Topics on Causal Structures in Quantum Theory

该项目属于EPSRC物理科学研究领域。它的目的是扩展我的硕士论文中发展的结果和想法，该论文研究了一个过程理论框架的范畴结构，该框架模拟了时空中因果尊重和因果违反区域之间的相互作用。在这个过程理论量子力学框架中，有一个假定的不对称性，一个假定的时间之箭的方向。然而，在广义相对论中，时空的结构被认为是一个动力学变量，因此，将量子力学能级从时间之箭的假设方向中解放出来是一个重要的任务。该项目旨在开发一个过程理论框架，该框架考虑了单向时间方向局部违反的可能性。1949年，戈德尔提出了爱因斯坦广义相对论方程的解，该方程承认存在封闭的类时间曲线(CTC)，即时空中违反时序的区域。这些解被许多人认为是非物理的：在经典物理学中，CTC的存在施加了一些自相矛盾的约束。然而，原则上不能因为科学考虑而排除它们[2]。事实上，David Deutsch已经提出，在量子力学的层面上，没有任何经典的病理发生。Deutsch在1991年的论文《类封闭时间线附近的量子力学》[1]中使用了源自计算的量子理论的方法来构建一种理论，该理论可以解释如果我们将一个违反时序的区域包括在‘传统’量子电路中会发生什么，表示假设的CTC的存在。二十年后，赛斯·劳埃德提出了一种不同的方法，试图修复Deutsch提出的优雅解决方案的一些有问题的方面[2]。劳埃德模型基于一种现象(后选择隐形传态)，这一现象特别适合于使用范畴量子力学提供的方法和途径进行分析。因此，该项目的目标之一将是将这两种方法转化为一种明确的和过程理论环境。这将允许我们构建一个正式的数学框架，使对CTC附近的量子现象和两位作者所描述的悖论的分析正式化，否则往往过于夸张。此外，我们可以尝试在科克和基辛格发展的过程理论中的因果关系理论的基础上，看看因果理论的想法可以在哪里以及如何保留，同时考虑到CTC的存在。一个特别有希望的方面在于，Deutsch和Lloyd[1，2]开发的模型是相互不兼容的，然而，Lloyd并没有设法排除旧模型的物理性。对这些现象的成分和分类方法应该有助于阐明这一方面，甚至可能允许我们排除这两种模型中的一种，更重要的是，从成分观察出发构建一种可能的替代方案。该项目的主要优势之一在于方法的内在新颖性，在上面提到的论文[1，2]中，没有一个正式的设置允许我们谈论量子过程的因果结构，这实际上是由量子力学的范畴方法开发的工具提供的。参考文献[1]封闭的类时间线附近的量子力学-Deutsch，David Phys。D44 3197-3217修订版(1991)[2]通过后选择隐形传态的时间旅行的量子力学-Lloyd等人。物理牧师。D84 025007(2011年)

This project falls within the EPSRC Physical Science research area. It is aimed at expanding the results and ideas developed in my master's thesis, which studied the categorical structure of a process theoretical framework modelling the interactions between causally respecting and causally violating regions of spacetime.In the process theoretical Quantum mechanical framework there is an assumed asymmetry, an assumed direction of the arrow of time. In general relativity the structure of the spacetime is however taken to be a dynamical variable, it is therefore an important task to free the quantum mechanical level from an assumed direction of the arrow of time. The project aims to develop a process theoretical framework which takes into account the possibility of local violations of the one-way time direction.In 1949, Godel proposed a solution to the Einstein's equations of general relativity which admits the existence of closed timelike curves (CTC), chronology-violating regions of the space time. Those solutions are by many considered to be unphysical: in classical physics the existence of the CTCs imposes some constraints which are paradoxical. However they are not in principle ruled out by scientific considerations [2]. In fact, it has been proposed by David Deutsch that at the level of quantum mechanics none of the 'classical' pathologies occur. Deutsch in the 1991 paper "Quantum mechanics near closed timelike lines" [1] uses the methods derived from the quantum theory of computation to construct a theory that would explain what happens if we include to a 'traditional' quantum circuit a chronology-violating region, representing the existence of an hypothetical CTC. Twenty years later Seth Lloyd proposed a different approach in an attempt to fix some problematic aspects of the otherwise elegant solution proposed by Deutsch [2]. Lloyd model is based on a phenomenon (post-selected teleportation) which is particularly amenable to an analysis using the methods and the approach provided by categorical quantum mechanics. One of the aims of the project will therefore be to translate the two approaches into a categorical and process theoretical setting. This will allow us to construct a formal mathematical framework which would formalise the otherwise often too discursive analysis of the quantum phenomena near CTCs and of the paradoxes described by the two authors. Moreover, we can try to build on the theory of causality in process theories developed by Coecke and Kissinger to see where and how the idea of a causal theory can be retained while taking into account the existence of CTCs. A particularly promising aspects lies in the fact that the models developed by Deutsch and Lloyd[1,2] are mutually incompatible, however, Lloyd doesn't manage to rule out the physicality of the older model. A compositional and categorical approach of the phenomena should shed light on this aspect and may even allow us to either rule out one of the two models and more importantly construct a possible alternative starting from compositional observations. One of the main strength of the project lies in the inherent novelty of the approach, in the above mentioned papers [1,2] there isn't a formal setting that would allow us to speak about the causal structure of quantum processes, this is in fact provided by the tools developed by the categorical approach toquantum mechanics.Bibliography[1] Quantum mechanics near closed timelike lines - Deutsch, David Phys. Rev. D44 3197-3217(1991)[2] The quantum mechanics of time travel through post-selected teleportation - Lloyd et al. Phys.Rev. D84 025007 (2011)