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?"
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?"
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.
Unlike classical mechanics, quantum mechanics assumes the famous Heisenberg uncertainty relations. One of these concerns time: the energy–time uncertainty relation. Unlike the canonical position–momentum uncertainty relation, the energy–time relation is not reflected in the operator formalism of quantum theory. Indeed, it is often said and taken as problematic that there is not a so-called “time operator” in quantum theory. This chapter sheds light on these questions and others, including the absorbing matter of whether quantum mechanics allows for the existence of ideal clocks. The second section notes that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry which already exists in classical physics. The third section studies time operators in detail. The fourth section discusses various uncertainty relations involving time.
The question of time in quantum mechanics has been a topic of intense discussion since its inception.Dating back to the works of the founders of the subject, the notion of time and how it fit in the theory was already a point with a lack of a formal explanation, especially in aspects such as the uncertainty principle and canonical commutation relations.A theorem by Pauli set in stone that any attempt at trying to understand time with the formalism in use at the time for quantum mechanics was impossible.Through studies on the logical basis of the subject and establishing different sets of axioms, more modern works were able to relax sufficiently the impositions on the concept of observables so that time could then be understood as a 'generalized' observable.Not only that, but it was shown that the correct way to interpret eventual time measurements was through the formalism of Positive Operator Valued Measures.Latest works then were focused in the direction of constructing either direct measures and time operators, or, in the case that was followed throughout this work, to establish a procedure to construct measures.This work was then an in-depth study of the work done by Brunetti and Fredenhagen, with the intent to do the first steps in generalizing their ideas by establishing their method to be covariant with respect to Poincaré transformations.
The Energy‐Time Uncertainty (ETU) has always been a problem‐ridden relation, its problems stemming uniquely from the perplexing question of how to understand this mysterious Δt. On the face of it (and, indeed, far deeper than that), we always know what time it is. Few theorists were ignorant of the fact that time in quantum mechanics is exogenously defined, in no ways intrinsically related to the system. Time in quantum theory is an independent parameter, which simply means independently known. In the early 1960s Aharonov (1961–64) and Bohm (1961–64) mounted a spirited attack against the ETU, which sealed its fate to the present date. By emphasising that time is always “well‐defined” in quantum theory, they were led to the conclusion that no ETU should exist, a view shared by many in the 1990s, if Busch (1990) is to be believed. In a similar vein, I emphasize that (a) physical systems occupy a particular energy state at a particular instant of time, if at all; (b) even in absence of all time‐measuring instruments, it is still trivially warranted that one can measure a system's energy as accurately as one pleases, and simply announce “The system's energy is exactly E NOW!”, a possibility which no quantum mechanics of any sort, or any physical theory whatsoever, can afford to tamper with or change, except circularly. One never loses one's own perception of time, when one measures the energy, a fact which no measurement conceivable can interfere with or affect. Both (a) and (b) uniquely entail that energy and time are compatible, if not indeed intimately interconnected, contrary to what the relevant uncertainty seems to affirm. In response to Aharonov's and Bohm's initial problem, I reinterpret ΔEΔt ≥ h, as directly derived from authentic quantum principles, without however having to assume a direct incompatibility between its related concepts, attributing their complementarity to conditions other than ordinarily assumed.
This article presents a novel interpretation of quantum mechanics. It extends the meaning of “measurement” to include all property-indicating facts. Intrinsically, space is undifferentiated: There are no points on which a world of locally instantiated physical properties could be built. Instead, reality is built on facts, in the sense that the properties of things are extrinsic, or supervenient on property-indicating facts. The actual extent to which the world is spatially and temporally differentiated (that is, the extent to which spatiotemporal relations and distinctions are warranted by the facts) is necessarily limited. Notwithstanding that the state vector does nothing but assign probabilities, quantum mechanics affords a complete understanding of the actual world. If there is anything that is incomplete, it is the actual world, but its incompleteness exists only in relation to a conceptual framework that is more detailed than the actual world. Two deep-seated misconceptions are responsible for the interpretational difficulties associated with quantum mechanics: the notion that the spatial and temporal aspects of the world are adequately represented by sets with the cardinality of the real numbers, and the notion of an instantaneous state that evolves in time. The latter is an unwarranted (in fact, incoherent) projection of our apparent “motion in time” into the world of physics. Equally unwarranted, at bottom, is the use of causal concepts. There nevertheless exists a “classical” domain in which language suggestive of nomological necessity may be used. Quantum mechanics not only is strictly consistent with the existence of this domain but also presupposes it in several ways.
We consider the situation in which an observer internal to an isolated system wants to measure the total energy of the isolated system (this includes his own energy, that of the measuring device and clocks used, etc.). We show that he can do this in an arbitrarily short time, as measured by his own clock. This measurement is not subjected to a time-energy uncertainty relation. The properties of such measurements are discussed in detail with particular emphasis on the relation between the duration of the measurement as measured by internal clocks versus external clocks.