Structures of quantum space-time

许多试图将引力物理学与其他基本力统一起来的尝试都追求将四维时空扩展到更高维度的想法。从Kaluza-Klein规范势理论的引入开始，这种扩展通常是通过增加时空的空间维度来实现的，同时保留了时间所起的特殊作用。在这类方法中，不能说基本力的量子力学特征已充分地纳入高维时空的结构中。也不能说，在任何有用的意义上，时空被视为一个量子实体。如果从某种意义上说，宇宙本质上是量子力学的，那么最自然的时空向更高维度的延伸与卡鲁扎-克莱因理论及其推广所提出的特征完全不同。在量子力学中，希尔伯特空间的维数与厄米算符对应空间的维数直接相关。因此，如果要将量子理论与时空物理学统一起来，就很自然地将表示时空点的矩阵空间扩展到更高的维度，从而假设时空的维度大于4，甚至可能是无限的。以这种方式对时空进行更高维的扩展，也与经常提出的“旋量和时空点一样基本”的理论相一致。这种思路源于这样一种观点，即在更深层次上，时空的对称性和量子理论的对称性是无法区分的。这个广义的“等效原理”反映了这样一个概念，即我们在自然界中观察到的基本对称性或近似对称性应该有一个共同的起源，这些对称性的破坏也应该有一个共同的原因。根据广义等价原理，可以假设时空事件本身是无限维厄米矩阵。由这种构造产生的模型被称为量子时空模型。该项目的目标是研究量子时空的特性，其中一些重点是(i)宇宙学方面（例如，宇宙常数在对称破缺中的作用）；和（ii）量子方面（纠缠的作用，缺乏交换性等）的理论。

Many attempts to unify gravitational physics with other fundamental forces have pursued the idea of extending four-dimensional space-time to higher dimensions. Beginning with the introduction of the Kaluza-Klein theory of gauge potentials, such extensions have typically been carried out by increasing the spatial dimension of the space-time, while retaining the special role played by time. In methodologies of this sort it cannot be said that quantum mechanical characteristics of the fundamental forces are adequately incorporated into the structure of the higher dimensional space-time. Nor can it be said that the space-time is being treated in any useful sense as a quantum entity. If the universe is intrinsically quantum mechanical in an appropriate sense, then the most natural extension of space-time into higher dimensions has a completely different character from that suggested by the Kaluza-Klein theory and its generalizations.In quantum mechanics, the dimensionality of the Hilbert space is directly related to the dimensionality of the corresponding space of Hermitian operators. Thus, if quantum theory is to be unified with space-time physics, it is natural to extend the space of matrices representing space-time points to higher dimensions, and hence to assume that the dimensionality of space-time is larger than four, perhaps infinite. A higher-dimensional extension of space-time in this manner is also consistent with the philosophy often put forward that spinors are just as fundamental as space-time points.This line of thinking is motivated by the idea that the symmetries of space-time and the symmetries of quantum theory are, at the deeper level, indistinguishable. This generalized 'equivalence principle' reflects the notion that the fundamental symmetries or approximate symmetries we observe in nature should have a common origin, and that the breakdown of these symmetries should also have a common cause. In view of the generalized equivalence principle, one can postulate that space-time events are themselves infinite-dimensional Hermitian matrices. The model that results from such a construction is known as a quantum space-time model.The objective of the project is to investigate properties of quantum space-time, with some focus on (i) cosmological aspects (e.g., role of cosmological constants in symmetry breaking); and (ii) quantum aspects (role of entanglement, lack of commutativity, etc.) of the theory.