#### A. Neumaier

Science Advisor

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The problem is that you need to assume the positivity of the quantum measure. This cannot be proved for the functional integrals used in QFT - else they would produce finite results without the need for regularization.It seems that it is possible, in some way, to recover a Hilbert space from the path integral formulation (and in non-relativistic QM, this Hilbert space is the standard Hilbert space):

Even in quantum mechanics, proving positivity requires somewhere a Hilbert space argument....