21 May 2007 Quantum properties that are extended in time
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Abstract
This paper focuses on "time-extended" properties that quantum mechanics represents as existing at a given moment, but which cannot be measured at a given moment. Several examples are presented in which the energy cannot be measured in a short time. The essence of these examples is an attempt to measure the momenta (or the energy) of an ideal quantum clock by having an interaction that lasts only a short time, where this short time is defined with respect to the internal time which is conjugate to this momenta. However, this momenta and this time cannot both be definite at once, even though quantum mechanics claims that a definite energy should be definite at a definite time. However, the momenta can be definite at a definite external parameter time, rather than this internal time. From the internal perspective, however, it is shown that the energy cannot be defined at a given internal time and therefore, this aspect of "time-extension" is completely quantum in origin, unlike the classical aspect of "energy is frequency." An additional consequence of time extended properties is that the connection between the internal time before and after the energy measurement is made uncertain. This suggests that the complementarity between energy and time is deeper than the notion that precise measurements of energy take a long time. Going beyond the "negative" statements about what cannot be measured (which characterize most of the discussions of the energy-time uncertainty relation), a positive aspect of ΔEΔt > 1 is demonstrated in a closed system based on causality. In this example, if one were to argue that energy really existed at a definite moment in time, then causality could be called into question. The standard understanding of the relationship between energy and time is that if the energy is conserved then we can calculate what the energy is at any point in time and thus we should be able to speak about energy as actually existing at that definite moment in time. This section challenges that assertion and is motivated by the question: "is there an example in physics of a property that the formalism tells us exists at a given moment, but which cannot be checked at a given time?"
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Jeff Tollaksen, Jeff Tollaksen, } "Quantum properties that are extended in time", Proc. SPIE 6573, Quantum Information and Computation V, 65730Y (21 May 2007); doi: 10.1117/12.719323; https://doi.org/10.1117/12.719323
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