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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Compound objects as particles in quantum mechanics

The property of fundamental mechanical theories which allows one to treat compound objects as particles under suitable conditions is considered. It is argued that such a property, called composition invariance, is a nonreleasable property of any fundamental mechanical theory. The proof that standard quantum mechanics enjoys composition invariance is reviewed. Finally, it is shown that the requirement of composition invariance allows one to choose between two alternative forms of quantum mechanics with spontaneous localization.     

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.     

Identical motion in relativistic quantum and classical mechanics

The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and a Hamilton-Jacobi equation is obtained.     

Gibbs ensembles in relativistic classical and quantum mechanics

Classical and quantum Gibbs ensembles are constructed for equilibrium statistical mechanics in the framework of an extension to many-body theory of a relativistic mechanics proposed by Stueckelberg. In addition to the usual chemical potential in the grand canonical ensemble, there is a new potential corresponding to the mass degree of freedom of relativistic systems. It is shown that in the nonrelativistic limit the relativistic ensembles we have obtained reduce to the usual ones, and mass fluctuations for the free-particle gas approach the fluctuations in N. The ultrarelativistic limit of the canonical ensemble for the free-particle gas differs from the corresponding limit of the ensemble proposed by Jüttner and Pauli. Due to the mass degree of freedom, the quantum counting of states is different from that of the nonrelativistic theory. If the mass distribution is sufficiently sharp, the thermodynamical effects of this multiplicity will not be large. There may, however, be detectable effects such as a shift in the Fermi level and the critical temperature for Bose-Einstein condensation, and some change in specific heats.     

On classical representations of finite-dimensional quantum mechanics

In the case of a finite-dimensional Hilbert space, it is shown that quantum mechanics can be embedded into discrete classical probability theory. In particular, states can be represented as stochastic vectors and observables as random variables such that all probabilities and expectation values are given in classical terms.     

Properties of nonautonomous classical systems and quantum mechanics

We show that the relativistic equation of the Brownian motion of a classical particle near a trajectory, stable in Lyapunov's sense, is identical with the Klein-Gordon equation. The conditions which make motions of this type possible are connected with cosmology.