Topics, G

g-Factor > see under Gyromagnetic Ratio.

$ Def: A transformation consisting in a rotation by π around the z-axis in isospace, followed by C conjugation.

G2 > s.a. cosmological models in general relativity [G2 cosmologies].
* Idea: A group that has a fibration with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions.
@ References: Cacciatori et al JMP(05)ht [Euler-type parametrization, Haar measure]; Cacciatori JMP(05).

Gabriel Graph
* Idea: Given a set of points {pi} in a manifold, the Gabriel graph defined by those points has an edge between points pi and pj a distance d apart if the ball of radius d/2 centered around the midpoint contains no other pk; It is a subgraph of the edge graph of the Delaunay triangulation.
@ References: Gabriel & Sokal AZ(69); Bertin et al AAP(02) [site percolation]; Bose et al CG(13) [k-Gabriel graphs, properties].
> Online resources: see Wikipedia page; Mauro Cherubini's page; Takasho Ohyama's page.

Gaia Mission
* 2011: An ESA astrometric mission scheduled for launch in late 2012, that will continuously scan the entire sky for 5 years, yielding positional and velocity measurements with the accuracies needed to produce a stereoscopic and kinematic census of about one billion stars up to V = 20 mag throughout our galaxy and beyond, with a precision of about 25 μas at 15 mag.
* 2013: The mission was successfully launched in December 2013.
@ References: Jordi a1105-proc, Brown a1310-proc [overview]; Cacciari in(14)-a1409-proc [mapping stellar populations]; Sozzetti et al IAU(15)-a1508 [rev]; Cacciari et al AN(16)-a1512 [rev]; Gaia Collaboration A&A(16)-a1609.

Galaxies > s.a. distances and distribution; formation and evolution; types of galaxies.

Galerkin Approximation / Method > s.a. variational methods.
* Idea: A class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem.
> Online resources: see Wikipedia page.

Galerkin Duality > see lattice gauge theory.

Galilean Group / Relativity / Transformations > s.a. lorentz group; spacetime models; types of quantum field theories.
* Idea: The coordinate transformations between two inertial reference frames in non-relativistic physics; They include rotations R(φ, θ, ψ), displacements T(a, b, c), and constant velocity transformations V(vx, vy, vz); The non-relativistic limit of the Lorentz transformations.
@ In classical mechanics: Rosen AJP(72)may [Galilean invariance of non-relativity physics]; Shariati & Jafari a1401 [Galilean relativity and special relativity]; > s.a. physics teaching.
@ In quantum mechanics: Dieks FPL(90) [Galilean boosts and Sagnac's phase]; Giulini AP(96); Greenberger PRL(01).

Galileon Field > s.a. Horndeski Theory [generalized Galileon].
* Idea: A scalar field whose field equations in flat spacetime are strictly of second order (they do not contain the undifferentiated or once differentiated field, nor derivatives of order higher than 2); It is motivated by a theory one gets from the decouling limit of DGP braneworld gravity.
* In curved spacetime: If the galileon is assumed to be minimally coupled to the metric, both galileon and metric field equations involve derivatives up to third order; However, there is a unique non-minimal coupling of the galileon to curvature which eliminates all higher derivatives and yields second-order field equations, without any extra propagating degrees of freedom; The resulting theory breaks the generalized "Galilean" invariance of the original model.
* Rem: The name refers to a group of symmetries that looks like a generalization of the Galileo group.
* Rem: A vector version has been proposed, and one could look for higher-spin generalizations; Skyrmion theory could be a group-valued analog.
* Phenomenology: Basically dead after the multimessenger observations of GW170817.
@ General references: Nicolis et al PRD(09)-a0811; Deffayet et al PRD(09)-a0901 [special non-minimal coupling and second-order equations]; Deffayet et al PRD(10) [p-form fields]; Curtright et al a1212 [primer]; Deffayet et al PRD(11); Trodden JPCS(15)-a1503-proc [rev]; Deffayet et al PRD(15)-a1506 [degrees of freedom]; Noller et al PRD(15)-a1506 [extended symmetries]; Klein et al PRD(16)-a1510 [Galilean counterparts of the Levi-Civita connection and Riemann tensor]; Ezquiaga & Zumalacárregui a1710 [constraints from GW170817].
@ Variants, generalizations: Deffayet et al PRD(09)-a0906 [extension to curved backgrounds]; Charmchi et al PRD(16)-a1511 [vector theory, effective action]; Deffayet et al PRD(17)-a1704 [p-form theories, classification]; Petrov a1812 [vector theories].
@ And cosmology: Chow & Khoury PRD(09)-a0905; Silva & Koyama PRD(09)-a0909; De Felice & Tsujikawa PRL(10)-a1007; Appleby & Linder JCAP(12)-a1204.
@ Other phenomenology: Burrage & Seery JCAP(10)-a1005 [and constraints on a fifth force]; Chagoya et al JCAP(14)-a1407 [and strong gravity]; Asvathaman et al MNRAS(16)-a1506 + news pt(17)jan [constraints from strong equivalence principle violation with supermassive black holes].
@ Related topics: Curtright & Fairlie PLB(12)-a1206 [and scalar geons]; Rubakov TMP(16)-a1509 [and Lorentzian wormholes]; > s.a. rainbow gravity.

Gallai's Conjecture > see graph theory.

Galois Algebras > see Galois Theory.

Galois Group

Galois Theory, Field (Extension)
* Idea: A Galois field is a finite field (one that contains a finite number of elements); They are classified by size, there is exactly one finite field up to isomorphism of size pk for each prime p and positive integer k.
@ Galois fields: Lidl & Niederreiter 94, 08; Snaith 03; Wan 11 [Galois rings].
@ Galois theory: Artin 44; Bastida 84; Garling 87; Tignol 01 [theory of algebraic equations]; Bewersdorff 06 [historical]; Weintraub 08 [intro]; in Roman 06; Khovanskii 14 [topological Galois theory].
@ Related subjects: Snaith AMS(94) [Galois modules]; Denecke et al 04 [Galois connections]; Stachowiak PhD(08)-a0810 [differential Galois group theory, integrability and chaos]; Futorny & Ovsienko 12 [Galois algebras]; Mazur BAMS(11) [abelian Galois extensions of basic number fields].
@ Quantum mechanics over a Galois field: Lev ht/02, TMP(04)ht/02, ht/02, ht/02, FFTA(06)ht/03, IJMPB(06)ht; Vourdas JPA(05), AAM(06)qp, JPA(07); Schumacher & Westmoreland a1010-conf; Hanson et al a1104; Chang et al MPLB(13)-a1205 [discrete quantum mechanics]; Hanson et al JPA(14)-a1305 [and computation]; Chang et al IJMPA(14)-a1312-proc [super-quantum correlations]; > s.a. modified quantum mechanics; modified quantum field theories.
> Online resources: see MathWorld page; PlanetMath page; Wikipedia page.

Game Theory

Gamma Distribution > s.a. measure / probability.
$ Def: The distribution Γθ(ν, λ) generalizing the Poisson distribution, defined by the probability density function

f(x) = θ λν/θ xν−1 exp{−λ xθ} / Γ(ν/θ) ,   for x ≥ 0 .

@ References: in Santaló 76.

Gamma Function > s.a. Beta Function; Digamma Function.
$ Def: The function

\[ \Gamma(x):= \int_0^\infty t^{x-1}{\rm e}^{-t}\, {\rm d}t
= \lim_{n\to\infty} {n!\,n^{x-1}\over x\,(x+1)\,(x+2)\dots(x+n-1)}\;. \]

* Properties: Γ(n+1) = n Γ(n), and, for integer argument, Γ(n+1) = n!; For half-integers, use Γ(1/2) = π1/2.
@ Generalized: Jurzak LMP(05).
> Online resources: see MathWorld page; Wikipedia page.

Gamma Matrices > s.a. clifford algebra; spinors in field theory.
$ Def: Four matrices γa, a = 0, 1, 2, 3 (a.k.a. Dirac matrices), defined by {γa, γb} = 2 gab I.
* Form: They generate a representation of the Clifford algebra; In the "real" representation, γ1, γ2, γ3 are real, γ4 = i γ0 pure imaginary, and they are all hermitian; γ5:= γ1 γ2 γ3 γ4 is imaginary hermitian.
* Properties: They satisfy the identities

[γm, γn] = 4 γmnab γ5 − 4 (gma gnbgmb gna) + 2 (−gma [γn, γb] + gmb [γn, γa] + gna [γm, γb] − gnb [γm, γa]) ,

and, if we define γab...d := γ[a γb... γd ],

γabc γmn = 2 gc[m gn]b γa + 2 gm[b gn]a γc + 2 ga[m gn]c γb + 2 gc[m γn]ab + 2 gb[m γn]ca + 2 ga[m γn]bc + γabcmn .

@ References: Delbourgo & Prasad NC(74) [n-dimensional]; Veltman NPB(89); Gran ht/01 [Mathematica package]; Bondarev NPB(06), Izaurieta et al AIP(12)-a1106 [trace calculations]; Kuusela a1905 [Mathematica package].

Gamma Metric > see axisymmetric spacetimes.

Gamma Rays > see gamma-ray astronomy; gamma-ray bursts [GRBs].

Gamow Functional / State / Vector > see resonance.

Gannon's Theorem > see singularities.

Gardner Method > see integrable systems.

Gas > s.a. ideal gas.

* Idea: Term used by cosmologists to denote the messy combination of turbulence, shock waves, magnetism, and nuclear reactions that rules the evolution of ordinary matter, as opposed to the simplicity of dark matter.

Gauge Group / Transformation / Symmetry > s.a. gauge choice or fixing.

Gauge Network > see spin networks.

Gauge Theory > s.a. lattice gauge theory; solutions; types of gauge theories; yang-mills theories.

Gauge Theories of Gravitation

Gauge / Gravity Duality > see approaches to quantum gravity; duality.

Gauge × Gauge = Gravity
@ References: Borsten et al a2104 [in homogeneous spaces].

Gauss' Law (Gauss' Flux Theorem) > s.a. electromagnetic field equations; lattice gauge theories; solutions of gauge theories.
* Idea: A result for the electrostatic field and the Newtonian gravitational field stating that the flux through any closed surface is proportional to the net enclosed charge (or mass); It does not hold in general for other gravitational theories; The differential form states that the divergence of the electric field is the charge density.
@ References: Donohoe AJP(08)oct [for the electric field]; Balachandran & Reyes-Lega a1807 [role in gauge theories].

Gauss Map
* Idea: The chaotic "return" map on the interval I = [0,1], defined by v \(\mapsto\) 1/v − [1/v]; It appears in mixmaster model dynamics.
@ References: in Barrow PRP(82); in Motter PRL(03)gq.

Gauss' Theorem (Gauss' Divergence Theorem) > s.a. integration on manifolds; vector calculus.
* Idea: An identity in vector calculus relating the flux of a vector field through a closed surface and the integral of its divergence over the enclosed volume.

Gauss-Bonnet Gravity > s.a. gauss-bonnet theorem; holography.
* Idea: A higher-order gravity theory in which the action includes a function of the Gauss-Bonnet combination of quadratic curvature terms; It arises in an expansion of the effective gravitational action in superstring theory, and is used to study curvature corrections to the Einstein action in supersymmetric string theories, while avoiding ghosts and keeping second-order field equations.
* Rem: A pure Gauss-Bonnet term in the action would not modify the classical field equations, because it is a topological invariant, but that term multiplied by a scalar-field dependent factor, for example, does make a difference even classically.
* Rem: It features possible superluminal propagation of gravitons due to the non-canonical kinetic terms in the action.
@ General references: Neupane & Dadhich CQG(09)-a0808 [as incorporating features of quantum gravity in classical theory]; Montelongo et al PRD(11)-a1011, JPCS(11)-a1012 [f(G) theories and energy conditions]; Schmidt PRD(11) [G ln(G) Lagrangians]; Izumi PRD(14)-a1406 [causal structure]; Baker & Kuzmin IJMPD(19)-a1811 [linearized, and classical electrodynamics]; Glavan & Lin PRL(20)-a1905, comment Gürses et al EPJC(20)-a2004 [Einstein-Gauss-Bonnet theory in the D = 4 limit, bypassing Lovelock's theorem].
@ Solutions and phenomenology: Dotti & Gleiser CQG(05), PRD(05)gq [tensor perturbations and stability]; Gleiser & Dotti PRD(05) [vector and scalar perturbations]; Kobayashi GRG(05)gq [Vaidya-type solution]; Dadhich ht/06-proc; Gürses GRG(08) [with scalar field, solutions]; Dotti et al PRD(10)-a1004 [static solutions]; Quirós & Teste a1005 [stability]; Bajardi et al a1911 [spherically symmetric]; > s.a. black holes and thermodynamics; wormhole solutions.
@ Solar-system tests: Sotiriou & Barausse PRD(07)gq/06 [+ scalar field, post-newtonian]; Davis a0709 [f(G) theories].
@ Gravitational collapse: Maeda CQG(06)gq/05 [effect on collapse]; Taves et al CQG(12)-a1110 [D-dimensional scalar field, Hamiltonian].
@ Bounces: Bamba et al PLB(14)-a1404; Oikonomou PRD(15)-a1509.
@ And dark energy: Koivisto & Mota PLB(06)ap; Sanyal PLB(07)ap/06; Amendola et al JCAP(07)ap, Davis AIP(07)-a0708, a0709 [solar-system tests].
@ With cosmological constant: Torii & Maeda PRD(05)ht [neutral static solutions], PRD(05)ht [charged static solutions]: > s.a. de sitter space.
@ Other cosmology: Kanti et al PRD(99) [+ scalar field, singularity-free]; Neupane & Carter JCAP(06)ht/05; Leith & Neupane JCAP(07)ht; Li et al PRD(07) [modified]; Chingangbam et al PLB(08)-a0711 [viability]; Neupane proc(08)-a0711 [constraints]; Andrew et al GRG(07)-a0708; De Felice et al PRD(10)-a0911 [small-scale matter instability]; Myrzakulov et al GRG(11)-a1009; Farajollahi & Salehi ApSS(12)-a1111 [generalized second law]; García-Aspeitia & Hernández-Almada a2007 [constraints from cosmological observations]; > s.a. cosmological acceleration; friedmann equation.
@ Quantum: Boernsen et al a0709 [dimensional regularization]; Niu & Pak a0709 [coupled to torsion]; Charmousis & Padilla JHEP(08)-a0807 [vacuum instability]; Haro et al PRD(15)-a1506 [Gauss-Bonnet extension of lqc]; Cheung & Remmen PRL(17)-a1608 [positivity of coefficient in quantum correction to Einstein gravity]; Narain a2101 [Lorentzian path integral].
@ Other related theories: Julié & Berti PRD(20)-a2004 [Einstein-scalar-Gauss-Bonnet theory, d+1 formalism]; Arrechea et al ChPC-a2004 [4D Einstein-Gauss-Bonnet gravity, inconsistencies]; > s.a. brans-dicke theory; higher-order gravity; teleparallel theories.

Gauss-Bonnet Operator
* The operator D:= d + δ.
@ References: Anné & Torki-Hamza AMP(14)-a1301 [on infinite graphs].

Gauss-Bonnet Theorem (a.k.a. Gauss-Bonnet-Chern Theorem)

Gauss-Codazzi Equations (note that the spelling is not "Gauss-Codacci") > s.a. dirac fields.
$ Def: Equations relating the curvature of a manifold to that of a (d−1)-dimensional submanifold embedded in it,

(d−1)Rabcd = Rabcd + ε (Kad KbcKac Kbd) ,       (d−1)Rabcd nd = Db KacDa Kbc .   [need to check these!]

* Applications: Constraint equations in the initial-value formulation of general relativity; Matching conditions for the metric across a hypersurface; Brane world (> see branes).
@ General references: Codazzi AdM(1869); in Eisenhart 26; in Schouten 54.
@ Generalizations: Gemelli JGP(02) [for null surfaces]; Bertrand et al a1412 [fermionic supersymmetric extension].
> Online resources: see Wikipedia page.

Gaussian Curvature > see riemann tensor.

Gaussian Functions [including Gaussian integrals]

Gaussian Integers > see numbers.

Gaussian Normal Coordinates > see coordinates.

Gaussian States > see under SqueezedStates.

> And cosmological observations: see cmb anisotropy and features; cosmological perturbations and phenomenology; Trispectrum.
> From cosmological theories: see cosmological tests of general relativity; brane-world cosmology.
> For quantum states: see semiclassical quantum states; states in quantum field theory.

Gay-Lussac Law > s.a. gas [ideal-gas law].
* Combining volumes: In a chemical reaction between gases and at constant temperature and pressure, the ratio between the volumes of the reactant gases and the products can be expressed in simple whole numbers.
* Pressure-temperature law: The pressure of a gas of fixed mass and volume is directly proportional to the gas's absolute temperature.
@ References: Holbrow & Amato AJP(11)jan [history and details].
> Online resources: see Wikipedia page.

Gedankenexperiment > s.a. Einstein Boxes; tests of quantum mechanics.
@ References: Sorensen AS(91); Schlesinger FP(96); Gendler BJPS(98) [Galileo]; Bishop PhSc(99)dec [not valid arguments]; Cucić a0812; Simsek a2003 [special relativity and the Doppler effect].

Gegenbauer Polynomials
@ Generalized: De Bie & Sommen JPA(07)-a0707 [in superspace].

Gegenbauer Transform
$ Def: A map between functions, depending on r > −1/2 and n ∈ \(\mathbb Z\); If Cnr are the Gegenbauer polynomials,

F \(\mapsto\) T{F(t)} ≡ fnr := −1+1 (1−t2)r−1/2 Cnr(t) F(t) dt .

* Inversion formula: For −1 < t <1,

\[ F(t) = \sum_{n=0}^\infty {n!\,(n+\rho)\,\Gamma^2(\rho)\,2^{2\rho-1}\over\pi\,\Gamma(n+2\rho)}\, C_n^\rho(t)\,f_n^\rho\;.\]

* Property: It reduces the differentiation R[F(t)]:= (1−t2) F'' − (2r−1)t F'' to T{R[F(t)]} = −n (n+2r) fnr.

Gel'fand Transform or Representation
$ Def: The map x: a \(\mapsto\) x(a) = â(x) from a commutative Banach algebra A to the functions on the space X of maximal ideals of A; There is a 1–1 correspondence between X and {homomorphisms: A → \(\mathbb C\)}.
* Relationships: If A is the group algebra of a locally compact Abelian group, the Gel'fand transform coincides with the Fourier transform.
* Applications: Used to prove Wiener's Theorem.
@ References: Gel'fand MathSB(41).

Gel'fand Triplet > see hilbert space [rigged].

Gel'fand-Kolmogorov, Gel'fand-Naimark Theorem > see manifolds.

Gell-Mann Matrices
* Idea: The 8 matrices that form a possible basis for the defining representation of the Lie algebra su(3).
@ References: Gell-Mann PR(62).

Gell-Mann-Low Function / Theorem > s.a. Adiabatic Approximation; Beta Function.
@ References: Molinari JMP(07)mp/06 [new proof of theorem].

General-Boundary Formulation of Field Theory > see boundaries in field theory; approaches to quantum field theory.

General Covariance > see covariance.

General Relativity > s.a. 3D general relativity; action; canonical formulation; other formulations; modifications; tests.

Generalized Functions > see under distributions.

Generalized Uncertainty Principle > see deformed quantum uncertainty relations.

Generating Function > s.a. legendre polynomials.
* Idea: A function that allows the determination of a sequence of quantities as coefficients in a series expansion; A bridge between discrete mathematics and continuous analysis.
* Enumeration: A representation of a Counting Function as an element of some algebra.
@ In combinatorics: in Comtet 74; Wilf 06; Poinsot et al JNSA(10)-a0910 [exponential formula].

Generation > see standard model of particle physics; beyond the standard model.

Generator of an R-Module > see module.

Generic Property
* Idea: A property defined for elements of a topological space holds generically if it holds on an open dense subset of that space.
@ References: Barrow a1503-ch [the issue of generality in cosmology]; Saraykar a1612 [for spacetime properties].

Generic Spacetime > see types of spacetimes.

Genus of a 2-Surface > see 2D manifold.

Geocentric Model > see history of astronomy.

Geodesics > s.a. types of geodesics.

Geodesic Completeness > see differential geometry.

Geodetic Effect / Precession
* Idea: The precession of the spin axis of an orbiting gyroscope around a gravitating mass; a.k.a. geodesic precession; Tested by Gravity Probe B.
@ Self-force corrections: Akcay PRD(17)-a1705, Bini et al PRD(18)-a1809 [Kerr spacetime].
> Online resources: see Wikipedia page.

Geodetic Set, Number > see graph theory.

* Idea: A surface of constant gravitational potential.
@ References: Oltean et al CQG(16)-a1510 [geoid quasilocal frames].

Geometric Algebra > s.a. clifford algebra; special relativity.
* Idea: Clifford algebra over the field of real numbers.
@ General references: Doran & Lasenby 03; Hestenes AJP(03)feb, AJP(03)jul; Henselder et al AP(05)mp/04; Chappell et al a1101 [introduction]; Chisolm a1205 [intro]; Sobczyk 13 [and approach to elementary and advanced mathematics].
@ Geometric calculus: Alho AACA(17)-a1509-conf [evaluating integrals without introducing coordinates].
@ And physical systems: Doran et al AIEP(96)qp/05 [spacetime algebra and electron physics]; Sobczyk a1507 [and Dirac spinors].
> Online resources: see Wikipedia page; Cambridge University page.

Geometric Flow > s.a. Ricci Flow.
@ References: Petropoulos FdP(10)-a1011 [applications in string theory and gravity].

Geometric Number Theory > see number theory.

Geometric Optics Approximation
* Idea: An approach to optics that describes light propagation in terms of "rays".
* Basic concepts: Light propagates along rectilinear paths in homogeneous media, curved paths in media in which the refractive index changes, and has a discontinuous change in the direction of propagation at the interface between two media with different indices of refraction; It may be absorbed or reflected.
* Applications: Optical instruments, whose imaging properties (magnification, optical aberrations, ...) are studied using ray tracing.
@ References: Corrente a1110-thesis.
> Gravity-related aspects: see gravitational phenomenology.
> Online resources: see Wikipedia page.
blue bullet Related concepts: see refraction; wave phenomena.
blue bullet Related areas: see electromagnetism; light; optics.

Geometric Phase

Geometric Quantization

Geometric Series > see series.

Geometric Topology > s.a. combinatorics.
@ References: Moise 77 [2D and 3D].

Geometrically Independent Points > see affine structures.

Geometrization Conjecture
@ References: McMullen BAMS(11) [overview, and approaches to its proof].

Geometrization of a Matter Field > see solutions of einstein's equation with matter.

Geometrodynamics > see canonical formulation of general relativity [classical]; gravitating matter; quantum geometrodynamics.

Geometrothermodynamics > s.a. thermodynamics [geometry of state space].
* Idea: An approach to the differential geometry of the thermodynamic state space of a system.
@ General references: Quevedo & Vázquez AIP(08)-a0812; Quevedo & Quevedo a1111 [basic elements]; Pineda et al a1704 [physical significance].
@ And cosmology: Luongo & Quevedo a1302-MG13; Bravetti & Luongo a1306, IJGMP(14) [universal acceleration]; Quevedo & Quevedo G&C(14); Gruber & Quevedo JCAP(17)-a1611 [model including the main features of inflation].
@ Other applications: Quevedo & Tapias a1301 [chemical reactions]; Bravetti et al JMP(13)-a1302 [conformal metric structure].
@ Examples: Gutiérrez-Piñeres et al AHEP(13)-a1303 [systems with constant thermodynamic curvature].
> Gravity-related systems: see black-hole thermodynamics; hořava-lifshitz gravity phenomenology.

Geometry > s.a. 2D; 3D; 4D; euclidean, lorentzian, riemannian geometry and geometry of the universe.


Gepner Model
@ References: Naka & Nozaki JHEP(00)ht [boundary states].

Gerbe > s.a. bundle; holonomy.
* Idea: Gerbes were introduced by Jean Giraud in 1971 as a tool for non-commutative cohomology in degree 2 and can be seen as a generalization of principal bundles to the setting of 2-categories; Gerbes or sheaves of groupoids provide a geometric realisation of three-dimensional integral cohomology through their Dixmier-Douady class.
$ Def: A gerbe on a topological space X is a stack G of groupoids over X which is locally non-empty and transitive.
@ General references: Lupercio & Uribe m.AT/01, CM-m.AT/01 [over orbifolds]; Vacaru mp/05 [non-holonomic].
@ Differential geometry: Breen & Messing m.AG/01; Laurent-Gengoux et al AiM(09)m.DG/05 [non-abelian].
@ And physics: Larsson mp/02 [p-form gauge theory]; Isidro IJGMP(06)ht/05 [over phase space, and uncertainty principle]; Mickelsson emp(06)mp [quantum field theory]; > s.a. lattice gauge theory.
> Online resources: see Wikipedia page.

Germ of an Object in a Topological Space
$ For a function at a point: Given a point x in a manifold X, it is the equivalence class of functions on X, any two of which coincide on a neighborhood of x (they are said to have the same germ at x).
@ References: Euh et al DG&A(13)-a1301 [transplanting geometrical structures].
> Online resources: see Wikipedia page.

Geroch Group > see solution methods for Einstein's equation.

Gerstenhaber Structures > see symplectic structures.

Ghost Fields in Field Theory > s.a. path-integral quantization of gauge theory and general relativity; types of quantum field theories [higher-derivative].
* Idea: In general, fields that have no real physical meaning, of which there are different kinds; Faddeev-Popov ghosts are fictitious fields that were introduced in the construction of a manifestly Lorentz-covariant quantization of the Yang-Mills field, for example in electrodynamics, to get rid of unphysical degrees of freedom; Others are states of a quantum field theory with negative norm, which classically correspond to instabilities in interacting theories with higher derivatives, and theories that have them are considered physically unacceptable.
* In gravity theories: In order to ensure renormalizability for theories with matter fields coupled to gravity, the gravitational action should include fourth derivative terms; These introduce ghosts, which violate stability both at the classical and quantum level; Several approaches have been proposed to overcome this problem.
@ General references: Smilga NPB(05) [benign and malicious]; Slavnov JHEP(08)-a0807 [Yang-Mills theory with gauge-invariant ghost field Lagrangian]; Garriga & Vilenkin JCAP(13)-a1202 [in Lorentz-invariant theories]; Tóth IJMPA(14)-a1309 [arbitrary-spin generalizations of the Faddeev-Popov ghost field]; Sbisà EJP(15)-a1406 [rev, classical and quantum]; Canarutto RPMP(16)-a1504 [geometry].
@ Ghosts in gravity theories: Krause & Ng IJMPA(06)ht/04 [ghost modification of general relativity and cosmology]; Biswas et al PRL(12)-a1110 [ghost-free theories]; Koivisto & Tamanini PRD(13)-a1304; Shapiro et al MPLA(14)-a1410 [instabilities for gravitational waves on classical backgrounds]; Salles & Shapiro Univ(18)-a1808 [higher-derivative quantum gravity]; > s.a. Gauss-Bonnet Gravity; theories of gravitation.
@ Faddeev-Popov ghosts: Faddeev IJMPA(10); Eichhorn PRD(13)-a1301 [in quantum gravity beyond perturbation theory]; > s.a. Faddeev-Popov Procedure.
> Related topics: see graviton; massive gravity [Boulware-Deser ghost]; Pais-Uhlenbeck Model; Pauli-Fierz Theory.

Ghost Fields in Quantum Mechanics
@ References: Wódkiewicz CP(95) [quantum correlations and locality].

GHP Formalism > see spin coefficients.

GHZ Experiment / Theorem > see experiments in quantum mechanics.

Gibbons-Hawking Effect
@ References: Fedichev & Fischer PRL(03)cm [1+1 de Sitter acoustic analog].

Gibbs Free Energy > see Free Energy.

Gibbs Ensemble / Measure / State > see Canonical Ensemble.

Gibbs Paradox > s.a. particle statistics; quantum entropy.
* Idea: The fact that in classical statistical mechanics, if we do not take into account the correct Boltzmann counting factor for identical particles, the entropy increases when we take away the separation between two parts of a box containing the same gas at the same density and T.
@ General references: Notes from PHY 731, p10b; Pešić AJP(91)nov [and quantum mechanics]; Peters JSP(10); Versteegh & Dieks AJP(11)jul, comment Corti AJP(12)feb [and particle distinguishability]; Dong et al a1201 [with few particles]; Ihnatovych a1305, a1306 [logical foundations, and classical thermodynamics]; Dieks FP(14)-a1405 [and quantum physics]; Saunders Ent(18)-a1808 [conceptual]; Xu a2105 [and information entropy].
@ Criticism of conventional argument: Dieks & van Dijk AJP(88)may [and quantum mechanics]; Swendsen JSP(02), Nagle JSP(04) [counterargument], response Swendsen JSP(04); Allahverdyan & Nieuwenhuizen PRE(06)qp/05; Dieks a1003-proc [classical particles are always distinguishable, and quantum ones can be]; Ainsworth PhSc(12); Etkin a1312 [proposed thermodynamic resolution]; Peters EJP(14); Unnikrishnan IJQI(16)-a1811 [physical criteria for indistinguishability].
@ Theories and topics: Kiefer & Kolland GRG(08)-a0707 [for black-hole entropy].
> Online resources: see Shu-Kun Lin's list of references.

Gibbs Theorem
* Idea: The entropy of the mixture of ideal gases is equal to the sum of the entropies of the components of the mixture.
@ References: Ihnatovych a1804-proc [on Gibbs' proof].

Gibbs-Duhem Relation
* Idea: A relationship between thermodynamic quantities for a homogeneous system, which follows from the fact that the entropy must be a first-order homogeneous function, E = TSPV + ∑i μi Ni; Other relations, such as G = ∑i μi Ni, can be obtained using the relationships between different thermodynamic potentials; There is also a differential version, which can be obtained from the other one using the first law of thermodynamics, S dTV dP + ∑i Ni dμi = 0.
> Online resources: see Wikipedia page.

GINGER Experiment (Gyroscopes IN General Relativity)
@ References: Capozziello et al EPJP-a2103 [and tests of gravity theories].

Ginzburg-Landau Equation
* Complex Ginzburg-Landau equation: An equation describing generically the dynamics of oscillating, spatially extended systems close to the onset of oscillations; It may be the most celebrated non-linear equation in physics.
@ References: García-Morales & Krischer CP(12) [intro].

Ginzburg-Landau Model > see under Landau-Ginzburg model.

Girth > see graph invariants.

Gisin's Theorem > see bell inequalities.

Glass > s.a. solid matter [including glass transition] / Disordered Systems; fluctuations [FD theorem for glassy systems]; spin models; Topological Glass.

Glauber Dynamics
* Idea: A method for sampling a given probability distribution via a Markov chain.
@ References: Martinelli LPTS(99) [for discrete spin models].

Gleason's Theorem > s.a. experiments in quantum mechanics; foundations of quantum mechanics.
* Idea: Any quantum state is given by a density operator.
$ Def: If \(\cal H\) is a (real or complex) Hilbert space of dimension greater than 2 and μ a probability measure on the subspace lattice \(L({\cal H})\), then there exists a density operator W on \(\cal H\) such that for all \(E \in L({\cal H})\), μ(E) = tr(WE).
@ References: Gleason JMM(57); Drisch IJTP(79) [without positivity and separability conditions]; Busch PRL(03)qp/99 [simple proof]; Buhagiar et al FP(09) [consequences]; Rajan & Visser a1904 [explicit construction of the density matrix].
@ Generalizations: Edalat IJTP(04) [extension for quantum computation]; Barnett et al NJP(14); Moretti & Oppio AHP(18)-a1803 [in quaternionic Hilbert spaces].
> Online resources: see Wikipedia page.

$ Def: (Souriau) An element of a recueil R acting on a space E.

Global Dimension of a Ring > see dimension.

Global Hyperbolicity > in causality conditions.

Global Positioning System > see under GPS.

Glueballs > s.a. QCD phenomenology.
* Idea: Bound states of gluons that can be color singlets.
* 1991: Not confirmed yet; Probably m > mprot, but hard to recognize.
* 1995: Not confirmed yet; Evidence from lattice calculations that the lightest one is fJ (1710 MeV).
* 2005: First analytical results for the glueball spectrum in the 3D case by R Leigh et al (> see quantum gauge theories).
* 2015: Report of discovery in candidate f0(1710 MeV).
@ General references: Ishikawa SA(82)nov; Sexton et al PRL(95) [numerical evidence]; Close CP(97), SA(98)nov; Niemi ht/03-in [as twisted closed strings]; Kondo et al JPA(06) [mass from topological knot soliton in Faddeev model]; Vandersickel PhD(11)-a1104 [from propagators, Gribov-Zwanziger framework]; Brünner & Rebhan PRL(15) + news scial(15)oct [report of discovery].
@ Spectrum: Morningstar & Peardon PRD(99); Bugg et al PLB(00); Frasca a0704 [strong coupling and lattice calculations].
> Online resources: see Wikipedia page.

Gluino > see particle types.

Gluons > see QCD.

GNS Construction > s.a. observable algebras.

Gödel Solution

Gödel's (Second Incompleteness) Theorem > see logic.

Goldbach Conjecture > see conjectures.

Goldberg-Sachs Theorem > s.a. petrov classification.
* Idea: A result about the vacuum Einstein equation, which relates algebraic properties of the Weyl tensor with the existence of a null, geodesic, shear-free congruence in spacetime; It is very useful in constructing algebraically special exact solutions.
@ References: Goldberg & Sachs APP(62), re GRG(09); Apostolov JGP(98) [4D pseudo-Riemannian]; Dain & Moreschi JMP(00)gq/02 [linearized]; Batista GRG(13)-a1205 [any 4D manifold with a torsion-free connection compatible with the metric]; Van den Bergh a1605 [and the alignment condition for Einstein-Maxwell fields].
@ 3D: Nurowski & Taghavi-Chabert CQG(15)-a1502.
@ Higher-dimensional: Durkee & Reall CQG(09)-a0908; Taghavi-Chabert CQG(11)-a1011 [5D, in terms of optical (or Robinson) structures], JGP(12) [complex]; Ortaggio et al CQG(12)-a1205 [5D], CQG(13)-a1211 [D > 5, non-twisting case]; Batista & Carneiro da Cunha JMP(13)-a1212 [6D]; Batista a1311-PhD [all dimensionalities and signatures].

Golden Mean / Ratio
* Idea: In a golden rectangle, the ratio (longer side)/(shorter side) = (sum of sides)/(longer side).
* History: In art, it has generally been considered to be the most pleasing to the eye, and has been used in works from the Pyramids to paintings by Rembrandt; Present in the shapes of hurricanes, spiral galaxies, and some biological structures such as the chambered nautilus, it describes a logarithmic spiral.
$ Def: The ratio x = a/b such that (a+b)/a = a/b, or x = (x+1)/x; Alternatively, lim an+1 / an as n → ∞, where {an} is the Fibonacci sequence; Equal to 1.618...
@ References: Dunlap 97 [and Fibonacci numbers]; Livio 02; Kak Foarm(06)phy/04 [physics of aesthetics]; Moorman & Goff EJP(07) [in a coupled-oscillator problem]; Posamentier & Lehmann 11; Cruz et al EPJC(17)-a1701 [null geodesics in Schwarzschild-Kottler black holes].

(Fermi's) Golden Rule
@ References: Dragoman PLA(00) [in phase space]; Debierre & Lassalle a2008 [onset time].

Goldman Bracket
@ References: Nelson & Picken ATMP(05)mp/04 [quantum deformed version], JPA(08)-a0711, a0903-conf [and 3D quantum gravity]; Chowdhury JHEP-a1310 [derivation].

Goldstone Boson / Theorem > see symmetry breaking.

Gonihedric Action / System > see path integrals.

Good Cut Equation > see geodesics [null geodesic congruences].

* Idea: A number equal to 10100, a 1 followed by 100 zeros.

* Idea: A number equal to 10^{10100}, a 1 followed by a googol of zeros, and the second largest number with a name.
> Online resources: see; Wikipedia page.

* Idea: A number equal to 10^{10^{10100}}, a 1 followed by a googolplex of zeros, and the largest number with a name.
> Online resources: see

Goos-Hänchen Effect
* Idea: A spatial shift along an interface resulting from an interference effect that occurs for total internal reflection; The phenomenon was suggested by Sir Isaac Newton, but it was not until 1947 that it was experimentally observed by Goos and Hänchen.
@ References: de Haan et al PRL(10) [for neutrons].

Gordon Ansatz > s.a. massive gravity; bimetric gravity.
* Idea: An Ansatz for the form of the spacetime metric which is general enough to include almost all spacetimes commonly considered to be physically interesting, and restricted enough to greatly simplify calculations.

Gouy Phase
@ References: da Paz et al NJP(11)-a1012 [for matter waves].

Gowdy Spacetime

GPS (Global Positioning System) > see Positioning Systems; coordinates.

Graded Geometry
* Idea: The theory of \(\mathbb Z\)-graded manifolds.
@ References: Qiu & Zabzine ArchM-a1105 [Batalin-Vilkovisky formalism, intro]; Salnikov JGP(14)-a1411 [in gauge theories]; Fairon EJM-a1512 [intro].

Gradient > see vector calculus.

GRAIL Mission
* Idea: (Gravity Recovery and Interior Laboratory) A 2011-2012 NASA mission to fly twin spacecraft in tandem orbits around the moon for several months and measure its gravity field in unprecedented detail.
@ References: Turyshev et al PRD(13)-a1212 [general relativistic observables].
> Online resources: see NASA site; Wikipedia page.

Gram-Schmidt Orthogonalization Procedure > see Orthogonalization.

Grand Canonical Ensemble > see states in statistical mechanics; quantum statistical mechanics.

Grand Potential
* Idea: A thermodynamic potential, defined as Ω:= UTSμN.
> Online resources: see Wikipedia page.

Grand Unified Theories

Grandfather Paradox
* Idea: A paradox that arises in causality-violating spacetimes Specific form: If you went back in time to a period before your parents were conceived and, while there, killed your grandfather before he had a chance to father your parent, logically this would result in your never having been born, which means that you could never have existed to go back in time and kill your grandfather, which means that you were born and thus could go back in time and kill your grandfather, which means…; General form: Backward time travel will necessarily interfere with the future path of the thing which travelled, making time travel inconsistent.
@ References: news geek(14)sep [resolution using quantum theory, including talk by Seth Lloyd].

Granular Materials > see gas; metamaterials.

Granularity of Spacetime > see discrete geometries; Discrete Models in Physics.

Graph Theory > s.a. graphs in physics; graph invariants, and types and embeddings.

Graph-like Space
@ References: Richter DM(11) [introduction].

Graphene > see carbon.

Graphon > see graph theory / phase transitions [in combinatorial systems]

Grassmann Structures

Grassmannian > s.a. Amplituhedron.
* Positive Grassmannian: A region in an N-dimensional space bounded by intersecting planes (it generalizes the interior of a triangle)

Gravastar > s.a. born-infeld theory.
* Idea: One of a very small number of serious challenges to our usual conception of a black hole; In the gravastar picture there is effectively a phase transition at/near where the event horizon would have been expected to form; The interior of what would have been the black hole is replaced by a segment of de Sitter space, separated from the exterior by a shell of small, but finite proper thickness of exotic fluid.
* Properties: They have no event horizons; They are thermodynamically stable (Mazur & Mottola), and some are dynamically stable (Visser & Wiltshire); They are stable under non-radial axial perturbations (Rezzolla & Chirenti).
@ References: Mazur & Mottola gq/01, PNAS(04)gq; Visser & Wiltshire CQG(04)gq/03 [dynamical stability]; Cattoën et al CQG(05)gq [anisotropic pressures]; Carter CQG(05) [stable solutions]; DeBenedictis et al CQG(06) [solutions]; Lobo & Arellano CQG(07) [and non-linear electromagnetism]; Horvat et al CQG(09) [electrically charged]; Rahaman et al PLB(12)-a1108, PLB(12)-a1205 [2+1 dimensional]; Rutkowski a2009 [rotating gravastar not a source for the Kerr metric].
@ Phenomenology: Broderick & Narayan CQG(07)gq [observational constraints]; Chirenti & Rezzolla CQG(07)-a0706 [stability and distinguishability from black holes]; Rocha et al JCAP(08)-a0803 [formation from collapse]; Harko et al CQG(09)-a0905 [accretion disks]; Pani et al PRD(09)-a0909, PRD(10)-a1001 [gravitational-wave signatures]; Antoniou a2010 [the GW190521 case].
@ Related topics: Lobo & Garattini JHEP(13)-a1004 [in non-commutative geometry, and stability].

* Idea: A device used to measure the gravitational field (of the Earth).
@ References: Debs et al PRA(11)-a1011 [with a Bose-Einstein condensate]; Poli et al PRL(11) [cold atoms in an optical lattice].
> Online resources: see Wikipedia page.

Graviscalar > s.a. scalar-tensor gravity.
* Idea: Scalar components of the gravitational field, e.g., graviscalar Kaluza-Klein excitations.

Gravitation > s.a. theories of gravitation; 2D, 3D, and higher-order theories.

Gravitational Bag > s.a. geons.
* Idea and history: A stable structure that accounts for the existence of particles in a classical field theory of gravitation and other interactions, due to equilibrium between forces; Einstein attempted a realization in 1919, even changing the field equations, but failed; The modern version of the concept has become that of geon.
@ References: Einstein SPAW(19); Davidson & Guendelman PRD(86).

Gravitational Collapse > s.a. critical collapse.

Gravitational Constant

Gravitational Lensing

Gravitational Memory > s.a. black-hole binaries; gravitational-wave propagation; non-local field theories / Memory Effects.
* Idea: A memory effect represented by a net change in the relative positions of test particles after the passage of a gravitational wave.
* Types: The original effect was seen in the weak-field, slow-motion approximation and has been called "linear", as opposed to a "non-linear" effect discovered later, related to the energy carried away in gravitational radiation; More recently it was realized that electromagnetic waves and neutrinos can contribute to the latter as well, and it has been renamed the "null" gravitational memory effect, as opposed to the "ordinary" one.
* Nature: It is understood at future null infinity as a transition induced by null radiation from a Poincaré vacuum to another vacuum, related by a supertranslation; It is a hereditary effect, in which the value of the field at a point depends on the whole history of the system generating it.
* Visualization: If a gravitational wave is visualized using a circle of test particles that is deformed into an oscillating ellipse shape, the effect of the gravitational memory would be to leave the circle in a non-circular configuration after it has passed.
* Christodoulou effect: The non-linear part of the gravitational memory, due to the fact that a gravitational wave produces itself new gravitational waves.
* Electromagnetic version: An effect represented by a momentum kick on test charges after passage of an electromagnetic wave; 2014, It appears to have been overlooked and never directly measured.
* Rem: The supertranslation freedom is related to the E mode component of the gravitational memory.
* And detection: There can be 'orphan memories' in a gravitational-wave detector, from signals at higher frequencies that are not detected.
@ General references: Winicour CQG(14)-a1407 & CQG+(14) [global aspects, gravitational and electromagnetic]; Bieri et al a1505-in; Hollands et al CQG(17)-a1612 [and supertranslations, in four and higher dimensions]; Pate et al JHEP(18)-a1712 [in higher dimensions]; Steinbauer CQG(19)-a1811 [in impulsive plane waves]; Bart JHEP(20)-a1908 [method applicable in the bulk]; Mao a2008 [infinite tower of gravitational memories].
@ Christodoulou effect: Christodoulou PRL(91); Favata ApJ(09)-a0902 [from binary-black-hole mergers]; Bieri & Garfinkle a1308 [and neutrino radiation]; Bieri & Garfinkle PRD(14)-a1312 [perturbative, gauge-invariant treatment]; Garfinkle & Rácz GRG(15)-a1406 [resolving a paradox]; Tolish et al PRD(14) [simple example].
@ Experimentally: Aggarwal et al ApJ(20)-a1911 [limits from NANOGrav]; Boersma et al PRD(20)-a2002 [with Advanced LIGO and Virgo].
@ In cosmological spacetimes: Bieri et al PRD(16)-a1509 [de Sitter spacetime]; Tolish & Wald PRD(16)-a1606, Chu CQG(17)-a1611 [spatially flat FLRW cosmology]; Bieri et al CQG(17)-a1706 [\(\Lambda\)CDM cosmology].
@ In modified gravity theories: Oikonomou ApSS(14)-a1403 [f(R) theories and scalar-tensor theories, effect on primordial black holes]; Du & Nishizawa PRD(16)-a1609 [scalar-tensor theories, new types of memory]; Kiliçarslan TJP(19)-a1811 [higher-derivative gravity]; Hou & Zhu a2005 [scalar-tensor theories, BMS group]; Hou a2011-proc [Brans-Dicke theory].
@ Other effects and examples: Tolish et al PRD(14)-a1405 [simple example]; Susskind a1507 [electromagnetic version]; Lasky et al PRL(16)-a1605 [and GW150914]; Compère IJMPD(16)-a1606-GRF [and supertranslations]; Garfinkle et al CQG(17)-a1702 [particle scattering]; Kolekar & Louko PRD(17)-a1703 [for uniformly accelerated observers]; Zhang et al PLB(17)-a1704, PRD(17)-a1705 [for an exact plane wave solution, and soft gravitons]; Talbot et al PRD(18)-a1807 [waveforms and phenomenology]; Johnson et al PRD(19)-a1810 [non-linear, detection prospects]; Banks & Fischler a1810 [for a galaxy, rotation of frames at infinity]; Mao & Wu JHEP(19)-a1812 [for the Robinson-Trautman solution, and observational time-delay effect for spin memory]; Flanagan et al PRD(19)-a1901 + news cosmos(19)may [persistent observables]; Bieri a2010 [new effects]; > s.a. angular momentum.
@ Variations: Pasterski et al JHEP(15)-a1502 [gravitational spin memory]; Nichols PRD(18)-a1807 [center-of-mass memory effect]; Rahman & Wald PRD(20)-a1912 [black-hole memory].

Gravitational Phenomenology > s.a. gravitating objects; lensing; thermodynamics.

Gravitational Radiation / Waves > s.a. detection; interferometers; propagation; sources; analysis [including other theories of gravity].

Gravitational Redshift

Gravitational Slip > see phenomenology of gravity [cosmological].

Gravitino > s.a. Rarita-Schwinger Equation; supergravity.
* Idea: The spin-3/2 supersymmetric partner of the graviton, with dynamics described by the Rarita-Schwinger equation.
@ Mass: Tkach et al MPLA(99); Takahashi et al PLB(08)-a0803 [from gravitational-wave background].
@ Interactions: Brignole et al JHEP(97)ht; Bjerrum-Bohr & Engelund PRD(10)-a1002 [from Yang-Mills Theory].
@ As dark matter candidate: Gorbunov et al JHEP(08)-a0805; Graefe JPCS(12)-a1111 [indirect searches]; Benakli et al PRD(17)-a1701; Dudas et al PRL(17)-a1704 [EeV mass gravitino].
@ In Schwarzschild spacetime: Fernández IJMPD(11)-a1006; Chen et al a1504.
@ In FLRW spacetime: Schenkel & Uhlemann PRD(12)-a1109 [spatially flat, quantization].
@ Other phenomenology: Brignole et al NPB(98)ht [at \(e^+ e^-\) colliders], JHEP(99) [muon anomalous magnetic moment]; Feruglio APPB(97)hp-proc; Lemoine PRD(99)hp [and inflation]; Kirchbach & Ahluwalia PLB(02)ht, ht/02-proc; Boubekeur et al JCAP(10)-a1004 [degenerate-gravitino scenario]; Dimastrogiovanni et al PRD(18)-a1706 [constaints on gravitinos and reheating]; Capozziello & Lambiase a2012 [gravitino problem in extended gravity cosmologies].
> Online resources: see Wikipedia page.



Gravity's Rainbow > see under rainbow gravity.

Gravity Probe A > s.a. gravitational redshift.
* Idea: A space-based experiment to test the equivalence principle, launched by NASA on June 18, 1976; It remained in space for 1 hour and 55 minutes at a height of 10,000 km (as intended), and the time measured by the hydrogen-maser atomic clock on board was compared to the time measured by an identical clock on the ground.
@ References: Vessot et al PRL(80).
> Online resources: see Wikipedia page.

Gravity Probe B > s.a. Gyroscope; tests of general relativity with orbits; tests of lorentz invariance.
* Idea: A space-based experiment launched in 2004, with four ultra-precise gyroscopes to measure the geodetic effect (the warping of space and time around a gravitational body), and frame dragging (the pulling of space and time by a spinning object with it as it rotates), two key predictions of the general theory of relativity.
@ References: Bencze et al NPPS(07); Focus issue CQG(15); Conklin CQG+(15) [data analysis].
> Online resources: see Gravity Probe B page; Wikipedia page.

Gravity Waves
* Idea: Propagating perturbations of the density field of a fluid. (Not related to gravitational waves.)

Gravity-Fluid Correspondence > see solution-generating methods for einstein's equation.

Greechie Diagram > s.a. non-commutative geometry [Greechie logic].
@ References: Mckay et al IJTP(00)qp [algorithms].
> Online resources: see Brendan McKay page.

Green Functions > s.a. feynman propagator; green functions in quantum field theory.

Green's Identities, Theorem > see integration on manifolds; vector calculus.

Greenberger-Horne-Zeilinger Experiment / Theorem > see experiments in quantum mechanics.

Gregory-Laflamme Instability > see black-hole geometry [black strings].

Greybody Factor > s.a. black-hole radiation.
* Idea: A frequency-dependent function that modifies the naive Planckian spectrum predicted for Hawking radiation when working in the limit of geometrical optics.
@ References: Boonserm PhD(09)-a0906 [rigorous bounds].

Gribov Ambiguity / Effect / Problem > s.a. BRST quantization; gauge choice; lattice gauge theory [Gribov copies]; quantum gauge theory.
* Idea: The non-existence of global cross sections of the principal fiber bundle of a gauge theory; It implies that one can't make a global gauge choice.
@ General references: Schön & Hájíček CQG(90) [quadratic constraints]; McMullan CMP(94); Langfeld hl/03 [toy model]; Esposito et al IJGMP(04)ht [intro]; Vandersickel & Zwanziger PRP(12)-a1202 [and the Gribov-Zwanziger framework, rev]; Pereira & Sobreiro EPJC(13)-a1308 [elimination by soft breaking of the BRST symmetry]; Lechtenfeld PPNL(14)-a1312-conf [rev]; Canfora et al PRD(14) [and degenerate symplectic structures].
@ On specific manifolds: Langmann & Semenoff PLB(93) [U(N) and SU(N) gauge theory on a circle]; Zhou PLB(17)-a1611 [3-torus].
@ And phenomenology: Ilderton eConf-a0709 [confinement]; Holdom PRD(09)-a0901 [infrared behavior, confinement].
@ In euclidean Yang-Mills theory: Killingback PLB(84) [on the 4-torus]; Sobreiro & Sorella ht/05-ln.
@ Related topics: Fleischhack CMP(03) [generalized connections]; de Cesare et al PRD(13)-a1308, de Cesare IJGMP(14) [in curved spacetime].

Gribov-Zwanziger Framework > see Glueballs.

Groenewold-Moyal Plane > see non-commutative geometry.

Groenewold-Van Hove Theorem > s.a. canonical quantum theory.
@ References: Gotay et al TAMS(96)dg/95 [S2]; Segre mp/05 [and general relativty].

Gromov-Hausdorff Distance / Space > s.a. distances between metric spaces; distances [quantum metric spaces].
* Idea: The space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance.
@ General references: Rouyer T&A(11) [properties].
@ Generalizations: Latrémolière a1506 [Gromov-Hausdorff propinquity, a non-commutative analog].

Gromov-Lawson-Rosenberg Conjecture
* Idea: A conjecture on the obstruction to the existence of a metric of positive scalar curvature on a manifold.
@ References: Schick Top(98) [counterexample].
> Online resources: see Encyclopedia of Mathematics page.

Gromov-Witten Invariants > s.a. Frobenius Manifold.
* Idea: In symplectic geometry, they are invariants of certain symplectic manifolds, following Gromov's fundamental work, which allows us to deal with them in a remarkably flexible way; In algebraic geometry, following Kontsevich, one considers certain compact varieties parametrizing maps from algebraic curves to a projective variety X, and the invariants are calculated as intersection numbers on the parameter variety; If they have a classical algebra-geometric interpretation in terms of X, the results obtained can be quite spectacular.
@ References: Fukaya & Ono Top(99) [and Arnold conjecture]; Ionel & Parker AM(03)m.SG/99 [relative]; Ionel & Parker AM(04) [symplectic sum formula]; Grunberg ht/06 [and string theory compactifications]; Maulik & Pandharipande Top(06) [relative].

Gross-Neveu Model > s.a. quantum phase transition.
@ 1+1: Schnetz et al AP(04), Thies JPA(06)ht-in [phase diagram]; Andersen ht/05 [long-range phase coherence, and 2+1]; Boehmer et al PRD(07) [near tricritical point].
@ 2+1, solvability: de Calan et al PRL(91); Wightman in(94); Christiansen et al PRD(00)ht/99 [thermodynamics].
@ 2+1, other: Charneski et al JPA(07)ht/06 [non-commutative]; Kneur et al PLB(07)-a0705 [massless, tricritical point].
@ D-dimensional: Khanna et al PRD(12)-a1204 [massive, in toroidal topologies, phase transition].
@ Related topics: Feinberg PRD(95) [kinks and bound states]; Miele & Vitale NPB(97) [on curved space]; Brzoska & Thies PRD(02), Thies & Urlichs PRD(03) [phase transition]; Schnetz et al AP(06)ht/05 [massive, phase diagram]; Gracey et al PRD(16)-a1609 [4-loop renormalization]; > s.a. renormalization.

Gross-Pitaevskii Equation > s.a. bose-einstein condensate; composite quantum systems.
* Idea: A non-linear model equation for the order parameter or single-particle wavefunction of a Bose-Einstein condensate.
@ References: Gravejat CMP(03) [no superluminal traveling waves]; Konotop & Kevrekidis PRL(03) [Bohr-Sommerfeld quantization]; Erdős et al PRL(07)mp/06, AM(10) [derivation]; Erdős et al CMP(09)-a0808 [and formation of correlations]; Béthuel et al CMP(09) [traveling wave solutions]; Pickl JSP(10)-a0907 [time-dependent, new derivation]; Benedikter et al a1208, a1404-proc [quantitative derivation].
> Online resources: see Wikipedia page.

Grothendieck Construction > s.a. K-Theory.
* Idea: A way to obtain an Abelian group G(A) from any additive semigroup G [@ Varadarajan notes, p63 ff].
* Examples: It is used to construct K-theory, Burnside rings,...

Grothendieck Topology > s.a. posets.
* Idea: A structure on a category C which makes the objects of C act like the open sets of a topological space.
@ References: Fioresi & Zanchetta a1509 [and representability in supergeometry].
> Online resources: see Wikipedia page.

Ground State > see types of quantum states.

Group Action

Group Algebra > s.a. lie algebra.
@ Generalizations: Grundling m.OA/04 [for non-locally-compact groups], JLMS(05)m.OA/04.

Group Averaging > see dirac quantization.

Group Field Theory > see approaches to quantum gravity; types of quantum field theories.

Group Presentation > see under Presentation.

Group Theory > s.a. representations, types of groups.

Group Velocity > see velocity.

Groupoid > s.a. gauge transformations [generalized symmetry transformations]; group; Gyrogroup; lie group; principal fiber bundle; Semigroup.
$ Def: A non-empty set S with a binary operation +; In the language of category theory a groupoid is a category all of whose morphisms are invertible.
* And other structure: One can always obtain a group by considering Aut(S,+).
@ General references: Renault 80.
@ Applications and examples: Kellendonk CMP(97)cm/95 [for a tiling]; Landsman CMP(01) [operator algebras and Poisson manifolds]; Heller CoP(06) [groupoid over frame bundle of spacetime]; Ciaglia et al IJGMP(19)-a1905, IJGMP(20)-a2002-proc, a2104 [and Schwinger's picture of quantum mechanics].
> And physics: see topology in physics [fundamental groupoid].

Grover's Algorithm > see quantum computing [search algorithm].

Growth Process > see stochastic processes.

GUP > the generalized uncertainty principle.

Gupta-Bleuler Formalism > s.a. quantum electrodynamics in curved spacetime.
* Idea: A technique which allows to interpret physically quantum electromagnetism in the Lorentz-covariant gauge (the photon has only 2 degrees of freedom); One introduces a spurious degree of freedom in a canonical quantum gauge theory, which gives a Hilbert space with indefinite metric; Then one restricts attention to states satisfying the operator Lorenz gauge condition (on which the inner product is positive-definite).
* Procedure: Decompose the vector potential into its positive and negative frequency parts, Aa = Aa(+) + Aa(−); Require that quantum states satisfy ∂a Aa (+) |ψ\(\rangle\) = 0; This is equivalent to \(\langle\)ψ| ∂a Aa (−) = 0, and implies \(\langle\)ψ| ∂a Aa |ψ\(\rangle\) = 0.
* Limitations: It is a gauge-fixed procedure; It works only for the free Maxwell field, not with interactions.
@ References: Gupta PPS(50); Bleuler HPA(50); Loran ht/02, ht/02 [generalized]; Gottschalk mp/05 [rev].


GW170817 > s.a. astrophysical tests of general relativity; gravitational-wave propagation; scalar-tensor gravity phenomenology.
* Idea: The first multimessenger astronomy event, produced by the collision and merger of two neutron stars.

Gyraton > see gravitational wave solutions.

$ Def: A groupoid (S,+) with a unit for the composition (an element 0 such that for all x in S, 0 + x = x + 0 = x) and inverses (for all x in S, there exists an −x such that −x + x = x + (−x) = 0); The thing is that it need not be associative.
@ In special relativity: Smith & Ungar JMP(96).

Gyromagnetic Ratio > s.a. Landé g-Factor.
$ Def: The combination of constants γ = eg/2m appearing in μ = (e\(\hbar\)g/2mc) S, where μ is the magnetic dipole moment of a particle, e its charge, g its Landé g-factor, m its mass, and S its spin.
* Spin-1/2 particles: The Dirac theory predicts g = 2; For the electron and other charged leptons, g = 2 + small standard model corrections, that are understood (deviations from those are studied for hints of new physics); > s.a. electron; particle types [muon].
* Baryons: For the proton, g = 5.58, and for the neutron, g = −3.82, because of large QCD corrections.
@ Specific types of particles: Belinfante PR(53) [spin-3/2 particle]; > s.a. electron; higher-spin field theory; particle types.
@ In general relativity: Garfinkle & Traschen PRD(90) [black hole]; Aliev CQG(07)ht/06 [charged Kerr-AdS black hole], PRD(08)-a0711 [non asymptotically flat spacetimes].
@ Arguments for g = 2: Ferrara et al PRD(92); Pfister & King CQG(03) [in quantum mechanics and general relativity]; Holstein AJP(06)dec.
@ Anomalous magnetic moment: Novello & Bittencourt IJMPA(14)-a1111 [proposal for the origin]; > s.a. particle types [leptons].

Gyroscope > s.a. Geodetic Precession.
* Idea: A device using a spinning wheel or disk used for measuring or maintaining orientation and angular velocity; > see Wikipedia page.
* Directionality claim: According to results reported in 1989, gyroscopes spinning in one direction change weight; The claim was later proved to be wrong.
@ General references: Jonsson AJP(07)may-a0708 [precession, in special and general relativity].
@ In curved manifolds: Sławianowski et al a0802; Wohlfarth & Pfeifer PRD(13) [synchronization and spacetime curvature].
@ Directionality claim: Hayasaka & Takeuchi PRL(89) [claim]; Maddox Nat(90)jan; Salter Nat(90)feb; Baker Nat(90)feb; Quinn & Picard Nat(90)feb; Faller at al PRL(90), Nitschke & Wilmarth PRL(90) [null result]; MacCallum NS(90)feb; Harvey Nat(90)aug.
@ Based on atom interferometry: news pw(11)oct; Berg et al PRL(15) + focus Phy(15) [high-precision].
@ Related topics: Keiser CQG+(15) [building the Gravity Probe B gyroscopes]; Di Virgilio et al a2003 [Sagnac gyroscope with sub-prad/s sensitivity].

GZK Cutoff / Effect / Puzzle > s.a. ultra-high-energy cosmic rays.
* GZK effect: Cosmic rays of energies above E > 5 × 1019 eV interact with the cmb and produce pions, losing 20% of their energy with a mean free path of 30 Mpc; This led to the prediction of a sharp cutoff (knee) in the number of cosmic rays above that energy.
* GZK puzzle: Cosmic rays at energies above the Greisen-Zatsepin-Kuzmin cutoff are not expected, but had been observed; 2007, Cutoff seen by Pierre Auger and HiRes experiments.
* Possible explanations: (2005 assessment) Superheavy relic particle decay (speculative); Topological defects (speculative); "Z-burst", ν + νZ resonance → decay (would produce too many photons); Lorentz invariance violation (speculative); Neutron stars; Active galactic nuclei; In the latter two cases, one needs magnetic fields to bend the cosmic rays and make them isotropic.
@ Puzzle, evidence: Takeda et al PRL(98); Bahcall & Waxman PLB(03)hp/02; Anchordoqui & Goldberg PLB(04) [analysis]; Kopenkin & Fujimoto PRD(05) [1975 event above the knee]; Abbasi et HiRes PRL(08)ap/07, Abraham et Auger PRL(08) [cutoff seen where expected]; Harari CRP(14)-a1406.
@ And Lorentz invariance violation: Coleman & Glashow hp/98; Anchordoqui et al PLB(00), PLB(00); Bertolami GRG(02) [preferred frames]; Amelino-Camelia IJMPD(03)ap/02 [DSR]; Gupta PLB(04)ap/03; Scully & Stecker APP(09)-a0811; González-Mestres NPPS(09)-a0902 [Auger-HiRes].
@ Puzzle, other proposals: Berezinsky et al PRL(97) [massive particle decay]; Svetlichny FPL(04)ht/03 [non-linear quantum mechanics]; Zachos MPLA(04), reply Chen & Yang MPLA(04) [re non-commutative proposal]; Wibig & Wolfendale ap/04 [drop not sharp]; Parizot NPPS(04)ap [effect of turbulent B]; Albuquerque & Smoot APP(06)ap/05 [smearing]; Kersting ap/06 [discrete space]; Berezinsky et al PRD(06); Hess & Greiner ht/07 [non-Lorentz-violating extension]; Gelmini et al JCAP(07)-a0706; de Unánue et al MPLA(07) [ns rather than ps]; > s.a. quantum-gravity phenomenology [lqg]; topological defects.
@ Related topics: Aloisio & Berezinsky AIP(05)ap [anti-GZK effect]; Erlykin & Wolfendale JPG(06)ap/05 [knee]; Unger a0812-conf [above the knee].

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