广义相对论经典和量子哈密顿公式中的普朗克长度

IF 4.2 2区 物理与天体物理 Q2 PHYSICS, PARTICLES & FIELDS
Claudio Cremaschini
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引用次数: 0

摘要

在经典广义相对论(GR)的无约束同步变分公式框架下研究了普朗克长度(\(\ell _{P}\))的物理意义。这一理论设置允许建立明显协变拉格朗日和哈密顿理论为连续介质引力场的爱因斯坦场方程。结果表明,这种公式的特点是存在一种新的变分贡献,它由合适的4标量项的无穷级数和表示,其中耦合系数是普朗克长度的偶幂。然而,实现经典GR哈密顿理论的要求对保留可容许的普朗克长度幂项施加了严格的约束。事实上,除去平凡的规范常数,证明了最终只允许该级数的\(O\left( \ell _{P}^{0}\right) \)贡献,即与\(\ell _{P}\)无关的唯一贡献。因此,为了与哈密顿原理相一致,普朗克长度实际上不允许出现在经典水平上。这对相应的经典量子引力理论的数学建立产生了重要的影响,然后通过\(O\left( \ell _{P}^{2}\right) \)发现它是正确的。额外的含义涉及相关量子动量的物理意义及其在半经典极限中的意义,以及普朗克长度在相同量子引力领域中的作用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Planck length in classical and quantum Hamiltonian formulations of general relativity

The physical meaning of the Planck length (\(\ell _{P}\)) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the \(O\left( \ell _{P}^{0}\right) \) contribution of the series is ultimately permitted, namely the unique one which is independent of \(\ell _{P}\). Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through \(O\left( \ell _{P}^{2}\right) \). Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

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来源期刊
The European Physical Journal C
The European Physical Journal C 物理-物理:粒子与场物理
CiteScore
8.10
自引率
15.90%
发文量
1008
审稿时长
2-4 weeks
期刊介绍: Experimental Physics I: Accelerator Based High-Energy Physics Hadron and lepton collider physics Lepton-nucleon scattering High-energy nuclear reactions Standard model precision tests Search for new physics beyond the standard model Heavy flavour physics Neutrino properties Particle detector developments Computational methods and analysis tools Experimental Physics II: Astroparticle Physics Dark matter searches High-energy cosmic rays Double beta decay Long baseline neutrino experiments Neutrino astronomy Axions and other weakly interacting light particles Gravitational waves and observational cosmology Particle detector developments Computational methods and analysis tools Theoretical Physics I: Phenomenology of the Standard Model and Beyond Electroweak interactions Quantum chromo dynamics Heavy quark physics and quark flavour mixing Neutrino physics Phenomenology of astro- and cosmoparticle physics Meson spectroscopy and non-perturbative QCD Low-energy effective field theories Lattice field theory High temperature QCD and heavy ion physics Phenomenology of supersymmetric extensions of the SM Phenomenology of non-supersymmetric extensions of the SM Model building and alternative models of electroweak symmetry breaking Flavour physics beyond the SM Computational algorithms and tools...etc.
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