Why the wavefunction already is an object on space

Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the 3ndimensional configuration space rather than on the 3-dimensional space seemed uncanny to many, including Schrödinger, Lorentz, and Einstein. This continues to be seen as an important problem in the foundations of quantummechanics even today. Despite this, in Phys. Rev. A 100, 042115 (2019) (arXiv:1906.12229) it was shown that it is possible to represent the wavefunction as classical fields on space or spacetime, although in a rather complicated way, meant as a proof of concept. But in this article it will be shown that the wavefunction already is a genuine object on space. While this may seem surprising, the wavefunction has no qualitatively new features that were not previously encountered in the objects known from Euclidean geometry and classical physics. This will be shown to be true also in Felix Klein’s Erlangen Program. The relation with the classification of quantum particles by the representations of the spacetime isometries realized by Wigner and Bargmann is discussed. It is argued that overwhelming empirical support already shows that the wavefunction is an object on space.