On the problem of hidden variables for quantum field theory
  We show that QFT (as well as QM) is not a complete physical theory. We constructed a classical statistical model inducing quantum field averages. The phase space consists of square integrable functions, $f(\phi),$ of the classical bosonic field, $\phi(x).$ We call our model prequantum classical statistical field-functional theory -- PCSFFT. The correspondence between classical averages given by PCSFFT and quantum field averages given by QFT is asymptotic. The QFT-average gives the main term in the expansion of the PCSFFT-average with respect to the small parameter $\alpha$ -- dispersion of fluctuations of "vacuum field functionals.'' The Scr\"odinger equation of QFT is obtained as the Hamilton equation for functionals, $F(f),$ of classical field functions, $f(\phi).$ The main experimental prediction of PCSFFT is that QFT gives only approximative statistical predictions that might be violated in future experiments. 
Andrei Khrennikov