Theurel David
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
Phys Rev E. 2024 Aug;110(2-1):024124. doi: 10.1103/PhysRevE.110.024124.
Quantum theory famously entails the existence of incompatible measurements, pairs of system observables which cannot be simultaneously measured to arbitrary precision. Incompatibility is widely regarded to be a uniquely quantum phenomenon, linked to failure to commute of quantum operators. Even in the face of deep parallels between quantum commutators and classical Poisson brackets, no connection has been established between the Poisson algebra and any intrinsic limitations to classical measurement. Here I examine measurement in classical Hamiltonian physics as a process involving the joint evolution of an object-system and a finite-temperature measuring apparatus. Instead of the ideal measurement capable of extracting information without disturbing the system, I find a Heisenberg-like precision-disturbance relation: Measuring an observable leaves all Poisson-commuting observables undisturbed but inevitably disturbs all non-Poisson-commuting observables. In this classical uncertainty relation the role of h-bar is played by an apparatus-specific quantity, q-bar. While this is not a universal constant, the analysis suggests that q-bar takes a finite positive value for any apparatus that can be built. (Specifically: q-bar vanishes in the model only in the unreachable limit of zero absolute temperature.) I show that a classical version of Ozawa's model of quantum measurement [Ozawa, Phys. Rev. Lett. 60, 385 (1988)0031-900710.1103/PhysRevLett.60.385], originally proposed as a means to violate Heisenberg's relation, does not violate the classical relation. If this result were to generalize to all models of measurement, then incompatibility would prove to be a feature not only of quantum, but of classical physics too. Put differently: The approach presented here points the way to studying the (Bayesian) epistemology of classical physics, which was until now assumed to be trivial. It now seems possible that it is nontrivial and bears a resemblance to the quantum formalism. The present findings may be of interest to researchers working on foundations of quantum mechanics, particularly for ψ-epistemic interpretations. More practically, there may be applications in the fields of precision measurement, nanoengineering, and molecular machines.
量子理论著名地蕴含了不相容测量的存在，即系统可观测量的成对组合，它们不能同时被测量到任意精度。不相容性被广泛认为是一种独特的量子现象，与量子算符的对易性失效相关联。即使面对量子对易子和经典泊松括号之间的深刻相似性，泊松代数与经典测量的任何内在限制之间也尚未建立联系。在这里，我将经典哈密顿物理学中的测量视为一个涉及对象系统和有限温度测量装置联合演化的过程。与能够在不干扰系统的情况下提取信息的理想测量不同，我发现了一种类似海森堡的精度 - 干扰关系：测量一个可观测量不会干扰所有泊松对易的可观测量，但不可避免地会干扰所有非泊松对易的可观测量。在这个经典不确定性关系中，ħ 的角色由一个特定于装置的量 q̅ 扮演。虽然这不是一个普适常数，但分析表明，对于任何可以构建的装置，q̅ 都取一个有限的正值。（具体而言：在模型中，q̅ 仅在绝对温度为零的不可达到极限情况下才会消失。）我表明，最初作为违反海森堡关系的一种手段而提出的小泽量子测量模型 [小泽，《物理评论快报》60, 385 (1988)0031 - 900710.1103/PhysRevLett.60.385] 的经典版本，并不违反经典关系。如果这个结果能够推广到所有测量模型，那么不相容性将被证明不仅是量子物理学的一个特征，也是经典物理学的一个特征。换句话说：这里提出的方法为研究经典物理学的（贝叶斯）认识论指明了方向，而经典物理学的认识论直到现在都被认为是微不足道的。现在看来，它可能并非微不足道，并且与量子形式主义有相似之处。本研究结果可能会引起从事量子力学基础研究的人员的兴趣，特别是对于 ψ - 认知解释。更实际地说，在精密测量、纳米工程和分子机器领域可能会有应用。
