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.
Quantum field theory (QFT) presents a genuine example of the underdetermination of theory by empirical evidence. There are variants of QFT—for example, the standard textbook formulation and the rigorous axiomatic formulation—that are empirically indistinguishable yet support different interpretations. This case is of particular interest to philosophers of physics because, before the philosophical work of interpreting QFT can proceed, the question of which variant should be subject to interpretation must be settled. New arguments are offered for basing the interpretation of QFT on a rigorous axiomatic variant of the theory. The pivotal considerations are the roles that consistency and idealization play in this case.