Algebraic quantum field theory, following Haag and Kastler, shows that inequivalent representations of the CCRs are physically distinct, making the separable Hilbert space choice underdetermined and arbitrary.
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Inequivalent representations(describing multiple ways to set up the mathematics)
Different mathematical ways of describing the same physical rules that actually lead to different physical outcomes—like two different maps of the same territory that don't match up.
Separable Hilbert space(a particular choice of mathematical framework)
A specific type of mathematical space that is 'countable' or can be built up from a simple list of building blocks, like how all numbers can be built from counting numbers.
underdetermined(Used to contrast mathematical conjectures, whose correct answers are fixed by logical facts, with cases where no such determination exists.)
A decision or truth is underdetermined when logical facts alone do not fix a single correct answer.