Real-Time Continuum Dynamics from Regulated Computations

Regulated calculations can recover selected real-time observables, but no regulator gives generic continuum dynamics with cheap, uniformly controlled errors. Euclidean methods are strongest for spectra and inclusive quantities linked to well-conditioned integral transforms; Hamiltonian and tensor-network methods access time directly but face truncation and entanglement growth; quantum simulation offers a new direct route whose continuum and hardware errors must still be certified together.

Evidence cutoff. 11 August 2026.

Required background. Real-time evolution and observable extraction supplies the Hamiltonian target; spectral reconstruction as an inverse problem explains Euclidean limitations. Helpful background. Real-time tensor-network dynamics supplies entanglement-controlled evolution, observables and dynamics in truncated spaces supplies cutoff errors, and Kadanoff–Baym evolution supplies continuum nonequilibrium closure.

Recoverable real-time observables

Normative question. Which real-time observables can be recovered with controlled errors from Euclidean, Hamiltonian, tensor-network, or quantum-simulation regulators?

The scope includes spectral functions, transport coefficients, scattering or inclusive response, quenches, and finite-time correlation functions in theories with a specified continuum target. It excludes a generic promise to reconstruct pointwise Minkowski correlators from finite noisy Euclidean data and treats hardware execution as evidence only after state preparation, noise, finite volume, and continuum scaling are validated.

Exact Euclidean Schwinger functions satisfying the Osterwalder and Schrader 1973 axioms determine the Minkowski theory. Numerical data are finite and noisy, however, and the map

$$G_E(\tau)=\int_0^\infty d\omega\,K(\tau,\omega)\rho(\omega)$$

smooths the spectral density. Inverting it amplifies small errors. Maximum entropy, Bayesian priors, Backus–Gilbert resolution functions, and neural ansätze regularize the problem but do not create information absent from the data. A transport coefficient that depends on a narrow $\omega\to0$ feature is therefore much harder than a smeared spectral integral.

Route	Best-controlled targets	Dominant errors and failure modes
Euclidean correlators	Discrete spectra, moments, sum rules, inclusive or smeared response, finite-volume amplitudes below inelastic complexity	Ill-conditioned continuation, finite temporal extent, kernel resolution, finite volume, lattice spacing
Hamiltonian truncation	Low-energy spectra and finite-time correlators in low dimensions or sparse bases	Basis cutoff, induced operators, renormalization, high-energy leakage, finite volume
Tensor networks	One-dimensional and some quasi-one-dimensional quenches before entanglement saturation	Bond-dimension error grows with entanglement; gauge constraints and continuum extrapolation
Quantum simulation	Direct unitary evolution and local observables for encoded, finite systems	State preparation, Trotter or algorithmic error, noise, gauge leakage, resource scaling, continuum certification

Finite-volume methods can turn Euclidean energy levels and matrix elements into scattering amplitudes when the channel and kinematics satisfy a derived quantization condition (Lüscher 1991). This is a controlled bridge, not generic analytic continuation. Inclusive observables can sometimes be designed as smeared spectral integrals whose resolution is matched to Euclidean information; Hansen, Meyer, and Robaina demonstrate such a finite-volume route for inclusive rates (Hansen, Meyer, and Robaina 2017). Pointwise spectral features narrower than the achievable resolution remain prior dominated.

Tensor-network studies of lattice gauge theories demonstrate direct real-time phenomena in 1+1 dimensions, but global quenches generally generate entanglement linearly in time, forcing exponential bond-dimension growth. Quantum algorithms for lattice field theory provide polynomial procedures under explicit discretization, state-preparation, and fault-tolerance assumptions (Jordan, Lee, and Preskill 2012). Present analogue and digital experiments are valuable demonstrations, not yet generic continuum calculations with end-to-end error budgets; the trapped-ion Schwinger-model experiment is a representative bounded achievement (Martinez et al. 2016).

Independence and negative results

Agreement between a Euclidean reconstruction and a Hamiltonian calculation is strongest when they do not share the same spectral ansatz, bare-parameter tuning, or perturbative matching. Two Bayesian reconstructions of one correlator are not independent evidence if their resolution kernels and priors are effectively the same. Likewise, two quantum devices using the same small lattice and extrapolation model mainly test hardware reproduction, not continuum universality.

Assessment. The question is partially resolved by observable class. Finite-volume scattering data, low-lying spectra, suitably smeared response, and bounded low-dimensional real-time dynamics can have controlled errors. Generic pointwise spectral functions, long-time higher-dimensional dynamics, and transport peaks remain difficult. No current platform provides a universal, scalable, regulator-to-continuum solution.

What would establish continuum control?

For a claimed observable, the minimum evidence is a renormalized definition, a sequence of volumes and regulator cutoffs, a convergent extrapolation with covariance, and a resolution or truncation bound that shrinks as resources increase. A cross-method benchmark must compare the same smeared or finite-time observable and hold back at least one dataset from tuning. For quantum simulation, the error budget must include encoding, state preparation, noise mitigation or correction, and continuum scaling; matching a classically simulable finite lattice is necessary but not sufficient.

Related routes include lattice and Hamiltonian field theory, thermal and nonequilibrium field theory, Hamiltonian truncation, tensor networks, and quantum simulation, and real-time QFT, kinetic theory, and hydrodynamics.

Evidence cutoff and source selection

The finite set emphasizes reconstruction theorems, finite-volume/inclusive bridges, direct Hamiltonian demonstrations, and explicit resource or inverse-problem limits. Targeted arXiv, journal, lattice, tensor-network, and quantum-simulation searches covered public evidence through 11 August 2026. Results without an identifiable continuum target or error decomposition were treated as demonstrations, not continuum determinations.

References

• Hansen, Maxwell T., Harvey B. Meyer, and Daniel Robaina. “From Deep Inelastic Scattering to Heavy-Flavor Semileptonic Decays: Total Rates into Multihadron Final States from Lattice QCD.” Physical Review D 96 (2017): 094513. DOI.
• Jordan, Stephen P., Keith S. M. Lee, and John Preskill. “Quantum Algorithms for Quantum Field Theories.” Science 336 (2012): 1130–1133. DOI.
• Lüscher, Martin. “Two-Particle States on a Torus and Their Relation to the Scattering Matrix.” Nuclear Physics B 354 (1991): 531–578. DOI.
• Martinez, Esteban A., et al. “Real-Time Dynamics of Lattice Gauge Theories with a Few-Qubit Quantum Computer.” Nature 534 (2016): 516–519. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI.