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On the Ergodicity of Renormalized Translation-Invariant Nelson-Type Semigroups

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Pith's one-line read Negative coupling makes the semigroups of the ultraviolet-renormalized non-relativistic and semi-relativistic Nelson Hamiltonians positivity improving for every total momentum; the semi-relativistic case at $P \neq 0$ is new.

desk verdict A clean functional-integral proof that the renormalized semi-relativistic Nelson semigroup is positivity improving at every momentum; the main theorem is new and mostly sound, but a few technical gaps need patching. read the letter →

arxiv 2412.09708 v1 pith:F65SW66V submitted 2024-12-12 math-ph math.MP

Benjamin Hinrichs , Fumio Hiroshima This is my paper · ORCID
classification math-phmath.MP MSC 81S4047D08
keywords Nelsonmodelsemi-relativisticpositivityimprovingFröhlichconeergodicsemigroupFeynman–Kacformulaultravioletrenormalizationtotalmomentum
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 establishes Theorem 1: for negative coupling $\lambda < 0$, the semigroups $e^{-tH_{\#}(P)}$ generated by the ultraviolet-renormalized non-relativistic ($\# = \mathrm{nr}$) and semi-relativistic ($\# = \mathrm{sr}$) Nelson Hamiltonians are positivity improving with respect to the Fröhlich cone, for every total momentum $P \in \mathbb{R}^d$ and every $t > 0$. Positivity improving means the semigroup maps every nonzero vector in the cone to a vector that has strictly positive inner product with every cone vector. By the Perron–Frobenius–Faris theorem this implies that any maximal eigenvalue of $H_{\#}(P)$ is non-degenerate and has a unique strictly positive eigenvector. The non-relativistic renormalized case was known through different methods, while the semi-relativistic case at arbitrary total momentum is new; the proof here is a single functional-integral argument covering both cases.

What carries the argument

The key object is the Feynman–Kac representation (2.25), which writes the semigroup as an expectation over a Lévy process whose characteristic exponent is the particle dispersion. Positivity is read off from the integral kernels: the assumptions in Theorem 2 make the kernel $F^{\pi,i,\ell}_{k,p,t}$ nonnegative almost everywhere, and the Lévy property makes $E[e^{-iR}] > 0$ strictly. For the renormalized models, the new Trotter product formula (Proposition 6), proved via the Chernoff product formula, is what allows the limit from ultraviolet-regularized to renormalized semigroups to preserve positivity improving. The Fröhlich cone is the self-dual cone of Fock-space vectors whose $n$-particle components are nonnegative almost everywhere for every $n$.

What would settle it

Exhibit vectors $\psi,\varphi$ in the Fröhlich cone and some $P \neq 0$, $t>0$ with $\langle \psi, e^{-tH_{\mathrm{sr}}(P)}\varphi\rangle = 0$; that would directly contradict Theorem 1. A less direct but checkable failure point is the exponential moment bound (3.31b) for the semi-relativistic model: if it diverges for some $P$, the monotone-convergence step in Theorem 8 breaks and the functional-integral proof no longer goes through.

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Extended reading notes

Core claim

The central claim is that the renormalized Nelson semigroups are ergodic at negative coupling for all total momenta. The proof represents $e^{-tH_{\#}(P)}$ by the Feynman–Kac formula $e^{-tH_{\#}(P)} = \mathbb{E}\left[ e^{u_t} F^{\omega}_{t/2}(U^-) F^{\omega}_{t/2}(U^+)^{*} e^{i(P-\mathrm{d}\Gamma(\hat{p}))\cdot X_t} \right]$, and shows that every matrix element against Fröhlich-cone vectors can be written as an integral of a nonnegative kernel times $E[e^{-iR}]>0$, the characteristic function expectation of a Lévy process. The renormalized limit is controlled by convergence of the random objects $u_t, U^\pm$ in $L^p$ together with exponential moment bounds. A new Trotter product formula (Proposition 6), derived from a Chernoff-type theorem, lets the authors pass from the ultraviolet-regularized to the renormalized semigroup by decomposing Fock space onto bounded momentum subsets, where the quadratic dispersion satisfies the needed lower-boundedness assumption. The outcome is that every maximal eigenvalue of $H_{\#}(P)$ is simple and the eigenspace is spanned by a unique strictly positive vector.

Load-bearing premise

The proof stands on the Feynman–Kac representation (2.25) and on the imported convergence and exponential moment bounds holding uniformly in the total momentum $P$, together with the Trotter product formula of Proposition 6; if any of those imported bounds fails, the functional-integral argument for Theorem 1 collapses.

Editorial extensions

If this is right

  • For $\lambda < 0$, both $e^{-tH_{\mathrm{nr}}(P)}$ and $e^{-tH_{\mathrm{sr}}(P)}$ are positivity improving and hence ergodic for every total momentum $P$.
  • Every maximal eigenvalue of $H_{\mathrm{nr}}(P)$ and $H_{\mathrm{sr}}(P)$ at negative coupling is non-degenerate, with a unique strictly positive eigenvector.
  • The proof also covers the ultraviolet-cutoff-free Fröhlich polaron, giving a direct functional-integral proof of its ergodicity for any $P$.
  • The unified argument removes the earlier restriction to $P = 0$ in the semi-relativistic renormalized case.

Reading between the lines

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

  • The same positivity machinery should apply to other translation-invariant polaron-type models that admit a Feynman–Kac representation with real characteristic exponent and exponential moment bounds, such as Hamiltonians with subcritical ultraviolet singularities.
  • The bounded-subset reduction in Proposition 6 is needed only because Assumption A fails for the quadratic dispersion on unbounded sets; a Trotter formula tolerating an unbounded-from-below $L$ would allow $\Theta = \Omega$ and a more direct proof of Theorem 8.
  • A concrete test of the method's limits is the three-dimensional semi-relativistic model, where the paper notes a similar result is not expected; if the required moment bounds fail there, the functional-integral route would mark the boundary of its applicability.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper claims that for the ultraviolet-renormalized translation-invariant Nelson Hamiltonian, both in the three-dimensional non-relativistic and the two-dimensional semi-relativistic case, the semigroup e^{-tH_{\#}(P)} is positivity improving with respect to the Fröhlich cone for every total momentum P, provided the coupling constant is negative (Theorem 1). The proof is functional-analytic and probabilistic: it establishes a general Feynman-Kac-type functional-integral calculus (Sections 3.1-3.3), proves positivity of such integrals in progressively more singular situations (Theorems 2, 3, and 8), and then applies Theorem 8 to the renormalized Nelson semigroup using previously established Feynman-Kac representations and moment bounds from [MM18], [HM23], and [HM24a]. The authors also derive a new Trotter product formula (Proposition 6) via a Chernoff-type argument and use it to handle the self-energy renormalization. The paper explicitly notes that the semi-relativistic case at arbitrary total momentum is new, while the non-relativistic case has recent proofs by Miyao and Lampart.

Significance. If the proof is correct, Theorem 1 settles a natural open case: ergodicity, and hence uniqueness and strict positivity of maximal eigenfunctions by the Perron-Frobenius-Faris theorem, for the renormalized semi-relativistic Nelson model at arbitrary total momentum; it also gives a transparent, unified proof for the non-relativistic model. The paper's strategy has real virtues: the core positivity in Theorem 2 is derived from an explicit nonnegative integral kernel and the elementary positivity of E[e^{-iR}] via (3.2), with no fitted parameters and no circular use of the desired conclusion. The limitation statements are honestly placed: Remark 3.5 acknowledges the bounded-domain restriction in the Trotter step, and the dependence on external Feynman-Kac inputs is visible in Section 2.5. The main concerns are that the proof of Theorem 8 contains a nontrivial limit interchange that is not justified as written, and that the paper does not spell out the exact hypotheses under which the imported Feynman-Kac and moment bounds cover the required parameter range.

major comments (2)
  1. [Section 3.3, Eqs. (3.36)-(3.37)] The passage from (3.36) to (3.37) is not justified as written. From (3.34) and (3.35) one obtains, for each fixed N, norm convergence of the Trotter product as n -> infinity, but the natural telescoping estimate gives an error proportional to N times the per-factor error, so (3.35) does not imply uniformity in N. The subsequent interchange of the limits n -> infinity and N -> infinity is therefore a genuine gap. The gap is repairable within the paper's own framework: since Theta is bounded, for all sufficiently large n one has Theta subset Omega_n, and then Q^Omega_Theta S(n,0) Q^Omega_Theta equals Q^Omega_Theta S(infinity,0) Q^Omega_Theta on F(Theta), with the analogous statement for S(n,k); using this eventual containment, (3.37) follows directly from the n-large case of (3.36), without a double-limit argument. This repair should be written out explicitly, because (3.37) is the identity on which the positivity-improving conclusion of Theorem 8 rests.
  2. [Section 2.5 and Proof of Theorem 1] Theorem 1 depends on the Feynman-Kac representation (2.25), on the L^p convergences (2.26)-(2.28), and on the exponential moment bounds (3.31a,b), all of which are imported from [MM18, Theorem 7.6], [HM23, Theorem 7.4], and [HM24a, Theorems 6.6-6.7, Lemma B.1]. These external inputs must hold for every total momentum P in R^d and, in the non-relativistic case, also for massless bosons m=0. The manuscript neither states the precise hypotheses of those theorems nor verifies that they cover the full parameter range used in Theorem 1. If any of those bounds or interchanges fails at an allowed parameter, the identification of e^{-tH_#(P)} with the functional integral whose positivity is proven collapses. Please add a precise statement of the hypotheses and either a short verification or an explicit pointer to the theorem in the cited works that covers exactly the required parameter range, including the massless non-relativistic case.
minor comments (6)
  1. [Section 3.3, Eq. (3.28d)] The displayed formula for D_t(tau) contains unresolved TeX control sequences ('bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright') and should be cleaned up.
  2. [Example 3.4(1)] The dispersion is printed as Psi(p) = sqrt(|p|^2+M^2)-M^2; it should be sqrt(|p|^2+M^2)-M, which is the function with Psi(0)=0 and subadditivity used in the text.
  3. [Theorem 8, Eq. (3.33)] The symbol f(t) in (3.33) is never defined in the statement of Theorem 8; presumably it is the pointwise limit of f_n(t), and that hypothesis should be stated explicitly.
  4. [Proposition 5, Eq. (3.20)] The assertion that h(t,x) is selfadjoint for arbitrary g_+ and g_- requires an additional condition such as g_-(t)=g_+(t) (or a specified relation between the two test functions); in the Nelson application the two functions are indeed equal, so this is a statement-level gap rather than a flaw in the main application.
  5. [Theorem 3, proof] The notation P_{Omega_n} psi is used but not defined; Section 2.1 already uses P_n for Fock-space projections, and the projection onto F(Theta) is denoted Q^Omega_Theta in (2.3), so this should be aligned to avoid confusion.
  6. [Theorem 1 and Section 2.2] The statement 'P in R^d' should specify the dimension for each model: d=3 for #=nr and d=2 for #=sr, since those are the dimensions in which the models are defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: ergodicity is derived from Feynman–Kac representations and kernel positivity; external bounds are cited hypotheses, not the target result.

full rationale

Theorem 1 is obtained by applying the new Theorem 8 to the Nelson models, and Theorem 8 is proven from the general positivity Theorem 2 together with the Trotter product formula of Proposition 6; none of these steps assumes that e^{-tH_#(P)} is positivity improving. The Feynman–Kac identity (2.25), the convergence statements (2.26)–(2.28), and the exponential moment bounds (3.31a,b) are imported from [MM18, HM23, HM24a], and although two of these sources share an author with the present paper, they supply functional-integral representations and moment estimates, not the Fröhlich-cone ergodicity claimed here; they are therefore independent evidence rather than a circular load-bearing citation. The bounded-Θ restriction necessary for the non-relativistic dispersion relation is explicitly acknowledged in Remark 3.5 as a technical difficulty, not as a hidden assumption of the conclusion. Any failure of the imported bounds over the full parameter range would be a correctness risk in the hypotheses, not a circularity in the derivation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on imported Feynman-Kac and stochastic-analysis machinery (listed under axioms), on standard Lévy process facts, and on the Vuillermot Chernoff theorem. No free parameters are fitted; the only tuning-like ingredient, E_Lambda, is the standard self-energy counterterm. No new physical entities are introduced; the Fröhlich cone and the Nelson Hamiltonians are established objects from the cited literature.

assumptions (5)
  • standard math Lévy-Khintchine representation and independent stationary increments, in particular (3.2): the joint characteristic function factors as a product of e^{-(t_j-t_{j-1})Psi} with real-valued, nonnegative Psi.
    Used in Theorem 2 Step 2 to conclude E[e^{-iR}] > 0, which is the engine of the positivity-improving kernel argument. Standard Lévy process theory from [App09, LHB20].
  • domain assumption Feynman-Kac formula (2.25) for the renormalized semigroup, with convergence (2.26)-(2.28) and exponential moment bounds (3.31a,b), imported from [MM18, Thm 7.6] (nr), [HM23, Thm 7.4] and [HM24a, Thms 6.6-6.7, Lem B.1] (sr).
    This external machinery identifies e^{-tH_#(P)} with the functional integral that Theorem 8 analyzes, and supplies the integrability needed for the n to infinity passages. Partly authored by the present first author; not re-derived in this paper.
  • domain assumption The dispersion relations Psi_nr(p) = |p|^2/2 and Psi_sr(p) = sqrt(|p|^2 + M^2) - M are real-valued Lévy symbols; their processes (Brownian motion in d=3, inverse Gaussian in d=2) have the properties used in (3.2).
    Real-valued Psi makes the factors in (3.2) strictly positive, which is the mechanism behind strict kernel positivity and hence 'improving' rather than merely 'preserving'. These are features of the model choice in Section 2.2 and Section 3.
  • standard math Vuillermot's Chernoff-type product formula [Vui10] applies to Q_Theta S^Omega_{s,t}(P) Q_Theta with generator Psi(P - dGamma(p)) + h(t,0) + L, under Assumption A and (3.27).
    Proposition 6 is the new tool that lets Theorem 8 reduce the renormalized case to the regularized case; its hypotheses include the strong derivative condition (3.28d), verified by a compactness argument, and Assumption A, which for Psi_nr holds only on bounded momentum sets (Remark 3.5).
  • domain assumption Selfadjointness of h(t,x) = dGamma(omega f'(t)) + a(e^{-ip·x}g_-(t)) + a†(e^{-ip·x}g_+(t)) on D(dGamma(omega)), in the used case g_+ = g_- real.
    Proposition 5 states this for arbitrary g±, but a(f) + a†(g) is symmetric only when g equals the conjugate of f in this paper's linear-in-f convention. The application in Theorem 1 uses g±_n = -v_n real, where h is a genuine field operator, so the main theorem is unaffected; the proposition as stated is too general.

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Pith. "Pith review of On the Ergodicity of Renormalized Translation-Invariant Nelson-Type Semigroups." pith.science (2026). https://pith.science/paper/F65SW66V

@misc{pith2026241209708,
  author       = {Pith},
  title        = {Pith review of: On the Ergodicity of Renormalized Translation-Invariant Nelson-Type Semigroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F65SW66V}},
  note         = {Machine review of arXiv:2412.09708}
}
read the original abstract

We present a simple functional integration based proof that the semigroups generated by the ultraviolet-renormalized translation-invariant non- and semi-relativistic Nelson Hamiltonians are positivity improving (and hence ergodic) with respect to the Fr\"ohlich cone for arbitrary values of the total momentum. Our argument simplifies known proofs for ergodicity and the result is new in the semi-relativistic case.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dyson expansion for form-bounded perturbations and applications to the polaron problem

    math-ph 2025-12 conditional novelty 7.0 of 10

    For a general class of polaron models, the ground state energy is concave as a function of |P|², proved by a new Dyson expansion valid for form-bounded perturbations.

Reference graph

Works this paper leans on

55 extracted references · 17 canonical work pages · cited by 1 Pith paper

  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

    doi:10.1023/a:1011408618527

  2. [2]

    Applebaum

    D. Applebaum. L \' e vy Processes and Stochastic Calculus , volume 116 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2nd edition, 2009. doi:10.1017/CBO9780511809781

    doi:10.1017/cbo9780511809781 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

    doi:10.1142/10367 2018

  4. [4]

    Infraparticle states in the massless Nelson model -- revisited

    V. Beaud, W. Dybalski, and G. M. Graf. Infraparticle States in the Massless Nelson Model: Revisited. In Press, 2021, arXiv:2105.05723 http://arxiv.org/abs/2105.05723 , doi:10.1007/s00023-022-01261-2

    work page Pith review arXiv 2021

  5. [5]

    Bachmann, D.-A

    S. Bachmann, D.-A. Deckert, and A. Pizzo. The mass shell of the N elson model without cut-offs. J. Funct. Anal. , 263(5):1224--1282, 2012, arXiv:1104.3271 http://arxiv.org/abs/1104.3271 . doi:10.1016/j.jfa.2012.04.021

    arXiv 2012

  6. [6]

    V. Bach, J. Fr \"o hlich, and I. M. Sigal. Renormalization Group Analysis of Spectral Problems in Quantum Field Theory. Adv. Math. , 137(2):205--298, 1998. doi:10.1006/aima.1998.1733

    work page arXiv 1998

  7. [7]

    J. Cannon. Quantum field theoretic properties of a model of N elson: Domain and eigenvector stability for perturbed linear operators. J. Funct. Anal. , 8(1):101--152, 1971. doi:10.1016/0022-1236(71)90023-1

    doi:10.1016/0022-1236(71)90023-1 1971

  8. [8]

    T. N. Dam. Absence of Ground States in the Translation Invariant Massless N elson Model. Ann. Henri Poincar\'e , 21(8):2655--2679, 2020, arXiv:1808.00088 http://arxiv.org/abs/1808.00088 . doi:10.1007/s00023-020-00928-y

    arXiv 2020

Show all 55 references
  1. [9]

    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

    2022 arXiv

  2. [10]

    Deckert and A

    D.-A. Deckert and A. Pizzo. Ultraviolet Properties of the Spinless, One-Particle Y ukawa Model. Commun. Math. Phys. , 327(3):887--920, 2014, arXiv:1208.2646 http://arxiv.org/abs/1208.2646 . doi:10.1007/s00220-013-1877-9

    2014 arXiv

  3. [11]

    Dybalski and A

    W. Dybalski and A. Pizzo. Coulomb Scattering in the Massless N elson model I . F oundations of Two-Electron Scattering. J. Stat. Phys. , 154(1):543--587, 2014, arXiv:1302.5001 http://arxiv.org/abs/1302.5001 . doi:10.1007/s10955-013-0857-y

    2014 arXiv

  4. [12]

    W. G. Faris. Invariant Cones and Uniqueness of the Ground State for Fermion Systems. J. Math. Phys. , 13(8):1285--1290, 1972. doi:10.1063/1.1666133

    1972 doi

  5. [13]

    Fr \"o hlich

    H. Fr \"o hlich. Electrons in Lattice Fields. Adv. Phys. , 3(11):325--361, 1954. doi:10.1080/00018735400101213

    1954 doi

  6. [14]

    Fr \" o hlich

    J. Fr \" o hlich. On the infrared problem in a model of scalar electrons and massless scalar bosons. Ann. de l'Inst. Henri Poincar\' e , 19(1):1--103, 1973

    1973

  7. [15]

    Fr \" o hlich

    J. Fr \" o hlich. Existence of dressed one-electron states in a class of persistent models. Fortschr. Phys. , 22(3):159--198, 1974. doi:10.1002/prop.19740220304

    1974 doi

  8. [16]

    Gubinelli, F

    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

    2014 arXiv

  9. [17]

    Glimm and A

    J. Glimm and A. Jaffe. The ( 4 ) 2 quantum field theory without cutoffs: II . The field operators and the approximate vacuum. Ann. Math. , 91:362--401, 1970. doi:10.2307/1970582

    1970 doi

  10. [18]

    G \" u neysu, O

    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:10.100...

    2017 arXiv

  11. [19]

    L. Gross. A noncommutative extension of the P erron- F robenius theorem. Bull. Amer. Math. Soc. , 77:343--347, 1971. doi:10.1090/S0002-9904-1971-12686-1

    1971 doi

  12. [20]

    L. Gross. Existence and uniqueness of physical ground states. J. Funct. Anal. , 10(1):52--109, 1972. doi:10.1016/0022-1236(72)90057-2

    1972 doi

  13. [21]

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

    1973 doi

  14. [22]

    Griesemer and A

    M. Griesemer and A. W \"u nsch. On the domain of the N elson H amiltonian. J. Math. Phys. , 59(4):042111, 2018, arXiv:1711.10916 http://arxiv.org/abs/1711.10916 . doi:10.1063/1.5018579

    2018 arXiv

  15. [23]

    Hasler, B

    D. Hasler, B. Hinrichs, and O. Siebert. Non- F ock ground states in the translation-invariant N elson model revisited non-perturbatively. J. Funct. Anal. , 286(7):110319, 2024, arXiv:2302.06998 http://arxiv.org/abs/2302.06998 . doi:10.1016/j.jfa.2024.110319

    2024 arXiv

  16. [24]

    Hiroshima

    F. Hiroshima. Ground states of a model in nonrelativistic quantum electrodynamics. II . J. Math. Phys. , 41(2):661--674, 2000. doi:10.1063/1.533158

    doi

  17. [25]

    Hiroshima

    F. Hiroshima. Fiber H amiltonians in non-relativistic quantum electrodynamics. J. Funct. Anal. , 252(1):314--355, 2007, arXiv:math-ph/0607014 http://arxiv.org/abs/math-ph/0607014 . doi:10.1016/j.jfa.2007.06.006

    2007 arXiv

  18. [26]

    Hiroshima

    F. Hiroshima. Translation invariant models in QFT without ultraviolet cutoffs. Preprint, 2015, arXiv:1506.07514 http://arxiv.org/abs/1506.07514

    2015 arXiv

  19. [27]

    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: Applications in Rigorous Quantum Field Theory , volume 34/2 of De Gruyter Studies in Mathematics . De Gruyter, Berlin, 2nd edition, 2020. doi:10.1515/9783110403541

    2020 doi

  20. [28]

    Hinrichs and J

    B. Hinrichs and J. Lampart. A Lower Bound on the Critical Momentum of an Impurity in a Bose--Einstein Condensate. C.\,R. Math. , 362:1399--1412, 2024, arXiv:2311.05361 http://arxiv.org/abs/2311.05361 . doi:10.5802/crmath.652

    2024 arXiv

  21. [29]

    Hiroshima and O

    F. Hiroshima and O. Matte. Ground states and their associated path measures in the renormalized N elson model. Rev. Math. Phys. , 34(2):2250002, 2022, arXiv:1903.12024 http://arxiv.org/abs/1903.12024 . doi:10.1142/S0129055X22500027

    2022 arXiv

  22. [30]

    Hinrichs and O

    B. Hinrichs and O. Matte. F eynman-- K ac formula for fiber H amiltonians in the relativistic N elson model in two spatial dimensions. Preprint, 2023, arXiv:2309.09005 http://arxiv.org/abs/2309.09005

    2023 arXiv

  23. [31]

    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. Ann. Henri Poincar\'e , 25(6):2877--2940, 2024, arXiv:2211.14046 http://arxiv.org/abs/2211.14046 . doi:10.1007/s00023-0...

    2024 arXiv

  24. [32]

    Hinrichs and O

    B. Hinrichs and O. Matte. F eynman-- K ac formulas for semigroups generated by multi-polaron H amiltonians in magnetic fields and on general domains. Preprint, 2024, arXiv:2403.12147 http://arxiv.org/abs/2403.12147

    2024 arXiv

  25. [33]

    J. Lampart. The renormalized B ogoliubov- F r\" o hlich H amiltonian. J. Math. Phys. , 61(10):101902, 2020, arXiv:1909.02430 http://arxiv.org/abs/1909.02430 . doi:10.1063/5.0014217

    2020 arXiv

  26. [34]

    J. Lampart. The Resolvent of the N elson H amiltonian Improves Positivity. Math. Phys. Anal. Geom. , 24(1):2, 2021, arXiv:2010.03235 http://arxiv.org/abs/2010.03235 . doi:10.1007/s11040-021-09374-6

    2021 arXiv

  27. [35]

    J. Lampart. Hamiltonians for Polaron Models with Subcritical Ultraviolet Singularities. Ann. Henri Poincar \'e , 24(8):2687--2728, 2023, arXiv:2203.07253 http://arxiv.org/abs/2203.07253 . doi:10.1007/s00023-023-01285-2

    2023 arXiv

  28. [36]

    L o rinczi, F

    J. L o rinczi, F. Hiroshima, and V. Betz. Feynman- K ac-Type Theorems and G ibbs Measures on Path Space. Volume 1: Feynman-Kac-Type Formulae and Gibbs Measures , volume 34/1 of De Gruyter Studies in Mathematics . De Gruyter, Berlin, 2nd edition, 2020. doi:10.1515/9783110330397...

    2020 doi

  29. [37]

    T.-D. Lee, F. E. Low, and D. Pines. The motion of slow electrons in a polar crystal. Phys. Rev. , 90(2):297--302, 1953. doi:10.1103/PhysRev.90.297

    1953 doi

  30. [38]

    Lampart and J

    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

    2019 arXiv

  31. [39]

    E. H. Lieb and K. Yamazaki. Ground-State Energy and Effective Mass of the Polaron. Phys. Rev. , 111:728--733, 1958. doi:10.1103/PhysRev.111.728

    1958 doi

  32. [40]

    Y. Miura. On order of operators preserving selfdual cones in standard forms. Far East J. Math. Sci. , 8(1):1--9, 2003

    2003

  33. [41]

    T. Miyao. On the semigroup generated by the renormalized N elson H amiltonian. J. Funct. Anal. , 276(6):1948--1977, 2019, arXiv:1803.08659 http://arxiv.org/abs/1803.08659 . doi:10.1016/j.jfa.2018.11.001

    1948 arXiv

  34. [42]

    T. Miyao. On Renormalized H amiltonian Nets. Ann. Henri Poincar\' e , 22(9):2935--2973, 2021, arXiv:1810.12716 http://arxiv.org/abs/1810.12716 . doi:10.1007/s00023-021-01029-0

    2021 arXiv

  35. [43]

    Matte and J

    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

    2018 arXiv

  36. [44]

    J. S. M ller. The Translation Invariant Massive N elson Model: I . T he Bottom of the Spectrum. Ann. Henri Poincar\' e , 6(6):1091--1135, 2005. doi:10.1007/s00023-005-0234-8

    2005 doi

  37. [45]

    J. S. M ller. On the Essential Spectrum of the Translation Invariant N elson Model. In Mathematical Physics of Quantum Mechanics , volume 690 of Lecture Notes in Physics , pages 179--195. Springer, Berlin, 2006. doi:10.1007/3-540-34273-7\_15

    2006 doi

  38. [46]

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

    1964 doi

  39. [47]

    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

    1992 doi

  40. [48]

    A. Pizzo. One-particle (improper) States in N elson's Massless Model. Ann. Henri Poincar\' e , 4(3):439--486, 2003. doi:10.1007/s00023-003-0136-6

    2003 doi

  41. [49]

    A. Pizzo. Scattering of an I nfraparticle : The One Particle Sector in N elson's Massless Model. Ann. Henri Poincar\' e , 6(3):553--606, 2005. doi:10.1007/s00023-005-0216-x

    2005 doi

  42. [50]

    S. Polzer. Renewal approach for the energy--momentum relation of the F r\" o hlich polaron. Lett. Math. Phys. , 113(4):90, 2023, arXiv:2206.14425 http://arxiv.org/abs/2206.14425 . doi:10.1007/s11005-023-01711-w

    2023 arXiv

  43. [51]

    Reed and B

    M. Reed and B. Simon. Analysis of Operators , volume 4 of Methods of Modern Mathematical Physics . Academic Press, San Diego, 1978

    1978

  44. [52]

    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

    2019 arXiv

  45. [53]

    B. Simon. Ergodic Semigroups of Positivity Preserving Self-Adjoint Operators. J. Funct. Anal. , 12:335--339, 1973. doi:10.1016/0022-1236(73)90084-0

    1973 doi

  46. [54]

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

    1974 doi

  47. [55]

    Vuillermot

    P.-A. Vuillermot. A generalization of C hernoff's product formula for time-dependent operators. J. Funct. Anal. , 259(11):2923--2938, 2010. doi:10.1016/j.jfa.2010.07.018

    2010 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.