Topics, T

T-Duality > see M-theory; principal fiber bundles.

T0, T1, T2, T3, T4 Spaces > see types of topological spaces.

T Symmetry / Violation > see CPT symmetry.

Tables of Integrals > see integration.


Tails > see gravitational phenomenology; huygens' principle; quantum localization [Hegerfeldt theorem and infinite tails]; wave phenomena.

Tajmar Effect
* Idea: An unexplained acceleration observed by accelerometers and laser gyroscopes close to rotating supercooled rings; The observed ratio between the gyroscope and ring accelerations was 3 ± 1.2 × 10−8 (for clockwise rotations, and about half this size for anticlockwise ones).
@ References: Tajmar et al AIP(07)gq/06 [rotating superconductors]; McCulloch EPL(10)-a0912, EPL(11)-a1106 [and modification of inertia].

Takagi Factorization > see metric tensor.

Takagi Function
* Idea: A continuous, nowhere-differentiable function whose graph is a fractal, related to the Weierstrass functions.
@ References: Allaart & Kawamura RAE-a1110.
> Online resources: see Wikipedia page [Blancmange curve].

Talbot Effect > s.a. light.
* Idea: A diffraction phenomenon by which a wave imprinted with transverse periodicity, such as in a diffraction grating, reconstructs itself at regular intervals; It occurs in many physical systems.
@ References: McMorran & Cronin NJP(09)-a0812 [for electrons].

Tangent Bundle, Map, Vector > see tangent structures.

Tangherlini Solutions > see higher-dimensionsional black holes [Schwarzschild-Tangherlini].

Tangle > s.a. knot theory; lattice field theory [random].
@ References: Baez & Langford LMP(98) [2-tangles]; Zinn-Justin & Zuber JKTR(00)mp, Zinn-Justin CMP(03)mp/01 [counting, and matrix models]; Kravchenko & Polyak LMP(11) [Milnor's μ-invariants and diassociative algebras].

Tau Lepton > see particle types.

Taub Function > see initial-value formulation of general relativity [densitized lapse].

Taub Numbers
* Idea: A set of tensorial conservation laws derivable from curves of solutions to the Einstein equation, i.e., solutions of the linearized equation; For asymptotically flat backgrounds with Killing symmetries, when the field equations and the linearized field equations for a metric perturbation are solved, such perturbed space-times admit 0th-, 1st-, and 2nd-order Taub numbers; 0th-order Taub numbers are Komar constants of the background; For each Killing symmetry of the background, first-order Taub numbers give the contribution of the perturbation to the associated Komar constant, such as the perturbing mass; second-order Taub numbers give the rate of gravitational radiative loss of the background conserved quantity.
@ Asymptotically flat backgrounds with Killing vectors: Glass PRD(93); Glass & Naber JMP(94) [extended to stationary electrovac]; Naber & Glass JMP(94) [at future null infinity]; Glass & Naber CQG(97) + Glass gq/97-MG8.

Taub Solution / Model > s.a. path-integral approach to quantum gravity.
* Idea: A locally rotationally-symmetric homogeneous solution of Bianchi type IX; Resembles a radiation filled Robertson-Walker solution which contains long-wavelength gravitational radiation rather than electromagnetic radiation.
@ General references: Matzner JMP(68); Jantzen GRG(05) [comment on rediscovery], Meléndez & Chauvet GRG(05) [reply].
@ Quantum, with gup: Battisti & Montani PRD(08)-a0707; Battisti et al a0903-proc [and polymer, fate of singularity].
@ Quantum, polymer approach: Battisti et al PRD(08)-a0806; Cascioli et al a1903 [with massless scalar, WKB approximation].

Taub-Bolt Solution > see gravitational instantons.

Taub-NUT Solution > s.a. dirac fields in curved spacetime; modified kerr solutions [Kerr-Taub-NUT-de Sitter]; wormhole solutions.
* Idea: A spatially homogeneous vacuum solution of Bianchi type IX with topology S3 × \(\mathbb R\); It describes a black hole which, in addition to the mass parameter (gravito-electric charge) as in the Schwarzschild solution, depends on a "gravito-magnetic charge", also known as NUT parameter; The NUT metric does not violate the uniqueness and no-hair theorems because it is not asymptotically flat.
* History: Initially proposed by Taub, with coordinate singularities; Newman, Unti & Tamburino gave an extension outside the horizon; It was once called by Charles Misner "a counterexample to almost anything"; Its physical interpretation is challenging and it has often been considered as unphysical.
* Line element: If we define U(t):= −1 + 2 (mt + l 2)/(t 2 + l 2),

ds2 = −U −1 dt2 + (2l)2U (dψ + cosθ dφ)2 + (t 2 + l 2) (dθ2 + sin2θ dφ2) .

@ General references: Taub AM(51); Newman, Tamburino & Unti JMP(63); Misner JMP(63); Misner in(67) [has the 2D example]; Misner & Taub JETP(69); Baleanu & Codoban GRG(99)gq/98 [dual, symmetries]; Chanda et al a1503; Clément et al PLB(15)-a1508 [physical relevance, version without time periodicity condition, geodesically complete]; Jefremov & Perlick CQG+(17); Yohannes & Giulini a2102 [group-theoretic characterisation].
@ Higher-dimensional: Astefanesei et al JHEP(05)ht/04 [Taub-NUT-AdS], JHEP(06)ht/05 [nuttier bubbles]; Awad CQG(06).
@ Other generalizations: Moncrief JGP(84); Van Holten PLB(95) [supersymmetric extension]; Beyer CQG(08)-a0804 [with cosmological constant]; Beyer & Hennig CQG(14)-a1401 [exact, smooth Gowdy-symmetric generalized Taub-NUT solution].
@ Thermodynamics: Mann & Stelea PRD(05); Holzegel CQG(06)gq [and instability]; Durka a1908 [first law].
@ Related topics: Visinescu JPA(00) [Killing-Yano tensors]; Catren & Ferraro PRD(01)gq/00 [quantization]; Bini et al CQG(02)gq [holonomy]; González gq/02 [thin disks as sources]; Jezierski & Lukasik CQG(07) [conformal Killing-Yano tensors].

Tavis-Cummings Model > s.a. Dicke Model.
* Idea: A model which describes N two-level atoms interacting with one mode of the quantized electromagnetic field containing an arbitrary number of excitations; It is the integrable approximation of the Dicke Model.
@ References: Fujii et al IJGMP(04)qp [evolution operator], IJGMP(05); Castaños et al PS(09)-a0909 [ground state, coherent-state description].

Taylor Series > see series.

Teaching > see physics teaching.

Technicolor > see composite models.

Technology > s.a. electronic technology; microscopes; nuclear technology; optical technology; quantum technology.

Teichmüller Space > see 2D manifolds.

Teleology > s.a. causality; Final Cause.
* Idea: The notion that there is a purpose or ultimate end of physical laws.
* Example: Used as motivation for the strong anthropic principle.
@ References: Carlin HSPSA(12) [Robert Boyle and immanent teleology]; Sols a1301-in [the presence or absence of finality in nature is fundamentally outside the scope of the scientific method]; Visser a1707-FQXi.
> Online resources: see Wikipedia page.

Teleparallel Structures > see teleparallel structures and gravity; Elasticity.

Teleportation > see quantum communication technology.


Temperley-Lieb Algebra
@ References: Bytsko JMP(15)-a1505 [tensor space representations].
> Online references: see Wikipedia page.

10j Symbols > s.a. SU(2).
@ References: Christensen & Egan CQG(02)gq/01 [Riemannian, calculation]; Baez et al CQG(02)gq, Freidel & Louapre CQG(03)ht/02 [asymptotics]; Christensen CQG(06)gq/05 [Lorentzian, finiteness].

Tendex Lines > see spacetime; spacetime structure.

Tenfold Way
* Idea: The point of view that there are ten fundamentally different kinds of matter, based on the fact that there are ten real super division algebras.
@ References: Baez NAMS-a2011 [expository article].

Tensile Strength
* Idea: The force required to pull something such as rope, wire, or a structural beam to the point where it breaks.
* Example: Spider silk has higher tensile strength than steel.
> Online references: see ScienceDaily page; Wikipedia page.

Tension > see force [gravitational vacuum tension or maximum tension]; matter dynamics in gravity; strings.

Tensor (including tensor product) > s.a. tensor fields.

Tensor Density > see tensor fields.

Tensor Models > a generalization of Matrix Models.
@ General references: Sasakura JMP(11)-a1104; Gurau a1209-proc; Narain et al JHEP(15)-a1410 [physical states]; Bonzom et al NPB(15)-a1502 [mixing melonic and planar maps]; Sasakura & Sato a1506 [canonical, formal continuum limit and general relativity constraint algebra]; Dartois a1512-PhD; Klebanov et al a1811-ln [large-N limits]; Gurau a1907-ln [intro].
@ Tensor field theories: Ben Geloun a1601-conf [renormalizable]; Rivasseau Sigma(16)-a1603 [constructive].
@ And quantum gravity: Rivasseau Sigma(16)-a1603 [informal introduction]; Banks & Fischler a1606-wd [in d-dimensional Minkowski space]; Eichhorn & Koslowski a1701 [3D, functional renormalization group and continuum limit]; Eichhorn et al JHEP(20)-a1912 [critical behavior]; Banks & Fischler a2003; > s.a. approaches to quantum gravity; asymptotic safety.
> Related topics: see causal dynamical triangulations.

Tensor Networks

Tensor-Vector-Scalar Theory of Gravity (TeVeS) > s.a. MOG (scalar-vector-tensor MOdified Gravity theory); MOND.
* Idea: A tensor-vector-scalar theory of gravitation proposed by Bekenstein in 2004; It can be written as a bimetric theory, with the two metrics related by a disformal transformation defined by a dynamical vector field and a scalar field; 2009, It seems to have trouble simultaneously fitting lensing and rotation curves without any dark matter.
@ General references: Bekenstein PoS-ap/04; Skordis et al PRL(06)ap/05 [large-scale structure]; Bekenstein & Sanders EAS(06)ap/05-proc [intro]; Zlosnik et al PRD(06) [equivalence to Einstein-ether-type theories]; Contaldi et al PRD(08)-a0802 [caustic problem]; Tamaki PRD(08)-a0803 [PN parameters]; Sagi PRD(10)-a1001 [gravitational waves]; Bekenstein PTRS(11)-a1201 [review]; Chaichian et al PLB(14)-a1402 [as a ghost-free, viable theory of gravity].
@ And cosmology: Skordis PRD(06)ap/05 [evolution and perturbations]; Díaz-Rivera et al PRD(06)ap [inflation]; news pw(06)jun; Bourliot et al PRD(07)ap/06; Dodelson & Liguori PRL(06) + pw(06)dec; Zhao IJMPD(07), MPLA(08)-a0802 [dark matter and dark energy]; Skordis PRD(08)-a0801, CQG(09)-a0903; Xu et al PRD(15)-a1412 [and observations, ΛCDM].
> Other phenomenology: see gravitomagnetism.

Tensorgluons > see QCD.

Ternary Operations
@ References: Kerner IJGMP(08) [in theoretical and mathematical physics, including non-associative]; Curtright et al PLB(09)-a0903 [ternary algebras and physics, classical and quantal]; Bocharov et al QIC-a1512 [quantum ternary arithmetics].

TESS (Transiting Exoplanet Survey Satellite) > see extrasolar systems.

Tessellations > see tilings.

* Idea: A hypercube, or 4D cube.
@ References: Aravind AJP(01)mar [toy and Bell's theorem].

Test Body / Particle > s.a. geodesic; particle models; test-body motion.
* Idea: An idealized object used to define and study properties of physical systems, usually fields; The test body couples to the system (through its mass, electric charge, or other coupling parameter) and its response measures properties of the system, but the effect of the test body itself on the system is considered to be negligible.
> Online resources: see Wikipedia page.

Test Space
$ Def: A test space (X, Σ) consists of a set X together with a set of subsets \(\Sigma \subseteq 2^X\) such that the members of Σ cover \(X\), i.e., \(\sum_{T\in\Sigma} T = X\).
* Related concepts: A test is an element of Σ, while an element of X is an outcome; A subset of a test T is an event.
@ References: in Fritz & Leifer a1505/QPL.

Testing (of a Theory) > see theory of physical theories.

Tetrad Formalism

Tetrahedron > s.a. Packings; simplex; spin-foam models [quantum tetrahedron]; SU(2) [6j-symbol].
* For tiling: There exist three kinds of monogenic filling of \(\mathbb R^3\) by tetrahedra, but only one has a dual lattice with monogenic filling (truncated octahedra).
@ References: Korepanov n.SI/00 [5 tetrahedra on 5 points, and curvature]; Chirco et al PRD(19)-a1811 [equilibrium state, from maximum entropy principle]; Crane & Yetter a1906 [geometry of a Euclidean tetrahedron].
> Online resources: see Wikipedia page.

Tetraquarks > see hadrons.

Teukolsky Equation > see black-hole perturbations [around the Kerr solution].

TeVeS > see under Tensor-Vector-Scalar Theory.

Textures > see topological defects.

Theia > s.a. moon.
* Idea: The proto-planet which collided with the Earth in the Giant Impact that formed the Moon according to the most widely accepted scenario.
@ References: Meier et al Icarus(14)-a1410 [origin and composition]; Quarles & Lissauer Icarus(15)-a1410 [dynamical evolution]; Kaib & Cowan Icarus(15)-a1506.

Theorem > see proof theory; also MathWorld page.

Theory > s.a. physical theories [including Theories of Everything]; physics and theory of physical theories [including space of theories, confirmation].
@ References: Mielczarek & Trześniewski PLB(16)-a1601 [phase spaces of field values that are not affine spaces].

Thermal Bath / Reservoir > s.a. heat; thermal radiation.
* Idea: (A.k.a. heat bath) A thermodynamic system with a heat capacity so large that can be considered to be infinite, so that when it is coupled to a system of interest the two will reach thermal equilibrium without affecting the temperature of the bath/reservoir.
@ Finite thermal baths: Potiguar & Costa PhyA(04) [thermodynamic relations]; Gemmer & Michel EPL(06)qp/05 [thermalization]; Ford & O'Connell PhyE(05)qp/06 [different models, and quantum oscillator]; Nechita & Pellegrini CM(09)-a0908 [statistical model]; Brody & Hughston JPA(16)-a1406 [quantum].
@ Coupled to quantum system: Ghosh et al PLA(05) [noise-induced transitions]; Hilt et al PRE(11)-a1106 [of harmonic oscillators, Hamiltonian of mean force]; Sergi JChemP(13)-a1306 [classical spin bath]; Ghesquière & Dorlas PLA(13) [effect on the entanglement of a bipartite Gaussian state].
> Online resources: see Wikipedia page.

Thermal Conductivity, Diffusivity > see Heat Flow.

Thermal Dimension > see dimensionality of spacetime.

Thermal Expansion > s.a. heat.
* Idea: The phenomenon by which matter changes size in response to a change in temperature through heat transfer.
* Thermal expansion coefficient: The volume and linear expansion coefficients are defined by α = V−1V/∂T|P and β = P−1P/∂T|V, respectively.
@ Thermal expansion coefficient: Dounas-Frazer et al AJP(13)may [simple experiment, and measurement uncertainty].
@ Negative thermal expansion: Mary et al Sci(96)apr [Zirconium Tungstate ZrW2O8]; news livesci(04)dec; Li et al PRL(11) + focus; Handunkanda et al PRB(15) + news ea(15)oct [scandium trifluoride]; Wang et al PRL(16) + focus [lightweight metamaterials]; news sn(19)nov [scandium fluoride].
> Online resources: see Wikipedia page.

Thermal Field Theory > see finite-temperature field theory.

Thermal Radiation

Thermalization > see statistical mechanical equilibrium.

Thermoacoustics > see sound.

Thermodynamic Geometry > see thermodynamics [geometry of state space].

Thermodynamic Limit
* Idea: The thermodynamic limit of a statistical mechanical system is the large-size limit of the system, in which the values of the extensive variables of the system tend to infinity with the intensive ones remaining constant, so the fractional fluctuations of statistical quantities go to zero and are ignored.
@ General references: Compagner AJP(89)feb [continuum limit]; Styer AJP(04)jan [paradoxes]; Batterman SHPMP(05) [validity of idealized limit]; Huang et al PhyA(09) [extensive and non-extensive systems, MaxEnt approach]; Snoke et al AP(12)-a1112 [from quantum field theory]; Kuzemsky IJMPB(14)-a1402 [rev]; De Pasquale et al nComm(16)-a1504 [local quantum thermal susceptibility].
@ Specific types of systems: Kalogeropoulos EPJB(14)-a1312 [systems with long-range interactions].
> Online resources: see Wikipedia page.

Thermodynamics > s.a. laws of thermodynamics; thermodynamic systems; generalized versions.

Thermoelectric Effect > see electricity.

Thermofield Dynamics > Equivalent to a restricted version of quantum statistical mechanics.

* Idea: The phenomenon by which certain materials that had previously absorbed energy that is trapped in electronic excited states, emit light when heated.
@ References: Martini & Meinardi RNC(97); Furetta RNC(98).
> Online resources: see Wikipedia page.

Theta Function
* Idea: Theta functions are special functions of several complex variables, with many applications in the theories of abelian varieties and moduli spaces, quadratic forms, soliton theory and, when generalized to a Grassmann algebra, they also appear in quantum field theory.
@ References: Gelca 14 [and knots].
> Online resources: see Wikipedia page.

Theta Sector / Vacuum

Thin Sandwich > see constraint equations in general relativity [solution method]; Sandwich Conjecture.

Third Degree Algebraic Equation > see elementary algebra.

Third Quantization
@ General references: Das in(87).
@ Of gravity: Gielen & Oriti proc(12)-a1102 [matrix models and group field theory]; Ohkuwa & Ezawa CQG(12)-a1203, CQG(13) [f(R) theories]; Faizal CTP-a1407 [and the multiverse]; Campanelli PRD(20)-a2007 [creation of universes]; > s.a. multiverse; quantum cosmology formalism and techniques.
@ Other theories: Prosen & Seligman a1007, Seligman & Prosen AIP(10)-a1011 [open many-body bosonic systems]; Franson a2103 [electromagnetic field].

Thirring Model > s.a. renormalization group.
* Idea: A theory, originally formulated in 2 spacetime dimensions, of a Dirac field with a quartic self-interaction.
@ General references: Thirring AP(58); Ilieva & Thirring ht/98-conf [rev]; Faber & Ivanov ht/03 [dynamical conformal symmetry breaking]; Benfatto et al ht/06 [rigorous]; Korenblit & Semenov JNMP(11)-a1108 [massless, canonical quantization]; Bufalo & Pimentel IJMPA(14)-a1408 [massive, gauged, non-perturbative aspects].
@ 3D theory: Gies & Janssen PRD(10)-a1006 [UV fixed-point structure]; Janssen & Gies PRD(12)-a1208 [chiral symmetry breaking, critical behavior].
> Online resources: see Wikipedia page.

Thomas Precession > s.a. atomic physics; geometric phase; Rapidity; special-relativistic kinematics.
* Idea: The spin precession for a particle in electromagnetic fields.
* Rem: Thomas precession is a Berry phase that the particle's wavefunction picks up when subjected to cyclic changes of reference frames [Oblak CQG+].
@ General references: Rindler & Perlick GRG(90); Urbantke AJP(90)aug; Samuel PRL(96); Philpott AJP(96)may; Hamilton AJP(96)sep; Muñoz AJP(01)may [in lab frame]; Bordovitsyn & Myagkii proc(01)qp [in classical and quantum theory]; Herrera & Di Prisco FPL(02)gq [approaches]; Behera ap/03, ap/03, ap/03 [gravitational]; Matolcsi & Matolcsi IJTP(05)mp; Chrysos EJP(06) [the 1/2 factor]; Matolcsi et al GRG(07) [and rotating reference frames]; Klioner a0803 [explicit exact expression]; Rębilas FP(11) [for the evolution of any vector quantity, and the Bargmann-Michel-Telegdi equation]; Dragan & Odrzygóźdź a1211, Lewulis & Dragan AJP(19)aug-a1902 [quick derivations]; Salas a2007 [transparent algebraic proof].
@ In curved spacetime: Jonsson AJP(07)may; Silenko PRD(16)-a1606; Oblak CQG(18) [gravitational generalization, BMS asymptotic symmetries].

Thomas Scattering
* Idea: The capture of a particle by a fast projectile.
@ References: Bitensky AJP(98)jul.

Thomas-Fermi Statistical Model / Theory > s.a. atomic physics.
@ References: Solovej MP(15)-a1601.
> Online resources: see Wikipedia page.

Thomas-Wigner Rotation
* Idea: The spatial rotation involved in the composition of two non-collinear boosts.
@ References: Beyerle Symm(17)-a1706 [visualization].

Thomson Scattering
* Idea: Scattering of an electromagnetic wave by a free electron.
@ References: in Jackson 75; Anderson gq/97 [in an Einstein-de Sitter universe].
> Online resources: see Wikipedia page.

Thought Experiment > see under Gedankenexperiment.

Threading > s.a. decomposition.
* Idea: An alternative approach to the decomposition of spacetime tensors, in which spacetime is decomposed not using a foliation by spacelike hypersurfaces, but a congruence of timelike curves representing a family of observers; It leads to an alternative evolution formulation for covariant field equations.
@ References: Ehlers in(59); Roy a1405; Park a1810.
> Related topics: see velocity [3-velocity].
> In gravity theory: see canonical formulation of general relativity and modified approaches.
> Other applications: see casimir effect; QED in curved spacetimes.

Three-Body Physics
@ Three-body forces: Hammer et al RMP(13) [colloquium]; Chapurin et al PRL(19) [limits to three-body interactions]; > s.a. Boltzmann Equation.
> Three-body problem in classical mechanics: see classical systems; gravitating bodies; orbits in newtonian gravity.
> Three-body states in quantum mechanics: see composite quantum systems; Efimov Effect.

3j-Symbols > s.a. SU(2) group [6j symbols, etc].
* Idea: Quantities arising in calculations with coupled angular momenta of two quantum systems, related to the Clebsch-Gordan coefficients.
@ References: Aquilanti et al JPA(07)qp [semiclassical analysis]; Yu a1104 [asymptotic analysis in the Bargmann Representation]; van Veenendaal EJP(11); Pain a2011 [non-trivial zeros and connections to hypergeometric functions].
> Online resources: see MathWorld page; Wikipedia page.

Throttling > see Joule-Thomson Effect.

Thurston Norm
@ References: Friedl & Kim Top(06) [fibered manifolds and twisted Alexander polynomials].

Tidal Acceleration / Force > s.a. geodesics [geodesic deviation]; gravitation.
@ References: Asenbaum et al PRL(17)-a1610 + Jaffe & Müller Phy(17) [on an individual particle's wave function]; Goswami & Ellis a1912 [as gravitational waves].
> Specific spacetimes: see dynamics of gravitating bodies; reissner-nordström spacetime.
> In specific theories: see Metric-Affine Theories; newtonian gravity [including tidal locking].

Tidal Tensor > see gravitomagnetism.

Tikhonov Theorem > see under Tychonoff Theorem.

Tiling > s.a. random tiling; voronoi tiling.

Tilt in Cosmology > s.a. bianchi models.
@ References: Coley et al PLB(06) [and phantom matter behavior].

Time > s.a. time in physical theories [and in classical gravity, quantum theory, quantum gravity].
> Online resources: see arrow of time; CPT symmetry [time reversal].

Time Crystal > see classical systems; crystals [quantum time crystals].

Time Delay > see tests of general relativity.

Time Dilation > see kinematics of special relativity; tests of gravity with light [gravitational time dilation]; clocks [quantum time dilation].

Time Evolution Operator > see time in quantum theory.

Time Machine / Travel > see causality violations.

Time-Ordered Product
* Idea: A tool in quantum field theory introduced by Wick.
$ Def: In the algebra of position-dependent (or often just time-dependent) boson and fermion operators on Fock space, it is defined on products by the time-ordering

T(A1(x1) A2(x2) ... An(xn)) :=  (−1)p Ap1(xp1) Ap2(xp2) ... Apn(xpn) ,

where {pi} is a permutation of {1, 2, ..., n} such that tpitpi+1, or xpiJ+(xpi+1), and p is the number of times two fermion operators were commuted; It is then extended to all other elements by linearity:

T(A+B) = T(A) + T(B) ,   T(cA) = c T(A) .

* Wick's theorem: The time-ordered product of n operators is equal to the sum of all possible normal products formed with all possible contractions.
@ General references: Wick PR(50); Hollands & Wald CMP(01)gq, CMP(02)gq/01 [in curved spacetime, covariant].
@ Time-ordered exponentials: Ebrahimi-Fard & Patras LMP(14) [and enveloping algebras of pre-Lie algebras].

Time Orientability > see orientation.

Time Reversal > see CPT theorem.

Time-Slice Axiom in Quantum Field Theory
* Idea: The observables which can be measured within an arbitrarily small time interval suffice to predict all other observables; Well known for free field theories, where the validity of the axiom is an immediate consequence of the field equations, and for certain superrenormalizable models in 2 dimensions; 2008, Extended to scalar field theories in globally hyperbolic spacetimes within formal renormalized perturbation theory.
@ References: Chilian & Fredenhagen CMP(09)-a0802 [for interacting scalar fields on globally hyperbolic spacetimes].

Time-Symmetric Quantum Theory > see modified formulations of quantum mechanics.

Time-Translation Invariance > see symmetries in quantum theory.

Timelike Curve > see lines; Worldlines.

Timescape > see cosmological acceleration and inhomogeneities.

Tipping Pencil / Rod Problem > see quantum effects.

Tired Light
* Idea: A 1929 proposed explanation by Zwicky for the cosmological redshift, based on the loss of energy by light to intergalactic gases rather than cosmological expansion; One objection to it is that it does not explain the time-dilation of supernova light curves.

Titius-Bode Law
* Idea: The most stable situation for a planetary system is achieved when each planetary orbit is roughly twice as far from the Sun as the preceding one; Empirically observed by Titius (1766) and Bode (1778).
@ And resonances: Bass ap/02 + Bass & Del Popolo IJMPD(05)ap/04; Christodoulou & Kazanas RAA(17)-a1507 [physical interpretation].

Titchmarsh's Theorem > see dispersion.

Toda Lattice

TOE (Theory of Everything) > see physical theories; unified theories.

Toeplitz Operators
@ References: Boutet de Monvel & Guillemin 80.

* Idea: Tokens allow to express a semigroup on one set via a semigroup convolution on another set; They are similar to intertwining operators, but more flexible.
@ References: Kisil m.FA/02-in.

Tolman Solutions / Wormholes > s.a. cosmological models in general relativity; cosmological perturbations; regge calculus; wormholes.
* History: Introduced by Tolman in 1934.
@ References: Tolman PNAS(34) – reprinted GRG(97); Ribeiro ApJ(92)-a0807, ApJ(92)-a0807 [and self-similar cosmology]; Hellaby PRD(94)gq/99 [Vaidya limit]; > s.a. Lemaître-Tolman-Bondi Solutions.

Tolman-Ehrenfest Effect
* Idea: The variation of temperature in space so that T (g00)1/2 remains constant.
@ References: Rovelli & Smerlak CQG(11)-a1005 [and the "speed of thermal time"].

Tolman-Oppenheimer-Volkoff Equations / Solutions > s.a. gravitating bodies / semiclassical general relativity; torsion phenomenology.
* Idea: The general relativistic equation for hydrostatic equilibrium in a static spherically symmetric spacetime supported by an isotropic perfect fluid, used to model stars.
@ References: Semiz SHPMP(16)-a1610; Carballo-Rubio PRL(18)-a1706 [semiclassical gravity corrections].

Tomboulis-Yaffe Inequality
* Idea: An inequality in lattice gauge theory stating that a system in a box that is sufficiently insensitive to boundary conditions has a non-zero mass gap.
@ References: Kanzawa AP(09) [generalized from SU(2) to SU(N)].

Tomimatsu-Sato Solutions > s.a. solutions in general relativity.
@ References: Manko et al G&C(99) [ring singularities]; Hikida & Kodama gq/03-proc; Kodama & Hikida CQG(03)gq; Gegenberg et al CQG(11)-a1010 [holography and quantum gravity].

Tomita Representation
@ References: in Chew et al a1703.

Tomography / Tomographic Representation
* Idea: An approach to the reconstruction/estimation of the state of a physical system based on a possibly incomplete set of data.
@ General references: Asorey et al PS(15)-a1510 [rev, classical and quantum].
> Classical systems: see earth; dirac fields; scalar fields.
> Gravity systems: see canonical quantum gravity; loop quantum cosmology.
> In quantum theory: see representations of quantum mechanics and quantum field theory; quantum states and mixed states.

Tonks Gas > see Cluster Expansion; Virial Expansion.

Tonks-Girardeau Gas > s.a. Fermionization.
* Idea: A 1D model of a gas of bosonic particles that cannot exchange places, with repulsive interactions.
@ References: Atas et al a2005 [non-equilibrium quantum thermodynamics].
> Online resources: see Wikipedia page.

Topological Algebra > see algebra / functional analysis.

Topological Censorship > see models of topology change.

Topological Defects > s.a. cosmic strings.

Topological Entropy / Order > s.a. mixed quantum states; Order.
* Idea: A measure of the complexity of a dynamical system or mapping, giving the exponential growth rate of the number of distinguishable orbits when the map is iterated.
@ References: Dikranjan et al T&A(12) [for mappings of uniform spaces and topological groups]; Giordano Bruno T&A(12), Dikranjan & Giordano Bruno T&A(12) [vs adjoint algebraic entropy].
> Online resources: see MathWorld page; PlanetMath page; Scholarpedia page; Wikipedia page.

Topological Field Theory

Topological Glass
* Idea: A disordered system in which the configurations are topological; For example, a model whose state space is given by all triangulations of a sphere with N nodes, half of which are red and half are blue.
@ References: Eckmann & Younan PMB(12)-a1104 [decay of correlations].

Topological Gravity > s.a. BF theory; black holes in 3D and in modified theories; 3D gravity.
* Idea: The reduction of general relativity to spacetimes with vanishing curvature, but with global topological degrees of freedom.
* Action: Given by Horowitz in terms of a Lie-algebra-valued 2-form Bab ij, and a connection ωaij in the BF form

S[ω, B] = M BijR ij ;

Variation of the action with respect to Bij gives that ωij is flat, so there are local degrees of freedom only if one admits torsion.
@ General references: Myers & Periwal NPB(91); Nakamichi et al PRD(91); Oda & Sugamoto PLB(91); Wu JGP(93); Waelbroeck & Zapata CQG(94)gq/93 [canonical]; Zapata CQG(96)gq [self-dual, lattice]; Thuillier JGP(98); Mignemi PRD(99)gq/98 [4D]; Constantinidis et al CQG(04)gq [approaches]; Salgado et al PLB(14)-a1311 [action in even dimensions as gauged Wess-Zumino-Witten term]; Palumbo EPJP(20)-a1905-GRF [and topologically massive gravity].
@ And general relativity: Montesinos CQG(01)gq [self-dual general relativity with topological term]; Smolin & Starodubtsev ht/03 [larger theory].
@ Variations: Mitskievich et al a0706-MGXI [with topological invariants for coefficients]; Mori & Nojiri Symm(18)-a1809 [motivated by renormalization group]; > s.a. 3D gravity; black holes in modified theories [quasi-topological gravity]; Topologically Massive Gravity.

Topological Group
$ Def: A group G such that the map m: G × GG by m(g, h) = gh is continuous (this combines requirements on the product and the inverse).
* Properties: Its fundamental group is always commutative.
* Amenable topological group: G is amenable if there is a positive G-invariant functional A (a "measure") on the Banach space of bounded Borel measurable functions on G such that A(1G) = 1; For example, SO(3) is not amenable with respect to the discrete topology, but it is as a compact Lie group; SO(n,1) is not amenable; Local gauge groups also have topologies different from their natural ones with respect to which they are amenable.
@ General references: Pontrjagin 66; Greenleaf 69 [means]; Arhangel'skii & Tkachenko 08.
@ Amenable groups: Pier 84; Carey & Grundling LMP(04)mp [gauge groups]; Seiler a1011 [theories with non-amenable symmetry groups].
@ Related topics and results: Megrelishvili T&A(08) [every Hausdorff topological group is a group retract of a minimal topological group].
@ Quasitopological groups: Hernández & Tkachenko T&A(06) [pseudocompact]; Li & Mou T&IA(14) [semimetrizability].
@ Other generalizations: Arhangel'skii & Reznichenko T&A(05) [paratopological and semitopological groups].

Topological Insulators > see under Insulators.

Topological Manifold > see types of manifolds.

Topological Materials > s.a. Topological Glass.
* Idea: A topological material is one with a property that is robust and insensitive to perturbations and impurities.
@ References: collection Phy(16) [topological phases of matter].

Topological Order > see Order.

Topological Particle Theory
@ References: Finkler & Jones PRD(85); Jones & Finkler PRD(85).

Topological Polariton > see Quasiparticles.

Topological Quantization > see formulations of quantum theory.

Topological Recursion
* Idea: A relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems.
@ References: Eynard a1412-proc [rev].

Topological Semigroup > see Semigroup.

Topological Space > s.a. types of topological spaces; topology in physics.

Topological Tangent Bundle
$ Def: A neighborhood of the diagonal in M × M.
@ References: Milnor Top(64).

Topological Transitivity
* Idea: A property of dynamical systems that is a form of mixing, defined using just the topology of the underlying space and not a measure.
@ References: Akin & Carlson T&A(12)-a1108 [relationships among definitions].
> Online resources: see Encyclopedia of Mathematics page; MathWorld page; Scholarpedia page; Wikipedia page on mixing systems.

Topological Vector Space > see vectors.

Topologically Massive Gravity > see 3D massive gravity theories.

Topologically Massive Gauge / Yang-Mills Theory > see types of gauge theories; types of yang-mills theories.

Topology > s.a. topology of the universe; topology in physics.

Topology Change > s.a. dynamical models.

Topos Theory > s.a. category theory in physics; history of mathematics.
* Idea: A branch of category theory, which generalizes set theory.
* In physics: Topos theory has been suggested first by Isham and Butterfield, and then by Isham and Döring, as an alternative mathematical structure within which to formulate physical theories; In this view, constructing a theory is equivalent to finding a representation in a topos of a certain formal language that is attached to the system; For example, classical physics uses the topos of sets, while other theories involve a different topos; Used to assign values to quantities in quantum theory (Isham & Butterfield).
* And quantum mechanics: There are two topos-theoretic approaches to quantum mechanics in the literature, the "contravariant approach", proposed by Isham and Butterfield, and later extended by Döring and Isham, and the "covariant approach", developed by Heunen, Landsman and Spitters.
@ General references: Bell 88; Wyler 91; Mac Lane & Moerdijk 92; Nishimura IJTP(96) [quantization]; Nakayama JMP(13) [topologies induced by quantization]; Eva EPTCS(15)-a1511 [connection with topos quantum theory]; Gauthier a1802 [Segal topos used to model natural phenomena].
@ In physics: Döring & Isham in(11)-a0803 ["what is a thing?"]; Tsatsos a0803-dipl; Isham a1004-in [intro]; Döring a1004-in [daseinisation]; Flori a1110, a1207-ln; Wolters JMP(14)-a1309; Flori 13 [textbook]; > s.a. poisson structures [Poisson algebras for non-linear field theories in the Cahiers topos].
@ And gravity / geometry: Grinkevich gq/96; Guts & Grinkevich gq/96; Isham & Butterfield FP(00)gq/99-in; Raptis gq/01-conf [review]; Zafiris FP(04), IJGMP(06) [quantum events]; Król FP(06), IJGMP(07); Döring & Isham JMP(08)qp/07, JMP(08)qp/07, JMP(08)qp/07, JMP(08)qp/07; Raptis IJTP(07); Król a0804/FP [and the cosmological constant]; > s.a. approaches to quantum gravity; causal sets; quantum spacetime.
@ And quantum theory: Isham & Butterfield IJTP(98), Butterfield & Isham IJTP(99)qp/98, Hamilton et al IJTP(00)qp/99, Butterfield & Isham IJTP(02)qp/01 [Kochen-Specker]; Isham & Butterfield FP(00) [and quantum gravity]; Fearns qp/02|IJTP; Heunen et al CMP(09)-a0709 [algebraic quantum mechanics]; Döring a0712-proc [rev]; Wolters CMP(13)-a1010 [comparison of two approaches]; Flori a1106 [rev]; Döring & Soares Barbosa a1107-proc [and 'unsharp values' of physical quantities]; Nakayama a1109; Brenna & Flori a1206; Corbett EPTCS(12)-a1210; Nakayama JMP(14)-a1404; Freytes et al FP(14)-a1412 [physical properties as modal operators]; Nakayama JMP(16)-a1507 [reduced sheaf-based quantum theory]; Miyazaki a1605; Harding & Heunen a1903; Zapatrin a1911-in [emergence of space-like structures]; Constantin & Döring a2006 [entropy]; > s.a. histories formulations; interpretations [modal]; logic; quantum computing; state-vector reduction.
> Online resources: see Wikipedia page.

Toronto Space
* Idea: A topological space that is homeomorphic to every one of its full-cardinality subspaces.
@ References: Brian T&IA(14) [Toronto problem].

Torsion Balance > s.a. Cavendish Experiment.
@ References: Adelberger a1308-conf [and probes of fundamental physics].

Torsion Invariant of a Manifold > see 3D manifolds.

Torsion Pendulum > see Pendulum.

Torsion Subgroup of an Abelian Group > see group types.

Torsion Tensor > s.a. torsion in physical theories; torsion phenomenology.

Tortoise Coordinate > s.a. schwarzschild geometry [Eddington–Finkelstein coordinates].
$ Def: The coordinate r*:= r + 2GM log(r/2GM − 1) used for Schwarzschild spacetime as part of the Eddington-Finkelstein coordinates, or in the Kruskal Extension.
@ References: Li & Zhao NCB(95) [in a non-static spacetime].

Torus > see differentiable manifolds.

Totalitarian Principle
* Idea: A metaphysical doctrine often associated with Murray Gell-Mann, which states that everything allowed by the laws of nature must actually exist; The principle is closely related to the much older "principle of plenitude".
@ References: Kragh a1907 [history, case studies, and the plenitude principle].

Totalitarian Property of Quantum Theory
* Idea: Any system capable of measuring a quantum observable must itself be quantized.
@ References: Marletto & Vedral a2005 [and exact symmetries].

Totally Disconnected Space > see connectedness.

Totally Geodesic Mapping > s.a. harmonic map [especially, condition on f].
$ Def: A mapping f : MN between metric manifolds, such that all M-geodesics are mapped to N-geodesics.

Totally Geodesic Submanifold
$ Def: A submanifold M of a manifold N such that a geodesic which is tangent to M at a point lies entirely in M.

Totally Vicious Spacetime > see types of spacetimes.

Trace-Class Operator > see operator theory.

Trace Dynamics > s.a. origin of quantum mechanics.
* Idea: A classical-like theory of non-commutating variables proposed as a model by Stephen Adler for the emergence of quantum mechanics; In it, statistical averages give the Schrödinger equation and operator algebra of quantum theory, while Brownian motion corrections give the low-level noise on which phenomenological reduction models are based.
@ References: Adler 04 + JPCS(12); Adler JPCS(14)-a1401 [the future of quantum mechanics].

Trace Formulas > s.a. quantum chaos.
* Idea: Formulas that establish links (dualities) between the energy spectra in the quantum description of a system and the spectrum of actions of periodic orbits in the Newtonian description; The dualities hold for chaotic as well as for integrable systems.
@ General references: Cohen et al AP(98) [and spectrum statistics]; Primack & Smilansky JPA(98), Smilansky JPA(00) [semiclassical]; Sugita AP(01) [and phase space path integrals].
@ Selberg's: Marklof LNP-m.SP/04 [intro].

Tractor Beam > s.a. sound [acoustic tractor beam].
@ References: Gorlach et al PRL(17) [pulling force from a quantum-mechanical matter wave].

Tractor Calculus > see Conformal Geometry.

Trajectories in Classical Mechanics > s.a. orbits of gravitating bodies, in newtonian gravity, of test particles.
* Rem: They can be obtained from the least action variational principle (> s.a. classical mechanics).
@ General references: Holland IJTP(12) [constructing field evolution using a continuum of trajectories].
@ From wave equations: Orefice et al a0705 [Helmholtz-like equations in electromagnetism and quantum mechanics]; > s.a. wave mechanics.

Trajectories in Qantum Mechanics > s.a. classical limit; histories-based quantum theory; pilot-wave interpretation.
* Idea: Trajectory-based approaches include the de Broglie-Bohm and Nelson's stochastic interpretations.
@ General references: Galehouse IJTP(81); Cufaro Petroni & Vigier FP(92) [and interference]; Griffiths PRL(93); Yamada PRA(96) [probabilities]; Holland FP(98), PRA(99); Floyd in(02)qp/00; Hiley et al qp/00; Bouda & Djama PS(02)qp/01; Brun AJP(02)jul [models]; Hall JPA(04) [incompleteness of trajectory-based interpretations]; Moser AP(08); John AP(10)-a1007, a1007-conf [complex trajectories, and dynamical origin of quantum probability]; Olivares et al PhyA(11) [from classical phase space]; Błaszak & Domański PLA(12) [based on Moyal description of quantum theory]; Hiley et al a1610 [Dirac, Moyal and Bohm]; Vink FP(18)-a1711 [trajectories and quantum field theory].
@ And measurement: Wiseman QSO(96)qp/03; Steinberg qp/03-ch; Bauer et al JPA(16)-a1512.
@ Past of a quantum particle: Vaidman PRA(14)-a1312; Tsang a1403 [inferring the past of quantum observables]; Svensson a1407; Englert et al PRA(17)-a1704; Paneru & Cohen IJQI(17)-a1709 [in an entangled state]; Hashmi et al a1710; Bhati & Arvind a1807 [and weak measurement]; > s.a. photons.
@ Quantum states as ensembles of trajectories: Schiff & Poirier JChemP(12); Poirier a1208 [relativistic particles]; Oaknin a1306.
@ Modified quantum theory: Wiseman & Gambetta PRL(08) [none for non-Markovian stochastic Schrödinger equations].
> Related topics: see ergodic theory; experiments in quantum theory; quantum fluctuations [fermions and classical paths].

Transcendental Functions
@ References: Yukalov & Yukalova PLA(07) [method of self-similar factor approximants].

Transcendental Number > see types of numbers.

Transfer-Matrix Method
* Idea: A technique for finding the partition function of certain types of systems in statistical mechanics.
@ References: Sánchez-Soto et al PRP(12) [1D lossless systems]; Pujol et al EJP(14) [1D quantum particle in a stationary potential].
> Online resources: see Wikipedia page.

Transition Amplitude > see path-integral quantum mechanics; vacuum.

Transition Function > see fiber bundles.

Transition Matrix
> see S-Matrix.

Transition Radiation > see radiation.

Transitive Action of a Group on a Manifold > see group action.

Transitive Relation > see Relation.

Transitivity, Topological > see under Topological Transitivity.

* Idea: An element of the translation subgroup of the Euclidean or Lorentz group.
> Translational symmetry: see gauge theories; Homogeneity; Position [tests of position invariance].

Transmission > see quantum systems [potential steps]; wave phenomena.

Transport Phenomena > s.a. chaos [anomalous]; graphs; networks; non-equilibrium statistical mechanics.
* Idea: Non-equilibrium (but possibly steady-state) statistical mechanics processes in which there is a net flow of some quantity inside a system; Examples are heat flow (by conduction), electric conduction, convection (related to viscosity), particle transport or diffusion.
@ Books: Lundstrom & Jeong 13 [near-equilibrium, and application to nano-devices]; Soto 16 [and kinetic theory]; Venerus & Öttinger 18.
@ Methods: Desloge AJP(64)oct, AJP(64)oct [coefficients from Boltzmann equation]; Mackay et al PhyD(84) [in Hamiltonian systems]; Arlotti et al m.AP/06 [semigroups]; Keanini PRS(07) [based on random walk]; Rossani PhyA(07) [electrons and phonons, integral equations for transport coefficients]; Cohen JSP(09) [failure of density expansion of transport coefficients]; Malyshev & Zamyatin a1110 [stochastic models]; Berkolaiko & Kuipers JMP(13)-a1307 [semiclassical treatment, combinatorial theory]; Engibaryan & Barseghyan TMP(14) [general theory, homogenous half-space and plane layer].
@ Turbulent: Bakunin PhyA(05) [weak compressibility, percolation approach].
@ Random medium: Panasyuk et al JPA(06)mp/05 [homogeneous]; Corngold JSP(05) [and diffusion].
@ In other media: Marklof a0909-proc [in crystals, failure of linear Boltzmann equation]; Halperin & Bergman PhyB(10)-a0910-conf [in inhomogeneous and disordered media, contributions by Landauer]; Denicol et al PLB(12) [relativistic fluid, viscous transport coefficients].
@ Quantum: Rau & Müller PRP(96) [irreversible]; Rammer 98 [disordered conductors, localization, ...]; Nazarov & Blanter 09; Buot 09 [in nanosystems]; Mahajan et al a1608 [for interacting quantum systems, based on entanglement structure]; Rams & Zwolak PRL(20)-a1904 [tensor network simulation].
@ Related topics: Suzuki PhyA(11), PhyA(12) [irreversibility and entropy production].
> Related topics: see Boltzmann Equation; Kadanoff-Baym Equations.
> Specific phenomena: see Conductivity; diffusion; Heat Flow; Percolation; viscosity.

Transversality Theory
@ References: in Banyaga & Hurtubise 04.

Transverse Gauge in General Relativity > see gauge.

Transverse Gravity > see unimodular relativity.

Trans-Planckian Fluctuations / Frequencies / Gravity
> In gravity theory: see modifications of general relativity; spin-foam quantum gravity.
> In astrophysics: see black-hole radiation and quantum black holes; quantum gravity phenomenology.
> In cosmology: see early-universe cosmology; cosmological acceleration and perturbations, in modified theories; dark-energy models.
> Other phenomenology: see hadrons [trans-planckian collisions].

Trapped Surface > s.a. event horizons; horizons [apparent horizon].
* Idea: A compact spacelike 2-surface without boundary, such that both families of null geodesics orthogonal to it (incoming and outgoing) converge, for each point on the surface; It is used in the proof of singularity theorems.
@ General references: Clarke CQG(88); Beig & Ó Murchadha PRL(91), CQG(94); Malec & Ó Murchadha PRD(96)gq/95 [cosmological]; Senovilla JHEP(03)ht [not with vanishing curvature scalars], gq/03-conf, m.DG/04-proc [trapped submanifold]; Mitra gq/05 [occurrence]; Krishnan CQG(08)-a0712 [review, isolated-dynamical horizons and applications]; Andersson AIP(09)-a0901; Hayward IJMPD(11)-a0906 [types]; Senovilla IJMPD(11)-a1107-ln [rev and applications]; Carrasco a1201-PhD [in spacetimes with symmetries, and applications to uniqueness theorems].
@ Spherical symmetry: Malec & Ó Murchadha PRD(94)gq; Iriondo et al PRD(96) [asymptotically flat].
@ Marginally trapped surfaces: Carrasco & Mars CQG(09) [stability]; > s.a. hawking radiation.
@ Other special cases: Galloway et al CMP(12)-a1005 [2+1, non-existence result]; Bengtsson a1112-ch [simple examples]; Jakobsson CQG(13)-a1208 [2+1].
@ Criteria for formation: Bizoń et al CQG(89) [spherical symmetry]; Mitra gq/98 [spherical]; Andersson et al PRL(05); Klainerman et al IM(14)-a1302 [fully anisotropic]; Bini & Esposito a1705 [intro].
@ And event horizons: Claudel gq/00; Dafermos CQG(05)gq/04 [spherical, and \(\cal I\)]; Åman et al JPCS(10)-a0912 [and apparent horizons].
@ Related topics: Ellis GRG(03)gq [compact cosmology]; Dain CQG(04)gq/03 [as inner boundary for constraints].
> Examples: see Vaidya Spacetime.

Trapping Horizon > see horizons.

Traveling Waves > see types of waves.

Tree > see graph.

Tree Diagram > see quantum field theory formalism.

* Idea: A relationship among three objects, often vector spaces.
@ References: Smolin IJTP(17)-a1503 [trialities and the foundations of dynamics].

Triangle > see simplex.

Triangle Inequality > see inequalities.

Triangulable Topological Space
$ Def: A topological space X homeomorphic to some polyhedron K.
* Remark: The polyhedron K (which is not unique) is a triangulation of X.
* Example: Every paracompact manifold is triangulable [@ Whitehead AM(40)].

Triangular Number > see number theory.

Triangulation > s.a. simplices [triangles]; tiling [including physics].
$ Def: A simplicial complex that covers (is homeomorphic to) a manifold; A special case of tiling.
* Irreducible: A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface.
* Special cases: The triangulations of the disc with n + 2 vertices on its boundary are counted by the nth Catalan number [@ in Benardi & Rué EJC(12)].
* Pseudo-triangulation: A partition of (the convex hull of a point set S in) the manifold into interior-disjoint pseudo-triangles (whose vertices are points of S).
@ General references: Lackenby G&T(00)m.GT [3D, taut]; Jaco & Rubinstein JDG(03) [3D, 0-efficient]; Effenberger JCTA(11) [any dimensionality, tight triangulations]; Alagic & Bering a1108 [quantum algorithms for invariants].
@ Statistics, enumeration: Aste JPA(98); Dumitrescu et al CG(01) [enumerating paths]; Bespamyatnikh CG(02); Aichholzer et al CG(04) [number of triangulations on a set of points in flat 2D]; King & Pelsmajer DM(10) [size of dominating sets]; Rivasseau EPL(13)-a1303 [number of triangulations of 3D homology spheres]; Adiprasito & Benedetti a1710 [Cheeger-type exponential bound].
@ Computational aspects: Castelli Aleardi et al IJCGA(11) [efficient representations of 2D triangulations].
@ Of spheres: King G&T(01) [of S3 with given link]; Brinkmann & McKay DM(05) [of S2, minimum degree 5]; Benedetti NPB(17)-a1608 [not all 3-balls are Mogami].
@ Of other spaces: Nakamoto & Tsuchiya DM(12) [of the Möbius band].
@ Different triangulations on the same space: Ontaneda Top(02) [space with non-equivalent triangulations]; Pournin & Liebling CG(07) [paths in the flip-graph of triangulations]; Wilson T&A(07) [convergence of cochain algebra under refinement]; Joret & Wood JCTB(10) [irreducible]; Jia et al DM(13) [balanced triangulations].
@ As a dynamical system: Collet & Eckmann JSP(05)mp/04, Eckmann JSP(07) [triangulations of S2]; > s.a. dynamical triangulations.
@ Causal triangulations: Durhuus et al JSP(10) [spectral dimension]; Sisko et al JSP(13) [growth process]; > s.a. dynamical triangulations.
@ Pseudo-triangulations: Aichholzer et al CG(04); Bereg CG(05) [of the plane]; Orden et al DM(07) [and rigid planar graphs].
@ Combinatorial aspects: Bagchi & Datta DM(05); > s.a. cell complex; graphs and graph types.
@ Random triangulations: Guitter AHP(17)-a1511 [distance-dependent two-point function]; > s.a. statistical geometry [and computational].
@ Other types: Datta & Singh JCTA(13) [vertex-transitive tight triangulations of manifolds].
> Types of triangulations: voronoi tiling [Delaunay triangulations]; Whitehead Triangulation.
> Related structures: see connection; distances [on the set of triangulations of M].

Tribology > see friction.


$ Def: A 3-punctured sphere.

* Idea: An object with three types of charges; 2011, For example, in semiconductors, trions consisting of one electron bound to two holes via Coulomb forces have been observed in a variety of systems, most recently in the optical spectra of doped carbon nanotubes.
@ References: Eto a1001; Hohenester & Goldoni Phy(11) [in carbon nanotubes].

Triple Point > see spin models [XXZ model].

Triple Systems > see Three-Body Physics

Triple-Product Rule
$ Def: The fact that, for three variables x, y and z with a functional dependence between them,

(∂z/∂y)x = –(∂x/∂y)z / (∂x/∂z)y .

Triple-Slit Interference > see interference; pilot-wave interpretation.

Triplectic Quantization > see quantization of constrained systems.

Triplet > see hilbert space [rigged].

* Of cosmological perturbations: The connected four-point correlation function; One motivation for studying it is that the trispectrum has weaker dependence on non-linear clustering than the bispectrum, and its measurement may provide useful additional constraints on primordial non-Gaussianity; > see cosmological matter distribution.
@ References: Verde & Heavens ApJ(01)ap [as a Gaussian test].

Trivialization, Local (Of a fiber bundle)
$ Def: Given a fiber bundle with base space B and fiber F, and a covering {Ui} of B, a trivialization is an assignment, for each j, of a map φj: π−1(Uj) → Uj × F, which is a homeomorphism.

Trojan State > see quantum states.

@ References: Livanios JGPS(07) [and spacetime].

Tropical Geometry / Mathematics
@ References: Litvinov a1005 [introduction and applications]; Angelelli a1701 [and the micro-macro correspondence].
> Online resources: see Wikipedia page.

Tropical Limit of Statistical Mechanics
* Idea: The theory one gets in the limit kB → 0.
@ References: Angelelli & Konopelchenko PLA(15)-a1502.

Trouser Spacetime > see models of topology change.

Trouton-Noble Paradox
* Idea: There is a 3D torque T in an inertial frame S in which a thin parallel-plate capacitor is moving, but there is no 3D torque T' in S', the rest frame of the capacitor.
@ References: in Panofsky & Phillips 62; Ivezić FP(07).

Truth > s.a. Explanation; logic [truth values in quantum theory]; physical theories.
* In science: We discover individual facts that are objectively true, but is our entire view of the universe, based on our current scientific theories, true? We can't even assume that we are making substantial progress toward knowing the truth about the universe, because we don't know how far our current theories are from the truth; However, we can measure the extent to which our present theories explain what we can currently examine [@ from Winkler letter PT(17)apr].
* About the universe: Can you arrive at the truth by a method other than science?
@ References: Margenau PhSc(34)oct [flexibility]; Breuer & Springer GRG(09) [in science].

Tsallis Non-Extensive Entropy, Statistics > see critical phenomena; non-extensive statistics.

Tsirelson Bound (a.k.a. Cirel'son Bound) > see types of quantum correlations.

Tsirelson's Problem
* Idea: The question whether all quantum correlation functions between two independent observers represented as commuting observables on a joint Hilbert space can also be expressed using observables defined on a Hilbert space of tensor product form; Tsirelson showed that the distinction is irrelevant if the ambient Hilbert space is finite-dimensional.
@ References: Scholz & Werner a0812 [finite vs infinite dimensionality]; Avis et al IEICE(09)-a0812 [Tsirelson's theorem, for non-specialists]; news sn(20)jan [implications of MIP* = RE].

Tube Formula > see under Weyl Tube Formula.

Tulczyjew Triples > s.a. geometric formulations of quantum theory.
@ References: Grabowska & Grabowski JGM(13)-a1306; Grabowska & Vitagliano JGM(15)-a1406 [in higher-derivative field theories].

Tully-Fisher Relation > s.a. galaxies / cosmological expansion; dark matter; Modified Gravity; MOND and astrophsyics/cosmology.
* Idea: The more massive (and therefore brighter) a spiral galaxy is, the faster it spins.
@ References: Tully & Fisher A&A(77); Giovanelli ap/96-proc; Kannappan et al AJ(02)ap [scatter]; Russell ApJ(04)ap [type dependence]; Cattaneo et al MNRAS(17)-a1706 + Lincoln livesci(17)jul [reproducing the Tully-Fisher relation with dark matter]; O'Brien et al PLB(18)-a1704 [from first principles]; Alexander & Smolin a1804 [and flat rotation curves, from equivalence principle and assumptions].


Turaev-Viro Theory > s.a. 3-manifolds; spin-foam models; topology-change models.
* Idea: 3D Riemannian quantum gravity with Λ > 0; A spin coupling theory, quantum-group generalization of the Ponzano-Regge model, giving a non-perturbative definition of the path integral for 3D gravity; The Turaev-Viro state sum invariant is known to give the transition amplitude for the 3D BF theory with a cosmological constant, related to the deformation parameter \(\hbar\) by \(\hbar = \Lambda^{1/2}\).
@ References: Turaev & Viro Top(92); Ionicioiu gq/96 [3-manifold partition function]; Girelli et al CQG(02)gq/01 [topological invariance]; García-Islas CQG(04)gq, Barrett et al JMP(07)m.QA/04 [observables]; Pranzetti PRD(14)-a1404 [amplitudes from 2+1 lqg].


Turing Machine > s.a. computation.
@ References: Turing PLMS(36); Fouché et al qp/07, Iriyama et al PLA(08) [quantum]; Kewming a2005 [noisy machines].
> Online resources: see Wikipedia page.

Turing's Thesis > see under Church-Turing Thesis.

Tutte Polynomial > see graph invariants.

12j Symbols > see SU(2); spin-foam models.

Twin Paradox > see special-relativistic kinematics [including curved-spacetime version].

Twist of a Vector Field > see vector calculus.

Twisted Geometry > s.a. discrete geometry; semiclassical quantum gravity.
@ References: Charles & Livine CQG(15)-a1501 [generalization to a q-deformed gauge group].

Twisted Light > see photons.


Two-Body Problem > see classical systems; orbits of gravitating bodies [also two-fixed center problem].

2dF Galaxy Redshift Survey > see galaxy distribution.

Two-Point Function > see correlations [correlation function]; green function.

Two-Slit Experiment > see interference.

Tychonoff Space > see types of topological spaces.

Tychonoff Theorem > see compactness.

Type Theory
@ References: Kamareddine et al 04 [overview].
> Online resources: see nLab page; Wikipedia page.

Typicality (in Cosmology) > s.a. multiverse cosmology.
* Idea: The question of how typical or special our place in the universe is; The spatial version is often called Copernican Principle.
@ References: Ćirković & Balbi IJAB(20)-a1907 [assumptions, conceptual issues].

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