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

TL;DR

This paper introduces symplectic quantization, a new method for simulating quantum field fluctuations directly in Minkowski space-time using a deterministic Hamilton-like dynamics, offering an alternative to traditional Euclidean-based sampling.

Contribution

The paper presents symplectic quantization, a novel functional approach enabling Minkowskian sampling of quantum fields through a Hamilton-like evolution, differing from conventional importance sampling methods.

Findings

Microcanonical correlation functions match Minkowskian canonical theory results.

The method provides a direct Minkowski space simulation of quantum fluctuations.

Ergodicity assumption links dynamical averages to quantum expectation values.

Abstract

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 $\exp(S/\hbar )$, with $S$ being the original relativistic action of the system.