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?"