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Wojciech Dybalski

Publications and source records attributed to Wojciech Dybalski.

At least 19 recordsLinked to original sources
arXiv2025

Exact Schwinger functions for a class of bounded interactions in $d\geq 2$

We consider a scalar Euclidean QFT with interaction given by a bounded, measurable function $V$ such that $V^{\pm}:=\lim_{w\to \pm\infty}V(w)$ exist. We find a field renormalization such that all the $n$-point connected Schwinger functions for $n\neq 2$ exist non-perturbatively in the UV limit. They coincide with the tree-level one-particle irreducible Schwinger functions of the $\mathrm{erf}(ϕ/\sqrt{2})$ interaction with a coupling constant $\frac{1}{2} (V^+ - V^-)$. By a slight modification of our construction we can change this coupling constant to $\frac{1}{2} (V_+ - V_-)$, where $V_{\pm}:= \lim_{w\to 0^{\pm}} V(w)$. Thereby non-Gaussianity of these latter theories is governed by a discontinuity of $V$ at zero. The open problem of controlling also the two-point function of these QFTs is discussed.

math-ph↗
arXiv2024

Stochastic quantization of two-dimensional $P(Φ)$ Quantum Field Theory

We give a simple and self-contained construction of of the $P(Φ)$ Euclidean Quantum Field Theory in the plane and verify the Osterwalder-Schrader axioms: translational and rotational invariance, reflection positivity and regularity. In the intermediate steps of the construction we study measures on spheres. In order to control the infinite volume limit we use the parabolic stochastic quantization equation and the energy method. To prove the translational and rotational invariance of the limit measure we take advantage of the fact that the symmetry groups of the plane and the sphere have the same dimension.

math-ph↗
arXiv2024

Quantum phase transition of infrared radiation

We describe a phase transition of infrared radiation, driven by quantum fluctuations, which takes place at the boundary of (the conformal diagram of) Minkowski spacetime. Specifically, we consider a family of states interpolating between the vacuum and the Kraus-Polley-Reents infravacuum. A state from this family can be imagined as a static source emitting flashes of infrared radiation in distant past. The flashes are in suitable squeezed states and the time intervals between them are controlled by a certain parameter r. For r<0 the states are lightcone normal, thus physically indistinguishable from local excitations of the vacuum. They suffer from the usual infrared problems such as disintegration of the Bloch-Nordsieck S-matrix and rotational symmetry breaking by soft photon clouds. However, for r>0 lightcone normality breaks down, the S-matrix is stabilized by the Kraus-Polley-Reents mechanism and the rotational symmetry is restored. We interpret these two situations as ordered (r<0) and disordered (r>0) phase of infrared radiation, and show that they can be distinguished by asymptotic fluctuations of the fields. We also determine the singular behaviour of some S-matrix elements near the critical point r=0.

hep-th↗
arXiv2024

The Bałaban variational problem in the non-linear sigma model

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Bałaban's approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the $O(4)$ non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Bałaban's approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.

math-ph↗
arXiv2023

Infrared problem in quantum electrodynamics

The infrared problem in quantum electrodynamics consists of intriguing difficulties in scattering theory appearing at large scales and low energies. Although they can be circumvented using ad hoc recipes, such as the inclusive collision cross sections, there have been continuing efforts to reach a conceptually clear and mathematically rigorous understanding. In this article we focus on such developments of the last two decades. We start from the Buchholz-Roberts approach in the setting of algebraic QFT, which is based on the idea that there should be no infrared problems inside the future lightcone. Then we move on to the setting of non-relativistic QED, where insights from Haag-Ruelle scattering theory and the Faddeev-Kulish formalism suggest concrete formulas for the physical electron. Finally, in a setting of perturbative QFT, we outline a recent proposal for an infrared finite scattering matrix, which is also in the spirit of the Faddeev-Kulish approach.

math-ph↗
arXiv2023

Lattice Green functions for pedestrians: Exponential decay

The exponential decay of lattice Green functions is one of the main technical ingredients of the Bałaban's approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The main sources of exponential decay are the Combes-Thomas method and the analyticity of the Fourier transforms. They are combined using a renormalization group equation and the method of images.

math-ph↗
arXiv2021

Asymptotic Completeness in a Class of Massive Wedge-Local Quantum Field Theories in any Dimension

A recently developed n-particle scattering theory for wedge-local quantum field theories is applied to a class of models described and constructed by Grosse, Lechner, Buchholz, and Summers. In the BLS-deformation setting we establish explicit expressions for n-particle wave operators and the S-matrix of ordered asymptotic states, and we show that ordered asymptotic completeness is stable under the general BLS-deformation construction. In particular the (ordered) Grosse-Lechner S-matrices are non-trivial also beyond two-particle scattering and factorize into 2-particle scattering processes, which is an unusual feature in space-time dimension d > 1 + 1. Most notably, the Grosse-Lechner models provide the first examples of relativistic (wedge-local) QFT in space-time dimension d > 1 + 1 which are interacting and asymptotically complete.

math-ph↗
arXiv2021

Interacting massless infraparticles in 1+1 dimensions

The Buchholz' scattering theory of waves in two dimensional massless models suggests a natural definition of a scattering amplitude. We compute such a scattering amplitude for charged infraparticles that live in the GNS representation of the 2$d$ massless scalar free field and obtain a non-trivial result. It turns out that these excitations exchange phases, depending on their charges, when they collide.

math-ph↗
arXiv2021

Local normality of infravacua and relative normalizers for relativistic systems

We revisit the problem of local normality of Kraus-Polley-Reents infravacuum representations and provide a straightforward proof based on the Araki-Yamagami criterion. We apply this result to the theory of superselection sectors. Namely, we extend the novel formalism of second conjugate classes and relative normalizers to the local relativistic setting.

math-ph↗
arXiv2021

Infraparticle states in the massless Nelson model -- revisited

We provide a new construction of infraparticle states in the massless Nelson model. The approximating sequence of our infraparticle state does not involve any infrared cut-offs. Its derivative w.r.t. the time parameter $t$ is given by a simple explicit formula. The convergence of this sequence as $t\to \infty$ to a non-zero limit is then obtained by the Cook method combined with stationary phase estimates. To apply the latter technique we exploit recent results on regularity of ground states in the massless Nelson model, which hold in the low coupling regime.

math-ph↗
arXiv2021

Relative normalizers of automorphism groups, infravacua and the problem of velocity superselection in QED

We advance superselection theory of pure states on a $C^*$-algebra $\mathfrak{A}$ outside of the conventional (DHR) setting. First, we canonically define conjugate and second conjugate classes of such states with respect to a given reference state $ω_{\mathrm{vac}}$ and background $a \in \mathrm{Aut}(\mathfrak{A})$. Next, for some subgroups $R \varsubsetneq S \varsubsetneq G\subset \mathrm{Aut}(\mathfrak{A})$ we study the family $\{\, ω_{\mathrm{vac}}\circ s\,|\, s\in S \}$ of infrared singular states whose superselection sectors may be disjoint for different $s$. We show that their conjugate and second conjugate classes always coincide provided that $R$ leaves the sector of $ω_{\mathrm{vac}}$ invariant and $a$ belongs to the relative normalizer $N_G(R,S):=\{\, g\in G\,|\, g\cdot S\cdot g^{-1}\subset R\,\}$. We study the basic properties of this apparently new group theoretic concept and show that the Kraus-Polley-Reents infravacuum automorphisms belong to the relative normalizers of the automorphism group of a suitable CCR algebra. Following up on this observation we show that the problem of velocity superselection in non-relativistic QED disappears at the level of conjugate and second conjugate classes, if they are computed with respect to an infravacuum background. We also demonstrate that for more regular backgrounds such merging effect does not occur.

math-ph↗
arXiv2021

Coulomb scattering in the massless Nelson model II. Regularity of ground states

For the massless Nelson model we provide detailed information about the dependence of the normalized ground states $\checkψ_{P,σ}$ of the fiber single-electron Hamiltonians $H_{P,σ}$ on the total momentum $P$ and the infrared cut-off $σ$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form \[ \|\partial_{P_i}\checkψ_{P,σ}\|,\ \| \partial_{P_i} \partial_{P_j}\checkψ_{P,σ} \|\leq \frac{c}{σ^{δ_{λ_0}}}, \] for some constant $c$ and a function of the maximal admissible coupling constant $λ_0\mapsto δ_{λ_0}$ s.t. $\lim_{λ_0\to 0}δ_{λ_0}=0$. These results hold both in the infrared-regular and infrared-singular case. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of Haag-Ruelle scattering theory. They should also be relevant for the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion.

math-ph↗
arXiv2021

Curing velocity superselection in non-relativistic QED by restriction to a lightcone

It is physically expected that plane-wave configurations of the electron in QED induce disjoint representations of the algebra of the electromagnetic fields. This phenomenon of velocity superselection, which is one aspect of the infrared problem, is mathematically well established in non-relativistic (Pauli-Fierz type) models of QED. We show that velocity superselection can be resolved in such models by restricting the electron states to the subalgebra of the fields localized in the future lightcone. This actually follows from a more general statement about equivalence of GNS representations for coherent states of the algebra of the future lightcone in free electromagnetism. Our analysis turns out to be meaningful in the non-relativistic setting and provides evidence in favour of the Buchholz-Roberts approach to infrared problems.

math-ph↗
arXiv2020

Coulomb scattering in the massless Nelson model IV. Atom-electron scattering

We consider the massless Nelson model with two types of massive particles which we call atoms and electrons. The atoms interact with photons via an infrared regular form-factor and thus they are Wigner-type particles with sharp mass-shells. The electrons have an infrared singular form-factor and thus they are infraparticles accompanied by soft-photon clouds correlated with their velocities. In the weak coupling regime we construct scattering states of one atom and one electron, and demonstrate their asymptotic clustering into individual particles. The proof relies on the Cook's argument, clustering estimates, and the non-stationary phase method. The latter technique requires sharp estimates on derivatives of the ground state wave functions of the fiber Hamiltonians of the model, which were proven in the earlier papers of this series. Although we rely on earlier studies of the atom-atom and electron-photon scattering in the Nelson model, the paper is written in a self-contained manner. A perspective on the open problem of the electron-electron scattering in this model is also given.

math-ph↗
arXiv2019

Asymptotic charges, large gauge transformations and inequivalence of different gauges in external current QED

In this paper we consider external current QED in the Coulomb gauge and in axial gauges for various spatial directions of the axis. For a non-zero electric charge of the current, we demonstrate that any two different gauges from this class correspond to quantum theories which are not unitarily equivalent. We show that the spacelike asymptotic flux of the electromagnetic field is the underlying superselected quantity. We also express the large gauge transformation linking two distinct axial gauges by the Wilson loop over a contour limited by the two axes. Thus the underlying physical mechanism appears to be related to the Aharonov-Bohm effect.

hep-th↗
arXiv2019

The Bisognano-Wichmann property for asymptotically complete massless QFT

We prove the Bisognano-Wichmann property for asymptotically complete Haag-Kastler theories of massless particles. These particles should either be scalar or appear as a direct sum of two opposite integer helicities, thus, e.g., photons are covered. The argument relies on a modularity condition formulated recently by one of us (VM) and on the Buchholz' scattering theory of massless particles.

math-ph↗
arXiv2019

Effective mass of the polaron -- revisited

Properties of the energy-momentum relation for the Fröhlich polaron are of continuing interest, especially for large values of the coupling constant. By combining spectral theory with the available results on the central limit theorem for the polaron path measure we prove that, except for an intermediate range of couplings, the inverse effective mass is strictly positive and coincides with the diffusion constant. Such a result is established also for polaron-type models with a suitable ultraviolet cut-off and for arbitrary values of the coupling constant. We point out a slightly stronger variant of the central limit theorem which would imply that the energy-momentum relation has a unique global minimum attained at zero momentum.

math-ph↗