Constrained Symplectic Quantization I: the Quantum Harmonic Oscillator

TL;DR

This work develops constrained symplectic quantization, an analytic-continuation-based framework that enforces stability via constraints on the generalized Hamiltonian dynamics to sample real-time quantum fluctuations. It proves the continuum equivalence of the constrained microcanonical measure to the Feynman path integral for quantum mechanics, and demonstrates its numerical efficacy by precisely reproducing the harmonic oscillator’s real-time two-point function, canonical commutator, and discrete Dirichlet-spectrum signatures on a Minkowskian lattice. By combining complex-field integration contours with a stable manifold constraint, the method overcomes limitations of prior real-time sampling approaches and provides a viable route to real-time nonperturbative calculations. The results indicate genuine real-time features—oscillatory propagators and excited-state towers—that Euclidean methods miss, highlighting the potential of constrained symplectic quantization for broader quantum-field-theoretic and many-body problems.

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 microcanonical ensemble similar to the one of the standard microcanonical approach to lattice field theory. In a previous paper we showed that, for an interacting scalar field theory in 1+1-dimensions, this formalism allows to capture numerically some crucial real-time features inaccessible to any Euclidean approach to lattice field theory. Yet, the new approach was plagued by two main limitations: an ill-defined non-interacting limit and the absence of a direct formal correspondence between its correlation functions and those generated by the Feynman path integral approach. In this paper, we introduce the new \emph{"constrained symplectic quantization"} approach, for which the perfect equivalence with the Feynman path integral is proved and which is perfectly well defined for the free theory. This new approach is characterized by the analytical continuation of all fields and of the action from $\mathbb{R}$ to $\mathbb{C}$ and the presence of some constraints which guarantee the stability of the generalized Hamiltonian dynamics and the convergence of the corresponding generalized microcanonical partition function, hence the name of the theory. We show the application of this formalism to the quantum harmonic oscillator on a Minkowskian-time lattice, finding perfect agreement between one- and two-point numerical correlators and the exact quantum-mechanical results. We observe genuine real-time features such as the oscillatory propagator and the discrete excited-state energy spectrum. Our results provide strong numerical evidence that constrained symplectic quantization can sample real-time quantum-mechanical observables, offering a concrete route to overcome the limitations of Euclidean-time importance sampling.

Figures (7)

• Figure 1: Integration contours in the complex $q_k$ plane. Each mode is rotated so that the holomorphic action becomes real and negative: contour (A) rotated by $+\pi/4$ for $k_0^2 < \Omega^2$, contour (B) rotated by $-\pi/4$ for $k_0^2 > \Omega^2$. This is equivalent to imposing the linear relations among Fourier components used in the symplectic dynamics.
• Figure 2: Time evolution of the kinetic ($E_{kin}=\mathbb{K}[\pi]$) and potential ($E_{pot}=\mathbb{V}[q]$) energy for $m=1.0$, $\Omega=2.5$, $a=0.1$, with integration parameters $d\tau=0.01$, and total simulation time $\Delta \tau=100$. The system size is $M=1024$, and we use periodic boundaries.
• Figure 3: Expectation value of $q(x_0^{(\ell)})$ with periodic boundary conditions and for different values of $\Delta\tau$. Simulation parameters: $M=128$, $a=0.1$, $\Omega=2.5$ and $m=1$.
• Figure 4: Harmonic-oscillator two-point function on the lattice with periodic boundary conditions. Left: coordinate-space correlator $\Im\,\langle q(\ell)\,q(\ell')\rangle_{\tau}$ compared with the analytic lattice prediction \ref{['eq:2pt-ho-lattice-PBC']}. Right: Fourier-spectrum correlator $\Im\,\langle \tilde{q}(n)\,\tilde{q}(-n)\rangle_{\tau}$ compared with the prediction \ref{['eq:2pt-ho-momentum-num-conv']}. Simulation parameters: $M=1024$, $a=0.1$, $m=1$, $\Omega=2.5$, $dt=0.01$, periodic boundaries.
• Figure 5: Numerical determination of the equal--time commutator from the constrained symplectic dynamics. The solid line is the exact prediction $C(\hbar)=\hbar$, while points are the intrinsic--time averages computed from Eq. \ref{['eq:comm-estimator-numerical-qREqRO']}.