First, I briefly review the different conceptions of time held by three rival interpretations of quantum theory: the collapse of the wave-packet, the pilot-wave interpretation, and the Everett interpretation (Section 2). Then I turn to a much less controversial task: to expound the recent understanding of the time-energy uncertainty principle, and indeed of uncertainty principles in general, that has been established by such authors as Busch, Hilgevoord and Uffink. Although this may at first seem a narrow topic, I point out connections to other conceptual topics about time in quantum theory: for example, the question under what circumstances there is a time operator (Section 4.3).
First, I briefly review the different conceptions of time held by three rival interpretations of quantum theory: the collapse of the wave-packet, the pilot-wave interpretation, and the Everett interpretation (Section 2). Then I turn to a much less controversial task: to expound the recent understanding of the time-energy uncertainty principle, and indeed of uncertainty principles in general, that has been established by such authors as Busch, Hilgevoord and Uffink. Although this may at first seem a narrow topic, I point out connections to other conceptual topics about time in quantum theory: for example, the question under what circumstances there is a time operator (Section 4.3).
The time energy uncertainty relation has been a controversial issue since the advent of quantum theory, with respect to appropriate formalisation, validity and possible meanings. A comprehensive account of the development of this subject up to the 1980s is provided by a combination of the reviews of Jammer (1974), Bauer and Mello (1978), and Busch (1990). More recent reviews are concerned with different specific aspects of the subject. The purpose of this chapter is to show that different types of time energy uncertainty relation can indeed be deduced in specific contexts, but that there is no unique universal relation that could stand on equal footing with the position-momentum uncertainty relation. To this end, we will survey the various formulations of a time energy uncertainty relation, with a brief assessment of their validity, and along the way we will indicate some new developments that emerged since the 1990s.
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.
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.
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 purpose of my Thesis is to gain a better understanding of the nature of time and the problems associated with time. For example, I aim to explore the problem associated with the energy-time uncertainty relation due to the lack of a universal operator for time. Using Jan Hilgevoord's work, I will explore the idea that if a measure of time is to be obtained in quantum mechanics, then time has to be a property of physical systems that can only be measured in relation to other systems. As such, time cannot be independent of physical systems. This implies that there cannot be time without physical systems. As a result, I will show how the philosophical position of "Relationism with Respect to Time" provides an accurate account of the nature of time that is consistent with observations made in physics. In order to achieve this, I will: 1) present Hilgevoord's argument about the problems of time in quantum mechanics; 2) provide an account of the ontology of time based on Hilgevoord's work; and 3) provide an analysis of the implications of what I am proposing in connection to Einstein's theory of special and general relativity in hopes of showing that his insights can and should be applied to quantum mechanics.
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.