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Temporal Ordering in Quantum Mechanics

Jonathan Oppenheim, Reznik, B, William George Unruh

Published Jan 1, 2000• 4 citations
OpenAlex

Abstract

We examine the measurability of the temporal ordering of two events, as well as event coincidences. In classical mechanics, a measurement of the order-of-arrival of two particles is shown to be equivalent to a measurement involving only one particle (in higher dimensions). In quantum mechanics, we find that diffraction effects introduce a minimum inaccuracy to which the temporal order-of-arrival can be determined unambiguously. The minimum inaccuracy of the measurement is given by 1/E where E is the total kinetic energy of the two particles. Similar restrictions apply to the case of coincidence measurements.

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Temporal ordering in quantum mechanics

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We examine the measurability of the temporal ordering of two events, as well as event coincidences. In classical mechanics, a measurement of the order-of-arrival of two particles is shown to be equivalent to a measurement involving only one particle (in higher dimensions). In quantum mechanics, we find that diffraction effects introduce a minimum inaccuracy to which the temporal order-of-arrival can be determined unambiguously. The minimum inaccuracy of the measurement is given by δ t = ℏ/Ē where Ē is the total kinetic energy of the two particles. Similar restrictions apply to the case of coincidence measurements. We show that these limitations are much weaker than limitations on measuring the time-of-arrival of a particle to a fixed location.

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arXiv:quant-ph/0312179•Dec 21, 2003•0 citations
We examine the measurability of the temporal ordering of two events, as well as event coincidences. In classical mechanics, a measurement of the order-of-arrival of two particles is shown to be equivalent to a measurement involving only one particle (in higher dimensions). In quantum mechanics, we find that diffraction effects introduce a minimum inaccuracy to which the temporal order-of-arrival can be determined unambiguously. The minimum inaccuracy of the measurement is given by 1/E where E is the total kinetic energy of the two particles. Similar restrictions apply to the case of coincidence measurements.
We examine the measurability of the temporal ordering of two events, as well as event coincidences. In classical mechanics, a measurement of the order-of-arrival of two particles is shown to be equivalent to a measurement involving only one particle (in higher dimensions). In quantum mechanics, we find that diffraction effects introduce a minimum inaccuracy to which the temporal order-of-arrival can be determined unambiguously. The minimum inaccuracy of the measurement is given by δ t = ℏ/Ē where Ē is the total kinetic energy of the two particles. Similar restrictions apply to the case of coincidence measurements. We show that these limitations are much weaker than limitations on measuring the time-of-arrival of a particle to a fixed location.
In quantum mechanics, time plays a role unlike any other observable. We find that measuring whether an event happened, and measuring when an event happened are fundamentally different - the two measurements do not correspond to compatible observables and interfere with each other. We also propose a basic limitation on measurements of the arrival time of a free particle given by 1/Ek where Ek is the particle's kinetic energy. The temporal order of events is also an ambiguous concept in quantum mechanics. It is not always possible to determine whether one event lies in the future or past of another event. One cannot measure whether one particle arrives to a particular location before or after another particle if they arrive within a time of 1/E of each other, where E is the total kinetic energy of the two particles. These new inaccuracy limitations are dynamical in nature, and fundamentally different from the Heisenberg uncertainty relations. They refer to individual measurements of a single quantity. It is hoped that by understanding the role of time in quantum mechanics, we may gain new insight into the role of time in a quantum theory of gravity.
After discussing the role of time in both General Relativity and Quantum Gravity, it is argued that even in ordinary Quantum Mechanics, the role of time is ambiguous. A rather surprising result is that measurements of the time-of-arrival of a free particle must always be more inaccurate than 1/E where E is the kinetic energy of the particle. This inaccuracy is not related to the Heisenberg uncertainty relations. It is a dynamical relation which applies to individual measurements of a single quantity.
It is argued that the time of arrival cannot be precisely defined and measured in quantum mechanics. By constructing explicit toy models of a measurement, we show that for a free particle it cannot be measured more accurately then $\ensuremath{\Delta}{t}_{A}\ensuremath{\sim}{1/E}_{k},$ where Ek{E}_{k}Ek​ is the initial kinetic energy of the particle. With a better accuracy, particles reflect off the measuring device, and the resulting probability distribution becomes distorted. It is shown that a time-of-arrival operator cannot exist, and that approximate time-of-arrival operators do not correspond to the measurements considered here.
The theoretical analysis of time observables in quantum mechanics has been controversial and is still subject to debate. In the present work, a measurement-based approach to quantum arrival times is studied in detail. It is based on the emission time of the first spontaneous photon from a a two-level atom moving into a laser-illuminated region. This quantum-optical model is shown to give an operational understanding of various ideal proposals for quantum arrival-time distributions as well as for local kinetic energy densities. As a main result, a first measurement-based procedure to Kijowski`s arrival-time distribution is given and the approach is used to generalize this distribution to the case of particles interacting with an external potential.