Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions

Benjamin Hinrichs; Oliver Matte

arxiv: 2309.09005 · v1 · submitted 2023-09-16 · 🧮 math-ph · math.MP

Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions

Benjamin Hinrichs , Oliver Matte This is my paper

Pith reviewed 2026-05-24 06:16 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Feynman-Kac formulafiber HamiltonianNelson modelrelativistic particletranslation invariancetwo dimensionssemigroupquantum field theory

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The pith

New Feynman-Kac formulas are derived for the fiber Hamiltonians attached to fixed total momenta in the two-dimensional relativistic Nelson model.

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

This paper derives Feynman-Kac formulas for the fiber Hamiltonians of a translation-invariant relativistic Nelson model in two spatial dimensions. The model describes a scalar relativistic particle interacting with a massive radiation field, and the full Hamiltonian is defined with energy renormalization. The authors first review a prior Feynman-Kac formula for the complete Hamiltonian and then apply technical key relations from a preprint to obtain the fiber versions in a self-contained manner. This also produces an alternative derivation of the Feynman-Kac formula for the full translation-invariant Hamiltonian. A sympathetic reader would care because these formulas supply probabilistic representations of the semigroups that can be used to study the spectrum and dynamics in fixed-momentum sectors.

Core claim

We present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian.

What carries the argument

The Feynman-Kac formula for a fiber Hamiltonian, which represents its semigroup via an expectation with respect to a stochastic process that incorporates the particle and field degrees of freedom at fixed total momentum.

If this is right

The fiber Hamiltonians admit Feynman-Kac representations that parallel the one already known for the full Hamiltonian.
An alternative derivation of the Feynman-Kac formula for the full translation-invariant Hamiltonian follows immediately from the fiber versions.
The same technical estimates suffice for the fiber case, so no separate verification step is required.
The formulas extend the probabilistic treatment to the fixed-total-momentum sectors of the translation-invariant system.

Where Pith is reading between the lines

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

The representations could support Monte Carlo sampling to approximate ground-state energies or correlation functions at fixed total momentum.
Similar reductions from full Hamiltonian to fiber Hamiltonians might apply in other translation-invariant models with ultraviolet renormalization.
The two-dimensional setting leaves open whether the same relations carry over to three or more spatial dimensions.

Load-bearing premise

The technical key relations and estimates obtained in the preprint hold and can be directly applied to the fiber Hamiltonians without additional verification.

What would settle it

Direct verification that the stated Feynman-Kac formula for a concrete fiber Hamiltonian reproduces the correct action of the semigroup on a dense set of test functions.

read the original abstract

In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian.

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.

Desk Editor's Note private letter to a colleague

This proceeding derives FK formulas for fiber Hamiltonians in the 2D relativistic Nelson model by reusing estimates from the authors' prior preprint, with an alternative derivation for the full Hamiltonian.

read the letter

The main thing here is a derivation of new Feynman-Kac formulas for the fiber Hamiltonians at fixed total momentum in this translation-invariant relativistic Nelson model in two dimensions. The authors review their earlier semigroup formula, then use a handful of technical relations from their recent preprint to get the fiber versions in an otherwise self-contained way, and finally recover an alternative route to the full Hamiltonian formula. Standard direct-integral techniques handle the momentum fixing, which fits the setting they already treated before. That part reads as a straightforward extension rather than a major methodological shift. The paper does well at keeping the argument compact and explicit once the prior estimates are accepted. It stays within the usual toolkit for these models and avoids overclaiming. The soft spot is the dependence on those key relations and estimates from the preprint. The derivation stands or falls with them, and this proceeding does not re-check or expand on that material. Because it is a short proceeding, the scope stays narrow to this specific 2D case with massive field, so it does not address broader open questions in relativistic QFT. No internal contradictions or unjustified steps appear once the cited relations are granted. This is for specialists already working on Nelson-type models or Feynman-Kac representations in mathematical physics. A reader outside that small circle will not find much of use. It deserves a serious referee because the formulas are new, the argument is presented cleanly, and the claims are modest enough that checking the transfer of the prior estimates is feasible. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript reviews a Feynman-Kac formula for the semigroup of the energy-renormalized Hamiltonian in the two-dimensional relativistic Nelson model (with translation invariance) from the authors' recent preprint. It then applies a small number of technical key relations and estimates from that preprint to derive new Feynman-Kac formulas for the fiber Hamiltonians at fixed total momenta via standard direct-integral decomposition, and concludes by recovering an alternative derivation of the formula for the full translation-invariant Hamiltonian.

Significance. If the derivations hold, the new fiber formulas supply path-integral representations for the momentum-fixed operators, which are expected to be useful for spectral and dynamical analysis in the translation-invariant relativistic Nelson model. The work credits the prior preprint for the underlying estimates and positions the fiber derivation as otherwise self-contained.

major comments (1)

[main derivation section (post-review of full Hamiltonian)] The central derivation of the fiber FK formulas (described after the review of the full Hamiltonian formula) applies the key relations from the preprint directly to the fiber operators without an explicit verification step that the estimates remain valid under the fixed-momentum projection; this transfer is load-bearing for the claim that the derivation is otherwise self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We address the major comment below.

read point-by-point responses

Referee: [main derivation section (post-review of full Hamiltonian)] The central derivation of the fiber FK formulas (described after the review of the full Hamiltonian formula) applies the key relations from the preprint directly to the fiber operators without an explicit verification step that the estimates remain valid under the fixed-momentum projection; this transfer is load-bearing for the claim that the derivation is otherwise self-contained.

Authors: The key relations and estimates from the preprint are formulated for the translation-invariant operators before decomposition and rely on properties of the interaction and free Hamiltonians that are diagonal with respect to total momentum. The direct-integral decomposition therefore carries these relations over to each fiber verbatim, as the fixed-momentum sectors are invariant under the relevant operators. We agree that an explicit sentence noting this transfer would improve clarity and reinforce the self-contained character of the fiber derivation. We will add such a remark in the main derivation section. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper explicitly describes its derivation of new Feynman-Kac formulas for the fiber Hamiltonians as 'otherwise self-contained' after employing a few technical key relations and estimates from a prior preprint. This is standard citation of previous results rather than a reduction of the central claim to self-citation by construction. The abstract and structure indicate that the new formulas and their derivation steps are presented independently, with fiber decomposition handled via standard direct-integral techniques. No self-definitional relations, fitted inputs renamed as predictions, or load-bearing self-citations that force the result are exhibited. The derivation chain remains self-contained against the external benchmark of the cited preprint's estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on the validity of energy renormalization (stated as well-known) and on estimates from the authors' own preprint whose details are unavailable here.

axioms (1)

domain assumption The corresponding Hamiltonian can be defined with the help of an energy renormalization.
Explicitly stated in the abstract as well-known.

pith-pipeline@v0.9.0 · 5654 in / 1078 out tokens · 27107 ms · 2026-05-24T06:16:22.372076+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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A Feynman-Kac formula is proved for the ultraviolet-renormalized spin-boson model, showing that ground state existence for infrared-regular versions survives removal of the ultraviolet cutoff.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper · 4 internal anchors

[1]

Z. Ammari. Asymptotic Completeness for a Renormalized Nonrelativistic H amiltonian in Quantum Field Theory: The N elson Model. Math. Phys. Anal. Geom. , 3(3):217--285, 2000. doi:10.1023/A:1011408618527

work page doi:10.1023/a:1011408618527 2000
[2]

Applebaum

D. Applebaum. L \' e vy Processes and Stochastic Calculus , volume 116 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, second edition, 2009. doi:10.1017/9780511809781

work page doi:10.1017/9780511809781 2009
[3]

A. Arai. Analysis on F ock Spaces and Mathematical Theory of Quantum Fields . World Scientific, New Jersey, 2018. doi:10.1142/10367

work page doi:10.1142/10367 2018
[4]

T. N. Dam and B. Hinrichs. Absence of ground states in the renormalized massless translation-invariant N elson model. Rev. Math. Phys. , 34(10):2250033, 2022, arXiv:1909.07661 http://arxiv.org/abs/1909.07661 . doi:10.1142/S0129055X22500337

work page doi:10.1142/s0129055x22500337 2022
[5]

Ultraviolet Renormalization of the Nelson Hamiltonian through Functional Integration

M. Gubinelli, F. Hiroshima, and J. L o rinczi. Ultraviolet renormalization of the N elson H amiltonian through functional integration. J. Funct. Anal. , 267(9):3125--3153, 2014, arXiv:1304.6662 http://arxiv.org/abs/1304.6662 . doi:10.1016/j.jfa.2014.08.002

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.jfa.2014.08.002 2014
[6]

Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

B. G \" u neysu, O. Matte, and J. S. M ller. Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields. Probab. Theory Relat. Fields , 167(3-4):817--915, 2017, arXiv:1402.2242 http://arxiv.org/abs/1402.2242 . doi:10.1007/s00440-016-0694-4

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00440-016-0694-4 2017
[7]

L. Gross. The relativistic polaron without cutoffs. Commun. Math. Phys. , 31(1):25--73, 1973. doi:10.1007/BF01645589

work page doi:10.1007/bf01645589 1973
[8]

Hiroshima and J

F. Hiroshima and J. L o rinczi. Feynman- K ac-type theorems and G ibbs measures on path space. V olume 2 , volume 34/2 of De Gruyter Studies in Mathematics . De Gruyter, Berlin, second edition, 2020. doi:10.1515/9783110403541

work page doi:10.1515/9783110403541 2020
[9]

Hinrichs and O

B. Hinrichs and O. Matte. F eynman-- K ac formula and asymptotic behavior of the minimal energy for the relativistic N elson model in two spatial dimensions. 2023, arXiv:2211.14046 http://arxiv.org/abs/2211.14046 . Preprint, to appear in Ann. Henri Poincar\' e

work page arXiv 2023
[10]

On Nelson-type Hamiltonians and abstract boundary conditions

J. Lampart and J. Schmidt. On N elson-Type H amiltonians and Abstract Boundary Conditions. Commun. Math. Phys. , 367(2):629--663, 2019, arXiv:1803.00872 http://arxiv.org/abs/1803.00872 . doi:10.1007/s00220-019-03294-x

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00220-019-03294-x 2019
[11]

Feynman-Kac formulas for the ultra-violet renormalized Nelson model

O. Matte and J. S. M ller. F eynman-- K ac Formulas for the Ultra-Violet Renormalized N elson Model. Ast\' e risque , 404, 2018, arXiv:1701.02600 http://arxiv.org/abs/1701.02600 . doi:10.24033/ast.1054

work page internal anchor Pith review Pith/arXiv arXiv doi:10.24033/ast.1054 2018
[12]

E. Nelson. Interaction of Nonrelativistic Particles with a Quantized Scalar Field. J. Math. Phys. , 5(9):1190--1197, 1964. doi:10.1063/1.1704225

work page doi:10.1063/1.1704225 1964
[13]

E. Nelson. Schr\" o dinger particles interacting with a quantized scalar field. In W. T. Martin and I. Segal, editors, Analysis in Function Space: Proceedings of a Conference on the Theory and Application of Analysis in Function Space Held At Endicott House in Dedham, Massachusetts, June 9-13, 1963 , pages 87--120, Cambridge, 1964. MIT Press

work page 1963
[14]

K. R. Parthasarathy. An Introduction to Quantum Stochastic Calculus , volume 85 of Monographs in Mathematics . Birkh\" a user, Basel, 1992. doi:10.1007/978-3-0348-0566-7

work page doi:10.1007/978-3-0348-0566-7 1992
[15]

J. Schmidt. On a direct description of pseudorelativistic N elson H amiltonians. J. Math. Phys. , 60(10):102303, 2019, arXiv:1810.03313 http://arxiv.org/abs/1810.03313 . doi:10.1063/1.5109640

work page doi:10.1063/1.5109640 2019
[16]

Siakalli

M. Siakalli. Stability properties of stochastic differential equations driven by L \'e vy noise . PhD thesis, University of Sheffield, 2009. ://etheses.whiterose.ac.uk/15019/

work page 2009
[17]

A. D. Sloan. The polaron without cutoffs in two space dimensions. J. Math. Phys. , 15:190--201, 1974. doi:10.1063/1.1666620

work page doi:10.1063/1.1666620 1974

[1] [1]

Z. Ammari. Asymptotic Completeness for a Renormalized Nonrelativistic H amiltonian in Quantum Field Theory: The N elson Model. Math. Phys. Anal. Geom. , 3(3):217--285, 2000. doi:10.1023/A:1011408618527

work page doi:10.1023/a:1011408618527 2000

[2] [2]

Applebaum

D. Applebaum. L \' e vy Processes and Stochastic Calculus , volume 116 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, second edition, 2009. doi:10.1017/9780511809781

work page doi:10.1017/9780511809781 2009

[3] [3]

A. Arai. Analysis on F ock Spaces and Mathematical Theory of Quantum Fields . World Scientific, New Jersey, 2018. doi:10.1142/10367

work page doi:10.1142/10367 2018

[4] [4]

T. N. Dam and B. Hinrichs. Absence of ground states in the renormalized massless translation-invariant N elson model. Rev. Math. Phys. , 34(10):2250033, 2022, arXiv:1909.07661 http://arxiv.org/abs/1909.07661 . doi:10.1142/S0129055X22500337

work page doi:10.1142/s0129055x22500337 2022

[5] [5]

Ultraviolet Renormalization of the Nelson Hamiltonian through Functional Integration

M. Gubinelli, F. Hiroshima, and J. L o rinczi. Ultraviolet renormalization of the N elson H amiltonian through functional integration. J. Funct. Anal. , 267(9):3125--3153, 2014, arXiv:1304.6662 http://arxiv.org/abs/1304.6662 . doi:10.1016/j.jfa.2014.08.002

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.jfa.2014.08.002 2014

[6] [6]

Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

B. G \" u neysu, O. Matte, and J. S. M ller. Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields. Probab. Theory Relat. Fields , 167(3-4):817--915, 2017, arXiv:1402.2242 http://arxiv.org/abs/1402.2242 . doi:10.1007/s00440-016-0694-4

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00440-016-0694-4 2017

[7] [7]

L. Gross. The relativistic polaron without cutoffs. Commun. Math. Phys. , 31(1):25--73, 1973. doi:10.1007/BF01645589

work page doi:10.1007/bf01645589 1973

[8] [8]

Hiroshima and J

F. Hiroshima and J. L o rinczi. Feynman- K ac-type theorems and G ibbs measures on path space. V olume 2 , volume 34/2 of De Gruyter Studies in Mathematics . De Gruyter, Berlin, second edition, 2020. doi:10.1515/9783110403541

work page doi:10.1515/9783110403541 2020

[9] [9]

Hinrichs and O

B. Hinrichs and O. Matte. F eynman-- K ac formula and asymptotic behavior of the minimal energy for the relativistic N elson model in two spatial dimensions. 2023, arXiv:2211.14046 http://arxiv.org/abs/2211.14046 . Preprint, to appear in Ann. Henri Poincar\' e

work page arXiv 2023

[10] [10]

On Nelson-type Hamiltonians and abstract boundary conditions

J. Lampart and J. Schmidt. On N elson-Type H amiltonians and Abstract Boundary Conditions. Commun. Math. Phys. , 367(2):629--663, 2019, arXiv:1803.00872 http://arxiv.org/abs/1803.00872 . doi:10.1007/s00220-019-03294-x

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00220-019-03294-x 2019

[11] [11]

Feynman-Kac formulas for the ultra-violet renormalized Nelson model

O. Matte and J. S. M ller. F eynman-- K ac Formulas for the Ultra-Violet Renormalized N elson Model. Ast\' e risque , 404, 2018, arXiv:1701.02600 http://arxiv.org/abs/1701.02600 . doi:10.24033/ast.1054

work page internal anchor Pith review Pith/arXiv arXiv doi:10.24033/ast.1054 2018

[12] [12]

E. Nelson. Interaction of Nonrelativistic Particles with a Quantized Scalar Field. J. Math. Phys. , 5(9):1190--1197, 1964. doi:10.1063/1.1704225

work page doi:10.1063/1.1704225 1964

[13] [13]

E. Nelson. Schr\" o dinger particles interacting with a quantized scalar field. In W. T. Martin and I. Segal, editors, Analysis in Function Space: Proceedings of a Conference on the Theory and Application of Analysis in Function Space Held At Endicott House in Dedham, Massachusetts, June 9-13, 1963 , pages 87--120, Cambridge, 1964. MIT Press

work page 1963

[14] [14]

K. R. Parthasarathy. An Introduction to Quantum Stochastic Calculus , volume 85 of Monographs in Mathematics . Birkh\" a user, Basel, 1992. doi:10.1007/978-3-0348-0566-7

work page doi:10.1007/978-3-0348-0566-7 1992

[15] [15]

J. Schmidt. On a direct description of pseudorelativistic N elson H amiltonians. J. Math. Phys. , 60(10):102303, 2019, arXiv:1810.03313 http://arxiv.org/abs/1810.03313 . doi:10.1063/1.5109640

work page doi:10.1063/1.5109640 2019

[16] [16]

Siakalli

M. Siakalli. Stability properties of stochastic differential equations driven by L \'e vy noise . PhD thesis, University of Sheffield, 2009. ://etheses.whiterose.ac.uk/15019/

work page 2009

[17] [17]

A. D. Sloan. The polaron without cutoffs in two space dimensions. J. Math. Phys. , 15:190--201, 1974. doi:10.1063/1.1666620

work page doi:10.1063/1.1666620 1974