An algebraic approach to Koopman classical mechanics

Morgan, Peter

doi:10.1016/j.aop.2020.168090

Cited by 20 publications

(23 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Usually, they are understood to be non-obserbables or hidden variables [28]. However, it is tempting to consider them as observables because (1) they are related to ergodic properties [29], (2) to avoid superselection rules [12], and (3) they are useful to construct a classical measurement theory [16]. Moreover, these operator are related to the Bopp operators in the Wigner phase space representation of quantum mechanics [30].…”

Section: Classical Representation Of the Poincaré Algebramentioning

confidence: 99%

“…This old approach is due to Koopman [5] and von Neumann [6]. Whether for derivation of purely classical results or for comparison between quantum and classical mechanics, the Koopman-von Neumann formalism (hereafter abbreviated as KvN) has received increasing attention in the past two decades (see [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]), the possibility of formulating quantum-classical hybrid theories has also increased the interest in this formalism [22,23,24,25]. The existence of the KvN theory raises the question of the classification of the unitary representations of the groups of space-time symmetries in the context of classical mechanics.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Unitary representation of the Poincaré group for classical relativistic dynamics

Manjarres

2021

Annals of Physics

View full text Add to dashboard Cite

Section: Classical Representation Of the Poincaré Algebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unitary representation of the Poincaré group for classical relativistic dynamics

Manjarres

2021

Annals of Physics

View full text Add to dashboard Cite

“…They stopped short of a thorough analysis; their primary interest being the quantum case. Recently Morgan [4] and Katagiri [5] made use of KvN formalism in separate attempts to use quantum measurement theory to examine measurement in classical mechanics.…”

Section: Introductionmentioning

confidence: 99%

Measurement, information, and disturbance in Hamiltonian mechanics

Theurel¹

2021

Preprint

View full text Add to dashboard Cite

Measurement in classical physics is examined here as a process involving the joint evolution of object-system and measuring apparatus. For this, a model of measurement is proposed which lends itself to theoretical analysis using Hamiltonian mechanics and Bayesian probability. At odds with a widely-held intuition, it is found that the ideal measurement capable of extracting finite information without disturbing the system is ruled out (by the third law of thermodynamics). And in its place a Heisenberg-like precision-disturbance relation is found, with the role of ̵ h 2 played by k B T Ω; where T and Ω are a certain temperature and frequency characterizing the ready-state of the apparatus. The proposed model is argued to be maximally efficient, in that it saturates this Heisenberg-like inequality, while various modifications of the model fail to saturate it. The process of continuous measurement is then examined; yielding a novel pair of Liouville-like master equations-according to whether the measurement record is read or discarded-describing the dynamics of (a rational agent's knowledge of) a system under continuous measurement. The master equation corresponding to discarded record doubles as a description of an open thermodynamic system. The fine-grained Shannon entropy is found to be a Lyapunov function (i.e. Ṡ ≥ 0) of the dynamics when the record is discarded, providing a novel H-theorem suitable for studying the second law and non-equilibrium statistical physics. These findings may also be of interest to those working on the foundations of quantum mechanics, in particular along the lines of attempting to identify and unmix a possible epistemic component of quantum theory from its ontic content. More practically, these results may find applications in the fields of precision measurement, nanoengineering and molecular machines.

show abstract

“…Either joint probability measures can be handled relatively easily or incompatible probability measures can be handled relatively easily. The above is presented in a manner peculiar to the author in two recent articles [1,2], where references to a selection of other literature may also be found.…”

mentioning

confidence: 99%

Classical and Quantum Measurement Theory

Morgan¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Classical and quantum measurement theories are usually held to be different because the algebra of classical measurements is commutative, however the Poisson bracket allows noncommutativity to be added naturally. After we introduce noncommutativity into classical measurement theory, we can also add quantum noise, differentiated from thermal noise by Poincaré invariance. With these two changes, the extended classical and quantum measurement theories are equally capable, so we may speak of a single "measurement theory". The reconciliation of general relativity and quantum theory has been long delayed because classical and quantum systems have been thought to be very different, however this unification allows us to discuss a unified measurement theory for geometry in physics.

show abstract

An algebraic approach to Koopman classical mechanics

Cited by 20 publications

References 30 publications

Unitary representation of the Poincaré group for classical relativistic dynamics

Unitary representation of the Poincaré group for classical relativistic dynamics

Measurement, information, and disturbance in Hamiltonian mechanics

Classical and Quantum Measurement Theory

Contact Info

Product

Resources

About