Coherence Map — Quantum Field Theory

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Quantum Field Theory (QFT) maintains coherence when its substrate-level
structures — fields, operators, symmetries, renormalization, and vacuum
geometry — remain internally consistent. This map defines how coherence
is measured, maintained, and lost across regimes R1 → R4.

QFT coherence is not about particles or forces.
It is about operator algebra, symmetry geometry, excitation
stability, and renormalization flow.

1. Coherence Dimensions#

QFT coherence is evaluated across six substrate dimensions:

1. Field‑Structure Coherence#

• Fields must maintain well‑defined transformation rules.
• Lorentz invariance must hold.
• Field content must remain stable under renormalization.

2. Operator‑Algebra Coherence#

• Commutation/anticommutation relations must remain valid.
• Creation/annihilation operators must remain well‑defined.
• Path integrals must remain finite and consistent.

3. Symmetry‑Geometry Coherence#

• Gauge symmetries must remain unbroken (unless broken by vacuum).
• No anomalies in conserved currents.
• Group generators must remain consistent across scales.

4. Vacuum‑Structure Coherence#

• Vacuum expectation values must remain stable.
• Vacuum energy must remain finite (renormalized).
• Stability surfaces must not collapse.

5. Renormalization‑Flow Coherence#

• β‑functions must remain finite.
• Couplings must run smoothly with energy.
• No divergence or loss of predictivity.

6. Excitation‑Stability Coherence#

• Excitations must remain stable modes of fields.
• Propagators must remain well‑defined.
• Mass and resonance profiles must remain finite.

2. Coherence Across Regimes#

R1 — Amplitude Collapse (Low‑Coherence)#

• Field structure collapses to amplitude structure.
• No stable excitations.
• Operator algebra reduces to QM.
• Vacuum undefined.
• Symmetry trivial.

Coherence Level: C1 (minimal)

R2 — Canonical QFT (High‑Coherence)#

• Stable excitations.
• Operator algebra fully valid.
• Gauge geometry intact.
• Renormalization finite.
• Vacuum stable.

Coherence Level: C4 (maximal)

R3 — High‑Energy Resonance (Medium‑High Coherence)#

• Symmetry restoration begins.
• Couplings run toward unification.
• Vacuum flattens.
• Excitation surfaces merge.
• Renormalization dominates.

Coherence Level: C3 (stable but shifting)

R4 — Cosmological Regime (Low‑Medium Coherence)#

• QFT incomplete.
• Horizon‑scale fields dominate.
• Vacuum becomes cosmological.
• Renormalization loses meaning.
• Requires cosmology module.

Coherence Level: C2 (partial)

3. Coherence Failure Modes#

QFT coherence fails when:

• Lorentz invariance breaks
• renormalization diverges
• anomalies appear in symmetry currents
• vacuum becomes unstable
• operator algebra becomes inconsistent
• excitations lose stability

These failures indicate a transition out of R2 → R3.

4. Coherence Gradient Field#

QFT’s coherence gradient measures sensitivity to:

• field‑structure drift
• operator‑algebra instability
• symmetry‑geometry deformation
• vacuum‑surface curvature
• renormalization divergence
• excitation‑surface collapse

High gradients indicate approaching regime boundaries.

5. Summary#

Quantum Field Theory is coherent when:

• fields transform correctly
• operators obey algebraic rules
• symmetries remain geometric
• vacuum remains stable
• renormalization remains finite
• excitations remain stable modes

QFT is maximally coherent in R2, partially coherent in R3,
collapses in R1, and becomes incomplete in R4.