchecked assertion is in the proof of Theorem 4.13 (Section 4.2): 'One can easily observe, due to the multiplicativity of S_m(.)/S_m(.), that M*_m is multiplicative.' The displayed adjoint expression for M*_m does not visibly factor over monomials, and the multiplicativity claim is equivalent to the theorem's polynomial-form conclusion; the paper does not supply the requested proof, and Remark 4.14's coaction identity is also stated with 'one can see easily.' This is an omitted proof or correctness risk, not a circular reduction, because the multiplicativity is asserted rather than assumed as a premise and the preceding equivalence theorems do not depend on it. The only self-citation, [9] for the explicit multi-index coproduct, is backed by a proof in Proposition 3.7 and is not load-bearing. The Phi^4_3 example is checked against the external result of Berglund (Theorem 5.2.4 in [7]) and agrees, which is independent support. No fitted parameter is relabelled as a prediction and no uniqueness theorem is imported from the authors' own prior work.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces new mathematical structures, such as the multi-index Hopf algebra, the maps Phi and P, and the products and coproducts, but these are definitional constructions, not postulated physical entities with independent evidence requirements. There are no free parameters fitted to data: the counterterms in the example, beta and gamma_0, are computed analytically, not adjusted. The axioms listed are the domain restrictions of the Gaussian setting, the degree-based divergence criterion, the no-higher-order-Taylor simplification, and the assumption that the renormalised measure lies in the same polynomial class, plus standard background theorems.

assumptions (6)
  • domain assumption Gaussian free field setting: ill-posed products are replaced by Wick powers or Hermite polynomials, and covariance kernels are single symmetric Green's functions.
    Section 1.2 and Lemma 3.1; this sets the class of measures and diagrams under study.
  • domain assumption Divergence detection is fully captured by the degree map deg Gamma = ell |E| + d(|V|-1); only the valuation of divergent subgraphs is subtracted, i.e. renormalisation without higher-order Taylor terms.
    Section 2.2 and Section 1.2: 'we restrict ourselves in this paper to a renormalisation without higher-order Taylor series terms'.
  • domain assumption The space M is restricted to multi-indices that can be realised as images of Feynman diagrams under the counting map Phi; other multi-indices have zero valuation.
    Section 3.1: 'We also require multi-indices in the space M to be able to form meaningful Feynman diagrams by pairing free legs.' The restriction is stated for clarity, with valuation zero for the rest.
  • standard math The Guin-Oudom construction provides the adjoint of the extraction-contraction coproduct, namely simultaneous insertion; the universal property of the pre-Lie enveloping algebra is taken from the literature.
    Used in Definition 2.6 and Proposition 3.5 and Theorem 4.11; references are provided.
  • standard math The Linked Cluster Theorem, i.e. the cumulant expansion projects to connected diagrams, is assumed as a background theorem.
    Used in Lemma 4.12; references are provided.
  • domain assumption The renormalised measure is assumed to remain in the same class of Lagrangians, with the same vertex arities plus a possible vacuum term gamma_0; the BPHZ renormalised cumulants are matched by shifting alpha_k to alpha_k + gamma_k.
    Theorem 4.13 assumes the renormalised measure has the form exp of minus the integral of a sum over k in K(R) union {0} of (alpha_k + gamma_k) H_k; this is a modelling assumption that the counterterms are local in the same vertices plus a constant.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Renormalising Feynman diagrams with multi-indices." pith.science (2026). https://pith.science/paper/27LYVEZ2

@misc{pith2026250108151,
  author       = {Pith},
  title        = {Pith review of: Renormalising Feynman diagrams with multi-indices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27LYVEZ2}},
  note         = {Machine review of arXiv:2501.08151}
}
abstract

In this work, we provide a method to obtain the renormalised measure in quantum field theory directly from the renormalisation of the expansion of the original measure. Our approach is based on BPHZ renormalisation via multi-indices, a combinatorial structure extremely successful for describing scalar-valued singular SPDEs. We propose the multi-indices counterpart to the Hopf algebraic program initiated by Connes and Kreimer for the renormalisation of Feynman diagrams. This new Hopf algebra also bridges the gap between the analysis of "pre-Feynman diagrams" and traditional diagrammatic methods. The construction relies on a well-chosen extraction-contraction coproduct of multi-indices equipped with a correct symmetry factor. We illustrate our method by the $ \Phi^4 $ measure example.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 20 canonical work pages

  1. [42]

    W. D. van Suijlekom.Renormalization of Gauge Fields: A Hopf Algebra Approach. Comm. Math. Phys.276, (2007), 773–798. doi:10.1007/s00220-007-0353-9 . Example: Renormalisation of the Φ4 measure 45

    work page doi:10.1007/s00220-007-0353-9 2007

  2. [1]

    Abdesselam

    A. Abdesselam. Feynman diagrams in algebraic combinatorics.Semin. Lothar. Comb. B49c, (2002), 45 p., electronic only. http://eudml.org/doc/123420

    work page 2002

  3. [2]

    E. Abe. Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka

    work page 1980

  4. [3]

    Bailleul, Y

    I. Bailleul, Y. Bruned, Locality for singular stochastic PDEs . arXiv:2109. 00399

    work page

  5. [4]

    Bollob ´as A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs

    B. Bollob ´as A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs . Eur. J. Comb. 1, no. 4, (1980), 311–316. doi: 10.1016/S0195-6698(80)80030-8

    doi:10.1016/s0195-6698(80)80030-8 1980

  6. [5]

    Bruned, A

    Y. Bruned, A. Chandra, I. Chevyrev, M. Hairer. Renormalising SPDEs in regularity structures. J. Eur. Math. Soc. (JEMS), 23, no. 3, (2021), 869–947. doi:10.4171/JEMS/1025

    doi:10.4171/jems/1025 2021

  7. [6]

    Bruned, I

    Y. Bruned, I. Chevyrev, P. K. Friz, R. Preiss. A rough path perspective on renormalization. J. Funct. Anal. 277, no. 11, (2019), 108283. doi:10.1016/j. jfa.2019.108283. Example: Renormalisation of the Φ4 measure 43

    arXiv 2019

  8. [7]

    Berglund

    N. Berglund. An Introduction to Singular Stochastic PDEs. EMS Press (2022). doi:10.4171/ELM/34

    work page doi:10.4171/elm/34 2022

Show all 45 references
  1. [8]

    Bruned, K

    Y. Bruned, K. Ebrahimi-Fard, Y. Hou. Multi-indice B-series. J. Lond. Math. Soc. 111, no. 1, (2025), e70049. doi:10.1112/jlms.70049

    2025 doi

  2. [9]

    Bruned, Y

    Y. Bruned, Y. Hou. Multi-indices coproducts from ODEs to singular SPDEs. To appear in Transactions of the American Society. arXiv:2405.11314

  3. [10]

    Bruned, M

    Y. Bruned, M. Hairer, L. Zambotti. Algebraic renormalisation of regular- ity structures. Invent. Math. 215, no. 3, (2019), 1039–1156. doi:10.1007/ s00222-018-0841-x

    2019

  4. [11]

    Berglund, T

    N. Berglund, T. Klose. Perturbation theory for the Φ4 3 measure, revisited with Hopf algebras. arXiv:2207.085559

    arXiv

  5. [12]

    Bruned, F

    Y. Bruned, F. Katsetsiadis. Post-Lie algebras in Regularity Structures. Forum of Mathematics, Sigma 11, e98, (2023), 1–20. doi:10.1017/fms.2023.93

    2023 doi

  6. [13]

    Bruned, P

    Y. Bruned, P. Linares. A top-down approach to algebraic renormalization in regularity structures based on multi-indices. Arch. Ration. Mech. Anal. 248, 111 (2024). doi:10.1007/s00205-024-02041-4

    2024 doi

  7. [14]

    Bruned, D

    Y. Bruned, D. Manchon. Algebraic deformation for (S)PDEs. J. Math. Soc. Japan. 75, no. 2, (2023), 485–526. doi:10.2969/jmsj/88028802

    2023

  8. [15]

    N. N. Bogoliubow, O. S. Parasiuk. ¨Uber die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder . Acta Math. 97, (1957), 227–266. doi: 10.1007/BF02392399

    1957 doi

  9. [16]

    Quantum field theory meets Hopf algebra

    C.Brouder. Quantum field theory meets Hopf algebra. Math. Nachr. 282, no. 12, (2009), 1664–1690. doi:10.1002/mana.200610828

    2009 doi

  10. [17]

    K. A. Brueckner. Many-Body Problem for Strongly Interacting Particles. II. Linked Cluster Expansion. Phys. Rev.100, (1955), 36–45. doi:10.1103/PhysRev.100. 36

    1955 doi

  11. [18]

    Bruned, K

    Y. Bruned, K. Schratz. Resonance based schemes for dispersive equations via decorated trees. Forum of Mathematics, Pi, 10, E2.doi:10.1017/fmp.2021.13

    2021 doi

  12. [19]

    J. C. Butcher. An algebraic theory of integration methods. Math. Comp. 26, (1972), 79–106. doi:10.2307/2004720

    1972 doi

  13. [20]

    Calaque, K

    D. Calaque, K. Ebrahimi-Fard, D. Manchon. Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series.Adv. in Appl. Math. 47, no. 2, (2011), 282–308. doi:10.1016/j.aam.2009.08.003

    2011 doi

  14. [21]

    Chandra, M

    A. Chandra, M. Hairer. An analytic BPHZ theorem for regularity structures. arXiv:1612.08138

    arXiv

  15. [22]

    Chartier, E

    P. Chartier, E. Hairer, G. Vilmart. A substitution law for B-series vector fields. INRIA Report, no. 5498, (2005). inria-00070509

    2005

  16. [23]

    Chartier, E

    P. Chartier, E. Hairer, G. Vilmart.Algebraic structures of B-series. Found. Comput. Math. 10, no. 4, (2010), 407–427. doi:10.1007/s10208-010-9065-1

    2010 doi

  17. [24]

    Connes, D

    A. Connes, D. Kreimer. Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199, no. 1, (1998), 203–242. doi:10.1007/ s002200050499. Example: Renormalisation of the Φ4 measure 44

    1998

  18. [25]

    Connes, D

    A. Connes, D. Kreimer. Renormalization in quantum field theory and the Riemann- Hilbert problem I: the Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys. 210, (2000), 249–73. doi:10.1007/s002200050779

    2000 doi

  19. [26]

    W.D. Faris. Combinatorial species and Feynman diagrams. Semin. Lothar. Comb. 61A, (2011), Article B61An, electronic only. http://www.kurims.kyoto-u. ac.jp/EMIS/journals/SLC/wpapers/s61Afaris.pdf

    2011

  20. [27]

    Fosdick, D.B

    B.K. Fosdick, D.B. Larremore, J. Nishimura, J. Ugander. Configuring Random Graph Models with Fixed Degree Sequences. SIAM Review 60, no. 2, (2018), 315–355. doi:10.1137/16M1087175

    2018 doi

  21. [28]

    Fetter, J.D

    A.L. Fetter, J.D. Walecka. Quantum Theory of Many-Particle Systems. Dover Publications (2003) ISBN:978-0-486-42827-7

    2003

  22. [29]

    X. Gao, D. Manchon, Z. Zhu. Free Novikov algebras and the Hopf algebra of decorated multi-indices. arXiv:2404.09899

    arXiv

  23. [30]

    Goldstone Derivation of the Brueckner many-body theory

    J. Goldstone Derivation of the Brueckner many-body theory. Proceedings of the Royal Society of London. 239, no. 1217, (1957), 267–279. doi:10.1098/rspa. 1957.0037

    1957

  24. [31]

    D. Guin, J. M. Oudom, Sur l’alg`ebre enveloppante d’une alg`ebre pr´e-Lie, C. R. Math. Acad. Sci. Paris340 (2005), no. 5, 331–336. doi:10.1016/j.crma.2005. 01.010

    2005 doi

  25. [32]

    D. Guin, J. M. Oudom, On the Lie enveloping algebra of a pre-Lie algebra , J. K-Theory 2 (2008), no. 1, 147–167. doi:10.1017/is008001011jkt037

    2008 doi

  26. [33]

    M. Hairer. A theory of regularity structures. Invent. Math. 198, no. 2, (2014), 269–504. doi:10.1007/s00222-014-0505-4

    2014 doi

  27. [34]

    M. Hairer. An Analyst’s Take on the BPHZ Theorem.Comput. Combin. Dyn. Stoch. Control, pages 429-476, Cham, 2018. doi:10.1007/978-3-030-01593-0_16

    2018 doi

  28. [35]

    K. Hepp. On the equivalence of additive and analytic renormalization. Comm. Math. Phys. 14, (1969), 67–69. doi:10.1007/BF01645456

    1969 doi

  29. [36]

    Kreimer Anatomy of a gauge theory

    D. Kreimer Anatomy of a gauge theory . Ann. Phys. 321, (2006), 2757–2781. doi:10.1016/j.aop.2006.01.004

    2006 doi

  30. [37]

    P. Linares. Insertion pre-Lie products and translation of rough paths based on multi-indices. arXiv:2307.06769

    arXiv

  31. [38]

    Linares, F

    P. Linares, F. Otto, M. Tempelmayr.The structure group for quasi-linear equations via universal enveloping algebras. Comm. Amer. Math. 3, (2023), 1–64. doi: 10.1090/cams/16

    2023 doi

  32. [39]

    Linares F

    P. Linares F. Otto. A tree-free approach to regularity structures: The regular case for quasi-linear equations. arXiv:2207.10627

    arXiv

  33. [40]

    Linares, F

    P. Linares, F. Otto, M. Tempelmayr, P. TsatsoulisA diagram-free approach to the stochastic estimates in regularity structures.Invent. Math.237, (2024), 1469–1565. doi:10.1007/s00222-024-01275-z

    2024 doi

  34. [41]

    F. Otto, J. Sauer, S. Smith, H. Weber. A priori bounds for quasi-linear SPDEs in the full sub-critical regime. J. Eur. Math. Soc. (2024).doi:10.4171/JEMS/1574

    2024 doi

  35. [43]

    van der Hoorn, M

    P. van der Hoorn, M. Olvera-Cravioto. Typical distances in the directed configuration model . Ann. Appl. Probab., 28, no. 3, (2018), 1739–1792 . doi:110.1214/17-AAP1342

    2018

  36. [44]

    A. Wurm, M. Berg Wick calculus. American Journal of Physics vol. 76, no. 1, (2008), 65–72 doi:10.1119/1.2805232

    2008 doi

  37. [45]

    Zimmermann

    W. Zimmermann. Convergence of Bogoliubov’s method of renormalization in momentum space. Comm. Math. Phys. 15, (1969), 208–234. doi:10.1007/ BF01645676

    1969

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.