The Quantum Equivalence Principle and Quantum Coordinate Systems

基础物理学的两大支柱是广义相对论和量子理论。每个人都在自己的领域非常成功。然而，有一个深层次的问题：它们并不适合在一起。量子引力的问题是找到一个新的物理理论，在这些理论被发现是经验正确的情况下，可以简化为广义相对论和量子理论。 这是基础物理学的圣杯。 这一建议涉及到解决这一问题的新办法。 事实上，爱因斯坦自己也面临着类似的问题。 他在1907年意识到狭义相对论和牛顿引力是不相容的。他花了接下来的八年时间来解决这个问题，一路上有许多曲折沿着。他的出发点是他所谓的“我一生中最快乐的想法”--没有办法区分我们所在的电梯是在重力作用下坠落还是漂浮在太空中。在局部（在足够小的电梯中），这两种情况是等效的。这个概念被称为爱因斯坦的等效原理。从技术上讲，我们总能找到一个参照系，在这个参照系中，引力的影响消失了。 在量子理论中，任何动力学量都是不确定的。例如，一个特定的粒子射向一个有两条狭缝的屏幕，究竟是通过一条狭缝还是另一条狭缝，这是没有事实依据的。在广义相对论中，因果顺序（事件在时间上的顺序）本身受动力学定律的支配。因此，我们期待在量子引力理论中有不确定的因果结构。 这代表着与以前的物理理论的根本背离。通常我们认为一个状态是随时间演化的。 我们不能在量子引力理论中做到这一点，因为这样做需要有序的事件序列。为了取得进展，我们必须正面解决不确定因果结构的问题。 我将采用量子参照系的概念（由Aharonov和Susskind在20世纪60年代引入），它允许我们处于不同参照系的量子叠加中。 在这里，我提出了量子等价原理（我最近介绍过）：总是有可能找到一个量子参照系，使得在任何给定点附近，我们都有确定的因果结构。爱因斯坦的等效原理提供了一种驯服引力的方法，而不会将其从图片中删除。类似地，量子等效原理提供了一种驯服不确定因果结构的方法。 学生将学习如何首先在概念上，然后在数学术语框架的问题。 他们将理解关键概念，如因果关系，本体论和操作论，并学习数学方法，如张量分析，线性代数和概率方法。 这些技能是非常可转移的，并有应用程序，例如，量子信息和大数据集分析。 接受这些技术培训的学生将加强加拿大学术界内外的劳动力。

The two pillars of fundamental physics are General Relativity and Quantum Theory. Each is very successful in its own realm. However, there is a deep problem: they do not fit together. The problem of Quantum Gravity is to find a new physical theory that reduces to General Relativity and Quantum Theory in situations where these theories have been found to be empirically correct. This is the holy grail of fundamental physics. This proposal concerns a new approach to this problem. In fact, Einstein himself faced a similar problem. He realized in 1907 that Special Relativity and Newtonian Gravity were incompatible. He spent the next eight years resolving this problem with many twists and turns along the way. His starting point was what he called "the happiest thought of my life" - that there is no way to distinguish whether an elevator we are in is falling under gravity or floating in space. Locally (in a small enough elevator) these two situations are equivalent. This idea is called Einstein's equivalence principle. More technically, we can always find a frame of reference in which the effects of gravity disappear. In Quantum Theory any dynamical quantity is subject to indefiniteness. For example, there is no-matter-of-the-fact as to whether a particular particle fired at a screen with two slits goes through one slit or the other. In General Relativity causal order (the order of events in time) is itself subject to dynamical laws. Hence we expect indefinite causal structure in a theory of Quantum Gravity. This represents a radical departure from previous physical theories. Usually we think of a state evolving in time. We cannot do this in a theory of Quantum Gravity since to do so requires an ordered sequence of events. We have to tackle the problem of indefinite causal structure head on to make progress. I will employ the notion of a quantum frame of reference (introduced by Aharonov and Susskind in the 1960's) which allows us to be in a quantum superposition of different frames of reference. Here I propose the quantum equivalence principle (which I recently introduced): that it is always possible to find a quantum frame of reference such that, in the vicinity of any given point, we have definite causal structure. Einstein's equivalence principle provides a way to tame gravity without removing it from the picture. Analogously, the quantum equivalence principle offers a way to tame indefinite causal structure. Students will learn how to frame a problem first in conceptual, then in mathematical terms. They will understand key concepts such as causality, ontology, and operationalism and learn mathematical methods such as tensor analysis, linear algebra, and probabilistic methods. These skills are very transferable and have applications, for example, to Quantum Information, and to big data set analysis. Students trained trained in these techniques will strengthen the Canadian workforce both inside and outside academia.