Incompatible observables in classical physics: A closer look at measurement in Hamiltonian mechanics

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 finitetemperature 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)], 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 psi-epistemic interpretations. More practically, there may be applications in the fields of precision measurement, nanoengineering, and molecular machines.