Euclidean hyperspace and its physical significance

Summary

Contemporary approaches to quantum field theory and gravitation often use a four-dimensional space-time manifold of Euclidean signature (which we call “hyperspace”) as a continuation of the Lorentzian metric. To investigate what physical sense this might have we review the history of Euclidean techniques in classical mechanics and quantum theory. Schwinger’s Euclidean postulate gives a fundamental significance to such techniques and leads to a clearer understanding of the TCP theorem in terms of space-time uniformity. This is closely related to the principle of identicality that characterizes non-relativistic quantum theory. In quantum gravity we suggest that the Hartle-Hawking treatment of the wave function of the universe rests on a notion of space-time uniformity which can be related to the Euclidean postulate as a kind of “perfect cosmological principle” on the unobservable wave function of the universe which eliminates anya priori asymmetry between space and time. Euclidean hyperspace may mediate between the infinite-dimensional Hubert space of quantum theory (whose metric is Euclidean) and the four-dimensional Lorentzian space-time of physical observations.