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Unary Classical Mechanics

Peter Morgan

Published Jan 2, 2019• 1 citation
OpenAlex

Abstract

Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving an operator algebraic variant of the Koopman--von Neumann approach. A classically natural approach to the Poisson bracket in a Koopman--von Neumann Hilbert space formalism gives a classical logic that is the same as quantum logic: in both cases the logic of projections on Hilbert space. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical approach that can also be successfully applied to the measurement problem of quantum mechanics.

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Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving an operator algebraic variant of the Koopman--von Neumann approach. A classically natural approach to the Poisson bracket in a Koopman--von Neumann Hilbert space formalism gives a classical logic that is the same as quantum logic: in both cases the logic of projections on Hilbert space. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical approach that can also be successfully applied to the measurement problem of quantum mechanics.
Abstract Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used to develop a novel formulation of classical mechanics in Hilbert space. This formulation differs from the Koopman-von Neumann theory in many ways. Classical observables are represented by van Hove operators, which satisfy a commutation algebra isomorphic to the Poisson algebra of functions in phase space. Moreover, these operators are both observables and generators of transformations, which makes it unnecessary to introduce unobservable auxiliary operators as in the Koopman-von Neumann theory. In addition, for consistency with classical mechanics, a constraint must be imposed that fixes the phase of the wavefunction. The approach can be extended to hybrid mixed classical-quantum systems in Hilbert space. The formalism is applied to the measurement of a quantum two-level system (qubit) by a classical apparatus.
Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function ψKvN\psi _{KvN}ψKvN​ ψ KvN as well as the quantum-mechanical one ψQM\psi _{QM}ψQM​ ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle E{{\mathcal {E}}}E E follows from this. ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM are shown to be sections in E{{\mathcal {E}}}E E . The difference between ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM originates from a quantization map interpreted as “twisting” of sections over E{{\mathcal {E}}}E E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.
The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative ⋆\star⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The considered class of deformations and the corresponding ⋆\star⋆-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of ⋆\star⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆\star⋆-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. The equivalence of two formulations of quantum mechanics is proved by observing that the Wigner-Moyal transform has all properties of the tensor product. This observation allows writing many previous results found in the literature in a transparent way, from which the equivalence of the two formulations of quantum mechanics follows naturally.
Abstract Classical Koopman–von Neumann Hilbert spaces of states are constructed here by the action of classical random fields on a vacuum state in ways that support an action of the quantized electromagnetic field and of the U (1)—invariant observables of the quantized Dirac spinor field, allowing a manifestly Lorentz invariant classical understanding of the state spaces of the two field theories, generalizing the Quantum–Mechanics–Free Systems of Tsang&Caves and Quantum Non-Demolition measurements. The algebra of functions on a classical phase space is commutative but the algebra of classical observables associated with coordinate transformations is noncommutative, so that, for example, we can as much ask whether a classical state is an eigenstate of a rotation as we can in quantum mechanics and so that entangled states can be distinguished from mixed states, making classical random fields as weird as quantum fields.
Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally brings about uncertainty relations inherent in classical mechanics, for example between position and the generator of space translations, between momentum and the generator of momentum translations, and between dynamical time and the Liouvillian, to name a few. Further, it is shown that the Wigner representation produces a quasi-probability distribution that can take on negative values. Thus, two of the hallmark features of quantum mechanics are reproduced, and become apparent, in a Hilbert-space formulation of classical mechanics.