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arXiv:quant-ph/9912060

Relativistic Time of Arrival

A. Ruschhaupt

Published Dec 13, 1999• 0 citations
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Abstract

We propose a covariant algorithm for relativistic ideal measurements and for relativistic continuous measurements, its non-relativistic limit results the algorithm of the Event-Enhanced Quantum Theory. Therefore an additional intrinsic parameter, the proper time, is used. As an application we compute the time of arrival of a particle at a detector and find good agreement between the expected values of the time of arrival for weak detectors and the results of the relativistic point-mechanic over a wide range. For very high momentums there is a small probability for a negative time of arrival, so the expected times are a bit smaller than the results of the relativistic mechanics.

DOI
10.48550/arxiv.quant-ph/9912060

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•Aug 17, 2016•1 citation
We propose a covariant algorithm for relativistic ideal measurements and for relativistic continuous measurements, its non-relativistic limit results the algorithm of the Event-Enhanced Quantum Theory. Therefore an additional intrinsic parameter, the proper time, is used. As an application we compute the time of arrival of a particle at a detector and find good agreement between the expected values of the time of arrival for weak detectors and the results of the relativistic point-mechanic over a wide range. For very high momentums there is a small probability for a negative time of arrival, so the expected times are a bit smaller than the results of the relativistic mechanics.
This is the second paper on a new formalism for relativistic quantum measurements. Here, we construct a fully relativistic model for detectors that takes into account the detector's state of motion, intrinsics dynamics, initial states and couplings to the measured field. The dual classical/quantum description of the detector is implemented by using a master-equation type of approximation for the coarse-grained pointer variables. Then we identify the probabilities that correspond to ideal measurements, i.e., measurements that are largely insensitive to modeling details of the apparatus. The Unruh-Dewitt and Glauber detectors are recovered at the appropriate limits. We employ our results to models of particle detection, photodetection and relativistic spin measurements, and we derive an ideal distribution for relativistic time-of-arrival probabilities.
Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.
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.
Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by Kijowski and the event enhanced quantum theory (EEQT) method. It is suggested that for free particles the two methods may lead to the same result. On the other hand, the EEQT method can be easily geometrized within the framework of Galilei–Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.
Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by Kijowski and the event enhanced quantum theory (EEQT) method. It is suggested that for free particles the two methods may lead to the same result. On the other hand, the EEQT method can be easily geometrized within the framework of Galilei–Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.