Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the 3n-dimensional configuration space rather than on the $3$-dimensional space seemed uncanny to many, including Schrödinger, Lorentz, and Einstein. Even today, this continues to be seen as an important problem in the foundations of quantum mechanics. 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. This fits naturally in the classification of quantum particles by the representations of the spacetime isometries realized by Wigner and Bargmann, adding another layer of confirmation. Once we realize that all quantum experiments take place in space, they can be interpreted naturally and consistently with the wavefunction being an object on space.
Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the 3n-dimensional configuration space rather than on the 3-dimensional space seemed uncanny to many, including Schrödinger, Lorentz, and Einstein. Even today, this continues to be seen as an important problem in the foundations of quantum mechanics. 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. This fits naturally in the classification of quantum particles by the representations of the spacetime isometries realized by Wigner and Bargmann, adding another layer of confirmation. Once we realize that all quantum experiments take place in space, they can be interpreted naturally and consistently with the wavefunction being an object on space.
Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the 3n-dimensional configuration space rather than on the 3-dimensional space has seemed uncanny to many, including Schrödinger, 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.
One of the major concerns of Schrodinger, Lorentz, Einstein, and many others about the wavefunction is that it is defined on the 3N-dimensional configuration space, rather than on the 3-dimensional physical space. This gives the impression that quantum mechanics cannot have a 3D-space or spacetime ontology, even in the absence of quantum measurements. In particular, this seems to affect interpretations which take the wavefunction as a physical entity, in particular the many worlds and the spontaneous collapse interpretations, and some versions of the pilot wave theory.
Here, a representation of the many-particle states is given, as multi-layered fields defined on the 3-dimensional physical space. This representation is equivalent to the usual representation on the configuration space, but it makes it explicit that it is possible to interpret the wavefunctions as living in the physical space. As long as only unitary evolution is involved, the interactions are local. I intended this representation to capture and formalize the non-explicit and informal intuition of many working quantum physicists, who, by considering the wavefunction sometimes to be defined on the configuration space, and sometimes on the physical space, may seem to researchers in the foundations of quantum theory as adopting an inconsistent view about its ontology. This representation does not aim to solve the measurement problem, and it allows for Schrodinger cats just like the usual one. But it may help various interpretations to solve these problems, through inclusion of the wavefunction as (part of) their primitive ontology.
In an appendix, it is shown how the multi-layered field representation can be extended to quantum field theory.
One of the major concerns of Schr\"odinger, Lorentz, Einstein, and many others about the wave function is that it is defined on the 3N-dimensional configuration space, rather than on the three-dimensional (3D) physical space. This gives the impression that quantum mechanics cannot have a 3D space or space-time ontology, even in the absence of quantum measurements. In particular, this seems to affect interpretations which take the wave function as a physical entity, in particular, the many-worlds and the spontaneous collapse interpretations, and some versions of the pilot wave theory. Here, a representation of the many-particle states is given, as multilayered fields defined on the three-dimensional physical space. This representation is equivalent to the usual representation on the configuration space, but it makes it explicit that it is possible to interpret the wave functions as defined on the physical space. As long as only unitary evolution is involved, the interactions are local. I intended this representation to capture and formalize the nonexplicit and informal intuition of many working quantum physicists, who, by considering the wave function sometimes to be defined on the configuration space and sometimes on the physical space, may seem to researchers in the foundations of quantum theory as adopting an inconsistent view about its ontology. This representation does not aim to solve the measurement problem, and it allows for Schr\"odinger cats just like the usual one. But, it may help various interpretations to solve these problems, through inclusion of the wave function as (part of) their primitive ontology. In appendices, it is shown how the multilayered field representation can be extended to quantum field theory.
As shown in our work on spacetime structures of quantum particles, Schrödinger wavefunctions in quantum mechanics can be utilised to construct the geometric structures of quantum particles which are considered to be three-dimensional differentiable manifolds. In this work we will extend this kind of geometric formulation of quantum particles by showing that wavefunctions that are normally used to describe wave phenomena in classical physics can in fact also be utilised to represent three-dimensional differentiable manifolds which in turns are identified with quantum particles. We show that such identification can be achieved by using a three-dimensional wave equation to construct three-dimensional differentiable manifolds that are embedded in a four-dimensional Euclidean space. In particular, the dual character that is resulted from the identification of a wavefunction with a three-dimensional differentiable manifold may provide a classical basis to interpret the wave-particle duality in quantum mechanics.
Abstract There are now several, realist versions of quantum mechanics on offer. On their most straightforward, ontological interpretation, these theories require the existence of an object, the wavefunction, which inhabits an extremely high‐dimensional space known as configuration space. This raises the question of how the ordinary three‐dimensional space of our acquaintance fits into the ontology of quantum mechanics. Recently, two strategies to address this question have emerged. First, Tim Maudlin, Valia Allori, and her collaborators argue that what I have just called the ‘most straightforward’ interpretation of quantum mechanics is not the correct one. Rather, the correct interpretation of realist quantum mechanics has it describing the world as containing objects that inhabit the ordinary three‐dimensional space of our manifest image. By contrast, David Albert and Barry Loewer maintain the straightforward, wavefunction ontology of quantum mechanics, but attempt to show how ordinary, three‐dimensional space may in a sense be contained within the high‐dimensional configuration space the wavefunction inhabits. This paper critically examines these attempts to locate the ordinary, three‐dimensional space of our manifest image “within” the ontology of quantum mechanics. I argue that we can recover most of our manifest image, even if we cannot recover our familiar three‐dimensional space.
The possibility that quantum mechanics is foundationally the same as classical theories in explaining phenomena in space and time is postulated. Such a view is motivated by interpreting the experimental violation of Bell inequalities as resulting from questions of geometry and algebraic representation of variables, and thereby the structure of space, rather than realism or locality. While time remains Euclidean in the proposed new structure, space is described by Projective geometry. A dual geometry facilitates description of a physically real quantum particle trajectory. Implications for the physical basis of Bohmian mechanics is briefly examined, and found that the hidden variables pilot-wave model is local. Conceptually, the consequence of this proposal is that quantum mechanics has common ground with relativity as ultimately geometrical. This permits the derivation of physically meaningful quantum Lorentz transformations. Departure from classical notions of measurability is discussed.