A Summary

From the perspective of Natural Quantum Theory (NQT), the Wheeler–DeWitt equation is not the "fundamental equation of the cosmic wavefunction." Instead, it is merely a formal "spectralization/operatorization" of the classical general relativity constraint equations, yielding a global spectral equation.

Mathematically, it parallels the relationship between the Schrödinger equation and classical mechanics. However, physically, it encounters two fundamental problems:

1. Gravity itself typically lacks well-defined spatial boundary constraints, making it unsuitable for "spectralization = quantization" in the NQT sense.

2. Treating such a global spectral equation as the "ontological description of the universe" (timeless, without classical background) is another classic case of mistaking mathematical tools for physical reality.

Within the NQT framework, the Wheeler–DeWitt equation is better understood as:
"A formal tool obtained by spectral representation of general relativity’s constraint equations, which may provide modal information in a few restricted cases—but is by no means a fundamental description of gravity or the universe."

1. The Wheeler–DeWitt Equation in the Traditional Framework

In conventional quantum gravity and quantum cosmology, the logic proceeds as follows:

• Starting from the ADM decomposition (3+1 split) of general relativity, we have:

• Hamiltonian constraint: H=0 (energy constraint)

• Momentum constraints: Hi=0 (momentum/coordinate transformation constraints)

• Then, so-called "canonical quantization" is performed:

• The three-dimensional spatial metric $)$ is treated as a "generalized coordinate."

• Its conjugate momentum $)$ is replaced by the operator −iℏδ/δhij(x).

• Applying operatorization to the Hamiltonian constraint H=0 yields:

  H^Ψ[hij(x),matter fields]=0

  This is the Wheeler–DeWitt equation, often touted as:

• The equation of the "cosmic wavefunction,"

• A "timeless equation" (the right-hand side is 0, not iℏ∂Ψ/∂t),

• A framework where time "emerges" from some "internal clock" (e.g., the scale factor) or "semiclassical background."

In mainstream quantum gravity research, it is regarded as one of the candidate "fundamental equations of quantum general relativity."

2. Core Tenets of Natural Quantum Theory (NQT)

NQT makes several foundational judgments directly relevant to the Wheeler–DeWitt equation:

1. "Quantization" = Spectralization:
   Canonical quantization is essentially the spectral representation of a classical Hamiltonian (operatorization is merely a mathematical notation). This requires the system to possess clear boundary constraints and eigenmodes.

2. Quantum mechanics = Spectral expression of classical mechanics:
   The Schrödinger equation is not a new ontology but a global solution method for classical dynamics in the spectral domain.

3. Gravity generally cannot be quantized:
   In cosmological or general relativistic contexts, gravity typically lacks a well-defined "constraint cavity," making the definition of global eigenmodes impossible. Thus, "spectralization" loses its physical foundation.

4. Global methods naturally lose local dynamical information:
   Just as the Schrödinger formalism loses trajectories and spin directions (local causality), the Wheeler–DeWitt equation—as an extreme global equation—loses even more.

Under these premises, the Wheeler–DeWitt equation takes on a completely different significance.

3. NQT’s Interpretation of the Wheeler–DeWitt Equation

3.1 Mathematically: It Is Merely the Spectral Representation of GR’s Constraint Equations

• In the ADM formalism of general relativity, the Hamiltonian constraint $\mathcal{H}=0$ is itself a constraint equation, not an "evolution equation."

• The so-called canonical quantization, πij→−iℏδ/δhij, is analogous to replacing $p$ with −iℏ∇ in particle mechanics.

• In NQT’s language, this step amounts to:
  Forcibly applying "global spectralization" to the classical dynamics of spacetime geometry, yielding a "wave equation for geometric modes."

Thus, from NQT’s perspective:

The Wheeler–DeWitt equation ≈ "A constrained version of the Schrödinger equation on geometric configuration space."

The difference from the Schrödinger equation lies not in essence but in the object:

• Traditional Schrödinger equation: Spectralizes the matter Hamiltonian.

• Wheeler–DeWitt equation: Attempts to spectralize the constraint structure of geometry itself.

3.2 Physically: Lack of Physical Justification for Spectralization

NQT emphasizes:

Spectralization only has physical meaning in systems with clear constraints and boundary conditions (natural quantization).

Examples:

• Atoms, molecules, resonant cavities: Boundaries + finite regions → discrete energy levels.

• Bound-state electrons: Coulomb potential wells → eigenmodes.

However, typical gravitational scenarios include:

• Cosmological scales: Space as a whole is expanding, with no fixed boundaries.

• Gravitational sources: Spacetime itself is curved, lacking a "strictly conserved cavity."

Therefore:

• Applying "spectralization" (i.e., the Wheeler–DeWitt equation) to the entire universe’s spacetime metric,
  From NQT’s view, is performing a formal operator substitution in the absence of physical constraints.

• Even if a functional differential equation can be written mathematically, it may not correspond to any physically realizable set of eigenstates.

NQT’s succinct evaluation:

The Wheeler–DeWitt equation is "spectralized" in form but lacks the boundary foundation for natural quantization in physics. It is more of a formal construct than a real dynamical equation.

4. The Problem of the "Timeless Cosmic Wavefunction"

One of the Wheeler–DeWitt equation’s major "philosophical selling points" is:

It contains no explicit time, H^Ψ=0,
Leading some to claim, "The universe is static; time is emergent."

From NQT’s perspective, this claim has two layers of problems:

4.1 Repeating the Error of "Mathematics as Ontology"

• In conventional quantum mechanics, treating the wavefunction as an ontology and collapse as a physical process has led to bizarre interpretations like wavefunction realism, parallel universes, and the many-worlds interpretation.

• The Wheeler–DeWitt equation takes this further by treating a spectral form of a constraint equation as the "fundamental equation of the universe," resulting in:

• Time being "erased,"

• The universe being "frozen" in the abstract object Ψ[hij,ϕ],

• All evolution forced to be explained as "emergent internal correlations."

NQT would say: This is another case of over-ontologizing mathematical tools, akin to the Copenhagen interpretation’s misreading of the wavefunction—just at a more abstract level.

4.2 Time Is Originally a Parameter of Macroscopic Geometry/Causal Structure

In NQT’s framework:

• Time is a parameter describing classical causal processes.

• Quantum spectral methods (e.g., the Schrödinger equation) are merely global modal analyses of these processes.

• Treating the spectral equation as "more fundamental" and then claiming "time has disappeared" is a semantic sleight of hand.

For the Wheeler–DeWitt equation:

• H^Ψ=0 is merely a spectral expression of a global constraint condition.

• The true "temporal evolution" should still be sought in the causal structure of classical geometry + matter fields (e.g., the FRW cosmological evolution equations).

• One should not expect "real time" to "emerge" from a timeless functional equation.

5. Demoting the Wheeler–DeWitt Equation from NQT’s Standpoint

In summary, NQT’s positioning of the Wheeler–DeWitt equation is as follows:

1. Origin:
   It is the result of applying Schrödinger-style spectralization to the ADM constraint equations of general relativity, equivalent to a "modal equation for the geometry-matter composite system."

2. Applicability Conditions (if physical meaning is to be assigned):

• Must have clear geometric constraints/boundaries (e.g., closed universes, bounded spacetime regions).

• Must specify appropriate approximations or simplifications (e.g., minisuperspace models with only a finite number of degrees of freedom).

3. Physical Content:

• A tool for finding "allowed geometric + matter eigenmodes,"

• Analogous to modal expansions in specific geometric backgrounds (e.g., scalar field fluctuations in the universe).

• Under these special conditions, the W–D equation can be viewed as:

• Beyond these applications, it degenerates into mathematical games or philosophical speculation.

4. What Should Not Be Done:

• Do not treat $\Psi [h_{ij},\varphi ]$ as the "true wavefunction ontology of the universe."

• Do not derive conclusions like "time does not exist/the universe is static" from the form $\hat{\mathcal{H}}\Psi =0$.

• Do not expect a single equation to unify all interactions—in NQT, gravity itself is not quantizable in the conventional sense.

A concise stance:

From the NQT perspective, the Wheeler–DeWitt equation can only be regarded as a "spectralized formulation" of general relativity’s constraints. In highly constrained, specific geometric models, it may serve as an analytical tool—but it cannot be treated as the fundamental dynamics or ontological description of the universe. To do so would repeat the same error as wavefunction realism and the Copenhagen misinterpretation.

6. Where Does True Value Lie, Following NQT?

If we adhere to NQT’s foundational approach:

• Matter = Electromagnetic/topological field objects with internal structure.

• Quantization = Eigenmode analysis of classical fields in constrained systems.

• Gravity = Large-scale geometric effects, typically unconstrained, and thus not amenable to "spectralization into quantum gravity."

Then, the truly worthwhile pursuits are not esoteric philosophical games with the Wheeler–DeWitt equation, but rather:

1. In specific constrained geometric backgrounds (e.g., near black holes, compact stars, finite closed universes), use classical GR + classical electromagnetic/NQT particle structures to derive real eigenmodes (quasi-normal modes, resonant structures, etc.).

2. Demote "quantum cosmology" from the "cosmic wavefunction" level back to the classical background + electromagnetic/matter eigenmode level—that is, the universe is a classical geometric stage, and quantum phenomena are the spectral manifestations of electromagnetic/matter on that stage.

3. Clearly distinguish:

• Geometric evolution: Governed by GR or its possible macroscopic modifications.

• Quantum behavior of matter/fields: Described by NQT (electromagnetic + topological).

• No longer forcibly "quantizing" geometry itself as Ψ[g].

7. Concluding Summary

From the perspective of Natural Quantum Theory, the Wheeler–DeWitt equation can be summarized as follows:

It is a "cosmic modal equation" obtained by Schrödinger-style spectralization of general relativity’s constraint equations. While mathematically elegant, it lacks the boundary foundation for natural quantization in physics and thus cannot serve as an ontological or fundamental dynamical description of gravity or the universe. At best, it may function as a tool for analyzing geometric + matter eigenmodes in highly constrained, specific geometric models. To treat it as an ontology is to repeat the same error as wavefunction realism and the Copenhagen misinterpretation.