There is no mathematically sound formulation of realistic QFT yet so at this point we have no real answer to your question. The QFT that physicists use to make predictions is in the so-called Lagrangian formulation, which is a heuristic framework for obtaining perturbative exp...
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I examine some problems standing in the way of a successful ``field interpretation'' of quantum field theory. The most popular extant proposal depends on the Hilbert space of ``wavefunctionals.'' But since wavefunctional space is unitarily equivalent to many-particle Fock space, two of the most powerful arguments against particle interpretations also undermine this form of field interpretation.