Why modal interpretations of quantum mechanics must abandon classical reasoning about physical properties

Modal interpretations of quantum mechanics propose to solve the measurement problem by rejecting the orthodox view that in entangled states of a system which are nontrivial superpositions of an observable's eigenstates, it is meaningless to speak of that observable as having a value or corresponding to a property of the system. Though denying this is reminiscent of how hidden-variable interpreters have challenged orthodox views about superposition, modal interpreters also argue that their proposals avoid any of the objectionable features of physical properties that beset hidden-variable interpretations, like contextualism and nonlocality. Even so, I shall prove that modal interpreters of quantum mechanics are still committed to giving up at least one of the following three conditions characteristic of classical reasoning about physical properties: (1) Properties certain to be found on measuring a system should be counted as intrinsic properties of the system. (2) If two propositions stating the possession of two intrinsic properties by the system are regarded as meaningful, then their conjunction should also correspond to a meaningful proposition about the system possessing a certain intrinsic property; and similarly for disjunction and negation. (3) The intrinsic properties of a composite system should at least include (though need not be exhausted by) the intrinsic properties of its parts. Conditions 1–3 are by no means undeniable. But the onus seems to be on modal interpreters to tell us why rejecting one of these is preferable to an ontology of properties incorporating contextualism and nonlocality.     

A new physical characterisation of classical systems in quantum mechanics

A physical characterisation of classical systems in quantum mechanics is given in terms of the set of ensembles in contrast to the well-known characterisations concerning the effects or observables: A quantum mechanical system is classical if and only if each two decompositions of every ensemble are compatible.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.     

Rotational quantum mechanics in hyperspherical coordinates

We present a new class of quantum-mechanical problems involving the rotation of a top in a spherically symmetric torque field. The eigenvalue problem is studied for the hyperspherical rotational oscillator. We introduce a new four-dimensional parity quantum number referring to integral spin values of the excited states of a spherical top.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 3–7, January, 1984.The author acknowledges useful discussions with Prof. A. G. Sitenko and with I. V. Simenog.     

Superposition in quantum and classical mechanics

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.Dedicated to the Memory of Charles H. Randall.     

Constitution of objects in classical mechanics and in quantum mechanics

The constitution of objects is discussed in classical mechanics and in quantum mechanics. The requirement of objectivity and the Galilei invariance of classical and quantum mechanics leads to the postulate of covariance which must be fulfilled by observable quantities. Objects are then considered as carriers of these covariant observables and turn out to be representations of the Galilei group. Individual systems can be defined in classical mechanics by their trajectories in phase space. However, in quantum mechanics the characterization of individuals can only be achieved approximately by means of unsharp observables.     

The simplified Fermi accelerator in classical and quantum mechanics

We review the simplified classical Fermi acceleration mechanism and construct a quantum counterpart by imposing time-dependent boundary conditions on solutions of the free Schrödinger equation at the unit interval. We find similiar dynamical features in the sense that limiting KAM curves, respectively purely singular quasienergy spectrum, exist(s) for sufficiently smooth wall oscillations (typically ofC ² type). In addition, we investigate quantum analogs to local approximations of the Fermi map both in its quasiperiodic and irregular phase space regions. In particular, we find pure point q.e. spectrum in the former case and conjecture that random boundary conditions are necessary to model a quantum analog to the chaotic regime of the classical accelerator.     

Transient chaos in quantum and classical mechanics

Bogolubov's classical example of statistical relaxation in a many-dimensional linear oscillator is discussed. The relation of the discovered relaxation mechanism to quantum dynamics as well as to some new problems in classical mechanics is considered.     

Time reversal in classical and quantum mechanics

A review of Wigner's time reversal is presented and some important aspects are emphasized. The subject is introduced via classical mechanics. Non-physical statements as time running backwards are avoided. Comments are made on the roles of time and of the operatori(/t) in quantum mechanics. The role of symmetries and conservation laws and some properties of the time-reversed states are discussed.Work supported by Instituto Nacional de Investigação Científica, Portugal.     

Relation between classical and quantum mechanics

Expressions for the Lie derivatives of functions of non-commuting variables are derived and used to reformulate classical mechanics. This is possible only if the phase space variables commute, or if they satisfy Heisenberg's commutation relations.This work was supported in part by the National Science Foundation Grant Number NSF GP-14803, and by the Air Force Office of Scientific Research, contract number AFOSR 68-1524.     

Pseudointegrable systems in classical and quantum mechanics

We study particles moving in planar polygonal enclosures with rational angles, and show by several methods that trajectories in the classical phase space explore two-dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply-handled spheres. The quantum mechanics of one such ‘pseudointegrable system’ is studied in detail by computing energy levels using an exact formalism. This system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side L. We find that neighbouring levels avoid degeneracies as L varies, and that the probability distribution for the spacing S of adjacent levels vanishes linearly as S→0 (‘level repulsion’). The Weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density. Oscillatory corrections to the mean level density are given as a sum over closed classical paths; for pseudointegrable systems these closed paths form families covering part of the phase-space invariant surfaces.     

Heating map in classical and quantum mechanics

Heating map of the classical probability-distribution function (in the phase space) and of density matrix (in the position representation) in quantum mechanics is introduced and its positivity is proved. The relation of the heating map to scaling transform and unitary squeezing transform of the momentum variable in the Wigner function is used to prove that noncanonical scaling transform of the position and momentum provides positive (but not completely positive!) map of density operator. The connection of momentum scaling transform with time scaling transform and Plancks constant scaling transform is discussed.     

The action-angle coordinates for particle motion in magnetic mirror systems

In this paper the problem of the possibility of determining the cyclic Hamiltonian function of a particle moving about a magnetic vessel of the mirror type is discussed. From the Hamiltonian of the typeH=H(P ₁,P ₂,P ₃,Q ₃), derived by the author in a previous paper, the Hamiltonian functionH=H(J₁,J ₂,J ₃,w ₃) was determined whereP _i ,Q _i are the generalized impulses and coordinates andJ _i ,w _k is the action-angle coordinate system. The determination of the form H=H(J₁,J ₂,J ₃) depends on the possibility of solving the quadrature of the Hamilton-Jacobi equation leading to an open form in the general case. Some approximative expressions, suitably replacing the mirror system, are discussed. The solution has been extended to the range of relativistic velocities. The question of the uniqueness of the expression of the Hamiltonian function in cyclic variables is analyzed; it follows from this that the class of canonical transformations leaving the Hamiltonian in a cyclic form does not allow any great simplification of the relatively complicated transformation expressions.     

Problem of measurements in classical and quantum mechanics

In the present paper, a phenomenological model of measurement process is suggested. It includes information on a measurable quantity and a hypothesis on a minimum measurement error. The use of this model and of the principle of the least action allows an equation of the information dynamics of a material point to be derived. This equation differs by the presence of an information force. By the example of a one-dimensional oscillator, the feasibility of solving the inverse problem of frequency reconstruction from the experimental data is demonstrated. The problem of quantum measurement is solved based on a classical analog. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 89–96, March, 2007.     

Aspects of classical and quantum Nambu mechanics

We present recent developments in the theory of Nambu mechanics, which include new examples of Nambu-Poisson manifolds with linear Nambu brackets and new representations of Nambu-Heisenberg commutation relations.     

A classical realization of quantum mechanics

A mechanism is presented by which a classical system could be described by the laws of quantum theory. Conflict with von Neumann's no-go theorem is avoided. Experimental predictions are made.