It is rather well-known that spacetime singularities are not covariant under field redefinitions. A manifestly covariant approach to singularities in classical gravity was proposed in [1]. In this paper, we start to extend this analysis to the quantum realm. We identify two types of covariant singularities in field space corresponding to geodesic incompleteness and ill-defined path integrals (hereby dubbed functional singularities). We argue that the former might not be harmful after all, whilst the latter makes all observables undefined. We show that the path-integral measure is regular in any four-dimensional theory of gravity without matter or in any theory in which gravity is either absent or treated semi-classically. This might suggest the absence of functional singularities in these cases, however it can only be confirmed with a thorough analysis, case by case, of the path integral. We provide a topological and model-independent classification of functional singularities using homotopy groups and we discuss examples of theories with and without such singularities.

众所周知，时空奇点在场重新定义下不是协变的。[1] 中提出了经典引力中奇点的明显协变方法。在本文中，我们开始将此分析扩展到量子领域。我们确定了场空间中两种类型的协变奇点，对应于测地线不完备性和定义不明确的路径积分（此处称为函数奇点）。我们认为前者毕竟可能没有害处，而后者使所有可观察量都未定义。我们表明，路径积分测度在任何没有物质的四维引力理论中或在任何引力不存在或半经典处理的理论中都是规则的。这可能表明在这些情况下不存在功能奇点，然而，它只能通过对路径积分的逐案分析来确认。我们使用同伦群提供了一个拓扑和模型独立的泛函奇点分类，我们讨论了有和没有这种奇点的理论例子。