Time-of-arrival probabilities for general particle detectors

Abstract

We develop a general framework for the construction of probabilities for the time of arrival in quantum systems. The time of arrival is identified with the time instant when a transition in the detector's degrees of freedom takes place. Thus, its definition is embedded within the larger issue of defining probabilities with respect to time for general quantum transitions. The key point in our analysis is that we manage to reduce the problem of defining a quantum time observable to a mathematical model where time is associated to a transition from a subspace of the Hilbert space of the total system to its complementary subspace. This property makes it possible to derive a general expression for the probability for the time of transition, valid for any quantum system, with the only requirement that the time of transition is correlated with a definite macroscopic record. The framework developed here allows for the consideration of any experimental configuration for the measurement of the time of arrival and it also applies to relativistic systems with interactions described by quantum field theory. We use the method in order to describe time-of-arrival measurements in high-energy particle reactions and for a rigorous derivation of the time-integrated probabilities in particle oscillations.Comment: 27 pages, latex. Changed title and added a conclusions section. Version to appear in PR

Equivalence principle for quantum systems: dephasing and phase shift of free-falling particles

“…(34) applies only to factorized initial states, and not to general states that are described by Eq. (24). This means that the dephasing is strictly derived only for such states.…”

“…Thus, the measured quantity is the time of arrival, while the location of the particle detector is fixed. Despite the existence of ambiguities in the definition for quantum time-of-arrival probabilities [22,23], it is now possible to construct time-ofarrival probability measures for general Hamiltonians [24], using a method that can be straightforwardly applied to free-falling particles. However, the method involves more complex techniques of quantum measurement theory.…”

Detection of a quantum particle on a lattice under repeated projective measurements

“…The measurement of the arrival time of a quantum mechanical particle in a given detection region is a longstanding and fundamental problem in quantum mechanics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In spite of much effort, the construction of a time operator has been found to be controversial [18].…”

A relativistic one-particle Time of Arrival operator for a free spin-1/2 particle in (1+1) dimensions

“…Going in a different direction, there are studies in both non-relativistic and relativistic quantum mechanics promoting the role of time to an observable [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Particularly, it was calculated that the TOA operator for a nonrelativistic particle with mass m 0 is −m 0T−1,1 [4,5], where theT m,n 's form a complete and linearly independent set called the Bender-Dunne operators [20,21].…”

“…In [15], a proper time operator was found for a relativistic electron and was used to study some properties of several position operators. And in [16], a method for constructing TOA probabilities valid for any experimental setup (including relativistic systems with interactions described by QFT) was developed. They associated the TOA of a particle as the time instant when there is a transition in the degrees of freedom of the detector.…”