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。从引入卡鲁扎-克莱因规范势理论开始，这种扩展通常是通过增加时空的空间维度来进行的，同时保留时间所扮演的特殊角色。在这类方法论中，不能说基本力的量子力学特性被充分地纳入了高维时空的结构中。也不能说时空在任何有用的意义上被当作一个量子实体。如果宇宙在某种意义上是内在量子力学的，那么时空向更高维度的最自然的扩展就具有与卡鲁扎-克莱因理论及其推广所提出的完全不同的性质。在量子力学中，希尔伯特空间的维数与相应的厄米特算子空间的维数直接相关。因此，如果量子理论要与时空物理学统一起来，自然要把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，甚至是无穷大。时空以这种方式的高维扩展也与哲学上经常提出的旋量和时空点一样基本的观点相一致，这种思路的动机是时空的对称性和量子理论的对称性在更深层次上是不可区分的。这个广义的“等效原理”反映了这样一种观念，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。根据广义等效原理，我们可以假设时空事件本身就是无限维的厄米特矩阵。该项目的目标是研究量子时空的性质，重点是（i）宇宙学方面（例如，宇宙学常数在对称性破缺中的作用）;（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.