  We present here the first numerical test of symplectic quantization, a new
approach to quantum field theory. Symplectic quantization is characterized by
the possibility to sample quantum fluctuations of relativistic fields by means
of a deterministic dynamics generated by Hamilton-like equations, the latter
evolving with respect to an additional time parameter $\tau$. In the present
work we study numerically the symplectic quantization dynamics for a real
scalar field in 1+1 space-time dimensions and with $\lambda \phi^4$ non-linear
interaction. We find that for $\lambda \ll 1$ the Fourier spectrum of the
two-point correlation function obtained numerically reproduces qualitatively
well the shape of the Feynman propagator. Within symplectic quantization the
expectation over quantum fluctuations is computed as a dynamical average along
the trajectories parametrized by the intrinsic time $\tau$. As a numerical
strategy to study quantum fluctuations of fields directly in Lorentzian
space-time, we believe that symplectic quantization will be of key importance
for the study of non-equilibrium relaxational dynamics in quantum field theory
and the phenomenology of metastable bound states with very short life-time,
something usually not accessible by numerical methods based on Euclidean field
theory.
