Four-Level Ontological Hierarchy

• Four-Level Ontological Hierarchy is a layered framework categorizing existence from quantum vacuum (Level 0) to empirical reality (Level 3) in quantum field theory.
• It interconnects off-shell virtual processes, on-shell quantum excitations, and classical measurement outcomes to explain phenomena like the Lamb shift and Casimir effect.
• The framework dissolves the traditional real/unreal dichotomy by assigning graded ontological statuses, enabling clear causal transitions and deeper philosophical insights.

A four-level ontological hierarchy is a rigorously stratified framework that partitions the modes and degrees of existence invoked in fundamental physics and philosophy of science. Across quantum field theory, cosmology, and interpretational schemes, this hierarchy dissolves the historically persistent dichotomy between “real” and “unreal” by introducing a spectrum of existence, each layer governed by distinct but interconnected criteria. The hierarchy’s core motivation is to elucidate how entities as disparate as quantum vacuum fluctuations, virtual particles, actualized quantum excitations, and classical empirical records are anchored within a unified ontological continuum.

1. Ontological Stratification in Quantum Field Theory

Mirzaee formalizes the four-level ontological hierarchy as a graded spectrum comprising: Level 0 (Quantum Vacuum), Level 1 (Virtual), Level 2 (Quantum/Actualizable), and Level 3 (Phenomenal Reality) (Mirzaee, 8 Nov 2025). Each level possesses precise existential and dynamical definitions:

• Level 0 – Quantum Vacuum: The foundational “sea” of field-theoretic zero-point fluctuations, defined as all states annihilated by creation and annihilation operators:

$$\hat a_{\mathbf p}\,|0\rangle = 0 \quad \forall\,\mathbf p$$

The vacuum state seeds all fluctuations via the path integral

$$Z[J] = \int \mathcal{D}\Phi \exp \left[ iS[\Phi] + i \int d^4x\,J(x)\Phi(x) \right]$$

yet contains no on-shell excitations or directly observable structures.

• Level 1 – Virtual: Populated by off-shell quanta (virtual particles), these entities appear only in the internal lines of Feynman diagrams and mediate force and loop corrections. The criterion is failure to satisfy the on-shell mass condition, i.e., $k^2 \neq m^2$. Their existence is encoded in propagators such as:

$$\Delta_F(x-y) = \int \frac{d^4k}{(2\pi)^4} \frac{i}{k^2-m^2+i\epsilon} e^{-ik\cdot(x-y)}$$

Virtuals are unobservable but induce measurable physical effects (e.g., Lamb shift, Casimir effect).

• Level 2 – Quantum (Actualizable): On-shell field quanta that may be detected as real particles, satisfying $p^2 = m^2$. Generated by creation operators acting on the vacuum, these excitations contribute coherent probability amplitudes for scattering or bound states:

$$\Psi(\mathbf x,t) = \langle \mathbf x, t\,|\, \Psi \rangle$$

which, under measurement, yield eigenvalues of macroscopic observables.

• Level 3 – Phenomenal Reality: The macroscopic empirical domain comprised of pointer states, detector outcomes, and classical records. Definite measurement outcomes correspond to eigenstates of Hermitian observables:

$\hat O |\omega\rangle = \omega |\omega\rangle$

Stabilized by decoherence and manifest empirical registration, this is the ultimate layer of actualization.

The levels are dynamically continuous; vacuum fluctuations (Level 0) feed virtual processes (Level 1), which themselves can become on-shell (Level 2) and are finally realized as empirical phenomena (Level 3).

2. Axiomatic Foundations and Formal Characterization

The hierarchy is axiomatized by a sequence of existence and criterion rules:

Level	Defining Criterion	Mathematical Formalism
0 (Vacuum)	$\hat a_{\mathbf p}\,|0\rangle = 0$ for all $\mathbf p$	Path integral, kernel of annihilation
1 (Virtual)	$k^2 - m^2 \neq 0$	Internal lines in Feynman graphs, off-shell propagators
2 (Quantum)	$p^2 - m^2 = 0$	Creation operators, Born probabilities
3 (Phenomenal)	Measurement: $\hat O|\omega\rangle = \omega|\omega\rangle$	Decoherence, pointer-basis eigenstates

Each axiom defines the necessary and sufficient conditions for an entity’s ontological status:

• Axiom I: Unique vacuum state with $\hat a_{\mathbf p}\,|0\rangle=0$ underlies all Green’s functions.
• Axiom II: Virtuality is confined to off-shell particles in internal lines, violating $k^2 = m^2$.
• Axiom III: Actualizable quantum excitations are on-shell and endow observable probability amplitudes.
• Axiom IV: Phenomenal states correspond to measured eigenvalues, robustly classical by environmental decoherence.

Underlying this is the path-integral structural equation

$Z[0] = \int \mathcal{D}\Phi\, \exp[iS[\Phi]]$

which encodes vacuum-to-virtual (Level 0 to 1) and, via the analytic structure of propagators, the emergence of on-shell amplitudes (Level 1 to 2).

3. Virtual Particles, Empirical Implications, and Dynamical Transitions

Level 1 virtual particles are central to physical predictions in QFT, as exemplified by their appearance in Feynman diagrams and their measurably influential role in higher-level phenomena:

• Electron–positron annihilation: Virtual photons mediate processes as internal propagators, with $k^2 \neq 0$.
• Loop corrections: Vacuum polarization and self-energy corrections derived from loop integrals critically depend on virtual entities:

$\Pi^{\mu\nu}(q) = \int \frac{d^4k}{(2\pi)^4} \frac{\mathrm{Tr}[\gamma^\mu(\slashed k+m)\gamma^\nu(\slashed k+\slashed q +m)]}{(k^2-m^2+i\epsilon)[(k+q)^2-m^2+i\epsilon]}$

Empirical manifestations include:

• Lamb Shift: Alters atomic spectral levels via virtual photon exchange, producing a measurable energy shift:

$$\Delta E_{n\ell} = \frac{\alpha}{\pi} \frac{(Z\alpha)^4 m}{n^3} \ln \frac{1}{(Z\alpha)^2} + \ldots$$

• Casimir Effect: The vacuum’s virtual modes induce a measurable force between conducting plates separated by distance $a$:

$$\frac{E_{\rm Cas}}{A} = -\frac{\pi^2 \hbar c}{720 a^3}$$

These phenomena demonstrate that virtuals, though unobservable per se, possess causal efficacy and cannot be dismissed as purely formal artifacts.

Transitions between levels are driven by well-characterized physical mechanisms:

• In cosmological expansion, off-shell modes acquire effective mass and become on-shell quanta.
• In strong fields (e.g., Schwinger effect, Hawking radiation), virtual pairs are promoted to on-shell status via external “work” overcoming mass thresholds:

$$\Gamma \propto E^2 \exp\left(-\frac{\pi m^2}{qE}\right)$$

4. Dissolution of the Real/Unreal Dichotomy

Traditional ontological schemes adopted a binary assignment: real (observable, classical, on-shell) vs. unreal (theoretical, unobservable, off-shell). This is insufficient for quantum field phenomena, in which unobservables (virtuals, vacuum modes) yield real, empirical effects. The four-level hierarchy resolves this by implementing:

• Ontological Principle of Continuity: Each level emerges causally and dynamically from the preceding one, with no arbitrary leap.
• Causal Efficacy Criterion: Entities at every level are causally relevant; neglecting virtual reality disrupts the explanation of observed phenomena.
• Modal Stratification: The status of “reality” is context-sensitive and graded, not digital—potential, semi-real, actualizable, and phenomenal are all recognized as legitimate modes of being.

This philosophical stance draws from process ontology and structural realism, situating the whole edifice as an ontological spectrum in which every level owns a distinctive mode of existence.

5. Generalization and Methodological Implications

The four-level ontology connects mathematical formalism (path integrals, operators) with physical observables and philosophical interpretation:

• Every component of QFT can be allocated a unique ontological stratum.
• The physical origins of empirical phenomena are linearly traceable to the quantum potential embedded in the vacuum state.
• This hierarchy links unobservable but necessary theoretical ingredients (virtuals, vacuum) to manifest classicality—a conceptual advance over binary metaphysics.
• The hierarchy’s structure has analogues in multiverse cosmologies (e.g., Level I–IV multiverses (0905.1283)) and interpretational schemes for quantum mechanics (classification levels: formalism, assignment, principles, ontology (Krizek, 2017)); in all cases, stratification aids in clarifying the overlap and divergence of explanatory frameworks.

A key methodological implication is that ontological commitments in physical theory must account not just for what is empirically registered, but for the entire spectrum of causally efficacious entities involved in actualization. The four-level stratification offers a principled roadmap for such commitments, moving beyond the limitations of classical metaphysics toward a continuous and dynamic ontology adapted to the structure and predictions of modern quantum field theory.