Effective Propagators in Theory & Computation

• Effective propagators are specialized operators modified via renormalization, regularization, or learning to capture critical dynamics in various systems.
• They are applied in quantum field theory as non-perturbative Green functions, in numerical simulations for stable time evolution, and in constraint programming for efficient domain contraction.
• Their design balances exact reformulation and controlled approximation, optimizing accuracy, convergence, and computational efficiency across diverse applications.

Across the literature surveyed here, effective propagators are propagators or propagator-like operators that are transformed, renormalized, regularized, derived, or learned so that they encode the physically or algorithmically relevant degrees of freedom of a reduced description. In quantum field theory and many-body physics, they appear as non-perturbative Green functions, renormalized resolvents, or EFT amplitudes; in numerical analysis they appear as time-advancement operators tailored to structure, stability, or parallel efficiency; and in constraint programming the term denotes contracting operators that propagate domain information rather than field excitations. This breadth is explicit in finite-temperature QCD, threshold EFT, AdS and de Sitter constructions, cosmological resummation, time-dependent Schrödinger solvers, learned wave simulators, and view-based or half-checking propagators in Gecode (Fukushima et al., 2012, Suzuki et al., 2012, Xu et al., 2024, Costa et al., 2014, Bernardeau et al., 2011, Bader et al., 2018, 0806.1806).

1. Domains and meanings of the term

In the surveyed literature, the word propagator does not designate a single object. In gauge theory and EFT it refers to two-point Green functions or effective amplitudes. In effective-interaction theory it refers to renormalized resolvents entering the $\widehat Q$ box. In numerical PDE and quantum-dynamics work it denotes the operator that advances a state by one coarse, exact-split, exponential, or learned time step. In constraint programming it denotes a contracting function on domains, with “derived propagators” and “half-checking propagators” being formal classes rather than Green functions. This suggests that the encyclopedia notion of effective propagators is best understood operationally: the propagator is made effective by incorporating otherwise implicit structure into a tractable representation (Suzuki et al., 2012, Bader et al., 2018, Jin et al., 2023, Lagerkvist et al., 2020).

Domain	Representative construction	Stated role
Finite-temperature QCD	Non-perturbative gluon and ghost propagators in Landau gauge	Polyakov-loop effective potential and thermodynamics
EFT and many-body theory	Renormalized propagators; $G_X(E)$ for threshold states	Effective interactions and lineshape analysis
AdS, de Sitter, cosmology	Split, celestial, and regularized propagators	Witten diagrams, effective actions, and polyspectra
Numerical simulation	CF, Strang-split, optimized coarse, and learned wave propagators	Stable or accelerated time evolution
Constraint programming	Perfect derived and half-checking propagators	Domain contraction under exact or incomplete propagation

A recurrent distinction is between exact reformulation and controlled approximation. Some constructions are formally exact if a recursion or truncation converges, as in the renormalized $\widehat Q$-box formulation (Suzuki et al., 2012). Others are deliberately approximate but designed to preserve key asymptotics, such as regularized cosmic propagators that match low-$k$ perturbation theory and large-$k$ resummation (Bernardeau et al., 2011), or coarse propagators optimized for parareal convergence (Jin et al., 2023).

2. Finite-temperature QCD and deconfinement effective potentials

A central non-perturbative use of effective propagators is the construction of the Polyakov-loop effective potential from finite-temperature Landau-gauge gluon and ghost propagators measured on the lattice. At finite temperature, the gluon propagator is decomposed into transverse and longitudinal components relative to the heat bath,

$$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$

with $\xi\to0$ in the Landau gauge. The gluon and ghost propagators are parametrized with Gribov-Stingl forms, and these non-perturbative inputs are inserted into the leading term of the 2-particle-irreducible formalism. The resulting glue potential is approximated by

$$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$

or, after decomposition,

$$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$

The Polyakov loop is introduced through the Matsubara shift $p_4\to(2\pi nT+gA_4)$, with

$G_X(E)$0

For Gribov-Stingl propagators, analytic Matsubara summation yields terms such as $G_X(E)$1, where $G_X(E)$2 carries the Polyakov-loop dependence through coefficients $G_X(E)$3 (Fukushima et al., 2012).

Within this construction, the effective potential exhibits a first-order deconfinement phase transition for pure SU(3) Yang-Mills theory, with critical temperatures $G_X(E)$4 MeV and $G_X(E)$5 MeV for the two lattice-fit parameter sets, while the actual critical temperature with fully $G_X(E)$6-dependent propagators would fall in this range and is consistent with the empirically known value $G_X(E)$7 MeV. The same framework yields a second-order transition for SU(2). When plotted against $G_X(E)$8, the Polyakov loop and thermodynamic quantities are nearly universal, supporting the phenomenological use of a single scale $G_X(E)$9 in Polyakov-loop potentials. The pressure agrees well with lattice data near $\widehat Q$0, while the interaction measure is less accurately reproduced above $\widehat Q$1; the stated sources of discrepancy are missing temperature dependence in the propagator parametrization and the neglect of subleading $\widehat Q$2 terms. When the same potential is used as input to a chiral model such as the $\widehat Q$3-flavor NJL model, the chiral and deconfinement order parameters undergo simultaneous crossover, and the thermodynamic quantities become sensitive to the detailed temperature dependence of the propagators, whereas the order parameters are less sensitive (Fukushima et al., 2012).

Near the transition temperature, the propagators themselves display sector-dependent sensitivity. Landau-gauge studies in SU(3) gluodynamics and in full QCD with $\widehat Q$4 show that the longitudinal propagator $\widehat Q$5 is strongly phase sensitive in the infrared, whereas the transverse propagator $\widehat Q$6 is weakly temperature dependent for $\widehat Q$7. The electric screening mass is extracted from a Stingl-like fit,

$\widehat Q$8

yielding $\widehat Q$9 and $k$0 in gluodynamics at $k$1 and $k$2, and $k$3 and $k$4 in full QCD at the same temperatures. The paper explicitly concludes that $k$5 shows almost no discontinuity across $k$6 and that the electric mass does not act as an order parameter; the parameter $k$7 is more sensitive to the transition (Bornyakov et al., 2011).

3. Infrared structure, confinement, and gauge-covariant dressing

Effective propagators in Yang-Mills theory are frequently organized around infrared suppression, positivity violation, and confinement criteria. In SU(3) lattice QCD in the maximally Abelian gauge with $k$8 Landau gauge fixing, the diagonal and off-diagonal gluon propagators are separated in coordinate and momentum space. Effective masses extracted from both spaces are $k$9 and $k$0 in the relevant fit windows. In momentum space, the off-diagonal propagator is suppressed in the infrared and appears finite at zero momentum, while the diagonal propagator is enhanced. The propagators are well fitted by

$k$1

for $k$2 GeV, and the analysis concludes that all spectral functions of diagonal and off-diagonal gluons would have negative regions. A common misconception is that an effective propagator of this type must reduce to a simple Proca pole; the MA-gauge fits explicitly favor the more general $k$3-Ansatz over either the simple massive form $k$4 or the four-dimensional Yukawa form $k$5 (Gongyo et al., 2013).

In Coulomb gauge, lattice studies in the Hamiltonian limit organize the effective description around static correlators and self-energies. The gluon propagator, Coulomb potential, and ghost form factor are reported as

$k$6

$k$7

Here $k$8 GeV is identified as the Gribov mass. The inverse static gluon propagator defines an infrared-enhanced gluon effective energy $k$9, and the quark propagator

$$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$0

induces an effective quark energy

$$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$1

which is also infrared divergent. The Coulomb string tension is extracted as $$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$2, summarized in the abstract as $$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$3 (Burgio et al., 2013).

A further infrared diagnostic is the scale-dependent spectral dimension derived from a dressed inverse propagator $$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$4 through the return probability

$$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$5

For the class of propagators that display a maximum at Euclidean momenta and therefore violate positivity, the large-$$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$6 limit approaches $$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$7. The paper emphasizes that the longest diffusion times are not related to the deep infrared but to the momentum scale defined by the position of the maximum. This directly constrains how effective infrared behavior is inferred from a non-perturbative propagator (Kern et al., 2019).

Functional calculations make the dependence of effective propagators on higher vertices explicit. In Landau-gauge Yang-Mills theory, Dyson-Schwinger calculations show that the ghost-gluon vertex leads only to minor modifications, while the three-gluon vertex has a sizeable impact on the mid-momentum regime of the gluon propagator. An effective three-gluon-vertex model is introduced to incorporate contributions from neglected two-loop diagrams, and this allows propagators in good agreement with lattice data; the same setup also produces a first self-consistent calculation including all two- and three-point functions (Huber et al., 2014). In QED, gauge covariance imposes an operator constraint on spectral Schwinger-Dyson kernels,

$$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$8

so that solutions in arbitrary covariant gauges remain consistent with the Landau-Khalatnikov-Fradkin transformation. The paper explicitly shows that the Gauge Technique, dimensionally regularized in 4D, does not satisfy this covariance requirement (Jia et al., 2016).

4. Renormalized propagators in effective interaction theory and threshold EFT

In nuclear effective-interaction theory, effective propagators arise from a reorganization of the model-space resolvent. The $$D_{A\,\mu\nu}(p^2)=\delta^{ab}\left(D_T^{(T)}P^T_{\mu\nu}+D_T^{(L)}P^L_{\mu\nu}+\xi D_L L_{\mu\nu}\right),$$9 box in the Bloch-Horowitz or Feshbach form,

$\xi\to0$0

is made tractable by introducing a new basis that transforms the Hamiltonian into a block-tridiagonal form whose blocks have dimension at most that of the model space $\xi\to0$1. This produces two equivalent representations. The first is a continued-fraction recursion,

$\xi\to0$2

leading to

$\xi\to0$3

The second is a series expansion in renormalized vertices and propagators,

$\xi\to0$4

The paper states that this yields an exact $\xi\to0$5 box if the calculation converges as the Hilbert-space dimension tends to infinity, and that the resulting renormalized propagators are non-perturbative, computationally tractable, and avoid inversion of the full $\xi\to0$6 matrix (Suzuki et al., 2012).

Near S-wave thresholds, the effective propagator becomes an EFT amplitude that carries compositeness information. The general form is

$\xi\to0$7

with

$\xi\to0$8

The compositeness relations include

$\xi\to0$9

This propagator is presented as a general formula for S-wave near-threshold states that can be used to fit lineshapes and extract the $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$0 factor. Its limiting cases reproduce Breit-Wigner for $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$1, low-energy scattering for $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$2, and the Flatté-type regime for $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$3. A common misconception addressed in the paper is that a single standard parameterization is structurally sufficient; instead, the EFT propagator is claimed to remain valid for all $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$4 and to avoid assuming that a state is purely compact or purely molecular (Xu et al., 2024).

5. Curved spacetime, AdS/CFT, celestial transforms, and cosmic resummation

In AdS, effective propagators are often obtained by representation changes that expose symmetry or factorization. The embedding formalism for symmetric traceless tensors in AdS$$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$5 encodes rank-$$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$6 fields by polynomials $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$7 with $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$8 and $$\beta\Omega_{\text{glue}}\simeq -\frac12 \operatorname{tr}\ln D_A^{-1}+\operatorname{tr}\ln D_C^{-1},$$9, and it yields an explicit bulk-to-bulk propagator for massive spin-$$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$0 fields together with a split representation that writes the propagator as an integral over the boundary of a product of two bulk-to-boundary propagators. This split representation is then used to obtain the conformal partial wave decomposition of Witten diagrams and the Mellin amplitude for AdS graviton exchange between minimally coupled scalars of general dimension, including the regular part of the amplitude (Costa et al., 2014).

A related transform maps AdS scalar propagators into the celestial basis. Starting from the Euclidean AdS bulk-to-boundary propagator in Schwinger parametrization, one constructs the boundary-to-boundary propagator and then transforms it with conformal primary wavefunctions. For massless scalars, the resulting celestial propagator reduces to an effectively two-dimensional boundary-to-boundary object on the celestial sphere that depends on the AdS/CFT conformal dimension $$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$1. For massive scalars, the celestial propagator contains a nontrivial kernel involving modified Bessel functions and closely resembles the momentum-space radial structure of AdS bulk-to-boundary propagators. The paper describes this as a structural translation from AdS propagators to celestial propagators (Srisangyingcharoen, 14 May 2026).

In de Sitter space, effective propagators depend on both patch and state. Time-ordered propagators between different $$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$2-states are constructed in the global manifold and in the Poincaré patch, with separate analysis of $$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$3-$$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$4, In-In, and In-Out correlators. The In-In propagators are real in both the Poincaré patch and the global manifold. The In-Out propagators at coincident points have finite imaginary contributions in even dimensions in both patches, but the two patch constructions are not equivalent; in odd dimensions the imaginary contributions vanish. The static-patch analysis identifies the state equivalent to the Bunch-Davies one in the Poincaré patch (Akhmedov et al., 2019).

In large-scale structure, regularized cosmic propagators are constructed to interpolate between low-$$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$5 perturbation theory and large-$$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$6 resummed behavior. For a multi-point propagator $$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$7, the generic one-loop regularized form is

$$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$8

with the high-$$\beta\Omega_{\text{glue}}=-\frac12 \operatorname{tr}\ln D_L^{-1}-\operatorname{tr}\ln D_T^{(T)-1}-\frac12 \operatorname{tr}\ln D_T^{(L)-1}+\operatorname{tr}\ln D_C^{-1}.$$9 damping controlled by the displacement variance $p_4\to(2\pi nT+gA_4)$0. The construction is stated to apply to any multi-point propagator, to match perturbative low-$p_4\to(2\pi nT+gA_4)$1 calculations to any number of loops, and potentially to extend to non-Gaussian initial conditions. Its validity is checked against previous prescriptions and measurements in numerical simulations, and it is used to give a consistent one-loop calculation of the matter bispectrum within the $p_4\to(2\pi nT+gA_4)$2-expansion (Bernardeau et al., 2011).

6. Numerical, learned, and algorithmic propagators

In numerical time evolution, effective propagators are often designed by exploiting the algebraic structure of the generator. For the time-dependent Schrödinger equation with Hamiltonian $p_4\to(2\pi nT+gA_4)$3, commutator-free exponential propagators are constructed as products of exponentials of linear combinations of $p_4\to(2\pi nT+gA_4)$4 and quadrature-sampled potentials. The paper proposes new fourth- and sixth-order CF propagators tailored to this structure, including a sixth-order method with a double commutator term $p_4\to(2\pi nT+gA_4)$5 that only depends on coordinates and is therefore treated as cost-free. The action of the exponentials is computed with the Lanczos method, and the reported performance is up to $p_4\to(2\pi nT+gA_4)$6 better than standard commutator-free methods and the exponential midpoint rule for the tested problems (Bader et al., 2018).

For parallel-in-time integration, the coarse propagator itself becomes an optimization target. In the parareal framework, the modewise convergence factor is written as

$p_4\to(2\pi nT+gA_4)$7

with $p_4\to(2\pi nT+gA_4)$8 and $p_4\to(2\pi nT+gA_4)$9 the stability functions of the fine and coarse propagators. The optimization strategy parametrizes $G_X(E)$00, imposes consistency and stability constraints, and minimizes the worst-case factor over the spectrum. The paper reports that conventional coarse propagators such as backward Euler or SDIRK-22 yield convergence factors around $G_X(E)$01–$G_X(E)$02, whereas optimized learned coarse propagators can reduce this to about $G_X(E)$03 in the tested settings, with higher parallel efficiency on linear diffusion, Allen-Cahn, and viscous Burgers models (Jin et al., 2023).

Learned propagators can also replace conventional PDE solvers directly. The generative wave propagator is a conditional diffusion-based model that advances seismic wavefields recursively from one time step to the next, conditioned on the five most recent wavefield snapshots, the velocity model, and the time-step index. It is trained for direct clean-snapshot prediction and uses a causal time-weighted loss based on exponential moving averages of per-snapshot errors to reduce long-rollout instability. Because the learned propagator is tied to the temporal spacing of the training snapshots rather than to the finite-difference stability limit, it advances the wavefield with a physical time step ten times larger than that required by the underlying solver. On Overthrust, SEG/EAGE, and Marmousi, the method accurately reproduces wavefield snapshots and shot gathers and reports an end-to-end speedup of $G_X(E)$04 over a GPU-accelerated tenth-order staggered-grid finite-difference implementation under matched hardware conditions (Cheng et al., 5 Jul 2026).

A closely related stability-oriented construction appears in coherent-state complex Langevin simulations. There the imaginary-time propagator is split by

$G_X(E)$05

so that the quadratic part is propagated exactly in the coherent-state basis. The resulting action,

$G_X(E)$06

is exact in the kinetic term and the algorithm enjoys guaranteed linear stability independent of the imaginary-time discretization (Kiely et al., 14 Aug 2025).

Lattice-QCD spectroscopy provides another computational use of effective propagators. LapH propagators smear quark fields by projecting onto the low-mode subspace of the gauge-covariant Laplacian, producing smeared timeslice-to-all propagators that enable two-particle correlators and phase-shift extraction. The stochastic LapH method adds diluted $G_X(E)$07 noise sources within the LapH subspace and is presented as effective for the $G_X(E)$08-to-$G_X(E)$09 diagrams needed in the isospin-0 channel, while the phase shift is extracted in the isospin-2 channel on $G_X(E)$10 dynamical anisotropic lattices with $G_X(E)$11 MeV (Bulava et al., 2010).

Constraint programming uses propagator in a domain-theoretic sense. “Perfect derived propagators” are constructed from views by functional composition,

$G_X(E)$12

and are proved to inherit correctness, domain and bounds consistency, idempotence, and related properties from the original propagator. The paper develops transformation, generalization, specialization, and channeling, and reports that without derived propagators Gecode would require 140000 rather than 40000 lines of code for propagators; each implementation is reused on average 3.6 times, and derived propagators are up to 6x faster and use half or less the memory compared to decomposition (0806.1806). A different relaxation is introduced by half-checking propagators, which are required to be contracting and to preserve the implication

$G_X(E)$13

but may remove actual solutions. They are therefore sound for identified solutions but incomplete for enumeration, and the paper proposes their use in portfolio solving processes and implements them in Gecode for the cost-circuit constraint (Lagerkvist et al., 2020).

A plausible unifying implication of these disparate constructions is that “effective” denotes not a fixed mathematical class but a design principle: the propagator is altered so that it exposes the part of the dynamics, spectrum, search space, or asymptotic behavior that is operationally decisive for the problem at hand.