{"title":"Stress Tensor Bounds on Quantum Fields","authors":"Ko Sanders","doi":"10.1007/s00220-024-05017-3","DOIUrl":null,"url":null,"abstract":"<p>The singular behaviour of quantum fields in Minkowski space can often be bounded by polynomials of the Hamiltonian <i>H</i>. These so-called <i>H</i>-bounds and related techniques allow us to handle pointwise quantum fields and their operator product expansions in a mathematically rigorous way. A drawback of this approach, however, is that the Hamiltonian is a global rather than a local operator and, moreover, it is not defined in generic curved spacetimes. In order to overcome this drawback we investigate the possibility of replacing <i>H</i> by a component of the stress tensor, essentially an energy density, to obtain analogous bounds. For definiteness we consider a massive, minimally coupled free Hermitean scalar field. Using novel results on distributions of positive type we show that in any globally hyperbolic Lorentzian manifold <i>M</i> for any <span>\\(f,F\\in C_0^{\\infty }(M)\\)</span> with <span>\\(F\\equiv 1\\)</span> on <span>\\(\\textrm{supp}(f)\\)</span> and any timelike smooth vector field <span>\\(t^{\\mu }\\)</span> we can find constants <span>\\(c,C>0\\)</span> such that <span>\\(\\omega (\\phi (f)^*\\phi (f))\\le C(\\omega (T^{\\textrm{ren}}_{\\mu \\nu }(t^{\\mu }t^{\\nu }F^2))+c)\\)</span> for all (not necessarily quasi-free) Hadamard states <span>\\(\\omega \\)</span>. This is essentially a new type of quantum energy inequality that entails a stress tensor bound on the smeared quantum field. In <span>\\(1+1\\)</span> dimensions we also establish a bound on the pointwise quantum field, namely <span>\\(|\\omega (\\phi (x))|\\le C(\\omega (T^{\\textrm{ren}}_{\\mu \\nu }(t^{\\mu }t^{\\nu }F^2))+c)\\)</span>, where <span>\\(F\\equiv 1\\)</span> near <i>x</i>.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-05017-3","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
The singular behaviour of quantum fields in Minkowski space can often be bounded by polynomials of the Hamiltonian H. These so-called H-bounds and related techniques allow us to handle pointwise quantum fields and their operator product expansions in a mathematically rigorous way. A drawback of this approach, however, is that the Hamiltonian is a global rather than a local operator and, moreover, it is not defined in generic curved spacetimes. In order to overcome this drawback we investigate the possibility of replacing H by a component of the stress tensor, essentially an energy density, to obtain analogous bounds. For definiteness we consider a massive, minimally coupled free Hermitean scalar field. Using novel results on distributions of positive type we show that in any globally hyperbolic Lorentzian manifold M for any \(f,F\in C_0^{\infty }(M)\) with \(F\equiv 1\) on \(\textrm{supp}(f)\) and any timelike smooth vector field \(t^{\mu }\) we can find constants \(c,C>0\) such that \(\omega (\phi (f)^*\phi (f))\le C(\omega (T^{\textrm{ren}}_{\mu \nu }(t^{\mu }t^{\nu }F^2))+c)\) for all (not necessarily quasi-free) Hadamard states \(\omega \). This is essentially a new type of quantum energy inequality that entails a stress tensor bound on the smeared quantum field. In \(1+1\) dimensions we also establish a bound on the pointwise quantum field, namely \(|\omega (\phi (x))|\le C(\omega (T^{\textrm{ren}}_{\mu \nu }(t^{\mu }t^{\nu }F^2))+c)\), where \(F\equiv 1\) near x.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.