The time of arrival quantum classical correspondence problem

Abstract

The quantum-classical correspondence (QCC) problem for dynamical observables is the problem of deriving the quantum image of a given classical observable and from the quantum image recover the classical observable via an appropriate limiting procedure involving the vanishing of the Planck's constant. Any solution to the QCC-problem for a given class of observables consists of (i) a prescription of obtaining the corresponding quantum observables, and (ii) a mapping of these to their respective classical counterparts, i.e. an implimentation of the correspondence principle. A satisfactory solution should solve the QCC-problem without conflict with the rest of quantum mechanics. At present quantum obsevables are constructed by the method of quantization. It is well-known though that there are obstructions to quantization. For example, in flat Euclidean space, Groenwold and van Hove have shown that there exists no quantization of all classical observables consistent with Schrodinger's quantization scheme. On the other hand, prescriptive mapping of classical observables to quantum observables, such as the Weyl prescription, is known to be generally inconsistent with Dirac's "Poisson bracket-commutator correspondence." Worse, quantization introduces circularity when one invokes the correspondence principle. This is unsatisfactory if quantum mechanics were to be internally coherent and autonomous from classical mechanics.
In this paper we address the time of arrival QCC-problem without quantization. Given a particle with mass μ in one dimension whose Hamiltonian is H(q,p). If the state of the particle at time t = 0 is given by the point (q,p) in the phase space, what is the time, T_x, that it will arrive at the point q(t = T_x) = x? The solution to this problem is straightforward and is given by
T_x(q,p) = sgn(p) (μ/2)^1/2 ∫_q^x [H(q,p) − V(q')]^−1/2 dq',
whenever the integral exists and is real valued. By virtue of T_x's dependence on the phase space points (q,p), T_x is a dynamical observable. Can we derive this equation from a quantum observable? Our answer is affirmative. But only by extension. Define the local time of arrival, t_x(q,p), at a given point x as the time of arrival at x in some small neigborhood of q. The local time of arrival is technically the expansion of the previous equation about the free time of arrival at x. For a given Hamiltonian H = (2/μ)⁻¹p² − V(q), the local time of arrival is given by
t_x(q,p) = ∑_k=0,∞ (−1)^k T_k(q,p;x),
where the T_k(q,p;x) are determined recursively. It can be shown that if p ≠ 0 and if V is continuous at q, then there exists a neighborhood of q such that for every x in the said neigborhood of q, t_x(q,p) converges absolutely and uniformly to T_x(q,p). Because T_x(q,p) is defined in the entire accesible region of the particle, we shall refer to it as the global time of arrival. t_x(q,p) converges to T_x(q,p) only in a small neighborhood so that T_x(q,p) is the analytic extension of t_x(q,p). In this paper we show that the local TOA, and thus the global TOA by extension, can be determined completely from pure quantum mechanical consideration. We shall limit though our discussion for the time of arrival at the origin.