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arxiv: 2605.28686 · v1 · pith:V3RJPK45new · submitted 2026-05-27 · ✦ hep-lat

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

Francesco Scardino , Martina Giachello , Giacomo Gradenigo This is my paper

Pith reviewed 2026-06-29 09:15 UTC · model grok-4.3

classification ✦ hep-lat
keywords symplectic quantizationconstrained quantizationMinkowski spacetimeFeynman generating functionalmicrocanonical ensemblereal-time quantum field theorylattice field theoryanalytic continuation
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The pith

Constrained symplectic quantization recovers the Feynman generating functional with the correct real-time prescription in the continuum limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops constrained symplectic quantization to formulate quantum field theory directly in Minkowski spacetime. It samples quantum fluctuations through Hamiltonian dynamics in an intrinsic time that generates a microcanonical ensemble, after fields and the action are continued analytically into the complex plane. Constraints are imposed to pick out stable trajectories and convergent integration cycles. In the continuum limit this construction is shown to reproduce the standard Feynman generating functional with its proper real-time contour, and the claim is checked by explicit lattice measurements on a free scalar field.

Core claim

By analytically continuing fields and the action from real to complex values and then imposing constraints that simultaneously select stable intrinsic-time trajectories and convergent integration cycles for the microcanonical partition function, the method yields, in the continuum limit, the Feynman generating functional equipped with the correct real-time prescription; the result is verified by direct measurement of real-time two-point functions and by confirmation of Dyson-Schwinger identities (including the proper contact term) for a free scalar field in 1+1 dimensions on a periodic lattice.

What carries the argument

The constraints that select stable intrinsic time trajectories while simultaneously defining convergent integration cycles for the microcanonical partition function.

If this is right

  • Real-time two-point functions can be measured directly without Wick rotation.
  • Dyson-Schwinger identities hold with the proper contact term.
  • The construction extends from quantum mechanics to relativistic quantum field theory on a periodic lattice.
  • In the continuum limit the standard Feynman path integral is recovered.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same constraint mechanism might be applied to interacting theories that suffer from sign problems in Euclidean formulations.
  • Lattice computations could be performed entirely in Minkowski space for a wider class of models.
  • The deterministic microcanonical dynamics may offer a classical underpinning for the quantum measure.

Load-bearing premise

The constraints that select stable intrinsic time trajectories and define convergent integration cycles are sufficient to produce the correct quantum theory.

What would settle it

Direct measurement on the lattice showing that the real-time two-point functions fail to match the expected continuum expressions, or that the Dyson-Schwinger identities are not satisfied with the correct contact term.

Figures

Figures reproduced from arXiv: 2605.28686 by Francesco Scardino, Giacomo Gradenigo, Martina Giachello.

Figure 1
Figure 1. Figure 1: Generalized action conservation along the constrained intrinsic–time evolution. The generalized Hamiltonian H𝜏 is conserved up to step-size errors, providing a global diagnostic of the reversible symplectic integrator under repeated Fourier-space projections. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Momentum-space propagator with periodic boundary conditions. Numerical estimate of Im⟨𝜑˜(𝑛0, 𝑛1)𝜑˜(−𝑛0, −𝑛1)⟩Δ𝜏 compared with the analytic lattice prediction Im 𝐶eP.B.(𝑘 (𝑛0 ) 0 , 𝑘 (𝑛1 ) 1 ) = 1/( ˆ𝑘 2 0 − ˆ𝑘 2 1 − 𝑚 2 ), with ˆ𝑘 𝜇 = 2 𝑎 sin 𝑘 𝜇𝑎/2  . The lightest singularity at 𝑛1 = 0 (𝑘1 = 0) identifies the mass gap. Simulation parameters: mass 𝑚 = 0.6, lattice size 𝑎 = 0.5 on a 𝑁0 × 𝑁1 = 65536 square … view at source ↗
Figure 3
Figure 3. Figure 3: Marginalized correlators on the periodic lattice. Left: spatial marginalization 𝐶𝑇 (ℓ0) = 1 𝑁1 Í ℓ1 𝐶(ℓ0, ℓ1) projects onto 𝑛1 = 0 (𝑘1 = 0) and isolates the temporal oscillation set by the mass gap. Right: temporal marginalization 𝐶𝐿 (ℓ1) = 1 𝑁0 Í ℓ0 𝐶(ℓ0, ℓ1) projects onto 𝑛0 = 0 (𝑘0 = 0) and governs the spatial correlation length. Simulation parameters: mass 𝑚 = 0.6, lattice size 𝑎 = 0.5 on a 𝑁0 × 𝑁1 = 6… view at source ↗
Figure 4
Figure 4. Figure 4: Dyson–Schwinger numerical verification with periodic boundary conditions. We measure ⟨𝛿𝑆/𝛿𝜑(ℓ) 𝜑(ℓ𝑖)⟩ and resolve the localized lattice contact term predicted by Eq. (23). 7. Conclusions and outlook Constrained symplectic quantization provides a stable deterministic dynamics in an intrinsic time variable and reproduces the Feynman weight through a suitable complexification and a con￾strained Hamiltonian fl… view at source ↗
read the original 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 Hamiltonian dynamics in an intrinsic time $\tau$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces constrained symplectic quantization as a generalization of a quantum-mechanical construction to relativistic QFT. Fields and the action are analytically continued from real to complex values; two classes of constraints are imposed that simultaneously select stable intrinsic-time trajectories and define convergent integration cycles for a microcanonical partition function. The central claim is that, in the continuum limit, this procedure recovers the Feynman generating functional equipped with the standard real-time prescription. The only numerical tests reported are for a free scalar field in 1+1 dimensions on a periodic lattice, where real-time two-point functions and Dyson-Schwinger identities (including the contact term) are examined.

Significance. If the construction can be shown to select the correct contour for interacting theories, the approach would supply a deterministic, real-time sampling method for QFT that derives the standard functional integral from constraints rather than from a fitted measure. The absence of free parameters and the explicit derivation from analytic continuation plus stability and convergence conditions are genuine strengths. At present, however, the supporting evidence remains confined to the free Gaussian case.

major comments (1)
  1. [Abstract] Abstract (and the section describing the numerical tests): the claim that the two classes of constraints together enforce the correct real-time integration cycle is verified only for the free scalar field. Because the free theory is quadratic, the microcanonical measure is Gaussian and the constraints may be satisfied identically once the lattice action is written; no demonstration is given that the same constraints continue to select the Feynman contour once non-linear interactions are present and multiple saddles or cycles become possible. This verification is load-bearing for the central claim.
minor comments (1)
  1. [Abstract] The abstract states that the tests are successful but supplies no lattice sizes, error bars, or quantitative measures of agreement for the two-point functions or Dyson-Schwinger identities; these details are required to judge the precision of the reported checks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the section describing the numerical tests): the claim that the two classes of constraints together enforce the correct real-time integration cycle is verified only for the free scalar field. Because the free theory is quadratic, the microcanonical measure is Gaussian and the constraints may be satisfied identically once the lattice action is written; no demonstration is given that the same constraints continue to select the Feynman contour once non-linear interactions are present and multiple saddles or cycles become possible. This verification is load-bearing for the central claim.

    Authors: We agree that the numerical verification of the constraints selecting the correct real-time cycle is performed only for the free scalar field. The construction itself, however, is formulated for a general action: analytic continuation of fields and action to complex values is performed without assuming a quadratic form, and the two classes of constraints (stability of intrinsic-time trajectories and convergence of the integration cycle) are defined independently of the specific interaction terms. The continuum-limit recovery of the Feynman generating functional with the standard prescription is argued at the level of the functional integral by construction. The free-field tests confirm consistency with known results, including the contact term in Dyson-Schwinger identities. We acknowledge that explicit numerical demonstration for interacting theories, where multiple saddles may appear, would provide stronger support. We will revise the abstract and numerical section to clarify the scope of the tests while retaining the general analytic argument. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation proceeds from analytic continuation plus explicit constraints to recover standard functional

full rationale

The paper defines constrained symplectic quantization via analytic continuation of fields/action to complex domain together with two classes of constraints (stable intrinsic-time trajectories and convergent microcanonical cycles). It then states that these yield the Feynman generating functional with standard real-time prescription in the continuum limit. No equation reduces a derived quantity to a fitted parameter or to a self-citation chain; the free-field lattice test verifies two-point functions and Dyson-Schwinger identities but does not constitute the derivation itself. The construction is therefore self-contained against external benchmarks and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, additional axioms, or invented entities are stated beyond the standard analytic continuation and constraint selection.

axioms (1)
  • domain assumption Analytic continuation of fields and action from real to complex numbers yields a well-defined theory whose constraints produce the correct quantum measure.
    Stated in the abstract as the basis for selecting stable trajectories and convergent cycles.

pith-pipeline@v0.9.1-grok · 5690 in / 1232 out tokens · 28416 ms · 2026-06-29T09:15:37.155456+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 13 canonical work pages · 7 internal anchors

  1. [1]

    Constrained Symplectic Quantization II: The Free Scalar Field

    F. Scardino, M. Giachello and G. Gradenigo,Constrained Symplectic Quantization II: The Free Scalar Field,2605.26892

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Gradenigo and R

    G. Gradenigo and R. Livi,Symplectic quantization i: dynamics of quantum fluctuations in a relativistic field theory,Found. Phys.51(2021) 66, [2101.02125]

    work page arXiv 2021

  3. [3]

    Gradenigo,Symplectic quantization ii: Dynamics of space-time quantum fluctuations and the cosmological constant,Found

    G. Gradenigo,Symplectic quantization ii: Dynamics of space-time quantum fluctuations and the cosmological constant,Found. Phys.51(2021) 64, [2101.01795]

    work page arXiv 2021

  4. [4]

    Gradenigo, R

    G. Gradenigo, R. Livi and L. Salasnich,Symplectic quantization iii: Non-relativistic limit,Found. Phys.54(2024) 50, [2401.12355]

    work page arXiv 2024

  5. [5]

    de Alfaro, S

    V. de Alfaro, S. Fubini and G. Furlan,Microcanonical ensemble and quantum field theory,Nuovo Cim. A74(1983) 365

    1983

  6. [6]

    A. E. Strominger,Microcanonical quantum field theory,Annals Phys.146(1983) 419

    1983

  7. [7]

    Iwazaki,Microcanonical formulation of quantum field theories,Phys

    A. Iwazaki,Microcanonical formulation of quantum field theories,Phys. Lett. B141(1984) 342–348

    1984

  8. [8]

    D. J. E. Callaway and A. Rahman,Lattice gauge theory in the microcanonical ensemble,Phys. Rev. D 28(Sep, 1983) 1506–1514

    1983

  9. [9]

    Duane, A

    S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth,Hybrid monte carlo,Phys. Lett. B195 (1987) 216–222

    1987

  10. [10]

    Giachello, F

    M. Giachello, F. Scardino and G. Gradenigo,Symplectic quantization: numerical results for the Feynman propagator on a 1+1 lattice and the theoretical relation with quantum field theory,JHEP09 (2025) 129, [2403.17149]

    work page arXiv 2025

  11. [11]

    Giachello, G

    M. Giachello, G. Gradenigo and F. Scardino,Symplectic quantization and minkowskian statistical mechanics: simulations on a 1+1 lattice,PoSLATTICE2024(2024) 359, [2412.09162]

    work page arXiv 2024

  12. [12]

    Giachello and G

    M. Giachello and G. Gradenigo,Symplectic quantization: A new deterministic approach to the dynamics of quantum fields inspired by statistical mechanics,EPJ Web Conf.314(2024) 00028

    2024

  13. [13]

    Constrained Symplectic Quantization I: the Quantum Harmonic Oscillator

    M. Giachello, F. Scardino and G. Gradenigo,Constrained Symplectic Quantization I: the Quantum Harmonic Oscillator,2601.16963

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Approaches to the sign problem in lattice field theory

    C. Gattringer and K. Langfeld,Approaches to the sign problem in lattice field theory,Int. J. Mod. Phys. A31(2016) 1643007, [1603.09517]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  15. [15]

    Alexandru, G

    A. Alexandru, G. Basar, P. F. Bedaque and N. C. Warrington,Complex paths around the sign problem, Rev. Mod. Phys.94(2022) 015006, [2007.05436]

    work page arXiv 2022

  16. [16]

    Creutz and B

    M. Creutz and B. Freedman,A STATISTICAL APPROACH TO QUANTUM MECHANICS,Annals Phys.132(1981) 427

    1981

  17. [17]

    Parisi and Y.-s

    G. Parisi and Y.-s. Wu,Perturbation Theory Without Gauge Fixing,Sci. Sin.24(1981) 483

    1981

  18. [18]

    High density QCD on a Lefschetz thimble?

    M. Cristoforetti, F. Di Renzo and L. Scorzato,New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble,Phys. Rev. D86(2012) 074506, [1205.3996]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  19. [19]

    The Lefschetz thimble and the sign problem

    L. Scorzato,The Lefschetz thimble and the sign problem,PoSLATTICE2015(2016) 016, [1512.08039]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  20. [20]

    Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence

    A. Behtash, G. V. Dunne, T. Schäfer, T. Sulejmanpasic and M. Ünsal,Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence,Ann. Math. Sci. Appl.02(2017) 95–212, [1510.03435]. 10 Constrained Symplectic Quantization: Quantum Field TheoryFrancesco Scardino

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  21. [21]

    Analytic Continuation Of Chern-Simons Theory

    E. Witten,Analytic Continuation Of Chern-Simons Theory,AMS/IP Stud. Adv. Math.50(2011) 347–446, [1001.2933]. 11

    work page internal anchor Pith review Pith/arXiv arXiv 2011