A mathematical analysis has derived conditional lower bounds on how quickly the ground-state norm can shrink with distance in two versions of the Pauli–Fierz model: the full model and its dipole approximation. The result does not give a measured decay rate. It sets a model-dependent lower floor under the norm’s spatial decay, subject to the assumptions used in the analysis.
The comparison uses a probabilistic Feynman–Kac approach for the full model and an Agmon-distance approach for the dipole model. An Agmon distance is a model-defined cost for travelling through space from one point to another; here, the resulting estimates describe how small the ground-state norm may become at large distance.
For the full model, the analysis first obtains a product-form lower bound for the positive vacuum component, written as (1, φg(x))F. It also gives an asymptotic result for potentials satisfying V(x)≤|x|^(2n) sufficiently far from the origin: the ground-state norm has a positive exponential lower bound with an exponent that scales as |x|^(n+1) for sufficiently large |x|.
Why the full model is harder to control
The main technical difficulty in the full model comes from a path-dependent stochastic integral in its Feynman–Kac representation. The analysis estimates an associated exponential moment, meaning an average of a random quantity after large values have been given extra weight. But the allowed range of the parameter ε narrows as the time parameter T increases.
The stated estimate has the form E[e^(εX)]≤C1(ε)e^(εC2T), with ε restricted below a threshold that depends on T. In the paper’s account, this prevents the estimates needed for a direct Agmon-distance argument from being uniform in T. The reported obstruction concerns those uniform direct estimates, while separate lower bounds are also stated for the full model.
A phase-rotation and positivity argument is central to the full-model result. Working with Θ⁻¹φg, the analysis establishes that the vacuum component (1, φg(x))F has a continuous, strictly positive version. The product-form lower bound for that component brings the potential in through Wa(x) and applies under the theorem’s stated path and parameter conditions.
The full model also has an Agmon-distance lower bound, but its exponent contains more than a linear term. Under the stated condition (5.1) and Assumption 4.2, the norm is bounded below by C exp[−(1+ε)G(γ)−εG(γ)^2] for |x|>Rε and every path γ from 0 to x. The quadratic correction distinguishes this estimate from the pure linear Agmon exponent stated for the dipole model.
The dipole result has a cleaner distance form
Under the dipole approximation, the relevant integrand becomes deterministic, and the admissible range of ε no longer depends on T. For 0≤ε<3/(8α²‖φ̂/ω‖²), the analysis gives E[e^(εα²Xdip)]≤e^(cεT) for all T>0, with cε independent of T. The exponential-moment calculation explicitly uses a Fredholm determinant, a determinant-based operator tool.
The dipole analysis states a pure Agmon-distance lower bound. Under the stated p,q restriction and Assumption 4.2, the dipole-model norm obeys ∥φg(x)∥F ≥ Ce−(p+ε)G(γ) for sufficiently large |x| along every path γ from 0 to x.
For a potential satisfying V(x)≤|x|^(2n), with n>0, the corresponding dipole lower bound has an exponent of (√2/(n+1))(p+ε)|x|^(n+1) when |x|>Rε, under the same stated conditions. The analysis does not establish that this exponent is optimal.
What the bounds do—and do not—say
This is a theoretical comparison of mathematical model variants, not an empirical study. It samples no participants or observations; the mathematical inputs are Brownian paths and Gaussian field variables used in the functional-integral representation. The explicit power-growth conclusions are asymptotic and conditional on the stated assumptions for the potential and the model.
The supplied work is an arXiv preprint, version 1, dated 26 Aug 2026. The analysis leaves open whether the path-dependent double stochastic integral in the full model can be controlled with an admissible ε range independent of T. It also says that the dipole theorem may not be optimal.
The reported conclusions remain conditional on the stated potential and model assumptions. The supplied material includes no empirical observations, and the results are not presented as evidence about observed physical or human outcomes.
Paper data and sources
Original title: Lower bounds on the spatial decay of the ground state of the Pauli-Fierz model
Authors: Fumio Hiroshima, Yuki Tsujimoto
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text