{"title":"量子场的应力张量约束","authors":"Ko Sanders","doi":"10.1007/s00220-024-05017-3","DOIUrl":null,"url":null,"abstract":"<div><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></div>","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":"https://link.springer.com/content/pdf/10.1007/s00220-024-05017-3.pdf","citationCount":"0","resultStr":"{\"title\":\"Stress Tensor Bounds on Quantum Fields\",\"authors\":\"Ko Sanders\",\"doi\":\"10.1007/s00220-024-05017-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><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. 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引用次数: 0
摘要
闵科夫斯基空间中量子场的奇异行为通常可以用哈密顿 H 的多项式来约束。这些所谓的 H 约束和相关技术使我们能够以数学上严谨的方式处理点量子场及其算子乘积展开。然而,这种方法的一个缺点是,哈密顿是一个全局而非局部算子,而且,它没有在一般的弯曲时空中定义。为了克服这一缺点,我们研究了用应力张量的一个分量(本质上是一种能量密度)来替代 H 的可能性,从而得到类似的边界。为了明确起见,我们考虑了一个大质量、最小耦合的自由赫米特标量场。利用关于正类型分布的新结果,我们证明了在任何全局双曲洛伦兹流形M中,对于在\(\textrm{supp}(f)\)上具有\(F\equiv 1\) 的任何\(f,F\in C_0^{\infty }(M)\) 和任何时间平滑矢量场\(t^{\mu }\) ,我们可以找到常数\(c,C>;0\) 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 \(\omega \)。这本质上是一种新型的量子能量不等式,它需要对熏染量子场进行应力张量约束。在(1+1)维度中,我们还建立了一个关于点量子场的约束,即 \(|\omega (\phi (x))|\le C(\omega (T^{\textrm{ren}}_{\mu \nu }(t^{\mu }t^{\nu }F^2))+c)\), 其中 \(F\equiv 1\) near x.
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.