Evading Quantum Mechanics á la Sudarshan: Quantum-Mechanics-Free Subsystem as a Realization of Koopman-Von Neumann Mechanics

Preprint

Brief Report

Evading Quantum Mechanics á la Sudarshan: Quantum-Mechanics-Free Subsystem as a Realization of Koopman-Von Neumann Mechanics

Altmetrics

Downloads

Views

Comments

A peer-reviewed article of this preprint also exists.

Zurab K. Silagadze^*

This version is not peer-reviewed

Submitted:

15 August 2023

Posted:

17 August 2023

You are already at the latest version

Alerts

Abstract

The notion of quantum-mechanics-free subsystem (QMFS) was introduced by Tsang and Caves in 2012. They showed that it is possible to design the dynamics of coupled quantum systems in such a way that one can construct a subsystem in which a given classical dynamics is realized, thus avoiding the measurement back-action of quantum mechanics. Although they mention that the Heisenberg picture equations of motion for QMFS was first proposed by Koopman in 1931 as a formulation of classical Hamiltonian dynamics in Hilbert space, in fact the connection between QMFS and Koopmann-von Neumann mechanics is deeper. Namely, quantum-mechanics-free subsystem is nothing but the realization of Sudarshan's perspective on the Koopman-von Neumann mechanics.

Keywords:

Subject: Physical Sciences - Quantum Science and Technology

The notion of quantum-mechanics-free subsystem (QMFS) was introduced by Tsang and Caves in [1]. They showed that it is possible to design the dynamics of coupled quantum systems in such a way that one can construct a subsystem in which a given classical dynamics is realized, thus avoiding the measurement back-action of quantum mechanics. Although they mention that the Heisenberg picture equations of motion for QMFS was first proposed by Koopman [2] as a formulation of classical Hamiltonian dynamics in Hilbert space, in fact the connection between QMFS and Koopmann-von Neumann mechanics [2,3] is deeper. Namely, quantum-mechanics-free subsystem is nothing but the realization of Sudarshan’s perspective [4,5] on the Koopman-von Neumann mechanics.

The idea of QMFS can be explained as follows [1]. Let us assume that the Hamiltonian of a quantum system is equal to (or to its symmetrized version with respect to non-commutative variables)

H = f (q, p, t) P + g (q, p, t) Q + h (q, p, t),

(1)

where

f (q, p, t)

g (q, p, t)

h (q, p, t)

are arbitrary functions, and

q, P

Q, p

are two pairs of quantum-mechanical conjugate variables subject to canonical commutation relations

[q, P] = i ℏ, [Q, p] = i ℏ,

(2)

with other commutators between them equal to zero. Then the Heisenberg picture equations of motion for commuting variables

q, p

\frac{d q}{d t} = \frac{\partial H}{\partial P} = f (q, p, t), \frac{d p}{d t} = - \frac{\partial H}{\partial Q} = - g (q, p, t),

(3)

do not contain "hidden" variables

Q, P

and will correspond to the classical Hamiltonian dynamics if there exists the classical Hamiltonian function

H_{c l} (q, p, t)

such that

f (q, p, t) = \frac{\partial H_{c l}}{\partial p}, g (q, p, t) = \frac{\partial H_{c l}}{\partial q} .

(4)

A non-trivial aspect of the QMFS proposal is that the quantum dynamics described by the Hamiltonian (1) can actually be realized. According to [1], a pairing of positive- and negative-mass oscillators can be used for this goal.1 Indeed, the quantum Hamiltonian in this case is

H = \frac{p_{1}^{2}}{2 m} + \frac{1}{2} m ω^{2} q_{1}^{2} - \frac{p_{2}^{2}}{2 m} - \frac{1}{2} m ω^{2} q_{2}^{2} .

(5)

In terms of new canonical variables

q = q_{1} + q_{2}, Q = \frac{1}{2} (q_{1} - q_{2}), p = p_{1} - p_{2}, P = \frac{1}{2} (p_{1} + p_{2}),

(6)

the Hamiltonian (5) takes the form

H = \frac{p P}{m} + m ω^{2} q Q,

(7)

and is exactly of the type (1) and (4) with

H_{c l} = \frac{p^{2}}{2 m} + \frac{1}{2} m ω^{2} q^{2}

An oscillator with an effective negative mass can be built, for example, using ensembles of atomic spins [7], and several experimental works have demonstrated the feasibility of creating QMFSs [7,8,9].

To the best of our knowledge, the QMFS literature fails to recognize the fact that QNFS is nothing more than an implementation of the Koopmann-von Neumann (KvN) mechanics. The same applies to literature related to KVN mechanics.

For modern overview of the KvN mechanics, see, for example, [10,11] and references therein. However, to substantiate the above statement about QMFS, it is enough to recall the very basics of KvN mechanics. The starting point is the Liouville equation of classical statistical mechanics:

\frac{\partial ρ (q, p, t)}{\partial t} = \frac{\partial H_{c l}}{\partial q} \frac{\partial ρ}{\partial p} - \frac{\partial H_{c l}}{\partial p} \frac{\partial ρ}{\partial q} .

(8)

Due to the linearity of this equation with respect to the derivatives

\frac{\partial ρ}{\partial p}

and

\frac{\partial ρ}{\partial q}

, we can introduce the classical wave function

ψ (q, p, t) = \sqrt{ρ (q, p, t)}

, which obeys the same Liouville equation (8), which can be rewritten in a Schrödinger-like form

i \frac{\partial ψ (q, p, t)}{\partial t} = L ψ, L = i (\frac{\partial H_{c l}}{\partial q} \frac{\partial}{\partial p} - \frac{\partial H_{c l}}{\partial p} \frac{\partial}{\partial q}) .

(9)

Based on this observation, it is possible to develop a Hilbert space formulation of classical mechanics that is completely reminiscent of the quantum formalism, except that, of course, all interference effects are absent [2,3].

Sudarshan gave a very interesting interpretation of KvN mechanics [4,5]. If we introduce Q and P operators as follows

Q = i ℏ \frac{\partial}{\partial p}, P = - i ℏ \frac{\partial}{\partial q},

(10)

then the equation (9) takes the form

i ℏ \frac{\partial ψ (q, p, t)}{\partial t} = H ψ, H = \frac{\partial H_{c l}}{\partial q} Q + \frac{\partial H_{c l}}{\partial p} P,

(11)

and it can be interpreted as the Schrödinger equation in the

(q, p)

-representation (with diagonal operators q and p) of a genuine quantum system with two pairs of canonical variables

(q, P)

and

(Q, p)

. The similarity with the QMFS idea is obvious, since the quantum Hamiltonian in the equation (11) has exactly the same type as given by (1) and (4), and hence

(q, p)

subsystem is nothing but QMFS.

I think that the identity of quantum-mechanics-free subsystems with Sudarshan’s interpretation of KvN mechanics, combined with the fact that such systems were actually implemented experimentally, is of great importance for Koopmann-von Neumann mechanics, as it makes KvN mechanics, in a sense, engineering science.

The renewed interest in KvN mechanics was caused by the need to build a suitable framework for hybrid classical-quantum systems (see, for example, [12,13] and references therein). I hope that the realization of the fact that quantum-mechanics-free subsystems are described by KvN mechanics will boost this interest in KvN mechanics.

There is one more curious aspect of QMFS-KvN mechanics connection. In [11], it was suggested that the modification of quantum mechanics supposedly expected from quantum gravity could lead to a deformation of classical mechanics, effectively destroying classicality, if Sudarshan’s views on KvN mechanics are taken seriously. In [11] this was of purely academic interest, since you are not required to accept the Sudarshan interpretation in order to develop the KvN mechanics. However, we now see that the existence of quantum-mechanics-free subsystems indicates that we should take Sudarshan’s interpretation of KvN mechanics seriously. Therefore, we expect that, due to the universal nature of gravity, if the effects of quantum gravity do modify quantum mechanics, these effects will destroy the classical dynamics in QMFS, since the hidden variables will no longer be hidden from the classical subspace. It may be worth looking for such effects in quantum-mechanics-free subsystems.

Note

1	I am tempted to recall here the maxim quoted in [6] and attributed to Sidney Coleman: “The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction."

References

Tsang, M.; Caves, C.M. Evading Quantum Mechanics: Engineering a Classical Subsystem within a Quantum Environment. Phys. Rev. X 2012, 2, 031016. [Google Scholar] [CrossRef]
Koopman, B.O. Hamiltonian Systems and Transformation in Hilbert Space. Proc. Natl. Acad. Sci. USA 1931, 17, 315–318. [Google Scholar] [CrossRef] [PubMed]
von Neumann, J. Zur Operatorenmethode In Der Klassischen Mechanik. Annals of Mathematics 1932, 33, 587–642. [Google Scholar] [CrossRef]
Sudarshan, E.C.G. Interaction between classical and quantum systems and the measurement of quantum observables. Pramana 1976, 6, 117–126. [Google Scholar] [CrossRef]
Sherry, T.N.; Sudarshan, E.C.G. Interaction between classical and quantum systems: A new approach to quantum measurement. II. Theoretical considerations. Phys. Rev. D 1979, 20, 857–868. [Google Scholar] [CrossRef]
Zhang, P.; Zhao, Q.; Horvathy, P.A. Gravitational waves and conformal time transformations. Annals of Physics 2022, arXiv:gr-qc/2112.09589]440, 168833. [Google Scholar] [CrossRef]
Møller, C.B.; Thomas, R.A.; Vasilakis, G.; Zeuthen, E.; Tsaturyan, Y.; Balabas, M.; Jensen, K.; Schliesser, A.; Hammerer, K.; Polzik, E.S. Quantum back-action-evading measurement of motion in a negative mass reference frame. Nature 2017, 547, 191–195. [Google Scholar] [CrossRef] [PubMed]
de Lépinay, L.M.; Ockeloen-Korppi, C.F.; Woolley, M.J.; Sillanpää, M.A. Quantum mechanics–free subsystem with mechanical oscillators. Science 2021, 372, 625–629. [Google Scholar] [CrossRef] [PubMed]
Ockeloen-Korppi, C.F.; Damskägg, E.; Pirkkalainen, J.M.; Clerk, A.A.; Woolley, M.J.; Sillanpää, M.A. Quantum Backaction Evading Measurement of Collective Mechanical Modes. Phys. Rev. Lett. 2016, 117, 140401. [Google Scholar] [CrossRef] [PubMed]
Mauro, D. Topics in Koopman-von Neumann Theory, 2003, [arXiv:quant-ph/quant-ph/0301172]. [CrossRef]
Chashchina, O.I.; Sen, A.; Silagadze, Z.K. On deformations of classical mechanics due to Planck-scale physics. Int. J. Mod. Phys. D 2020, arXiv:physics.class-ph/1902.09728]29, 2050070. [Google Scholar] [CrossRef]
McCaul, G.; Zhdanov, D.V.; Bondar, D.I. The wave operator representation of quantum and classical dynamics, 2023, [arXiv:quant-ph/2302.13208]. arXiv:quant-ph/2302.13208]. [CrossRef]
Morgan, P. An algebraic approach to Koopman classical mechanics. Annals of Physics 2020, 414, 168090. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Evading Quantum Mechanics á la Sudarshan: Quantum-Mechanics-Free Subsystem as a Realization of Koopman-Von Neumann Mechanics

Abstract

Note

References

MDPI Initiatives

Important Links

Subscribe