Symplectic Quantization and Minkowskian Statistical Mechanics: simulations on a 1+1 lattice

We introduce symplectic quantization, a novel functional approach to quantum field theory which allows to sample quantum fields fluctuations directly in Minkowski space-time, at variance with the traditional importance sampling protocols, well defined only for Euclidean Field Theory. This importance sampling procedure is realized by means of a deterministic dynamics generated by Hamilton-like equations evolving with respect to an auxiliary time parameter $\tau$. In this framework, expectation values over quantum fluctuations are computed as dynamical averages along the trajectories parameterized by $\tau$. Assuming ergodicity, this is equivalent to sample a microcanonical partition function. Then, by means of a large-M calculation, where M is the number of degrees of freedom on the lattice, we show that the microcanonical correlation functions are equivalent to those generated by a Minkowskian canonical theory where quantum fields fluctuations are weighted by the factor $\exp (S\mathrm{/}\mathrm{\hslash })$, with $S$ being the original relativistic action of the system.