The Euclidean $ϕ^4_2$ theory as a limit of an interacting Bose gas

Fröhlich, Jürg; Knowles, Antti; Schlein, Benjamin; Sohinger, Vedran

doi:10.48550/arxiv.2201.07632

Cited by 3 publications

(4 citation statements)

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“…We now conclude the proof of (2.41) by using (2.43) and a union bound as in [54, (A.18)-(A. 24)]. In order to apply the above union bound argument, one needs to show that, given a dyadic integer M , and…”

Section: )mentioning

confidence: 75%

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A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials

Sohinger

2021

International Mathematics Research Notices

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We study the derivation of the Gibbs measure for the nonlinear Schrödinger (NLS) equation from many-body quantum thermal states in the mean-field limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded $L^p$ interaction potentials on $\mathbb{T}^d$ for $d=1,2,3$. This extends the author’s earlier joint work with Fröhlich et al. [ 45], where the regime of defocusing and bounded interaction potentials was considered. When $d=1$, we give an alternative proof of a result previously obtained by Lewin et al. [ 69]. Our proof is based on a perturbative expansion in the interaction. When $d=1$, the thermal state is the grand canonical ensemble. As in [ 45], when $d=2,3$, the thermal state is a modified grand canonical ensemble, which allows us to estimate the remainder term in the expansion. The terms in the expansion are analysed using a graphical representation and are resummed by using Borel summation. By this method, we are able to prove the result for the optimal range of $p$ and obtain the full range of defocusing interaction potentials, which were studied in the classical setting when $d=2,3$ in the work of Bourgain [ 15].

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confidence: 75%

“…The method in [21] is based on a functional integral formulation. In [24], the d = 2 result of [21] was extended to the complex Euclidean Φ 4 2 theory, which corresponds to taking w = δ.…”

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confidence: 99%

A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials

Sohinger

2021

International Mathematics Research Notices

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“…Here, we are interested in the dynamics of mean-field bosons, initially prepared in an equilibrium state at positive temperature (Gibbs state). Just above the critical temperature for BEC, correlation functions of the many-body Gibbs state have been shown to converge, in the mean-field limit, to correlation functions of the invariant measure associated with the nonlinear Hartree equation, see [32,34,35,52,53]; observe that, in dimensions d = 2, 3, this requires an appropriate renormalization of the interaction. In [33], also timedependent correlations have been proven to converge.…”

Section: Background and Summarymentioning

confidence: 99%

Dynamics of mean-field bosons at positive temperature

Caporaletti¹,

Deuchert²,

Schlein³

2022

Preprint

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We study the time-evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in [24] that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in N, as N → ∞, by the one of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree-Fock-Bogoliubov equations to define a fluctuation dynamics.

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“…More recent reviews include [104][105][106] and a talk [107]. Through a lot of work starting from the late sixties, success has been achieved in rigorously constructing and studying 2d scalar theories with arbitrary polynomial interaction (the so-called P(φ) 2 theories) [88,[108][109][110][111][112][113][114][115][116][117][118][119][120][121][122][123][124][125][126][127] (see also book [81], recent papers [128,129], and [130][131][132][133] on other potentials), threedimensional λφ 4 theory [134][135][136][137][138][139][140][141][142] (see also more recent [143][144][145][146][147]), Gross-Neveu theory [148]…”

Section: Introductionmentioning

confidence: 99%

Snowmass White Paper: The Quest to Define QFT

Dedushenko¹

2022

Preprint

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This article provides a review of the literature on rigorous definitions and constructions in Quantum Field Theory, spanning the period of seven decades. Comparing with the ideas and constructions that exist in the modern physics literature, we conclude that none of the existing systems of QFT axioms can cover all the physical situations. Therefore, it is still an outstanding open problem to formulate a complete definition of QFT. We argue that the question is of relevance for both physicists and mathematicians.

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The Euclidean $ϕ^4_2$ theory as a limit of an interacting Bose gas

Cited by 3 publications

References 48 publications

A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials

A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials

Dynamics of mean-field bosons at positive temperature

Snowmass White Paper: The Quest to Define QFT

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