Topos Quantum Theory and Gravity

在物理学的基础上，寻找一种量子引力理论来统一或调和量子理论与广义相对论是一个长期存在的问题。尽管付出了巨大的努力，今天还没有量子引力的预言理论存在。乍一看，人们可能会认为，量子理论和广义相对论都必须在成功统一之前进行修正。然而，在实践中，大多数研究项目或多或少使用标准量子理论，包括主要的量子理论（弦理论和圈量子引力）。这在概念上很可能是错误的，因为人们通常认为，在非常小的距离上（这与量子引力有关），时空的连续统图像就崩溃了。因此，对于量子引力来说，最好有一种数学形式，这种数学形式不依赖于连续统的使用，即真实的或复数的形式。其次，量子理论的通常解释是工具主义的，即，它依赖于量子系统本身之外的测量和观察者。在未来的量子引力和宇宙学理论中，整个宇宙将被视为一个量子系统。显然，没有外部观察者，对量子系统的更实在的描述是必要的。在过去的几年里，我们在发展一个真正新的物理理论数学框架方面取得了重大进展，这个框架（a）是实在的，因为测量没有特殊的作用，（B）从根本上不依赖于连续统的使用，即，真实的或复数。主要的数学工具是拓扑理论，一个高度发展的范畴论的分支。通常，我们的数学结构是由集合和它们之间的函数建立的。一种说法是集合和函数形成一个范畴Set。对于一个给定的点x和一个子集S，命题x位于S中要么是真要么是假。一个拓扑提供了所有必要的结构，允许做数学：一个可以定义数学对象在拓扑就像在集（这是一个拓扑本身，当然），每个拓扑有一个内部逻辑。这种逻辑是直觉主义的，这意味着排中律不一定成立。人们可以使用内部逻辑在拓扑中进行证明。因此，每一个拓扑都是它自己的数学宇宙。有趣的是，数学基础的这种结构在物理学基础中也发挥着作用。今天，我们可以在内部描述物理系统的状态和物理量。主要的开放问题是物理空间和时空的描述。显然，任何量子引力的方法都与详细描述空间和时空是如何在数学上编码的，或者它们是如何从更基本的结构中产生的密切相关。拟议的研究将考虑解决这些问题的几种方法。在第一个较长的研究部分（A。类空结构的拓扑结构的预层，月1-18），我们将使用几个数学对象，已被证明是有用的描述拓扑量子理论，并将考虑从他们定义的类空结构。第二部分（B.其他相关理论，月19-30），我们将考虑相关的数学理论，并发展和澄清其物理内容。特别是，我们想考虑如何描述空间和时间从更基本的范畴结构中的出现。和B，我们将考虑物理模型（C。模型），以具体化和测试我们的想法。topos方法允许开发真正新的物理模型。目标之一是建立一个简单的量子宇宙学模型。

It is a long-standing problem in the foundations of physics to find a theory of quantum gravity that would unify or reconcile quantum theory with general relativity. Despite huge efforts, today no predictive theory of quantum gravity exists.At first sight, one might expect that both quantum theory and general relativity must be revised before a successful unification can be achieved. In practice, however, most research programmes use more-or-less standard quantum theory, including the major ones (string theory and loop quantum gravity). This may very well be conceptually wrong.It is commonly expected that at very small distances (which are relevant for quantum gravity) the continuum picture of space-time breaks down. Thus, for quantum gravity it is desirable to have a mathematical formalism that does not fundamentally depend on the use of the continuum in the form of the real or complex numbers. Secondly, the usual interpretation of quantum theory is instrumentalist, i.e., it depends on measurements and observers external to the quantum system itself. In a future theory of quantum gravity and cosmology, the whole universe will be treated as a quantum system. Clearly, there is no external observer, and a more realist description of quantum systems is necessary.In the last few years, we have made major progress in developing a genuinely new mathematical framework for physical theories which (a) is realist in the sense that measurements play no special rle and (b) does not fundamentally depend on the use of the continuum, i.e., the real or complex numbers. The main mathematical tool is topos theory, a highly developed branch of category theory.Usually, our mathematical structures are built from sets and functions between them. One says that sets and functions form a category Set. For a given point x and a subset S, the proposition x lie in S is either true or false. A topos provides all the necessary structures that allow to do mathematics : one can define mathematical objects in a topos just as in Set (which is a topos itself, of course), and each topos has an internal logic. This logic is intuitionistic, which means that the law of excluded middle need not hold. One can do proofs in a topos using the internal logic. Thus each topos is a mathematical universe of its own. It is intriguing that this structure from the foundations of mathematics plays a rle in the foundations of physics.Today, we can describe the states of a physical system and the physical quantities topos-internally. The main open problem is the description of physical space and space-time. Clearly, it is highly relevant for any approach to quantum gravity to describe in detail how space and space-time are encoded mathematically, or how they possibly arise from more fundamental structures. The proposed research will consider several approaches to these questions. There will be major extensions to our current framework.In the first, longer part of the research (A. Space-like structures in a topos of presheaves, months 1-18), we will use several mathematical objects that have proven useful in the description of topos quantum theory and will consider space-like structures defined from them. Other results of category theory will be important in this research.In the second part (B. Other relevant theories, months 19-30), we will consider related mathematical theories and develop and clarify their physical content. In particular, we want to consider how the emergence of space and time from more basic, categorical structures can be described.In parallel to A. and B., we will consider physical models (C. Models), in order to concretise and test our ideas. The topos methods allow the development of genuinely new physical models. One aim is a simple model of quantum cosmology.