Temporal aspects of quantum theory and emergent classicality

几千年来，关于时间本质的问题一直吸引着来自生活各个领域的科学家和思想家。物理学家的时间概念随着牛顿力学在18世纪的发展而第一次有了完全具体的形式，随着相对论在20世纪初的出现，我们的时间概念所包含的深度和微妙之处变得明显起来。量子理论出现在20世纪20年代，作为物质的基本理论，其核心是薛定谔著名的方程，描述了代表原子波动性质的波函数随时间的演化。量子理论在其所有应用中立即获得了巨大的成功。然而，一个没有得到解决的问题是关于薛定谔方程中时间参数的地位。它以一种不同于其他变量的方式出现，比如位置和动量。这些变量（由运算符在数学上表示）遵循不确定性原理，这限制了它们可以被指定的程度。然而，时间在薛定谔方程中表现为一个本质上经典的量，对应于牛顿力学描述的经典时钟所测量的时间，似乎不受测不准原理的限制。这是一种特殊的混合状态，对于一个所谓的基本理论来说，这是一个非常令人惊讶的状态。近年来，在量子理论中澄清时间的本质已经成为一个相当有趣的话题。这个问题是一个特别令人兴奋的问题，因为在量子理论中对时间的研究使我们更深入地理解时间实际上是什么。在量子理论中，人们想定义的一个特定的时间概念是到达时间：在给定的时间间隔内，处于给定量子态的粒子到达空间中某一点的概率是多少？时间在量子理论中的特殊地位意味着必须使用通常量子力学工具箱之外的方法来回答这些问题。许多这样的方法是间接的。例如，通过在两个不同时间测量空间区域中的概率量，可以推导出在给定时间间隔期间离开该空间区域的概率通量，由此可以定义到达某点的时间。但是像这样的间接方法有一些有趣的问题。例如，通过通量定义的到达时间概率，经典上为正，对于某些量子态可以是负的。这种奇怪的、很少被研究的非经典现象被称为回流，它困扰着量子理论中定义时间的许多尝试，定义到达时间的更复杂的方法包括在短时间间隔内重复测量位置，检查粒子是否还在那里。然而，这些方法也可能遭受有趣的问题。任何量子系统的一个基本特性就是测量会干扰它。如果测量太频繁，它会受到太多的干扰，以至于没有什么东西可以测量！这种非经典效应被称为量子芝诺效应，是另一种阻碍在量子理论中定义时间的现象。拟议的研究解决了量子理论中各种背景下的时间定义，并解决了由此产生的相关问题。将详细研究回流效应。事实证明，这与一种新形式的贝尔不等式有一些有趣的关系，贝尔不等式涉及时间分布的测量。量子芝诺效应也将被研究。特别是，一个有趣的问题是，这种高度非经典效应如何在经典极限中消失。这些和相关的问题可能会产生有趣的实验结果，也会揭示时间本身的性质。

Questions concerning the nature of time have fascinated scientists and thinkers from all areas of life for millenia. The physicists conception of time first took fully concrete form with the development of Newtonian mechanics in the eighteenth century and the depth and subtleties involved with our concepts of time became apparent with the advent of relativity at the beginning of the twentieth century. Quantum theory emerged in the 1920s as the fundamental theory of matter and at its heart lies Schrodinger's famous equation, describing the evolution, in time, of the wave function representing the wave-like properties of atoms. Quantum theory was immediately spectacularly successful in all its applications. Yet an issue that was not addressed concerns the status of the time parameter in the Schrodinger equation. It appears in a way which is different to other variables, such as position and momentum. These variables (which are mathematically represented by operators) obey the uncertainty principle, which limits the degree to which they may be specified. Yet time appears in the Schrodinger equation as an essentially classical quantity, corresponding to the time measured by a classical clock described by Newtonian mechanics, seemingly unrestricted by the uncertainty principle. This is a peculiarly hybrid state of affairs and a very surprising one for a supposedly fundamental theory. The question of clarifying the nature of time in quantum theory has become a topic of considerable interest in recent years.This question is a particularly exciting one since a study of time in quantum theory leads us to a deeper understanding of what time actually is.One particular notion of time one would like to define in quantum theory is the arrival time: what is the probability that a particle in a given quantum state arrives at a certain point in space during a given time interval? The peculiar status of time in quantum theory means that methods outside the usual quantum-mechanical toolbox have to be used to answer such questions. Many such methods are indirect. For example, by measuring the amount of probability in a spatial region at two different times one can deduce the flux of probability leaving that spatial region during the given time interval, from which one can define the arrival time at a point. But indirect methods such as this have interesting problems. For example, the arrival time probability defined through the flux, classically positive, can come out to be negative for certain quantum states. This curious and little-investigated non-classical phenomenon is called backflow and haunts many attempts to define time in quantum theory.More elaborate means of defining the arrival time involve repeated position measurements at short time intervals, checking to see if the particle is still there. Yet these methods may also suffer from interesting problems. A basicproperty of any quantum system is that measurement disturbs it. If measurements are made too frequently, it is disturbed so much that there is nothing left to measure! This non-classical effect is called the quantum Zeno effect and is another phenemonon that gets in the way of attempts to define time in quantum theory.The proposed research addresses the definition of time in quantum theory in a variety of contexts, and also addresses the associated problems that arise. The backflow effect will be investigated in detail. This turns out to have someinteresting relationships to a novel form of the Bell inequalities which involve measurements distributed in time. The quantum Zeno effect will also be investigated. In particular, an interesting question is how this highly non-classical effect goes away in the classical limit. These and related questions may have interesting experimental consequences and will also shed light on the nature of time itself.