Glossary

Status: living reference — comprehensive but not exhaustive; definitions are standard but each links to its home domain for the full treatment. Last updated: 2026-06-08

This page collects precise, cross-linked definitions for the mathematical objects, physical concepts, and named results used throughout this wiki. Entries are alphabetical. Each definition is meant to be correct as stated within its domain of validity; where a term carries genuine subtlety or dispute, an epistemic marker (ESTABLISHED, INFERENCE, SPECULATIVE, OPEN, CONTESTED) flags it (see EPISTEMICS.md for the calibration scheme). For the map of how domains relate, see THEORY_MAP.md; for the canonical-source pointers, see BIBLIOGRAPHY.md.

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A

Action — The functional $S[\gamma]=\int L\,dt$ (or $\int \mathcal L\,d^4x$ in field theory) assigning a number to each history $\gamma$. Physical trajectories make $S$ stationary, not necessarily least ESTABLISHED. See domains/classical-mechanics.md.

Action–angle variables — Canonical coordinates $(I,\phi)$ for an integrable system in which $\dot I=0$ and $\dot\phi=\omega(I)$; motion is linear winding on invariant tori. The cleanest picture of integrable order, against which chaos is the failure ESTABLISHED.

Adiabatic (cosmological) perturbations — Primordial fluctuations in which all species share a common local time-shift, so number-density ratios are spatially constant. Observed to high accuracy and predicted by single-field inflation ESTABLISHED as description. See domains/cosmology.md.

ADM formalism — Arnowitt–Deser–Misner $3+1$ decomposition of the metric into spatial metric $h_{ij}$, lapse $N$, and shift $N^i$, recasting general relativity as constrained Hamiltonian evolution of geometry ESTABLISHED. Foundation of numerical relativity and canonical quantization attempts. See domains/general-relativity.md.

AdS/CFT correspondence — Conjectured exact duality between a quantum-gravity theory in asymptotically anti-de Sitter spacetime and a conformal field theory on its boundary [INFERENCE: overwhelming evidence, no general nonperturbative proof]. See domains/information-theory.md, UNIFICATION_LANDSCAPE.md.

Algebraic QFT (Haag–Kastler) — Formulation of QFT as a net of local operator algebras $\mathcal O\mapsto\mathfrak A(\mathcal O)$ satisfying isotony, microcausality, and covariance, avoiding any privileged Hilbert-space representation ESTABLISHED structural framework. See domains/quantum-field-theory.md.

Anomaly (quantum/chiral) — A classical symmetry broken by quantization; e.g. the Adler–Bell–Jackiw anomaly $\partial_\mu j^{5\mu}=\frac{e^2}{16\pi^2}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$. Physical for global currents ($\pi^0\to2\gamma$); must cancel for gauged currents ESTABLISHED. See domains/particle-physics.md.

Asymptotic freedom — The decrease of a gauge coupling at high energy due to a negative beta function; in QCD $\beta_0=11-\tfrac23 n_f>0$ makes quarks weakly coupled at short distance ESTABLISHED. See domains/quantum-field-theory.md.

Asymptotic safety — Conjecture that gravity (or another theory) is UV-completed by a nontrivial fixed point of the renormalization-group flow SPECULATIVE/OPEN. See OPEN_PROBLEMS.md.

Atiyah–Singer index theorem — Equates the analytic index of an elliptic operator (e.g. net chiral zero modes of the Dirac operator) with a topological integral, $\operatorname{ind}(\slashed D)=\int_M \hat A(M)\,\mathrm{ch}(F)$ ESTABLISHED. Underlies anomalies and instanton fermion counting. See domains/mathematics.md.

B

Baryon acoustic oscillations (BAO) — Frozen-in sound-horizon scale in the matter distribution, a standard ruler probing cosmic geometry and expansion history ESTABLISHED. See domains/cosmology.md.

Bekenstein bound — Upper bound on the entropy in a weakly gravitating region of radius $R$ and energy $E$: $S\le 2\pi k_B R E/\hbar c$ INFERENCE; sharp only in weak gravity. See domains/information-theory.md.

Bekenstein–Hawking entropy — Black-hole entropy proportional to horizon area: $S_{\rm BH}=k_B c^3 A/(4G\hbar)=k_B A/4\ell_P^2$ ESTABLISHED theory; microstate counting [OPEN for generic black holes]. See domains/thermodynamics.md.

Bell's theorem — No local-hidden-variable theory satisfying measurement independence can reproduce all quantum correlations; the CHSH inequality $|S|\le2$ for LHV is violated up to the Tsirelson bound $2\sqrt2$ by quantum systems, confirmed loophole-free ESTABLISHED. The escape via denying measurement independence (superdeterminism) is OPEN/CONTESTED. See domains/quantum-mechanics.md.

Beta function — $\beta(g)=\mu\,dg/d\mu$, governing the renormalization-group running of a coupling; its sign and zeros determine asymptotic freedom, Landau poles, and fixed points ESTABLISHED. See renormalization group.

Bianchi identity (contracted) — The geometric identity $\nabla^\mu G_{\mu\nu}\equiv0$, which forces local energy–momentum conservation $\nabla^\mu T_{\mu\nu}=0$ and reduces the Einstein equations to 6 independent dynamical equations ESTABLISHED. See domains/general-relativity.md.

Black-hole information paradox — Apparent conflict between Hawking's thermal (pure→mixed) evaporation and quantum unitarity. Recent island/replica-wormhole computations reproduce a unitary Page curve INFERENCE within their assumptions; a complete mechanism for realistic black holes is OPEN/CONTESTED. See GAPS_AND_CONTRADICTIONS.md.

Bohmian mechanics (de Broglie–Bohm) — Interpretation restoring definite particle positions guided by the wavefunction; empirically equivalent to standard QM, paying with explicit nonlocality and contextuality ESTABLISHED equivalence; CONTESTED as preferred. See domains/quantum-mechanics.md.

Boltzmann entropy — $S_B=k_B\ln W$, the logarithm of the number of microstates compatible with a macrostate; the surface-entropy definition. Contrast the Gibbs (volume) entropy; their inequivalence for small/bounded systems is CONTESTED. See domains/statistical-mechanics.md.

Boltzmann equation — Kinetic evolution equation for the one-particle distribution $f$ with a collision integral; valid in the dilute-gas (Grad) limit. Its $H$-theorem yields irreversibility only after the molecular-chaos assumption ESTABLISHED. See domains/statistical-mechanics.md.

Born rule — Outcome probability $\Pr(A\in\Delta)=\operatorname{Tr}(\rho\,E_A(\Delta))$. Gleason's theorem makes it the unique noncontextual probability measure on projections for $\dim\ge3$ ESTABLISHED as form; whether it is derivable from unitary dynamics is CONTESTED. See domains/quantum-mechanics.md, EPISTEMICS.md.

Bousso (covariant entropy) bound — Holographic bound $S_{\rm light\text{-}sheet}\le A/4G\hbar$ on the entropy crossing a light-sheet of a surface of area $A$; the covariant, generally applicable form of holography INFERENCE. See domains/information-theory.md.

BRST symmetry — A global fermionic symmetry of the gauge-fixed action encoding gauge redundancy, used to prove unitarity and renormalizability of gauge theories ESTABLISHED. See domains/quantum-field-theory.md.

C

C*-algebra — A complex Banach $*$-algebra satisfying the C*-identity $\|a^*a\|=\|a\|^2$. The Gelfand–Naimark theorem represents any C*-algebra as bounded operators on a Hilbert space ESTABLISHED. The natural setting for algebraic QM/QFT and superselection. See domains/mathematics.md.

Canonical commutation relation (CCR) — $[\hat x,\hat p]=i\hbar\mathbb 1$. Has no bounded or finite-dimensional realization; its exponentiated (Weyl) form is essentially unique for finitely many degrees of freedom by Stone–von Neumann but admits inequivalent representations for infinitely many ESTABLISHED. See Stone–von Neumann theorem.

Canonical ensemble — Probability distribution $\rho=Z^{-1}e^{-\beta H}$ for a system in thermal contact with a reservoir at temperature $T=1/k_B\beta$; $F=-k_BT\ln Z$ ESTABLISHED for short-range interactions. See domains/statistical-mechanics.md.

Carnot bound — Maximum efficiency of a heat engine between reservoirs, $\eta=1-T_c/T_h$, a direct consequence of the second law ESTABLISHED. See domains/thermodynamics.md.

Chern number / characteristic class — Topological invariant of a bundle; e.g. the second Chern number $\frac{1}{8\pi^2}\int\operatorname{tr}(F\wedge F)\in\mathbb Z$ classifies instanton sectors, and the first Chern number gives the quantized Hall conductance (TKNN) ESTABLISHED. See domains/mathematics.md.

CHSH inequality — See Bell's theorem. The four-correlator form $S=|E(a,b)-E(a,b')+E(a',b)+E(a',b')|\le2$ for LHV, with quantum maximum $2\sqrt2$.

CKM matrix — Unitary mixing matrix relating quark mass and weak-interaction eigenstates; its single irreducible phase is the sole source of Standard-Model quark CP violation (Jarlskog $J\approx3\times10^{-5}$) ESTABLISHED. See domains/particle-physics.md.

Coarse-graining — Partition of microstate space into macroscopically distinguishable cells; thermodynamic entropy and irreversibility are defined relative to such a partition. The need for it is structurally fundamental; the specific choice has a conventional element CONTESTED interpretation. See domains/statistical-mechanics.md.

Coleman–Mandula theorem — Under reasonable assumptions (nontrivial S-matrix, mass gap, finitely many particle types), the only Lie-algebra symmetries of the S-matrix that combine spacetime and internal symmetries are direct products of the Poincaré group with internal symmetries — forbidding nontrivial mixing ESTABLISHED. Supersymmetry evades it by using a graded (super-)algebra. See UNIFYING_PRINCIPLES.md.

Conformal field theory (CFT) — A QFT invariant under the conformal group; sits at renormalization-group fixed points and anchors the UV/IR endpoints of RG flows. Most rigorously controlled interacting QFTs (especially in 2D) ESTABLISHED. See domains/quantum-field-theory.md.

Connection — On a fiber bundle, a rule for parallel transport / covariant differentiation; for a principal $G$-bundle it is a $\mathfrak g$-valued one-form whose curvature is the gauge field strength $F=dA+A\wedge A$ ESTABLISHED. The geometric content of the gauge principle. See domains/mathematics.md.

Constraint (Hamiltonian) — A phase-space relation that must vanish (weakly) on physical states, e.g. the GR momentum and Hamiltonian constraints. First-class constraints generate gauge transformations; their quantization yields the Wheeler–DeWitt equation $H\Psi=0$ ESTABLISHED. See problem of time.

Contextuality (Kochen–Specker) — No noncontextual assignment of definite values to all observables (for $\dim\ge3$) reproduces quantum predictions; measured values can depend on which compatible observables are measured alongside ESTABLISHED. See domains/quantum-mechanics.md.

Cosmological constant ($\Lambda$) — The unique term Lovelock's theorem permits beyond the Einstein tensor; drives late-time accelerated expansion. Its observed value lies $\sim10^{120}$ below naive QFT vacuum-energy estimates — the cosmological-constant problem OPEN. See GAPS_AND_CONTRADICTIONS.md, domains/cosmology.md.

Cosmological principle — Spatial homogeneity and isotropy on large scales ($\gtrsim100$ Mpc); the symmetry assumption underlying FLRW ESTABLISHED statistically; INFERENCE as exact. See ASSUMPTIONS_LEDGER.md.

CPT theorem — Any local, Lorentz-invariant QFT with a Hermitian Hamiltonian is invariant under the combined operation of charge conjugation, parity, and time reversal ESTABLISHED. See domains/quantum-field-theory.md.

Critical exponents / universality — Power-law behavior of thermodynamic quantities near a continuous phase transition (e.g. $\xi\sim|t|^{-\nu}$), with exponents depending only on dimension and symmetry (universality classes), explained by RG fixed points ESTABLISHED. See domains/statistical-mechanics.md.

Crooks fluctuation theorem — Detailed symmetry $P_F(+W)/P_R(-W)=e^{\beta(W-\Delta F)}$ between forward and reverse driving protocols; quantifies the exponential rarity of transient second-law violations ESTABLISHED. See domains/thermodynamics.md.

D

Dark energy — The component (constant or dynamical) driving cosmic acceleration; modeled as $\Lambda$ with equation of state $w=-1$. Acceleration is ESTABLISHED; whether dark energy is exactly $\Lambda$ is OPEN/CONTESTED. See domains/cosmology.md.

Dark matter — A non-baryonic, cold, gravitating component required by rotation curves, lensing, the CMB peaks, and structure formation. Its existence (beyond baryons + GR) is near-ESTABLISHED; its particle identity is OPEN. See domains/cosmology.md, domains/particle-physics.md.

Decoherence — Suppression of interference in a subsystem through entanglement with an uncontrolled environment, selecting a quasi-classical pointer basis and yielding effective classicality. The dynamics is ESTABLISHED; its foundational significance for the measurement problem is CONTESTED. See domains/quantum-mechanics.md.

Density matrix — Positive, unit-trace operator $\rho$ encoding pure ($\rho^2=\rho$) or mixed states; the reduced state $\rho_A=\operatorname{Tr}_B\rho_{AB}$ captures all locally accessible statistics ESTABLISHED. See domains/quantum-mechanics.md.

Diffeomorphism invariance — Invariance under smooth coordinate changes; in GR physical states are equivalence classes under diffeomorphisms (background independence), making spacetime points have no physical identity apart from fields (hole argument) ESTABLISHED. See domains/general-relativity.md, UNIFYING_PRINCIPLES.md.

Dirac–Bergmann constraint analysis — Hamiltonian treatment of singular (gauge) Lagrangians via first- and second-class constraints and Dirac brackets; the classical skeleton of gauge theory ESTABLISHED. See domains/classical-mechanics.md.

Domain of validity — The regime (energies, scales, couplings, idealizations) in which a theory is empirically reliable. Distinguishing a true inconsistency from a domain-of-validity mismatch is central to this wiki's epistemics. See EPISTEMICS.md, GAPS_AND_CONTRADICTIONS.md.

E

Effective field theory (EFT) — A theory valid below a cutoff $\Lambda$, organizing all symmetry-allowed operators by mass dimension into relevant/marginal/irrelevant, with $\mathcal L_{\rm eff}=\sum_i c_i\Lambda^{4-d_i}\mathcal O_i$. Renormalizability becomes an emergent low-energy property ESTABLISHED. See domains/quantum-field-theory.md, UNIFYING_PRINCIPLES.md.

Ehrenfest's theorem — $\frac{d}{dt}\langle\hat x\rangle=\langle\hat p\rangle/m$, $\frac{d}{dt}\langle\hat p\rangle=-\langle\nabla V\rangle$; recovers Newtonian motion for narrow packets but fails for chaotic systems beyond the (logarithmic) Ehrenfest time ESTABLISHED. See domains/quantum-mechanics.md.

Einstein field equations (EFE) — $G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$; ten coupled quasilinear second-order PDEs relating curvature to stress-energy ESTABLISHED. See domains/general-relativity.md.

Eigenstate thermalization hypothesis (ETH) — Ansatz that few-body observables in individual energy eigenstates of a generic non-integrable many-body Hamiltonian already take thermal values, explaining thermalization of isolated quantum systems [INFERENCE: strong numerical support, unproven; fails for integrable and many-body-localized systems]. See domains/statistical-mechanics.md.

Energy conditions — Pointwise positivity constraints on $T_{\mu\nu}$ (weak/null/strong/dominant) used in singularity and censorship theorems; violated by quantum fields and by a positive $\Lambda$, so the pointwise forms are now seen as too strong CONTESTED, replaced by averaged/quantum versions (ANEC, QNEC). See domains/general-relativity.md.

Entanglement — Nonseparability of a composite quantum state; the source of Bell violations and a quantifiable resource (teleportation, dense coding). Pure-state bipartite entanglement is measured by the entanglement entropy $S(\rho_A)$ ESTABLISHED. See domains/quantum-mechanics.md, domains/information-theory.md.

Entanglement entropy — Von Neumann entropy $S(\rho_A)=-\operatorname{Tr}\rho_A\ln\rho_A$ of a subsystem. In continuum QFT it is UV-divergent (area law); only universal/finite pieces are cutoff-independent, and local QFT algebras are Type III (no density matrices) ESTABLISHED. See domains/information-theory.md.

Envariance — Environment-assisted invariance; Zurek's symmetry argument attempting to derive the Born rule within a no-collapse (Everettian) framework CONTESTED. See domains/quantum-mechanics.md.

Equivalence principle — Universality of free fall (weak EP, tested to $\sim10^{-15}$) and local Lorentz/position invariance (Einstein EP); the basis for geometrizing gravity ESTABLISHED. See domains/general-relativity.md.

ER=EPR — Conjecture (Maldacena–Susskind) identifying entanglement (EPR) with geometric wormhole connections (Einstein–Rosen) SPECULATIVE. See UNIFICATION_LANDSCAPE.md.

Ergodicity — Equality of long-time and phase-space averages on the energy surface. Generic Hamiltonian systems are not ergodic (KAM); statistical mechanics is now justified more by typicality/measure concentration than by strict ergodicity ESTABLISHED that ergodicity is neither necessary nor sufficient as usually invoked. See domains/statistical-mechanics.md.

Euler–Lagrange equations — $\frac{d}{dt}\frac{\partial L}{\partial\dot q^i}-\frac{\partial L}{\partial q^i}=0$; coordinate-invariant equations of motion from stationary action ESTABLISHED. See domains/classical-mechanics.md.

Everett (many-worlds) interpretation — Retains only unitary evolution; measurement is branching into decoherent worlds. Pays with branching multiplicity and an unsettled account of probability ESTABLISHED equivalence to QM predictions; CONTESTED foundations. See domains/quantum-mechanics.md.

F

Faddeev–Popov / Gribov ambiguity — Gauge-fixing procedure introducing ghost fields; globally obstructed in non-abelian theories (Gribov copies; Singer's theorem shows no global gauge section exists), so it is only locally valid nonperturbatively ESTABLISHED. See domains/mathematics.md.

Fermi–Dirac / Bose–Einstein statistics — Occupation numbers $\langle n_k\rangle=[e^{\beta(\epsilon_k-\mu)}\pm1]^{-1}$ ($+$ fermions, $-$ bosons), following from identical-particle (anti)symmetrization ESTABLISHED. See domains/statistical-mechanics.md, spin–statistics theorem.

Feynman path integral — Propagator as a sum over histories $\int\mathcal D[q]\,e^{iS/\hbar}$; the classical equations arise by stationary phase as $\hbar\to0$. Rigorous via Euclidean (Wiener) measure in QM and some low-dimensional QFTs; the literal Lorentzian measure does not exist ESTABLISHED. See domains/mathematics.md.

Fiber bundle — A space locally a product $U\times F$ of base and fiber but possibly globally twisted; principal $G$-bundles host gauge connections, and the bundle topology is physical (Dirac quantization, Aharonov–Bohm) ESTABLISHED. See domains/mathematics.md.

First law of entanglement — Perturbative equality $\delta S_A=\delta\langle K_A\rangle$ (modular Hamiltonian $K_A$); in holographic setups it yields the linearized Einstein equations — a concrete "gravity from entanglement" result INFERENCE within AdS/CFT; SPECULATIVE as a route to full GR. See domains/information-theory.md.

Fluctuation–dissipation theorem — Relates a system's linear response to its equilibrium fluctuation spectrum; Einstein, Nyquist, and Green–Kubo relations are special cases ESTABLISHED. See domains/statistical-mechanics.md.

Fluctuation theorems — Exact relations (Jarzynski equality, Crooks theorem, Gallavotti–Cohen) constraining nonequilibrium work and entropy-production distributions, sharpening the second law into an equality over trajectory ensembles ESTABLISHED. See domains/thermodynamics.md.

FLRW metric — Friedmann–Lemaître–Robertson–Walker line element, the unique homogeneous-isotropic spacetime, with scale factor $a(t)$ and curvature $k$ ESTABLISHED. See domains/cosmology.md.

Free energy — Thermodynamic potentials obtained as Legendre transforms of internal energy: Helmholtz $F=U-TS$, Gibbs $G=U-TS+pV$, grand $\Omega=U-TS-\mu N$, each minimized at equilibrium under its control variables ESTABLISHED. See domains/thermodynamics.md.

Friedmann equations — Dynamical equations for $a(t)$ from the EFE under FLRW symmetry: $H^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}+\frac{\Lambda}{3}$ and the acceleration equation $\ddot a/a=-\frac{4\pi G}{3}(\rho+3p)+\Lambda/3$ ESTABLISHED. See domains/cosmology.md.

G

Gauge invariance / gauge principle — Invariance under local transformations of internal (or spacetime) symmetry; interactions follow from the covariant derivative $D_\mu=\partial_\mu-igA^a_\mu T^a$. Gauge symmetry is a redundancy of description, not a physical symmetry (Elitzur's theorem forbids its spontaneous breaking) ESTABLISHED; framing it as a fundamental "principle" is CONTESTED. See domains/quantum-field-theory.md, UNIFYING_PRINCIPLES.md.

Generalized probabilistic theories (GPT) — Operational framework reconstructing quantum theory from information-theoretic axioms (local tomography, purification, continuous reversibility); illuminating but with multiple inequivalent axiom sets OPEN. See domains/quantum-mechanics.md.

Generalized second law (GSL) — Conjecture that ordinary entropy plus black-hole horizon entropy never decreases INFERENCE, strong support. See domains/thermodynamics.md.

Geodesic — Extremal-proper-time worldline of a free-falling body: $\ddot x^\mu+\Gamma^\mu_{\alpha\beta}\dot x^\alpha\dot x^\beta=0$; "gravity = inertial motion in curved spacetime" ESTABLISHED. See domains/general-relativity.md.

Geodesic deviation — Relative acceleration of nearby geodesics governed by curvature: $D^2\xi^\mu/d\tau^2=-R^\mu{}_{\alpha\nu\beta}u^\alpha\xi^\nu u^\beta$; the operationally measurable tidal signature of gravity ESTABLISHED. See domains/general-relativity.md.

Gibbs entropy — $S_G=-k_B\int\rho\ln\rho\,d\Gamma$ (classical) or $-k_B\operatorname{Tr}(\rho\ln\rho)$ (von Neumann, quantum); the volume-entropy definition. Constant under exact dynamics; coarse-graining is required for increase ESTABLISHED. See domains/statistical-mechanics.md.

Gleason's theorem — For $\dim\ge3$, the only probability measures on the projection lattice of a Hilbert space are of the form $\operatorname{Tr}(\rho\,\cdot)$, fixing the Born rule's form given noncontextuality ESTABLISHED. See Born rule.

Goldstone bosons — Massless (or pseudo-) excitations accompanying spontaneous breaking of a continuous global symmetry (Goldstone's theorem); when the symmetry is gauged they are "eaten" via the Higgs mechanism ESTABLISHED. See domains/quantum-field-theory.md.

Grand unified theory (GUT) — Embedding of $SU(3)\times SU(2)\times U(1)$ into a simple group (SU(5), SO(10), Pati–Salam), explaining charge quantization and predicting proton decay INFERENCE/SPECULATIVE; minimal SU(5) excluded by proton-decay bounds. See UNIFICATION_LANDSCAPE.md.

Groenewold–van Hove theorem — No quantization map exists from the full classical Poisson algebra to operators respecting $\{\cdot,\cdot\}\mapsto[\cdot,\cdot]/i\hbar$ on all observables; operator ordering is a genuine ambiguity ESTABLISHED. See domains/mathematics.md.

H

Haag's theorem — In QFT the interaction picture does not exist: the free and interacting field representations are unitarily inequivalent, so the naive perturbative starting point is mathematically inconsistent (yet renormalized perturbation theory works) ESTABLISHED. See domains/quantum-field-theory.md.

Hamilton–Jacobi equation — $H(q,\partial S/\partial q,t)+\partial S/\partial t=0$; a single first-order PDE whose complete integral solves the dynamics, and the classical limit of the Schrödinger equation via $S=-i\hbar\ln\psi$ ESTABLISHED. See domains/classical-mechanics.md.

Hamiltonian mechanics — Phase-space dynamics $\dot q^i=\partial H/\partial p_i$, $\dot p_i=-\partial H/\partial q^i$ on a symplectic manifold ESTABLISHED. See domains/classical-mechanics.md.

Hawking radiation — Thermal emission from a black-hole horizon at temperature $T_H=\hbar c^3/(8\pi G k_B M)$, derived from QFT on a curved background [INFERENCE: not directly observed, but theoretically robust]. See domains/general-relativity.md.

Higgs (Brout–Englert–Higgs) mechanism — Spontaneous breaking of a gauge symmetry by a scalar vacuum expectation value, giving mass to gauge bosons ($M_W=\tfrac12 gv$, $v\approx246$ GeV) without breaking gauge invariance explicitly ESTABLISHED. See domains/particle-physics.md.

Hilbert space — Complete complex inner-product space; states of a quantum system are rays (unit vectors up to phase), and observables are self-adjoint operators on it ESTABLISHED. See domains/quantum-mechanics.md, domains/mathematics.md.

Holographic principle — Conjecture that the degrees of freedom in a region are bounded by its boundary area in Planck units, so a gravitational theory is encoded on a lower-dimensional boundary INFERENCE; realized concretely in AdS/CFT. See domains/information-theory.md, UNIFYING_PRINCIPLES.md.

Hubble tension — $\sim4$–$6\sigma$ discrepancy between the early-universe ($H_0\approx67$–68) and local distance-ladder ($H_0\approx73$ km/s/Mpc) determinations of the expansion rate [CONTESTED: systematics vs new physics]. See domains/cosmology.md, GAPS_AND_CONTRADICTIONS.md.

I

Identical particles — Indistinguishable quanta whose multiparticle states must be symmetric (bosons) or antisymmetric (fermions); in 2D the exchange structure is richer (anyons, braid group) ESTABLISHED. See domains/quantum-mechanics.md.

Inflation — A hypothesized epoch of slow-roll accelerated expansion solving the horizon, flatness, and monopole problems and seeding near-scale-invariant fluctuations INFERENCE bordering SPECULATIVE; no primordial B-mode detection, inflaton unidentified. See domains/cosmology.md.

Instanton — A finite-action solution of the Euclidean field equations representing tunneling between topologically distinct gauge vacua, labeled by an integer winding (second Chern) number ESTABLISHED. See domains/mathematics.md.

Integrability (Liouville–Arnold) — A Hamiltonian system with as many independent Poisson-commuting conserved quantities as degrees of freedom; motion confines to invariant tori ESTABLISHED. See domains/classical-mechanics.md.

J

Jacobson's derivation — Demonstration that the Einstein equations follow as an equation of state from horizon thermodynamics ($\delta Q=T\delta S$ on local Rindler horizons) SPECULATIVE/INFERENCE: suggestive of emergent gravity. See UNIFYING_PRINCIPLES.md.

Jarzynski equality — Exact nonequilibrium identity $\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}$, valid arbitrarily far from equilibrium, implying $\langle W\rangle\ge\Delta F$ ESTABLISHED. See domains/thermodynamics.md.

K

KAM theorem — Kolmogorov–Arnold–Moser: most invariant tori of an integrable system survive small perturbations, provided their frequencies satisfy a Diophantine (non-resonance) condition; resonances seed chaos ESTABLISHED. See domains/classical-mechanics.md.

KMS condition — Kubo–Martin–Schwinger characterization of thermal equilibrium states via analyticity of correlation functions in imaginary time; via Tomita–Takesaki it identifies "temperature" and modular "time" with a common algebraic origin ESTABLISHED. See domains/mathematics.md.

Kochen–Specker theorem — See contextuality.

Kolmogorov complexity — Length $K(x)$ of the shortest program generating a string $x$; machine-independent up to an additive constant but provably uncomputable ESTABLISHED. See domains/information-theory.md.

Kraus (operator-sum) representation — General quantum channel $\rho\mapsto\sum_k K_k\rho K_k^\dagger$ with $\sum_k K_k^\dagger K_k=\mathbb1$; via Stinespring/Naimark every channel and POVM is a unitary/projective measurement on a dilated space ESTABLISHED. See domains/quantum-mechanics.md.

L

Lagrangian — Function $L=T-V$ (or density $\mathcal L$) whose action governs the dynamics; the specific $T-V$ form is a low-energy shadow of deeper (relativistic/path-integral) structure ESTABLISHED, INFERENCE as to origin. See domains/classical-mechanics.md.

Landauer's principle — Erasing one bit of information dissipates at least $k_BT\ln2$ of heat, linking logical irreversibility to thermodynamic cost; experimentally confirmed near the bound ESTABLISHED; its independent status vs the second law is CONTESTED. See domains/information-theory.md.

Landau pole — Energy at which a coupling (e.g. in QED or $\phi^4$) formally diverges, signaling that the theory is at most an EFT with a finite UV cutoff (triviality) INFERENCE, strong. See domains/quantum-field-theory.md.

$\Lambda$CDM — The concordance cosmological model: GR with a cosmological constant and cold dark matter, specified by six base parameters, fitting the CMB, BBN, BAO, and supernovae ESTABLISHED as a fit; constituents [OPEN]. See domains/cosmology.md.

Legendre transform — Map exchanging a variable for its conjugate (e.g. $L\to H$, $U\to F$); underlies the Hamiltonian formalism and thermodynamic potentials ESTABLISHED. See domains/classical-mechanics.md, domains/thermodynamics.md.

Levi-Civita connection — The unique torsion-free, metric-compatible connection on a (pseudo-)Riemannian manifold, with Christoffel symbols $\Gamma^\lambda_{\mu\nu}=\tfrac12 g^{\lambda\sigma}(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu})$ ESTABLISHED; the torsion-free choice is conventional, cf. Einstein–Cartan. See domains/general-relativity.md.

Lindblad (GKSL) equation — Most general generator of completely-positive, trace-preserving Markovian dynamics: $\dot\rho=-\tfrac{i}{\hbar}[H,\rho]+\sum_k(L_k\rho L_k^\dagger-\tfrac12\{L_k^\dagger L_k,\rho\})$; governs open-system evolution and decoherence ESTABLISHED. See domains/quantum-mechanics.md.

Liouville's theorem — Hamiltonian phase-space flow preserves volume ($d\rho/dt=0$), so fine-grained entropy is conserved — the technical reason coarse-graining is needed for irreversibility ESTABLISHED. See domains/statistical-mechanics.md.

Lovelock's theorem — In 4D, the only divergence-free symmetric 2-tensor built from the metric and its first two derivatives (with second-order field equations) is $aG_{\mu\nu}+bg_{\mu\nu}$, making the Einstein equations plus $\Lambda$ essentially unique ESTABLISHED. See domains/general-relativity.md.

LSZ reduction — Lehmann–Symanzik–Zimmermann formula relating S-matrix elements to amputated correlation functions, justified for isolated mass shells with a mass gap; fails for confined or massless asymptotic states ESTABLISHED. See domains/quantum-field-theory.md.

Lyapunov exponent — Rate $\lambda$ of exponential divergence of nearby trajectories, $|\delta x(t)|\sim|\delta x(0)|e^{\lambda t}$; $\lambda>0$ is the signature of deterministic chaos ESTABLISHED. See domains/classical-mechanics.md.

M

Majorana mass / neutrino — A fermion identical to its own antiparticle; Majorana neutrino masses (via the seesaw or Weinberg operator) would violate lepton number, testable by neutrinoless double-beta decay. Dirac vs Majorana nature of neutrinos is OPEN. See domains/particle-physics.md.

Mandelstam–Tamm bound — Time–energy inequality bounding the rate of state evolution by the energy spread, reinterpreting $\Delta E\,\Delta t\gtrsim\hbar$ (which is not a Robertson relation, since there is no time operator) ESTABLISHED. See domains/quantum-mechanics.md.

Margolus–Levitin theorem — Quantum speed limit: a state of mean energy $E$ above the ground state needs at least $t_\perp\ge\pi\hbar/2E$ to reach an orthogonal state ESTABLISHED. See domains/information-theory.md.

Maxwell relations — Equalities among partial derivatives of thermodynamic potentials (e.g. $(\partial S/\partial V)_T=(\partial p/\partial T)_V$) following from exactness of the differentials ESTABLISHED. See domains/thermodynamics.md.

Maxwell's demon — Thought experiment apparently violating the second law by sorting molecules; resolved by the entropy cost of information acquisition and especially erasure (Szilard, Landauer, Bennett) ESTABLISHED. See domains/thermodynamics.md.

Measurement problem — The structural incompatibility between linear unitary evolution (which never selects a single outcome) and the stochastic projection postulate, with no rule specifying what triggers collapse ("the shifty split") OPEN/CONTESTED. The deepest internal problem of QM. See domains/quantum-mechanics.md, GAPS_AND_CONTRADICTIONS.md.

Microcausality — Vanishing of commutators of local observables at spacelike separation, $[\phi(x),\phi(y)]=0$ for $(x-y)^2<0$; the QFT encoding of relativistic causality and no-signaling ESTABLISHED. See domains/quantum-field-theory.md.

Microcanonical ensemble — Uniform (Liouville) measure on the energy shell; $S_B=k_B\ln\Omega(E)$. The equal-a-priori-probability postulate bridges micro to macro INFERENCE/OPEN as to first-principles derivation. See domains/statistical-mechanics.md.

Modular Hamiltonian — Generator $K_A=-\ln\rho_A$ of the Tomita–Takesaki modular flow of a region's state; central to relative entropy, energy conditions, and the entanglement first law ESTABLISHED. See domains/information-theory.md.

Molecular chaos (Stosszahlansatz) — Assumption that pre-collision velocities are uncorrelated; the precise time-asymmetric input that lets the Boltzmann $H$-theorem produce irreversibility ESTABLISHED as the locus of time-asymmetry. See domains/statistical-mechanics.md.

Moyal star product — Noncommutative product $f\star g=fg+\tfrac{i\hbar}{2}\{f,g\}+O(\hbar^2)$ deforming the Poisson algebra; the phase-space (deformation-quantization) route from classical to quantum mechanics ESTABLISHED. See domains/classical-mechanics.md.

Mutual information — $I(X;Y)=H(X)+H(Y)-H(X,Y)=D(p(x,y)\Vert p(x)p(y))\ge0$; the shared information between two systems, finite even when individual entropies diverge ESTABLISHED. See domains/information-theory.md.

N

Negative temperature — For systems with a bounded energy spectrum, population inversion gives $dS/dU<0$, hence $T<0$, which is "hotter" than any positive $T$; experimentally realized ESTABLISHED; admissibility tied to the CONTESTED Boltzmann-vs-Gibbs entropy debate. See domains/thermodynamics.md.

No-cloning theorem — Linearity of quantum mechanics forbids a universal device that copies an unknown state; underlies quantum cryptography and sharpens the information paradox ESTABLISHED. See domains/information-theory.md.

No-signaling theorem — Local marginal statistics are independent of a distant party's measurement choice, so entanglement cannot transmit usable information faster than light ESTABLISHED. See domains/quantum-mechanics.md.

Noether's theorem — Every continuous symmetry of an action yields a conserved current/charge; in Hamiltonian form, $\dot J=0\iff\{J,H\}=0$ ESTABLISHED. See domains/classical-mechanics.md, UNIFYING_PRINCIPLES.md.

O

Objective-collapse models (GRW/CSL, Diósi–Penrose) — Modifications of QM adding stochastic nonlinear terms that spontaneously localize macroscopic superpositions, predicting tiny, testable deviations from unitary QM OPEN, empirically constrained. See domains/quantum-mechanics.md, HYPOTHESES.md.

Osterwalder–Schrader reconstruction — Theorem relating Euclidean (reflection-positive) correlation functions to a Lorentzian Wightman QFT; underlies rigorous and lattice work but requires reflection positivity (which can fail at finite density — the sign problem) ESTABLISHED where conditions hold. See domains/quantum-field-theory.md.

P

Page curve — The entanglement entropy of Hawking radiation as a function of time, rising then falling for a unitarily evaporating black hole; recently reproduced via quantum-extremal-surface/island computations INFERENCE within assumptions. See domains/information-theory.md.

Partition function — Generating functional $Z=\operatorname{Tr}e^{-\beta\hat H}$ (statistical mechanics) or $\int\mathcal D\phi\,e^{-S_E}$ (Euclidean QFT); all equilibrium thermodynamics follows by differentiating $\ln Z$, and the Euclidean path integral is a partition function (Wick-rotation isomorphism) ESTABLISHED. See domains/statistical-mechanics.md, domains/quantum-field-theory.md.

Past hypothesis — The posit of an extraordinarily low-entropy initial cosmological state, required to ground the thermodynamic arrow of time given time-symmetric microdynamics INFERENCE that it is necessary; OPEN why it holds. See domains/statistical-mechanics.md, domains/cosmology.md.

Pauli's theorem (no time operator) — No self-adjoint operator is canonically conjugate to a Hamiltonian bounded below; time is treated as a c-number parameter, not an observable ESTABLISHED. See domains/quantum-mechanics.md.

PBR theorem — Pusey–Barrett–Rudolph: under the assumption that independently prepared systems have independent physical states, the quantum state cannot be a mere probability distribution over deeper "ontic" states (a $\psi$-epistemic model) — the wavefunction is "real" INFERENCE, conditional on the preparation-independence assumption. See domains/quantum-mechanics.md.

Penrose–Hawking singularity theorems — Under energy conditions and a trapped surface (or expansion), spacetime is geodesically incomplete, proving singularities are generic in gravitational collapse and cosmology ESTABLISHED. See domains/general-relativity.md.

Phase transition — Non-analyticity of a thermodynamic potential, occurring only in the thermodynamic limit (Lee–Yang); classified by order and universality class ESTABLISHED. See domains/statistical-mechanics.md.

PMNS matrix — Pontecorvo–Maki–Nakagawa–Sakata lepton mixing matrix governing neutrino oscillations; its large mixing angles and undetermined CP phase contrast with the nearly diagonal CKM matrix ESTABLISHED that mixing is large; CP phase OPEN. See domains/particle-physics.md.

Poincaré group — The isometry group of Minkowski spacetime (Lorentz transformations + translations); its unitary irreducible representations classify particles by mass and spin (Wigner) via the Casimirs $P^2=m^2$, $W^2=-m^2 s(s+1)$ ESTABLISHED. See domains/quantum-field-theory.md.

Poisson bracket — Lie-algebraic structure $\{f,g\}=\sum_i(\partial_{q^i}f\,\partial_{p_i}g-\partial_{p_i}f\,\partial_{q^i}g)$ on phase space; evolution is $df/dt=\{f,H\}+\partial_t f$. Survives (deformed) into the quantum commutator ESTABLISHED. See domains/classical-mechanics.md.

POVM — Positive operator-valued measure $\{E_i\ge0,\sum_i E_i=\mathbb1\}$; the most general quantum measurement, generalizing projective (von Neumann) measurements ESTABLISHED. See domains/quantum-mechanics.md.

Problem of time — In canonical gravity the Hamiltonian is a sum of constraints, so $H\Psi=0$ (Wheeler–DeWitt) describes a "timeless" quantum universe; reconciling this with observed time evolution is unresolved OPEN/CONTESTED. See domains/general-relativity.md, GAPS_AND_CONTRADICTIONS.md.

Q

QBism — Quantum Bayesianism: reads the wavefunction and Born rule as an agent's subjective degrees of belief, dissolving the measurement problem at the cost of a non-objective $\psi$ CONTESTED interpretation. See domains/quantum-mechanics.md.

Quantum error correction (QEC) — Encoding logical information redundantly across physical degrees of freedom to protect against noise; in AdS/CFT it models how bulk operators are reconstructed from boundary subregions (entanglement-wedge reconstruction) ESTABLISHED as code theory; INFERENCE as a model of holography. See domains/information-theory.md.

R

Reduced density matrix — See density matrix.

Reeh–Schlieder theorem — The vacuum is cyclic and separating for every local algebra: acting with operators in an arbitrarily small region densely spans the whole Hilbert space, exhibiting ubiquitous vacuum entanglement (no contradiction with no-signaling) ESTABLISHED. See domains/quantum-field-theory.md.

Relational quantum mechanics — Rovelli's interpretation in which physical facts are relative to the systems interacting, with no observer-independent absolute state CONTESTED. See domains/quantum-mechanics.md.

Renormalization — Procedure absorbing cutoff dependence into a finite set of measured parameters; modern Wilsonian view treats it as RG flow of an effective theory rather than infinity-subtraction ESTABLISHED. See renormalization group, domains/quantum-field-theory.md.

Renormalization group (RG) — The flow of couplings with energy scale, $\beta(g)=\mu\,dg/d\mu$; fixed points define scale-invariant (conformal) theories, and irrelevant operators are suppressed at low energy, explaining universality and emergent renormalizability ESTABLISHED. See domains/statistical-mechanics.md, UNIFYING_PRINCIPLES.md.

Ricci / Riemann curvature — $R^\rho{}_{\sigma\mu\nu}$ is the tensorial measure of gravity ($[\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma$); its contraction (Ricci) sources the Einstein tensor ESTABLISHED. See domains/general-relativity.md.

Rigged Hilbert space — Gelfand triple $\Phi\subset\mathcal H\subset\Phi'$ making rigorous the Dirac formalism of continuous-spectrum "eigenstates" (e.g. position, momentum) that lie outside $\mathcal H$ ESTABLISHED. See domains/mathematics.md.

Robertson uncertainty relation — $\Delta A\,\Delta B\ge\tfrac12|\langle[A,B]\rangle|$, a statement about ensemble spreads in a fixed state (not measurement disturbance), specializing to $\Delta x\,\Delta p\ge\hbar/2$ ESTABLISHED. See domains/quantum-mechanics.md.

Ryu–Takayanagi formula — In holography, the entanglement entropy of a boundary region equals the area of the minimal homologous bulk surface, $S(A)=\mathrm{Area}(\gamma_A)/4G_N$; covariant (HRT) and quantum-corrected (QES) versions extend it INFERENCE within AdS/CFT. See domains/information-theory.md.

S

Sakharov conditions — The three requirements for dynamically generating a baryon asymmetry: baryon-number violation, C and CP violation, and departure from thermal equilibrium ESTABLISHED as necessary conditions. See domains/particle-physics.md.

Schmidt decomposition — Canonical form $|\psi\rangle_{AB}=\sum_i\sqrt{p_i}\,|i\rangle_A|i\rangle_B$ of a bipartite pure state, with the $p_i$ the eigenvalues of either reduced density matrix ESTABLISHED. See domains/quantum-mechanics.md.

Schrödinger equation — $i\hbar\,\partial_t|\psi\rangle=H|\psi\rangle$; deterministic, linear, unitary evolution of a closed quantum system, with $U(t)=e^{-iHt/\hbar}$ ESTABLISHED. See domains/quantum-mechanics.md.

Second law of thermodynamics — Entropy of an isolated system does not decrease, $\Delta S\ge0$ (Clausius $dS\ge\delta Q/T$); phenomenologically absolute, but statistically a probabilistic statement refined by fluctuation theorems ESTABLISHED at macroscale; refined by statistical mechanics. See domains/thermodynamics.md.

Self-adjoint operator — A symmetric operator equal to its adjoint including domains; required (not mere symmetry) for a real spectrum, a spectral measure, and unitary dynamics (Stone). Inequivalent self-adjoint extensions encode different physics (boundary conditions) ESTABLISHED. See domains/mathematics.md.

Shannon entropy — $H(X)=-\sum_x p(x)\log_2 p(x)$, the unique measure of average uncertainty (up to base), equal to the optimal lossless compression rate ESTABLISHED. See domains/information-theory.md.

Spectral theorem — Every self-adjoint operator decomposes as $A=\int_{\sigma(A)}\lambda\,dE_A(\lambda)$ against a projection-valued measure on its spectrum; underwrites both the value set and the Born rule ESTABLISHED. See domains/mathematics.md.

Spin–statistics theorem — In relativistic QFT, integer-spin fields are bosonic (commuting) and half-integer-spin fields are fermionic (anticommuting), forced by microcausality, positive energy, and Lorentz invariance; an added postulate in non-relativistic QM ESTABLISHED. See domains/quantum-field-theory.md.

Spontaneous symmetry breaking (SSB) — A symmetric dynamics with a non-symmetric ground state; the symmetry is realized nonlinearly (Goldstone modes) rather than on the vacuum. Requires infinitely many degrees of freedom (inequivalent vacua) ESTABLISHED. See domains/quantum-field-theory.md, UNIFYING_PRINCIPLES.md.

Standard Model — The renormalizable, anomaly-free chiral gauge theory with group $SU(3)_c\times SU(2)_L\times U(1)_Y$ spontaneously broken to $SU(3)_c\times U(1)_{\rm em}$; the most precisely tested theory in science ESTABLISHED, with confirmed incompleteness via neutrino mass. See domains/particle-physics.md.

Stinespring dilation — Theorem that every completely-positive map is a compression of a $*$-homomorphism (unitary) on a larger space; the structural basis for Kraus/POVM purification ESTABLISHED. See domains/quantum-mechanics.md.

Stone's theorem — A strongly continuous one-parameter unitary group has a unique self-adjoint generator, tying unitary dynamics to self-adjointness of the Hamiltonian ESTABLISHED. See domains/mathematics.md.

Stone–von Neumann theorem — For finitely many canonical pairs, all irreducible representations of the (Weyl form of the) CCR are unitarily equivalent; uniqueness fails for infinitely many degrees of freedom, the technical root of inequivalent vacua, Haag's theorem, and SSB ESTABLISHED. See domains/mathematics.md.

Strong CP problem — The unexplained smallness ($\bar\theta\lesssim10^{-10}$) of the CP-violating QCD theta angle, naturally $O(1)$; the Peccei–Quinn axion is the leading proposed resolution OPEN; axion SPECULATIVE. See domains/particle-physics.md.

Strong subadditivity — The deep entropy inequality $S(\rho_{ABC})+S(\rho_B)\le S(\rho_{AB})+S(\rho_{BC})$, equivalent to nonnegativity of conditional mutual information; source of most holographic entropy constraints ESTABLISHED. See domains/information-theory.md.

Superselection sector — A subspace between which coherent superpositions are unobservable (e.g. different charge or, for infinite systems, distinct phases), arising from inequivalent representations ESTABLISHED. See domains/quantum-field-theory.md.

Supersymmetry (SUSY) — Proposed graded symmetry relating bosons and fermions (evading Coleman–Mandula via the Haag–Łopuszański–Sohnius extension), addressing the hierarchy problem and improving gauge unification SPECULATIVE; no superpartners observed. See UNIFICATION_LANDSCAPE.md.

Symplectic form — Closed, nondegenerate two-form $\omega=dq^i\wedge dp_i$ on phase space defining Hamiltonian vector fields via $\iota_{X_H}\omega=dH$; the rigid geometric core of classical mechanics (Darboux's theorem) ESTABLISHED. See domains/classical-mechanics.md.

T

Tensor product — The composition rule $\mathcal H_{AB}=\mathcal H_A\otimes\mathcal H_B$ for quantum subsystems, encoding local tomography and entanglement; singled out among operationally reasonable rules in GPT reconstructions ESTABLISHED operationally; INFERENCE as necessity. See domains/quantum-mechanics.md.

Thermodynamic limit — The idealization $N,V\to\infty$ at fixed density, in which extensivity holds, ensembles are equivalent, and genuine phase transitions exist ESTABLISHED idealization; fails for long-range/gravitating systems. See domains/statistical-mechanics.md.

Third law of thermodynamics — Entropy approaches a constant (conventionally zero) as $T\to0$ for a non-degenerate ground state, and absolute zero is unattainable; a quantum consequence of discrete spectra ESTABLISHED. See domains/thermodynamics.md.

Tomita–Takesaki modular theory — For a cyclic separating state, the operator $S=J\Delta^{1/2}$ generates a canonical modular automorphism flow $\sigma_t(a)=\Delta^{it}a\Delta^{-it}$ and identifies the commutant ($JMJ=M'$); thermal (KMS) states are modular states ESTABLISHED. See domains/mathematics.md.

Topological quantum field theory (TQFT) — A QFT whose observables depend only on topology; formalized as a symmetric monoidal functor $Z:\mathrm{Bord}_n\to\mathcal C^\otimes$ (Atiyah–Segal), with the cobordism hypothesis classifying fully extended TQFTs ESTABLISHED for topological theories. See domains/mathematics.md.

Triviality — The conjecture/theorem that the continuum limit of a theory (e.g. $\phi^4$ in 4D, proven by Aizenman–Duminil-Copin) is the free theory, so it survives only as a cutoff EFT [ESTABLISHED for $\phi^4_4$; INFERENCE for QED]. See domains/quantum-field-theory.md.

Tsirelson bound — The quantum maximum $2\sqrt2$ of the CHSH expression, exceeding the classical $2$ but below the no-signaling (PR-box) maximum $4$ ESTABLISHED. See domains/quantum-mechanics.md.

U

Unitarity — Conservation of total probability under time evolution / the S-matrix; non-negotiable within quantum mechanics, enforced order-by-order via Ward/BRST identities and the optical theorem ESTABLISHED. See domains/quantum-field-theory.md.

Universality — See critical exponents / universality.

Unruh effect — A uniformly accelerated observer perceives the Minkowski vacuum as a thermal bath at temperature $T=\hbar a/2\pi k_B c$; demonstrates the observer-dependence of "particle" in QFT [INFERENCE: theoretically robust, not directly observed]. See domains/quantum-field-theory.md.

V

Vacuum (QFT) — The lowest-energy state; for interacting theories it is not the free Fock vacuum (Haag's theorem), is entangled across regions (Reeh–Schlieder), and carries energy whose gravitational weight is the cosmological-constant problem ESTABLISHED. See domains/quantum-field-theory.md.

Von Neumann entropy — $S(\rho)=-\operatorname{Tr}(\rho\ln\rho)$; the quantum analogue of Shannon entropy, zero for pure states and maximal for the maximally mixed state, and the entanglement entropy of a bipartite pure state ESTABLISHED. See domains/information-theory.md.

Von Neumann algebra — A $*$-subalgebra of bounded operators closed in the weak operator topology; classified into types I, II, III. Local QFT algebras are Type III$_1$ (no trace, no local density matrix) ESTABLISHED. See domains/mathematics.md.

W

Ward identities — Relations among correlation functions enforcing a (gauge) symmetry at the quantum level; ensure renormalizability and unitarity ESTABLISHED. See domains/quantum-field-theory.md.

Weinberg operator — The unique dimension-5 Standard-Model EFT operator, $\frac{c_{ij}}{\Lambda}(\bar L_{Li}^c\tilde\phi^*)(\tilde\phi^\dagger L_{Lj})$, giving Majorana neutrino masses $m_\nu\sim v^2/\Lambda$ — the leading window to physics beyond the SM ESTABLISHED as the leading operator. See domains/particle-physics.md.

Weinberg–Witten theorem — Forbids, under broad assumptions, massless particles of spin $>1$ carrying a Lorentz-covariant conserved current, and spin $>2$ carrying a covariant stress-energy tensor — constraining how gravitons or composite gauge bosons can be emergent (the graviton must evade it, e.g. via the absence of a covariant boundary stress tensor) ESTABLISHED. See UNIFICATION_LANDSCAPE.md.

Wheeler–DeWitt equation — The quantum Hamiltonian constraint $H\Psi=0$ of canonical gravity, formally timeless; see problem of time ESTABLISHED as the constraint; interpretation OPEN. See domains/general-relativity.md.

Wick rotation — Analytic continuation $t\to-i\tau$ relating Lorentzian QFT to Euclidean statistical field theory; rigorous under reflection positivity (Osterwalder–Schrader) but problematic in generic curved/Lorentzian backgrounds ESTABLISHED where conditions hold. See domains/mathematics.md.

Wightman axioms — Axiomatic characterization of a relativistic QFT in terms of operator-valued distributions, Poincaré covariance, the spectrum condition, microcausality, and a cyclic vacuum ESTABLISHED framework; 4D interacting realizations OPEN. See domains/quantum-field-theory.md.

Wigner's theorem — Every symmetry preserving transition probabilities is implemented by a unitary or antiunitary operator (antiunitary for time reversal) ESTABLISHED. See domains/mathematics.md.

Wilson loop — Gauge-invariant trace of the holonomy of a connection around a closed loop; an order parameter for confinement and the natural gauge-invariant observable ESTABLISHED. See domains/quantum-field-theory.md.

WKB approximation — Semiclassical expansion in $\hbar$ with phase $e^{iS/\hbar}$; recovers classical mechanics by stationary phase and captures tunneling ($\sim e^{-S/\hbar}$), but the $\hbar\to0$ limit is singular and non-uniform ESTABLISHED. See domains/classical-mechanics.md.

Y

Yang–Mills theory — Non-abelian gauge theory with field strength $F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+gf^{abc}A^b_\mu A^c_\nu$ and self-interacting gauge bosons; the basis of QCD and the electroweak sector. Rigorous 4D existence with a mass gap is a Clay Millennium Problem OPEN. See domains/quantum-field-theory.md, OPEN_PROBLEMS.md.

See also

• THEORY_MAP.md — how the domains and their concepts interconnect
• EPISTEMICS.md — the meaning and calibration of the epistemic markers used here
• UNIFYING_PRINCIPLES.md — cross-cutting principles (symmetry, RG, locality, holography)
• GAPS_AND_CONTRADICTIONS.md — where theories clash or break down
• OPEN_PROBLEMS.md — the named open questions referenced above
• CONSTANTS_AND_SCALES.md — fundamental constants and characteristic scales
• Domain pages: quantum-mechanics · quantum-field-theory · general-relativity · classical-mechanics · thermodynamics · statistical-mechanics · cosmology · particle-physics · information-theory · mathematics

References

For canonical textbooks and landmark papers underlying these definitions, see BIBLIOGRAPHY.md.