Infinite Scale Skepticism: Probing the Epistemology of the Limit of Infinite Degrees of Freedom and Hilbert Space Non-Uniqueness

Abstract

There has been much debate about how the semantic content of quantum theories with infinitely many degrees of freedom ought to be understood. This paper focuses instead on the epistemological status of the limit of infinite degrees of freedom. Do we have strong grounds to believe that the actual world is an infinite quantum system? I consider two types of argument for thinking that we do: indispensability arguments, which appeal to the role that unitarily inequivalent Hilbert space representations play in accounting for physical phenomena like spontaneous symmetry breaking, and what I call extrapolation arguments, which support the physical significance of the limit of infinite degrees of freedom by pointing to infinite volume cosmological models. Neither of these strategies turn out to be conclusive as they stand leaving the relevance of novel features of infinite quantum systems to real-world physics murky. I go on to discuss ways of motivating foundational work on infinite quantum systems which do not turn on the representational veracity of these features, and point out that when it comes to the question of the epistemological status of unitarily inequivalent Hilbert space representations other sources of non-uniqueness than the limit of infinite degrees of freedom must also be considered.