The question is inspired from the Lubos Motl post: http://motls.blogspot.com/2015/05/inconsequential-irrational-incorrect.html which says:
In this approach, quantum mechanics is primary because a Hilbert space pre-exists and the field operators have to "adapt" to it. This is very different from the approach in which a Hilbert space is produced out of local fields, e.g. as the Fock space. In such an approach, the spacetime would be primary and fundamental. Such an approach is probably inapplicable in quantum gravity.
But we do know that unlike in quantum mechanics where the position operator and thinking about one particle are possible, in QFT, these are not possible. So the two frameworks seem not reconcilable, unless we can somehow embed QFT into quantum mechanics and allow talking about one particle, not just fields.
What exactly is the current state of physics for this matter? Is quantum mechanics more primary/fundamental, or is quantum field theory fundamental?