Classicized Dynamics of Unitary Theories with Fakeons

• Classicized dynamics is a framework where a nonlocal effective action, derived by integrating out fakeon modes from a higher-derivative Lagrangian, governs physical fields.
• It ensures that only the expected physical degrees of freedom remain by projecting out ghost-like modes via a carefully defined fakeon prescription.
• This approach benefits quantum gravity and field theory by maintaining unitarity and a well-posed initial-value problem despite involving infinitely many derivatives.

Classicized dynamics in unitary field theories with fakeons is a framework in which the classical action is not the original local higher-derivative Lagrangian, but rather a nonlocal "classicized" effective action. This effective action is obtained by projecting out—or "integrating out"—the fake degrees of freedom ("fakeons") at the tree level, encoding their effects into nonlocal interaction terms. An important aspect is that although the classicized equations are nonlocal (and typically involve infinitely many derivatives), the number of physical degrees of freedom and initial conditions matches that of an ordinary second-order system. The solution space is a well-defined subspace of the parent higher-derivative model, and the fakeon prescription must be constructed as a limiting case of a more general inversion to avoid overcounting. The classicization procedure fundamentally shapes the initial-value problem, the counting of degrees of freedom, and the physical content of the resulting theory.

1. Structure of Classicized Dynamics

In fakeon field theories, the starting point is a higher-derivative (HD), local Lagrangian where extra poles (corresponding to ghost or tachyonic degrees of freedom) appear in the propagators. These problematic excitations are excised from the physical spectrum by defining their quantum propagators through the Anselmi–Piva (AP) "average continuation" prescription, or similar methods, such that these degrees remain purely virtual—internal to diagrams and never present in the S-matrix or asymptotic states (Anselmi, 2018).

After quantization, the classical limit is not reached by discarding quantum corrections on the original HD Lagrangian but by "classicizing" the theory: integrating out the fakeon modes (i.e., removing them via their special prescription) at tree level. The resulting classicized action is nonlocal and defines classicized equations for the remaining (physical) fields. For example, for a system with fields $\phi$ (physical) and $\varphi$ (fakeon), the classicized action—schematically—is

$$\mathcal{L}_{\text{cl}} = \frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2}m^2\phi^2 - V(\phi) + \frac{g^2}{8}\phi^2\left[\frac{M^2}{(\Box + m^2)^2 + M^4} \right]_f \phi^2,$$

where the subscript $f$ indicates inversion using the fakeon prescription.

This nonlocal effective action governs the physical classical dynamics after the fakeon projection (Anselmi et al., 6 Oct 2025).

2. Initial Conditions and Solution Space

Despite the appearance of infinitely many derivatives, the number of initial conditions required by the classicized system is finite and matches physical expectations (two per field for standard kinetic terms). This outcome is directly related to the relationship between the classicized (nonlocal) system and its parent higher-derivative system.

The HD system, such as one governed by

$(\Box + m^2)^2 + M^4 \quad\text{(acting on %%%%3%%%%)},$

would formally require more initial conditions (e.g., four for a fourth-order operator). However, the fakeon projection removes the extra unphysical solutions by imposing appropriate constraints: the nonlocal classicized equations select a subspace of the HD solution space. Explicitly, initial data belonging to the kernel of the nonlocal operator (often associated with runaway or ghost modes) are projected out by the procedure. The general solution to the classicized equation retains only the physical degrees of freedom (Anselmi et al., 6 Oct 2025).

The correct parameter counting is rigorously established both by direct computation in linear models and by general arguments in the presence of nonlocal interactions. For instance, order-by-order in perturbation theory, each higher-order term in the classicized expansion is expressed in terms of a fixed set of initial data derived from the local part. Section 6 of (Anselmi et al., 6 Oct 2025) presents a general proof: starting from all possible initial data for the HD model, only those consistent with the fakeon-prescribed nonlocal equations survive in the classicized theory.

3. Nonlocality, the Fakeon Prescription, and Avoiding Overcounting

A critical technical feature is the procedure for integrating out the fakeon modes. Naïvely imposing the fakeon prescription (e.g., using a fixed nonlocal operator inversion) can give the impression that extra degrees of freedom remain. To avoid this, the projection must be realized as the limit of a generic inversion prescription—such as integrating over finite time intervals or using a general kernel, and then taking an appropriate limit (e.g., integrating over all time).

For example, a nonlocal equation may take the form: $$\Box\,\phi + m^2\phi = -\frac{\omega^2}{2\tau}\left[ \int_{a}^t f(t, t')\phi(t')dt' - \int_{t}^{b} f(t, t')\phi(t')dt' \right],$$ where $a, b$ are arbitrary endpoints. Only after extracting the necessary constraints and projecting out the unwanted directions does one take the limit $a\to-\infty$, $b\to+\infty$, corresponding to the fakeon prescription (half-advanced plus half-retarded Green function, or principal value in momentum space). This approach ensures that all unphysical modes are discarded, and the solution space is not artificially enlarged (Anselmi et al., 6 Oct 2025).

4. Degrees of Freedom and Ghost Elimination

The principal physical outcome is that fakeon removal projects out the extra (ghost) degrees of freedom while preserving the correct counting for the physical sector. Although the parent HD Lagrangian might support solutions with additional exponential (e.g., runaway or oscillatory) behavior corresponding to ghostlike modes, the classicized equations—by virtue of their specific nonlocal structure and the AP prescription—admit as general solution just those compatible with standard causality and stability.

Explicit propagators in the classicized theory take a form like: $D(k^2) = \frac{M^2}{[( -k^2 + m^2)^2 + M^4]},$ operating under the fakeon prescription (average continuation), with the result that the only physical poles participating in on-shell propagation are those associated with healthy degrees of freedom. The fakeon poles affect internal lines in Feynman diagrams but have no physical asymptotic states in the spectral decomposition.

5. Illustrative Examples and the Analogy with Runaway Removal

Analytical solutions for toy models (e.g., linear systems with HD operators acting on the physical field) confirm that the HD system supports solutions of the form: $x_{\text{HD}}(t) = \sum_{i} c_i e^{\lambda_i t},$ where the $\lambda_i$ may include both physical frequencies and unphysical runaways. Substituting this ansatz into the classicized nonlocal equation generated by the fakeon prescription, the necessary boundary conditions and convergence constraints unambiguously fix (usually) all but two of the $c_i$, leaving only the physical modes.

The procedure parallels Dirac’s removal of runaway solutions in classical electrodynamics. There, a higher-derivative equation for the electron's motion leads to nonphysical exponential solutions, which are removed by applying an appropriate (retarded or averaged) Green’s function, yielding a nonlocal—but well-defined—classicized equation depending only on physically meaningful initial data. The fakeon approach generalizes this logic to a full field-theoretic context (Anselmi et al., 6 Oct 2025).

6. Mathematical Representation

A key mathematical device is the inversion of higher-derivative kinetic operators using the fakeon prescription. For a general operator polynomial in $\Box$ (e.g., $P(\Box)$), the inversion is defined through the AP prescription: $$G_{\text{AP}}(x^2) = \int \frac{d^4k}{(2\pi)^4} \frac{ -i M^2 e^{-ik\cdot x} }{( -k^2 + m^2 )^2 + M^4 },$$ with the precise handling of branch cuts and pole structure ensuring the elimination of unphysical (ghost) contributions to the solution and S-matrix.

7. Consequences for Physical Theories

This approach significantly impacts physical applications, particularly in quantum gravity and nonlocal modifications of field theories. In the context of renormalizable or super-renormalizable gravity models, integrating out the spin-2 fakeon guarantees the absence of ghost degrees, renders the theory unitary, and keeps the solution space manageable despite the apparent presence of infinite-derivative terms. The classicized equations, albeit nonlocal, can be analyzed perturbatively and yield observables determined by a fixed, physically consistent set of initial data. The formalism also clarifies why the deterministic evolution problem in such theories is well-posed, despite their higher-derivative origin.

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In summary, the classicized dynamics of unitary field theories with fakeons is governed by a nonlocal effective action obtained by integrating out fake degrees of freedom at tree level using a limiting prescription. This procedure ensures that the nonlocal classicized equations require only the expected number of initial conditions, that the solution space contains only physical modes, and that no extra or ghost-like degrees of freedom survive. The mechanism is robustly illustrated in linear models and parallels classical constructions such as Dirac's handling of the Abraham–Lorentz force, thus establishing classicization as a necessary and sufficient method for producing physically consistent dynamics in fakeon theories (Anselmi et al., 6 Oct 2025).