On the transition from the quantum to the classical regime for massive scalar particles: A spatiotemporal approach

Abstract

If the classical structure of space-time is assumed to define an a-priori scenario for the formulation of the structure of quantum theory (QT), the coordinate representation of the solutions ψ( x, t) (ψ( x 1 , .., x N , t)) of the Schroedinger equation of a quantum system containing one (N ) massive scalar particle has a preferred status. It is then possible to perform a multipolar expansion of the density matrix ρ( x, t) = |ψ( x, t)| 2 (and more generally of the Wigner function) around a space-time trajectory x c (t) to be properly selected. A special set of solutions ψ EM W F ( x, t), named Ehrenfest monopole wave functions(EMWF), is characterized by the conditions that: (i) the quantum expectation value of the position operator coincides at any time with the searched classical trajectory, < ψ EM W F |ˆ x|ψ EM W F >= x c (t), and, (ii) Ehrenfest's theorem holds for the expectation values of the position and momentum operators. The first condition implies the vanishing of the 'dipole' term in the multipolar expansion of the density matrix with respect to such trajectory. Ehrenfest's theorem applied to EMWF leads then to a closed Newton equation of motion for the classical trajectory, where the effective force is the Newton force plus non-Newtonian terms (of order 2 or higher) depending on the higher multipoles of the probability distribution ρ. Note that the super-position of two EMWFs is not an EMWF, a result to be strongly hoped for, given the possible unwanted implications concerning classical spatial perception. These results can be extended to N particle systems and to relativistic quantum mechanics.Consequently, for the states of a quantum particle which are EMWF, we get the emergence of a corresponding classical 'effective' particle following a Newton-like trajectory in space-time.Note that this holds true in the standard framework of quantum mechanics, i.e. by assuming the validity of Born's rule and the individual system interpretation of the wave function (no ensemble interpretation). These results are valid without any approximation (like → 0, big quantum numbers,...). Moreover, we do not commit ourselves to any ontological interpretation of quantum theory (such as, e.g., the Bohmian one. It will be clear that our trajectories are not Bohm's trajectories). We will argue that, in substantial agreement with Bohr's viewpoint, the macroscopic description of the preparation, certain intermediate steps and the detection of the final outcome of experiments involving massive particles are dominated by these 'classical effective trajectories'.This approach can be applied to the point of view of de-coherence (in which positions turn out to be selected preferred robust bases) in the case of a diagonal reduced density matrix ρ red (an improper mixture) depending on the position variables of a massive particle and of a pointer. When both the particle and the pointer wave functions appearing in ρ red are EMWF, the expectation value of the particle and pointer position variables becomes a statistical average on a cl...

Conditions for the classicality of the center of mass of many-particle quantum states

“…In order to satisfy our classical intuition, we need to certify that the observable associated with r cm  follows a classical trajectory in each experiment (not in an average over several experiments). This problem could be analyzed within the orthodox formalism [4,5,7,[15][16][17]33]. The typical approach would be to construct a reduced density matrix of the center of mass by tracing out the rest of the degrees of freedom interpreted as the environment.…”

On relativistic entanglement and localization of particles and on their comparison with the nonrelativistic theory

“…It has the expression As shown in Ref. [18] 9 we can consider the multipolar expansion of the wave function g(τ, σ) around a classical trajectory σ cl (τ ). For all the wave functions with vanishing dipole moment with respect to the classical trajectory we get < g(τ )|ˆ σ|g(τ ) >= σ cl (τ ) and the Ehrenfest theorem implies…”

“…As shown in Ref. [18], if one assumes that the wave function describes the given quantum system (no ensemble interpretation), the statement of Bohr can be justified by noting that the wave functions used in the preparation of particle beams (semi-classical objects with a mean classical trajectory and a classical mean momentum determined with time-of-flight methods) are a special subset of the wave functions solutions of the Schroedinger equation for the given particles. Their associated density matrix, pervading the whole 3-space, admits a multi-polar expansion around a classical trajectory having zero dipole.…”

Non-inertial frames in Minkowski space-time, accelerated either mathematical or dynamical observers and comments on non-inertial relativistic quantum mechanics

“…If one would be able to quantize these systems, then the hybrid view would emerge due to notions like decoherence, suggesting that the macroscopic quantum system has a quasi-classical collective variable (the Pointer) following a semi-classical Newton-like trajectory. See [44] for a discussion of the emergence of this classical regime from the quantum one.…”