Symplectic Quantization: numerical results for the Feynman propagator on a 1+1 lattice and the theoretical relation with Quantum Field Theory

Abstract

We present here the first lattice simulation of symplectic quantization, a new functional approach to quantum field theory which allows to define an algorithm to numerically sample the quantum fluctuations of fields directly in Minkowski space-time, at variance with all other present approaches. Symplectic quantization is characterized by a Hamiltonian deterministic dynamics evolving with respect to an additional time parameter $\tau$ analogous to the fictious time of stochastic quantization. The difference between stochastic quantization and the present approach is that the former is well defined only for Euclidean field theories, while the latter allows to sample the causal structure of space-time. In this work we present the numerical study of a real scalar field theory on a 1+1 space-time lattice with a $\lambda\phi^4$ interaction. We find that for $\lambda\ll1$ the two-point correlation function obtained numerically reproduces qualitatively well the shape of the free Feynman propagator. Within symplectic quantization the expectation values over quantum fluctuations are computed as dynamical averages along the dynamics in $\tau$, in force of a natural ergodic hypothesis connecting Hamiltonian dynamics with a generalized microcanonical ensemble. Analytically, we prove that this microcanonical ensemble, in the continuum limit, is equivalent to a canonical-like one where the probability density of field configurations is $P[\phi]\propto\exp(zS[\phi]/\hbar)$. The results from our simulations correspond to the value $z=1$ of the parameter in the canonical weight, which in this case is a well-defined probability density for field configurations in causal space-time, provided that a lower bounded interaction potential is considered. The form proposed for $P[\phi]$ suggests that our theory can be connected to ordinary quantum field theory by analytic continuation in the complex-$z$ plane.