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3 or 3N Dimensions? Underdetermination about the Fundamental Space of Quantum Mechanics

Matthias Egg

Published May 20, 2016• 0 citations
OpenAlex

Abstract

The question concerning the dimensionality of our fundamental physical space is arguably one of the most profound cases of underdetermination in non-relativistic quantum mechanics. Those who take the wave function to represent fundamental reality think that the world fundamentally has 3N dimensions, where N is the total number of particles. Against this, various versions of quantum mechanics postulate certain entities in 3-dimensional space as fundamental. An important argument for the latter view suggests that this is not strictly a case of underdetermination of theory by evidence, because a view that does not include some entities in 3-dimensional space fails to explain how there can be empirical evidence for quantum mechanics in the first place. A critical assessment of this argument will clarify how the debate on dimensionality depends on the evaluation of non-empirical virtues which is well-known from other debates on underdetermination.

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The question concerning the dimensionality of our fundamental physical space is arguably one of the most profound cases of underdetermination in non-relativistic quantum mechanics. Those who take the wave function to represent fundamental reality think that the world fundamentally has 3N dimensions, where N is the total number of particles. Against this, various versions of quantum mechanics postulate certain entities in 3-dimensional space as fundamental. An important argument for the latter view suggests that this is not strictly a case of underdetermination of theory by evidence, because a view that does not include some entities in 3-dimensional space fails to explain how there can be empirical evidence for quantum mechanics in the first place. A critical assessment of this argument will clarify how the debate on dimensionality depends on the evaluation of non-empirical virtues which is well-known from other debates on underdetermination.
The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wave function is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the deeper topological explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence.
The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wave function is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the deeper topological explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence.
The existence of ontologically different but empirically indistinguishable versions of non-relativistic quantum mechanics is one of the most serious and widely discussed cases of metaphysical underdetermination. I argue that this problem is unlikely to disappear when we turn to the next more fundamental theory, namely relativistic quantum field theory. The problem is merely overshadowed by yet another interpretational controversy, but it persists regardless of the outcome of this latter debate. I further argue that one should not be too optimistic about transferring ontological lessons learned in the context of one physical theory to the context of a more fundamental theory.
Abstract The three major interpretations of quantum mechanics—Bohm, GRW, and many worlds—all treat the wave function as a physical entity. But this entity apparently resides in a high-dimensional configuration space, with three coordinates for each particle in the universe. Some conclude on this basis that the three-dimensionality of the world of experience is an illusion. Others respond by adding three-dimensional ontology to the theory or by reinterpreting the wave function either as a law or as an indirect representation of properties in three-dimensional space. It is argued that the latter possibility makes it unnecessary to postulate either additional ontology or a fundamentally high-dimensional world.
What does realism about the quantum world entail? If one takes the formalism of the theory at face value, the most plausible candidate for a fundamental entity is the quantum wave function. The wave function cannot be represented in standard three-dimensional space, but instead inhabits a space of very high dimensionality. This poses a problem for realists, because we ordinarily take the world to be three-dimensional. One response is to explain away the apparent three-dimensionality of the everyday world as an illusion. Another is to insist on a way of construing quantum reality as fundamentally three-dimensional, even if this means supplementing or replacing the standard quantum formalism. Neither option is attractive, and neither is warranted. Rather, close attention to the structure of the standard quantum formalism reveals a picture of the world that, although unfamiliar, does not contradict the three-dimensionality of the everyday world.
The quantum wave-function is at the heart of quantum mechanics. It is the physical state of a quantum system before a measurement is taken. This wave-function is described mathematically by the formulation of matrix mechanics and represents the evolution of the quantum wave-function. In contemporary philosophy of physics, the wave-function’s nature of space is hotly debated. Contemporaries such as David Albert, who posits an ontology which is based off the mathematical features of the wave-function, claim that the dimensionality of our physical space is proportional to 3N, where N is equal to the number of particles in the universe. Others such as Bradley Monton claim quantum mechanics to be a false theory from the outset due to its incompatibility with general relativity and posit a much more realist property-based-physics ontology. However, these ontologies have their problems. Albert’s ontology takes the mathematics of the quantum wave-function at face value and disregards the distance properties of metric space to misconstrue the phase space nature of the 3N-configuration space as a metric space. Monton, on the other hand, questionably grounds his thesis on the incompatibility of quantum mechanics with general relativity and on common sense rather than on observation and experimentation, as required by the scientific method. In this thesis, I discuss the viewpoints of Albert and Monton in detail and explore the nature of space in the quantum wave-function in an attempt to reach a better understanding of the fundamental nature of space in the quantum wave-function.