Searcharxiv⌕ Search

arXiv subjects

Paweł Duch

Publications and source records attributed to Paweł Duch.

17 recordsLinked to original sources
arXiv2026

Lecture notes on the flow equation approach to singular stochastic PDEs

The flow equation approach is a robust framework applicable to a broad class of singular SPDEs, including those with fractional Laplacians, throughout the entire subcritical regime. Inspired by Wilson's renormalization group, this method studies the coarse-grained process, which captures the behaviour of solutions across spatial scales. The corresponding flow equation describes how the nonlinear terms in the effective dynamics evolve with the coarse-graining scale, playing a role analogous to the Polchinski equation in quantum field theory. The renormalization problem is then solved inductively by imposing appropriate boundary conditions on the flow equation.

math.PR↗
arXiv2025

Parabolic stochastic quantisation of the fractional $Φ^4_3$ model in the full subcritical regime

We present a construction of the fractional $Φ^4$ Euclidean quantum field theory on $\mathbb{R}^3$ in the full subcritical regime via parabolic stochastic quantisation. Our approach is based on the use of a truncated flow equation for the effective description of the model at sufficiently small scales and on coercive estimates for the non-linear stochastic partial differential equation describing the interacting field. The measure is invariant under translations, reflection positive and has quartic exponential tails.

math.PR↗
arXiv2025

Ergodicity of infinite volume $Φ^4_3$ at high temperature

We consider the infinite volume $Φ^4_3$ dynamic and show that it is globally well-posed in a suitable weighted Besov space of distributions. At high temperatures / small coupling, we furthermore show that the difference between any two solutions driven by the same realisation of the noise converges to zero exponentially fast. This allows us to characterise the infinite-volume $Φ^4_3$ measure at high temperature as the unique invariant measure of the dynamic, and to prove that it satisfies all Osterwalder--Schrader axioms, including invariance under translations, rotations, and reflections, as well as exponential decay of correlations.

math.PR↗
arXiv2025

Flow equation approach to singular stochastic PDEs

We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on $\mathbb{R}_+\times\mathbb{T}$ with fractional Laplacian $(-Δ)^{σ/2}$, additive noise and polynomial non-linearity, where $\mathbb{T}$ is the $d$-dimensional torus. We consider the weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. We prove that the macroscopic scaling limit exists and has a universal law characterized by parameters of the relevant perturbations of the linear equation. We develop a new solution theory for singular SPDEs of the above-mentioned form using the Wilsonian renormalization group theory and the Polchinski flow equation. In particular, in the case of $d=4$ and the cubic non-linearity our analysis covers the whole sub-critical regime $σ>2$. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches.

math.PR↗
arXiv2024

Construction of Gross-Neveu model using Polchinski flow equation

The Gross-Neveu model is a quantum field theory model of Dirac fermions in two dimensions with a quartic interaction term. Like Yang-Mills theory in four dimensions, the model is scaling critical (i.e. renormalizable but not super-renormalizable) and asymptotically free (i.e. its short-distance behavior is governed by the free theory). We give a new construction of the massive Euclidean Gross-Neveu model in infinite volume. The distinctive feature of the construction is that it does not involve cluster expansion, discretization of phase-space or a tree expansion ansatz and is based solely on the renormalization group flow equation. We express the Schwinger functions of the Gross-Neveu model in terms of the effective potential and construct the effective potential by solving the flow equation using the Banach fixed point theorem. Moreover, we construct a random field in the probability space of the free field such that its moments coincide with the Schwinger functions of the Gross-Neveu model. This is the first construction of a strong coupling between the free and interacting fields for a scaling-critical QFT. Since we use crucially the fact that fermionic fields can be represented as bounded operators our construction does not extend to models including bosons. However, it is applicable to other asymptotically free purely fermionic theories such as the symplectic fermion model.

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↗
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↗
arXiv2021

Infrared problem in perturbative quantum field theory

We propose a mathematically rigorous construction of the scattering matrix and the interacting fields in models of relativistic perturbative quantum field theory with massless fields and long-range interactions. We consider quantum electrodynamics and a certain model of interacting scalar fields in which the standard definition of the scattering matrix is not applicable because of the infrared problem. We modify the Bogoliubov construction using the ideas of Dollard, Kulish and Faddeev. Our modified scattering matrix and modified interacting fields are constructed with the use of the adiabatic limit which is expected to exist in arbitrary order of perturbation theory. In the paper we prove this assertion in the case of the first- and the second-order corrections to the modified scattering matrix and the first-order corrections to the modified interacting fields. Our modified scattering matrix and modified interacting fields are defined in the standard Fock space. However, the particle interpretation of states in this space is non-standard. In particular, the electrons and positrons are always surrounded by irremovable clouds of photons. Moreover, the physical energy-momentum operators do not coincide with the standard ones and their joint spectrum does not contain the mass hyperboloid.

math-ph↗
arXiv2018

Weak adiabatic limit in quantum field theories with massless particles

We construct the Wightman and Green functions in a large class of models of perturbative QFT in the four-dimensional Minkowski space in the Epstein-Glaser framework. To this end we prove the existence of the weak adiabatic limit, generalizing the results due to Blanchard and Seneor. Our proof is valid under the assumption that the time-ordered products satisfy certain normalization condition. We show that this normalization condition may be imposed in all models with interaction vertices of canonical dimension 4 as well as in all models with interaction vertices of canonical dimension 3 provided each of them contains at least one massive field. Moreover, we prove that it is compatible with all the standard normalization conditions which are usually imposed on the time-ordered products. The result applies, for example, to quantum electrodynamics and non-abelian Yang-Mills theories.

math-ph↗
arXiv2017

Massless fields and adiabatic limit in quantum field theory

The thesis is devoted to a rigorous construction of the Wightman and Green functions in models of the perturbative quantum field theory in the four-dimensional Minkowski spacetime in the framework of the causal perturbation theory developed by Epstein and Glaser. In the first part of the thesis we give an overview of the Epstein-Glaser approach to the perturbative quantum field theory. In the second part we construct the Wightman and Green functions in a large class of models, generalizing the result due to Blanchard and Seneor. To this end, we show the existence of the so-called weak adiabatic limit. The proof of the existence of this limit is valid under the assumption that the time-ordered products satisfy certain normalization condition. We show that this normalization condition may be imposed in all models with interaction vertices of the canonical dimension equal to four as well as in all models with interaction vertices of the canonical dimension equal to three provided each of them contains at least one massive field. Moreover, we prove that the above-mentioned normalization condition is compatible with all the standard normalization conditions which are usually imposed on the time-ordered products. We consider in detail the case of the quantum electrodynamics with a massive or massless, spinor or scalar charged field and certain model of interacting scalar fields with the interaction vertex of dimension three which we call the scalar model. Our result is also applicable to non-abelian Yang-Mills theories. Using the method developed in the proof of the existence of the weak adiabatic limit, we also show the existence of the central splitting solution in the quantum electrodynamics with a massive spinor field.

math-ph↗
arXiv2016

Observer's observables. Residual diffeomorphisms

We investigate the fate of diffeomorphisms when the radial gauge is imposed in canonical general relativity. As shown elsewhere, the radial gauge is closely related to the observer's observables. These observables are invariant under a large subgroup of diffeomorphisms which results in their usefulness for canonical general relativity. There are, however, some diffeomorphisms, called residual diffeomorphisms, which might be "observed" by the observer as they do not preserve her observables. The present paper is devoted to the analysis of these diffeomorphisms in the case of the spatial and spacetime radial gauges. Although the residual diffeomorphisms do not form a subgroup of all diffeomorphisms, we show that their induced action in the phase space does form a group. We find the generators of the induced transformations and compute the structure functions of the algebras they form. The obtained algebras are deformations of the algebra of the Euclidean group and the algebra of the Poincaré group in the spatial and spacetime case, respectively. In both cases the deformation depends only on the Riemann curvature tensor and in particular vanishes when the space or spacetime is flat.

gr-qc↗
arXiv2016

The algebra of observables in Gaußian normal spacetime coordinates

We discuss the canonical structure of a spacetime version of the radial gauge, i.e. Gaußian normal spacetime coordinates. While it was found for the spatial version of the radial gauge that a "local" algebra of observables can be constructed, it turns out that this is not possible for the spacetime version. The technical reason for this observation is that the new gauge condition needed to upgrade the spatial to a spacetime radial gauge does not Poisson-commute with the previous gauge conditions. It follows that the involved Dirac bracket is inherently non-local in the sense that no complete set of observables can be found which is constructed locally and at the same time has local Dirac brackets. A locally constructed observable here is defined as a finite polynomial of the canonical variables at a given physical point specified by the Gaußian normal spacetime coordinates.

gr-qc↗
arXiv2015

Addendum: Observables for General Relativity related to geometry

In this addendum we clarify a point which strengthens one of the results from [the original paper]. Namely, we show that the algebra of the observables $F(r,θ)$ is yet simpler then it was described in [the original paper]. This is an important point, because with this simplification an important subalgebra becomes canonical, allowing for a natural reduction of the phase space.

gr-qc↗
arXiv2014

Massless asymptotic fields and Haag-Ruelle theory

We revisit the problem of the existence of asymptotic massless boson fields in quantum field theory. The well-known construction of such fields by Buchholz [2], [4] is based on locality and the existence of vacuum vector, at least in regions spacelike to spacelike cones. Our analysis does not depend on these assumptions and supplies a more general framework for fields only very weakly decaying in spacelike directions. In this setting the existence of appropriate null asymptotes of fields is linked with their spectral properties in the neighborhood of the lightcone. The main technical tool is one of the results of a recent analysis by one of us [11], which allows application of the null asymptotic limit separately to creation/annihilation parts of a wide class of non-local fields. In vacuum representation the scheme allows application of the methods of the Haag-Ruelle theory closely analogous to those of the massive case. In local case this Haag-Ruelle procedure may be combined with the Buchholz method, which leads to significant simplification.

math-ph↗
arXiv2014

Observables for General Relativity related to geometry

We present a new scheme of defining invariant observables for general relativistic systems. The scheme is based on the introduction of an observer which endowes the construction with a straightforward physical interpretation. The observables are invariant with respect to spatial diffeomorphisms which preserve the observer. The limited residual spatial gauge freedom is studied and fully understood. A full canonical analysis of the observables is presented: we analyze their variations, Poisson algebra and discuss their dynamics. Lastly, the observables are used to solve the vector constraint, which triggers a possible considerable reduction of the degrees of freedom of general relativistic theories.

gr-qc↗
arXiv2012

Quantum Schwarzschild space-time

Using new approach to construction of space-times emerging from quantum information theory, we identify the space of quantum states that generates the Schwarzschild space-time. No quantisation procedure is used. The emergent space-time is obtained by the Poincaré-Wick rotation and Fronsdal embedding of certain submanifold of the riemanian manifold of six-dimensional strictly positive matrices with the Bogolyubov-Kubo-Mori metric.

gr-qc↗