Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Mechanics

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 $τ$ 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.

Figures (5)

• Figure 1: Two-point correlation function in real space for a $\lambda\phi^4$ theory in $1+1$ dimensions simulated on a square lattice of size $L=128$, lattice spacing $a=1.0$ and mass $m=1.0$. The nonlinearity is set to $\lambda=0.001$). Left: oscillations along the $x_0=ct$ (temporal) axis. Right: exponential decay along the $x_1$ (spatial) axis, consistent with causal propagation.
• Figure 2: Left: coordinate-space correlator $\Im\,\langle q(\ell)\,q(\ell')\rangle_{\tau}$ compared with the analytic lattice prediction of Eq. \ref{['eq:2pt-ho-lattice-PBC']}. Right: Fourier-spectrum correlator $\Im\,\langle \tilde{q}(n)\,\tilde{q}(-n)\rangle_{\tau}$ compared with the prediction of Eq. \ref{['eq:2pt-ho-momentum-lattice']}. Simulation parameters: $N_t=1024$, $a=0.1$, $m=1$, $\Omega=2.5$, $d\tau=0.01$, periodic boundaries.
• Figure 3: Fourier spectra of fixed--fixed signals (here $q^j$ with $j=8,9$) for the harmonic oscillator. Simulation parameters: $m=1.0$, $\Omega=2.5$, $a=0.1$, $d\tau=10^{-2}$, $N_t=1024$; fixed boundaries $q(x_0^i)=1.021$, $q(x_0^f)=-2.414\times10^{-1}$.
• Figure 4: Fixed+free setup used for probability-density reconstruction: an ensemble of initial conditions $q_0^{(r)}$ is drawn from a chosen $P_n(q_0)$ and evolved to reconstruct $P(q,x_0^{(\ell)})$ by histogramming.
• Figure 5: Eigenstate probability-density reconstruction with fixed+free boundaries. Initial conditions are sampled from $P_n(q_0)=|\psi_n(q_0)|^2$ and histograms $P(q,x_0^{(\ell)})$ are shown at several intermediate times $x_0^{(\ell)}$. Dashed curves: analytic $|\psi_n(q)|^2$. Left to right: $n=0,1,2$.