# Quantum Field Theory and Hilbert space dimensionality

Much (All?) of quantum theory can be done in separable Hilbert spaces with a countable basis.
How about quantum field theory? Is it “quite happy” (mathematically consistent) if everything is countable, or does it “need” to use an uncountable, continuous space (e.g. rigged Hilbert space) for mathematical consistency, or some other reason?

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Quite generally, we want the Hilbert space to be separable in QFT, too. After all, the Hilbert space of a QFT isn't much different from the direct sum of the $N$-particle Hilbert spaces in (non-relativistic) QM (summed over non-negative integers $N$) and those are separable.