Tessarotto Massimo, Cremaschini Claudio
Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy.
Research Center for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo nám.13, CZ-74601 Opava, Czech Republic.
Entropy (Basel). 2020 Oct 26;22(11):1209. doi: 10.3390/e22111209.
The subject of this paper deals with the mathematical formulation of the Heisenberg Indeterminacy Principle in the framework of Quantum Gravity. The starting point is the establishment of the so-called time-conjugate momentum inequalities holding for non-relativistic and relativistic Quantum Mechanics. The validity of analogous Heisenberg inequalities in quantum gravity, which must be based on strictly physically observable quantities (i.e., necessarily either 4-scalar or 4-vector in nature), is shown to require the adoption of a manifestly covariant and unitary quantum theory of the gravitational field. Based on the prescription of a suitable notion of Hilbert space scalar product, the relevant Heisenberg inequalities are established. Besides the coordinate-conjugate momentum inequalities, these include a novel proper-time-conjugate extended momentum inequality. Physical implications and the connection with the deterministic limit recovering General Relativity are investigated.
本文主题涉及在量子引力框架下对海森堡不确定性原理的数学表述。起点是建立适用于非相对论和相对论量子力学的所谓时间共轭动量不等式。结果表明，量子引力中类似的海森堡不等式的有效性（其必须基于严格的物理可观测物理量，即本质上必然是4 - 标量或4 - 矢量）要求采用引力场的明显协变且幺正的量子理论。基于对希尔伯特空间标量积合适概念的规定，建立了相关的海森堡不等式。除了坐标共轭动量不等式外，这些还包括一个新颖的原时共轭扩展动量不等式。研究了其物理意义以及与恢复广义相对论的确定性极限的联系。
