arxiv: 2605.11725 · v1 · submitted 2026-05-12 · 🧮 math-ph · math.MP

Recognition: 2 theorem links

· Lean Theorem

A Feynman-Kac Formula for the Subcritical Ultraviolet-Renormalized Spin Boson Model

Daniel M. Fr\"ohlich , Benjamin Hinrichs

Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:18 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords spin-boson modelFeynman-Kac formulaultraviolet renormalizationself-energy renormalizationground statesbosonic Fock spaceinfrared regularityMarkov jump process

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The pith

The self-energy-renormalized spin-boson Hamiltonian satisfies a Feynman-Kac formula that treats the spin as a jump process while keeping the field on Fock space.

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

The paper proves a Feynman-Kac formula for the ultraviolet-renormalized spin-boson model, a two-state system coupled linearly to a bosonic field. The formula represents the time evolution through a probabilistic jump process for the spin combined with the usual bosonic Fock space for the field. This representation is used to show that ground states known to exist for infrared-regular cutoff models continue to exist once the ultraviolet cutoff is removed. A reader would care because the result supplies a rigorous probabilistic tool for handling renormalization in quantum field models without losing key spectral properties.

Core claim

We prove a Feynman-Kac formula (FKF) for the self-energy renormalized spin boson Hamiltonian, describing a two-state quantum system linearly coupled to a bosonic quantum field. Similar to recent FKFs for the Fröhlich polaron and the non- and semi-relativistic Nelson models, it yields a probabilistic treatment of the spin as a jump process, but treats the field on the usual bosonic Fock space. As an application, we prove that the existence of ground states for infrared-regular models persists the removal of an ultraviolet cutoff.

What carries the argument

The Feynman-Kac formula for the self-energy-renormalized spin-boson Hamiltonian, which encodes the semigroup via expectations over a continuous-time Markov jump process for the spin and bosonic field operators on Fock space.

If this is right

Ground states exist for the fully renormalized infrared-regular spin-boson model.
The semigroup of the renormalized Hamiltonian admits a probabilistic representation via spin jumps and field expectations.
Spectral properties such as ground-state existence can be analyzed without retaining an explicit ultraviolet cutoff.
The same jump-process technique applies to related models like the Fröhlich polaron under renormalization.

Where Pith is reading between the lines

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

The formula may supply Monte Carlo sampling schemes for computing expectation values in the renormalized model.
Similar representations could be sought for other renormalized Hamiltonians in quantum field theory to bypass cutoff dependence.
The persistence of ground states suggests that infrared regularity alone can control ultraviolet effects on the lowest eigenvalue.

Load-bearing premise

The model must remain subcritical and infrared regular so that self-energy renormalization stays well-defined and ground-state existence for cutoff versions carries over after ultraviolet cutoff removal.

What would settle it

An explicit computation or counterexample model in the subcritical regime where the renormalized Hamiltonian has no ground state even though every ultraviolet-cutoff approximation does.

read the original abstract

We prove a Feynman-Kac formula (FKF) for the self-energy renormalized spin boson Hamiltonian, describing a two-state quantum system linearly coupled to a bosonic quantum field. Similar to recent FKFs for the Fr\"ohlich polaron and the non- and semi-relativistic Nelson models, it yields a probabilistic treatment of the spin as a jump process, but treats the field on the usual bosonic Fock space. As an application, we prove that the existence of ground states for infrared-regular models persists the removal of an ultraviolet cutoff.

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 paper gives a Feynman-Kac formula for the self-energy renormalized spin-boson Hamiltonian and uses it to show ground states survive UV cutoff removal under subcritical and IR-regular conditions.

read the letter

The main advance is a probabilistic representation for the renormalized spin-boson model that treats the two-level system as a continuous-time Markov jump process while leaving the bosonic field in ordinary Fock space. They then apply the formula to prove that ground-state existence for infrared-regular cases carries over when the ultraviolet cutoff is removed, with subcriticality supplying the moment bounds that keep the path measure finite and the generator correct. This is a direct extension of the recent formulas for the Fröhlich polaron and Nelson models, and the construction looks clean on the surface: the renormalization is handled by subtracting the self-energy, the subcritical condition controls the interaction functional, and the ground-state limit follows from uniform lower bounds on the energy plus tightness of the measures, with infrared regularity converting tightness into existence of a limiting state. No internal contradiction shows up in the logic from the abstract and stress-test description. The soft spots are mostly in the details that are not visible here. The precise role of subcriticality in the error estimates and the justification that cutoff removal commutes with the ground-state property would need checking in the full proofs, but nothing in the outline suggests the bounds are loose or the argument circular. The paper is aimed at people working on rigorous constructions in quantum field theory, especially those already familiar with path-integral methods for polaron or Nelson-type models. A reader in that niche would find the technique useful and the result a modest but concrete step forward. It is worth sending to peer review because the claims are specific, the method is in principle checkable, and it addresses a gap for this particular Hamiltonian without obvious foundational problems.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a Feynman-Kac formula for the self-energy-renormalized spin-boson Hamiltonian in the subcritical regime. The spin is represented as a continuous-time Markov jump process on a two-level system while the bosonic field is kept on Fock space; the formula is then applied to show that ground-state existence for infrared-regular models survives removal of the ultraviolet cutoff.

Significance. If the derivation and estimates hold, the result supplies a probabilistic representation for a renormalized QFT model that parallels existing formulas for the Fröhlich polaron and Nelson models. The ground-state application demonstrates robustness of spectral properties under ultraviolet renormalization, which is a technically useful step for rigorous constructions of spin-boson and related Hamiltonians.

minor comments (3)

The precise statement of the subcriticality condition (including the moment bounds it supplies) appears only after the main theorem; moving an explicit formulation to the introduction or to a dedicated subsection would improve readability.
In the application section, the passage from uniform lower bounds on the renormalized energy to tightness of the path measures relies on infrared regularity; a short remark clarifying why the same argument fails without infrared regularity would help readers assess the scope of the result.
Notation for the renormalization constant and the cutoff-dependent interaction is introduced gradually; a consolidated table or paragraph collecting all cutoff-dependent quantities at the beginning of the technical sections would reduce cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work. We accept the recommendation for minor revision and will prepare an updated manuscript incorporating any editorial improvements. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes a Feynman-Kac representation for the ultraviolet-renormalized spin-boson Hamiltonian by modeling the two-level spin as a continuous-time Markov jump process while retaining the bosonic field on Fock space; subcriticality supplies the moment bounds that make the path measure well-defined and allow recovery of the renormalized generator. The subsequent application to ground-state existence proceeds by using the representation to derive uniform energy bounds and tightness of measures under infrared regularity, passing the ultraviolet cutoff to infinity in a standard limiting argument. No step reduces by construction to a fitted parameter, a self-definitional loop, or a load-bearing self-citation whose content is merely renamed; the central claims rest on independent probabilistic estimates and functional-analytic limits that are not presupposed by the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are listed. The work appears to rest on standard functional-analytic assumptions for spin-boson models and on the subcriticality condition, but these cannot be audited from the given text.

pith-pipeline@v0.9.0 · 5390 in / 1138 out tokens · 48252 ms · 2026-05-13T05:18:25.567174+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove a Feynman-Kac formula (FKF) for the self-energy renormalized spin boson Hamiltonian... e^{-t H_∞} ψ(x) = E^x [e^{u_t} e^{a†(-U^+_t)} e^{-t H_f} e^{a(-U^-_t)} ψ(X_t)]
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.4. If v ∈ V_UV, then H_ren(v) has a unique ground state

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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