Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Mechanics

TL;DR

This paper introduces constrained symplectic quantization, a novel approach that reformulates quantum field theory to directly sample real-time quantum fluctuations, overcoming limitations of previous methods and establishing a link with Feynman path integrals.

Contribution

The work presents a holomorphic reformulation of symplectic quantization with constraints, enabling direct sampling of real-time quantum observables and establishing equivalence with Feynman path integrals.

Findings

Numerical results agree with exact solutions for the quantum harmonic oscillator.

The method reconstructs real-time features like oscillatory propagators and energy spectra.

Constrained symplectic quantization provides a practical route beyond Euclidean importance sampling.

Abstract

Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable $\tau$ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in 1+1 dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time 1-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.