Abstract
Abstract We investigate measurement theory in classical mechanics in the formulation of classical mechanics by Koopman and von Neumann (KvN), which uses Hilbert space. We show a difference between classical and quantum mechanics in the “relative interpretation” of the state of the target of measurement and the state of the measurement device. We also derive the uncertainty relation in classical mechanics.
Highlights
In order to discuss the crucial difference between quantum and classical mechanics, it is essential to compare the two with the same formalism
If the new variable introduced is not included in the Hamiltonian, this formalism is equivalent to classical mechanics
Since Koopman and von Neumann (KvN) is a rewrite of classical mechanics to quantum mechanics formalism, it is natural that Hamiltonian H only include x and p
Summary
In order to discuss the crucial difference between quantum and classical mechanics, it is essential to compare the two with the same formalism. Quantum mechanics is described in terms of q-numbers, i.e., non-commutative physical quantities This difference is apparent in measurement theories, such as Heisenberg’s uncertainty principle[2]. In the KvN formalism, classical mechanics is described using non-commutative operators, as in quantum mechanics. If the new variable introduced is not included in the Hamiltonian, this formalism is equivalent to classical mechanics Under such an operator formalism, the difference between classical mechanics and quantum mechanics is not in the non-commutative nature of the operators, but in the form of the commutations relations. After reviewing the observation problem using the von Neumann model, we discuss in Section 5 the measurement theory in classical mechanics. In Appendix F, we describe in detail the evolution of time in the von Neumann model
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