Topos Quantum Theory and Gravity

寻找一种统一或调和量子理论与广义相对论的量子引力理论，是物理学基础中一个长期存在的问题。尽管付出了巨大的努力，但今天量子引力的预测性理论仍然不存在。乍一看，人们可能会认为，量子理论和广义相对论都必须修改，才能实现成功的统一。然而，在实践中，大多数研究程序或多或少地使用标准量子理论，包括主要的量子理论(弦理论和圈量子引力)。这很可能在概念上是错误的。人们普遍认为，在非常小的距离(这与量子引力相关)，时空的连续图景就会破裂。因此，对于量子引力来说，希望有一种数学形式，它基本上不依赖于实数或复数形式的连续体的使用。其次，通常对量子理论的解释是工具主义的，即它依赖于量子系统本身外部的测量和观察者。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，对量子系统的更现实的描述是必要的。在过去的几年里，我们在为物理理论开发一个真正新的数学框架方面取得了重大进展，它(A)是现实主义的，因为测量没有特殊的RLE作用，(B)从根本上不依赖于连续统的使用，即实数或复数。主要的数学工具是拓扑论，这是范式论的一个高度发展的分支。我们的数学结构通常是由它们之间的集合和函数建立起来的。有一种说法是，集合和函数构成一个范畴集合。对于给定的点x和一个子集S，位于S的命题x不是真就是假。Topos提供了允许进行数学运算的所有必要结构：可以像在set中一样在topos中定义数学对象(当然，这是一个topos本身)，并且每个topos都有一个内部逻辑。这种逻辑是直观的，这意味着排除中间法则不需要成立。人们可以使用内部逻辑在TOPO中进行证明。因此，每个TOPO都是一个自己的数学宇宙。耐人寻味的是，这种从数学基础出发的结构在物理学的基础上起着重要的作用。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，详细描述空间和时空是如何数学编码的，或者它们是如何从更基本的结构中产生的，这与量子引力的任何方法都是高度相关的。拟议的研究将考虑解决这些问题的几种方法。我们目前的框架将有重大的扩展。在研究的第一个更长的部分(A.预沉的拓朴中的类空结构，1-18个月)，我们将使用几个已被证明在描述拓朴量子理论中有用的数学对象，并将考虑由它们定义的类空结构。在第二部分(B.其他相关理论，19-30个月)，我们将考虑相关的数学理论，并发展和澄清它们的物理内容。特别是，我们想要考虑如何描述从更基本的范畴结构中出现的空间和时间。与A和B类似，我们将考虑物理模型(C模型)，以便具体化和检验我们的想法。TOOS方法允许开发真正新的物理模型。其中一个目标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.