Constrained Symplectic Quantization II: The Free Scalar Field

Abstract

Constrained symplectic quantization is a functional formulation of quantum field theory in which quantum fluctuations are sampled through a deterministic Hamiltonian flow in an auxiliary intrinsic time $τ$. In this paper we extend the quantum-mechanical framework introduced in [1] to a relativistic scalar quantum field theory in Minkowski space-time. The construction is based on the analytic continuation of fields and action from $\mathbb{R}$ to $\mathbb{C}$ together with constraints that select stable intrinsic-time trajectories and, at the same time, define convergent integration cycles for the corresponding microcanonical functional. We show that, in the continuum limit, the microcanonical generating functional reproduces the Feynman generating functional. For the free scalar field in $1+1$ dimensions we derive the constrained equations of motion, implement the resulting dynamics numerically, and verify real-time two-point correlators, equal-time commutator relations, and Dyson--Schwinger equations including the expected contact terms.