arXiv subjects

Martina Giachello

Publications and source records attributed to Martina Giachello.

7 recordsLinked to original sources

arXiv2026

Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Field Theory

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 Hamiltonian dynamics in an intrinsic time $τ$ which samples a microcanonical ensemble, in close analogy with the standard microcanonical approach to lattice field theory. In this contribution we present constrained symplectic quantization for relativistic quantum field theory, generalizing from the quantum mechanical case. The method is based on the analytic continuation of fields and action from $\mathbb{R}$ to $\mathbb{C}$ and on constraints that select stable intrinsic time trajectories and that simultaneously define convergent integration cycles for the microcanonical partition function. In the continuum limit we recover the Feynman generating functional with the correct real time prescription. We test the construction for a free scalar field in $1+1$ dimensions on a periodic lattice by measuring real time two point functions and by verifying Dyson Schwinger identities with the correct contact term.

hep-lat↗

arXiv2026

Constrained Symplectic Quantization II: The Free Scalar Field

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.

hep-th↗

arXiv2026

Constrained Symplectic Quantization I: the Quantum Harmonic Oscillator

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.

hep-th↗

arXiv2026

Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Mechanics

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.

hep-lat↗

arXiv2025

Localization and "classical entanglement'' in the Discrete Non-Linear Schrödinger Equation

We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the Discrete Non-Linear Schrödinger Equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called {\it ``classical entanglement''}. Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called $S_{\mathrm{ent}}$, grows with the system size $N$ in the localized phase as $S_{\mathrm{ent}}(N) \sim \log(N)$, therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of {\it ``classical entanglement''} beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations.

cond-mat.stat-mech↗

arXiv2025

Symplectic Quantization: numerical results for the Feynman propagator on a 1+1 lattice and the theoretical relation with Quantum Field Theory

We present here the first lattice simulation of symplectic quantization, a new functional approach to quantum field theory which allows to define an algorithm to numerically sample the quantum fluctuations of fields directly in Minkowski space-time, at variance with all other present approaches. Symplectic quantization is characterized by a Hamiltonian deterministic dynamics evolving with respect to an additional time parameter $τ$ analogous to the fictious time of stochastic quantization. The difference between stochastic quantization and the present approach is that the former is well defined only for Euclidean field theories, while the latter allows to sample the causal structure of space-time. In this work we present the numerical study of a real scalar field theory on a 1+1 space-time lattice with a $λϕ^4$ interaction. We find that for $λ\ll1$ the two-point correlation function obtained numerically reproduces qualitatively well the shape of the free Feynman propagator. Within symplectic quantization the expectation values over quantum fluctuations are computed as dynamical averages along the dynamics in $τ$, in force of a natural ergodic hypothesis connecting Hamiltonian dynamics with a generalized microcanonical ensemble. Analytically, we prove that this microcanonical ensemble, in the continuum limit, is equivalent to a canonical-like one where the probability density of field configurations is $P[ϕ]\propto\exp(zS[ϕ]/\hbar)$. The results from our simulations correspond to the value $z=1$ of the parameter in the canonical weight, which in this case is a well-defined probability density for field configurations in causal space-time, provided that a lower bounded interaction potential is considered. The form proposed for $P[ϕ]$ suggests that our theory can be connected to ordinary quantum field theory by analytic continuation in the complex-$z$ plane.

hep-lat↗

arXiv2024

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

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 $τ$. In this framework, expectation values over quantum fluctuations are computed as dynamical averages along the trajectories parameterized by $τ$. 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.

hep-lat↗