Theory Map of Physics

Status: Living synthesis — structural overview, cross-linking all domain pages. Last updated: 2026-06-08

This page is the atlas of the wiki: it situates every physical framework relative to every other, exhibits the reduction structure (which theory is a limit of which), and lays out the scale ladder from the Planck scale to the cosmological horizon. It is deliberately structural rather than exhaustive; per-framework depth lives on the linked domains/ pages, and the failures of the map — the places where frameworks clash rather than nest — are catalogued in GAPS_AND_CONTRADICTIONS.md. Epistemic markers follow EPISTEMICS.md; load-bearing modeling assumptions are tracked in ASSUMPTIONS_LEDGER.md.

A guiding caveat: the words "limit," "reduction," and "emergence" below denote controlled approximation in a specified regime, not logical derivation in general. Several of the most important limits are mathematically singular (e.g. $\hbar\to0$), so "X reduces to Y" almost always means "Y is the leading term of an asymptotic expansion of X, valid where a dimensionless ratio is small." INFERENCE

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1. Master table of frameworks

A cross-cutting principle: the action principle $\delta\!\int L\,dt=0$ / $\int\mathcal D\phi\,e^{iS/\hbar}$ is the single organizing structure shared from classical mechanics through QFT ESTABLISHED; the symplectic/Poisson structure survives (deformed) into QM via $\{\,,\}\to[\,,]/i\hbar$ ESTABLISHED; the renormalization group unifies QFT with critical phenomena in statistical mechanics ESTABLISHED. See UNIFYING_PRINCIPLES.md.

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2. The tower of effective theories (reduction structure)

Each arrow reads "A reduces to B in the indicated limit" — B is the effective/low-energy/coarse-grained description, A the more fundamental one. All limits are flagged for singularity.

• QFT $\xrightarrow{\,c\to\infty,\ \text{fixed }N\,}$ NRQM. Non-relativistic QM is the fixed-particle-number, low-energy sector of QFT ESTABLISHED. QFT supplies what NRQM postulates: spin–statistics, antiparticles, and the resolution of one-particle relativistic pathologies. The reduction is not representation-preserving: the Stone–von Neumann uniqueness that NRQM relies on fails for infinitely many dof (Haag's theorem, inequivalent vacua) ESTABLISHED. See domains/quantum-field-theory.md.

• QM $\xrightarrow{\,\hbar\to0\ (S\gg\hbar)\,}$ classical mechanics. Recovered via Ehrenfest, WKB/stationary phase, and the Wigner–Moyal map (Moyal bracket $\to$ Poisson bracket) ESTABLISHED. The limit is singular and non-uniform: $e^{iS/\hbar}$ oscillates without bound, tunneling $\sim e^{-S/\hbar}$ is non-analytic, and for chaotic systems the Ehrenfest time $t_E\sim\lambda^{-1}\ln(S/\hbar)$ truncates correspondence ESTABLISHED. Decoherence supplies the effective pointer basis; a fully general derivation of classicality is OPEN. Definite trajectories and noncontextual values are excluded by Bell / Kochen–Specker ESTABLISHED.

• SR / relativistic dynamics $\xrightarrow{\,c\to\infty\,}$ Galilean mechanics. The relativistic free Lagrangian $-mc^2\sqrt{1-v^2/c^2}\to\text{const}+\tfrac12 mv^2$ ESTABLISHED. Absolute time is the $c\to\infty$ contraction of Lorentzian causal structure.

• GR $\xrightarrow{\,\text{weak field, slow motion}\,}$ Newtonian gravity. $g_{00}\approx-(1+2\Phi/c^2)$, geodesics $\to\ddot{\mathbf x}=-\nabla\Phi$, and the field equation $\to\nabla^2\Phi=4\pi G\rho$; this limit fixes the coupling $8\pi G/c^4$ ESTABLISHED.

• GR (+ matter) $\xrightarrow{\,E\ll M_{\rm Pl}\,}$ GR as an effective field theory. GR quantized perturbatively is a valid EFT at low energy but perturbatively non-renormalizable at $M_{\rm Pl}$ ESTABLISHED. What UV-completes it is OPEN — the central problem of OPEN_PROBLEMS.md.

• Statistical mechanics $\xrightarrow{\,N\to\infty,\ \text{coarse-grain}\,}$ thermodynamics. Stat-mech derives thermodynamics: $F=-k_BT\ln Z$ reproduces the potentials, $S=k_B\ln W$ grounds entropy ESTABLISHED. The reduction has a subtle residue: the strict second law is the probabilistic statement's $N\to\infty$ shadow, and irreversibility requires the extra Past Hypothesis input that thermodynamics merely posits INFERENCE/OPEN. See domains/statistical-mechanics.md and GAPS_AND_CONTRADICTIONS.md.

• Hamiltonian mechanics $\xrightarrow{\,\text{+ Liouville measure}\,}$ statistical mechanics. Hamiltonian phase-space flow plus Liouville's theorem is the microscopic foundation of the microcanonical ensemble ESTABLISHED; the tension is that reversible, volume-preserving microdynamics cannot alone produce monotone entropy increase (Loschmidt, Zermelo) ESTABLISHED tension.

• Euclidean QFT $\;\cong\;$ classical statistical field theory (Wick rotation, Osterwalder–Schrader). Not a reduction but a formal isomorphism: the path integral is a partition function, and Wilsonian RG is literally shared ESTABLISHED, where reflection positivity holds. See domains/mathematics.md.

• SM $\xrightarrow{\,\text{leading operators}\,}$ SMEFT / low-energy EFTs (chiral perturbation theory, Fermi theory). Renormalizability is not a fundamental axiom but an emergent low-energy property: irrelevant operators are suppressed by powers of $E/\Lambda$ ESTABLISHED, Wilsonian reinterpretation.

Diagrammatic note. The tower is not a single linear chain. Three deformation parameters act quasi-independently — $\hbar$ (quantum), $1/c$ (relativistic), $GM/rc^2$ (gravitational) — plus a coarse-graining/$N$ axis. The "missing corner" where all of $\hbar$, $1/c$, and $G$ are simultaneously $O(1)$ is quantum gravity, which no current framework occupies OPEN. See UNIFICATION_LANDSCAPE.md.

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3. Relationship diagram

graph TD
    QG["Quantum Gravity<br/>(missing: ℏ, 1/c, G all O(1))<br/>[OPEN]"]

    QG -->|low energy, classical g| GR["General Relativity"]
    QG -->|fixed background| QFT["Quantum Field Theory"]

    GR -->|weak field, v≪c| NG["Newtonian Gravity"]
    GR -->|flat metric, local frame| SR["Special Relativity"]
    GR -->|FLRW symmetry + fluid| COSMO["Cosmology (ΛCDM)"]

    QFT -->|c→∞, fixed N| QM["Quantum Mechanics (NRQM)"]
    QFT -->|gauge group + reps| SM["Standard Model"]
    QFT -.Wick rotation.- STAT["Statistical Mechanics"]

    QM -->|ℏ→0, S≫ℏ| CM["Classical Mechanics"]
    QM -->|density matrices, ETH| STAT
    SR -->|c→∞| CM

    CM -->|+ Liouville measure| STAT
    STAT -->|N→∞, coarse-grain| THERMO["Thermodynamics"]

    QM -.operational substrate.- INFO["Information Theory"]
    GR -.Bekenstein–Hawking S=A/4.- THERMO
    GR -.holography, RT.- INFO
    THERMO -.Landauer, Maxwell demon.- INFO

    SM --> COSMO
    QFT -.cosmological constant problem.- COSMO

    classDef open fill:#fdd,stroke:#900;
    class QG open;

Solid arrows = controlled reduction (limit/coarse-graining). Dashed lines = structural correspondence or shared formalism, not a reduction. The triple point GR ∩ QFT/QM ∩ thermodynamics (black-hole thermodynamics, the information paradox, $S=A/4$) is the densest cluster of OPEN/CONTESTED problems in physics. See GAPS_AND_CONTRADICTIONS.md and domains/information-theory.md.

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4. The scale ladder: from Planck to the horizon

Energies in GeV, lengths in metres; values are standard constants, not invented. The "ruling framework" column states which theory is operationally correct at that scale; arrows indicate where one framework hands off to another.

Numerical anchors for the dimensionless ratios that switch frameworks live in CONSTANTS_AND_SCALES.md: $\hbar$, $c$, $G$, $k_B$, $M_{\rm Pl}=\sqrt{\hbar c/G}$, $\alpha\approx1/137$, $\Lambda\sim10^{-52}$ m$^{-2}$, and the Bekenstein–Hawking entropy $S_{\rm BH}=k_B c^3A/4G\hbar$.

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5. Where the map tears (pointers, not content)

The reductions above are the successes. The wiki's value concentrates in the failures — pairs of frameworks that do not nest cleanly. These are catalogued in full in GAPS_AND_CONTRADICTIONS.md; a structural index:

• QM/QFT ∩ GR — quantum gravity & the problem of time. QM/QFT presuppose a fixed classical background and external time; GR makes spacetime dynamical (Wheeler–DeWitt $H|\Psi\rangle=0$ is timeless). Perturbative quantization is non-renormalizable. The sharpest clash in physics ESTABLISHED that it is unresolved. See OPEN_PROBLEMS.md.
• QFT ∩ GR ∩ cosmology — the cosmological constant problem. Naive QFT vacuum energy exceeds observed $\rho_\Lambda$ by $\sim10^{60}$–$10^{120}$ OPEN. A domain-clash, possibly the deepest quantitative mismatch known.
• Reversible microdynamics ∩ thermodynamics — the arrow of time. Time-symmetric mechanics + thermodynamic arrow forces a low-entropy boundary condition (Past Hypothesis), whose origin is cosmological and OPEN.
• The measurement problem. Unitary evolution (A5) and the projection postulate (A6) are jointly applied with no rule demarcating their jurisdictions OPEN/CONTESTED. A genuine incompleteness, not a domain mismatch — distinct in kind. See domains/quantum-mechanics.md.
• Black-hole thermodynamics & information. $S_{\rm BH}=A/4$ and Hawking radiation place thermodynamics, QM (unitarity), and GR at a triple point; the information paradox and microstate counting are OPEN/CONTESTED. See domains/information-theory.md and domains/thermodynamics.md.
• Rigor gap. No physically realistic interacting 4D QFT is rigorously constructed (Yang–Mills mass gap, a Clay problem); the Lorentzian path-integral measure does not exist as a measure OPEN. See domains/mathematics.md.

A discipline of classification (per EPISTEMICS.md): distinguish a true logical inconsistency (none confirmed between accepted frameworks), a domain-of-validity mismatch (most "clashes" above — e.g. QFT-on-fixed-background vs dynamical GR), and an unsolved-but-consistent problem ($\Lambda$CDM dark sector, confinement proof, glass transition).

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See also

• README.md — wiki entry point and reading order
• THEORY_MAP.md (this page) ↔ UNIFICATION_LANDSCAPE.md — candidate UV completions and unification routes
• UNIFYING_PRINCIPLES.md — the action principle, symmetry↔conservation, RG, holography
• GAPS_AND_CONTRADICTIONS.md — full catalogue of inter-framework clashes
• OPEN_PROBLEMS.md · HYPOTHESES.md · ASSUMPTIONS_LEDGER.md
• CONSTANTS_AND_SCALES.md — the constants that set the scale-ladder transitions
• EPISTEMICS.md — meaning of the inline markers · GLOSSARY.md · FINDINGS.md
• Domain pages: classical-mechanics · quantum-mechanics · quantum-field-theory · particle-physics · general-relativity · thermodynamics · statistical-mechanics · cosmology · information-theory · mathematics

References

See BIBLIOGRAPHY.md for canonical sources underlying each framework (e.g. von Neumann and Reed–Simon for QM foundations; Weinberg, Haag, and Streater–Wightman for QFT; Wald and Misner–Thorne–Wheeler for GR; Callen and Jaynes for thermodynamics/statistical mechanics; Arnold for classical mechanics; Nielsen–Chuang for quantum information). Per wiki policy (AGENTS.md), only real, standard sources are cited; specific landmark results (Gleason, Stone–von Neumann, Haag, Lovelock, Coleman–Mandula, Penrose–Hawking, Aizenman–Duminil-Copin, Ryu–Takayanagi) are named on the relevant domain pages.