Is the Wavefunction Already an Object on Space?

TL;DR

This paper argues that the wavefunction in quantum mechanics can be understood as a genuine object in three-dimensional space, reconciling its mathematical form with classical geometric objects and clarifying its ontological status.

Contribution

It demonstrates that the wavefunction is equivalent to geometric objects on space using reinterpretations within Euclidean geometry and Klein's Erlangen Program, challenging traditional views.

Findings

Wavefunction can be reinterpreted as a geometric object on space.

The approach aligns with Euclidean geometry and Klein's Erlangen Program.

Supports the view that quantum experiments occur in space with a space-based wavefunction.

Abstract

Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the $3\mathbf{n}$-dimensional configuration space rather than on the $3$-dimensional space has seemed uncanny to many, including Schr\"odinger, Lorentz, and Einstein. Even today, this continues to be seen as a significant issue in the foundations of quantum mechanics. In this article, it will be shown that the wavefunction is, in fact, a genuine object on space. While this may seem surprising, the wavefunction does not possess qualitatively new features that were not previously encountered in objects known from Euclidean geometry and classical physics. The methodology used involves finding equivalent reinterpretations of the wavefunction exclusively in terms of objects from the geometry of space. The result is that we will find the wavefunction to be equivalent to geometric objects on space in the same way as was always the case in geometry and physics. This will be demonstrated to hold true from the perspective of Euclidean geometry, but also within Felix Klein's Erlangen Program, which naturally fits into the classification of quantum particles by the representations of spacetime isometries, as realized by Wigner and Bargmann, adding another layer of confirmation. These results lead to clarifications in the debates about the ontology of the wavefunction. From an empirical perspective, we already take for granted that all quantum experiments take place in space. I suggest that the reason why this works is that they can be interpreted naturally and consistently with the results presented here, showing that the wavefunction is an object on space.