**Topics, G**

** g-Factor** > see under Gyromagnetic
Ratio.

**G-Parity**

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

**G _{2}**
> s.a. cosmological models in
general relativity [G

*

@

**Gabriel Graph**

* __Idea__: Given a set
of points {*p _{i}*} in a manifold, the
Gabriel graph defined by those points has an edge between points

@

>

**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*(*v*_{x}, *v*_{y},
*v*_{z});
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.

@ __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 & Fairlie a1212 [primer]; Deffayet et al PRD(11); Trodden 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].

@ __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 a1704 [*p*-form theories, classification].

@ __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 *p*^{k} 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.

**Gamma Distribution** > s.a. measure / probability.

$ __Def__: The distribution
Γ_{θ}(*ν*, *λ*)
generalizing the Poisson distribution, defined by the probability density function

*f*(*x*) = *θ* *λ*^{ν/θ} *x*^{ν–1} exp{–*λ* *x*^{θ}}
/ 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)
= π^{1/2}.

@ __Generalized__: Jurzak LMP(05).

**Gamma Matrices** > s.a. clifford
algebra; spinors in field theory.

$ __Def__: Four matrices *γ*^{a}, *a* =
0, 1, 2, 3 (a.k.a. Dirac matrices), defined by {*γ*^{a}, *γ*^{b}}
= 2 *g*^{ab} I.

* __Form__: They
generate a representation of the Clifford algebra;
In the "real" representation, *γ*^{1}, *γ*^{2}, *γ*^{3} are
real, *γ*^{4} =
i *γ*^{0} pure
imaginary, and they are all hermitian; *γ*^{5}:= *γ*^{1} *γ*^{2} *γ*^{3} *γ*^{4} is
imaginary hermitian.

* __Properties__: They satisfy
the identities

[*γ*^{m}, *γ*^{n}]
= 4 *γ*^{mnab }*γ*_{5} – 4
(*g*^{ma}*g*^{nb} – *g*^{mb}*g*^{na})
+ 2 (–g^{ma}[*γ*^{n}, *γ*^{b}]
+ *g*^{mb}[*γ*^{n}, *γ*^{a}]
+ *g*^{na}[*γ*^{m}, *γ*^{b}] – *g*^{nb }[*γ*^{m}, *γ*^{a}])
,

and, if we define *γ*^{ab...
d}:= *γ*^{[a}*γ*^{b}... *γ*^{d ]},

*γ*^{abc }*γ*^{mn} =
2 *g*^{c[m }*g*^{n]b}*γ*^{a} +
2 *g*^{m[b}*g*^{n]a}*γ*^{c} +
2 *g*^{a[m}*g*^{n]c}*γ*^{b} +
2 *g*^{c[m}*γ*^{n]ab} +
2 *g*^{b[m}*γ*^{n]ca} +
2 *g*^{a[m}*γ*^{n]bc} + *γ*^{abcmn} .

@ __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].

**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.

**Gastrophysics**

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

**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.

@ __References__: Donohoe AJP(08)oct [for the electric field].

**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 Operator**

* The operator D:= d + δ.

@ __References__: Anné & Torki-Hamza AMP(14)-a1301 [on infinite graphs].

**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].

@ __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 a1004 [static solutions]; Quirós & Teste a1005 [stability]; > 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 a1408 [*D*-dimensional scalar field, Hamiltonian].

@ __Bounces__: Bamba et al PLB(14)-a1404; Oikonomou 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 MPLA-a0711-proc
[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 ASS(12)-a1111 [generalized second law]; > 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].

> __Related theories__: see brans-dicke theory; higher-order
gravity; teleparallel theories.

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

**Gauss-Codazzi Equations** (no, 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)}*R*_{abcd} =
*R*_{abcd} +
*ε* (*K*_{ad} *K*_{bc} –
*K*_{ac} *K*_{bd})
, ^{(}^{d}^{–1)}*R*_{abcd} *n*^{d}
= *D*_{b}
*K*_{ac} – *D*_{a} *K*_{bc} . [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 coherent states; types of quantum states.

**Gaussianity**

> __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.

**Gegenbauer Polynomials**

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

**Gegenbauer Transform
**$

*F* \(\mapsto\) *T*{*F*(*t*)}
≡ *f*_{n}^{r}
:= ∫_{–1}^{+1} (1–*t*^{2})^{r–1/2} *C*_{n}^{r}(*t*) *F*(*t*)
d*t* .

* __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–*t*^{2}) *F''* – (2*r*–1)*t* *F''* to *T*{*R*[*F*(*t*)]} = –*n* (*n*+2*r*) *f*_{n}^{r}.

**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 determination of some quantities as coefficients in a series
expansion.

* __Enumeration__: A
representation of a Counting Function as an element of some algebra.

@ __In combinatorics__: in Comtet 74; 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.

> __Online resources__: see Wikipedia page.

**Geodetic Set, Number** > see graph theory.

**Geoid**

* __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.

__Related concepts__: see refraction; wave phenomena.

__Related areas__: see electromagnetism; light; optics.

**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.

* __Idea__: 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 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.

**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 a1309 [arbitrary-spin generalizations of the Faddeev-Popov ghost field]; Sbisà a1406 [rev, classical and quantum]; Canarutto 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]; > s.a. 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].

@
__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).

@ __Theories and topics__: Kiefer & Kolland GRG(08)-a0707 [for
black-hole entropy].

> __Online resources__:
see Shu-Kun Lin's list of references.

**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* = *TS* –
*PV* + ∑_{i} *μ _{i} N_{i}*;
Other relations, such as

>

**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 of 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]; Edalat IJTP(04)
[extension for quantum computation]; Buhagiar et al FP(09)
[consequences]; Barnett et al NJP(14) [generalization].

> __Online resources__: see Wikipedia page.

**Glissement**

$ __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* > *m*_{prot}, but hard to recognize.

* __1995__: Not confirmed
yet; Evidence from lattice calculations that the lightest one is *f*_{J} (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 *f*_{0}(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'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 *a*_{n+1}/*a*_{n}
as *n* → ∞,
where {*a*_{n}}
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 a1701 [null geodesics in Schwarzschild-Kottler black holes].

**(Fermi's) Golden Rule**

@ __References__: Dragoman PLA(00)
[in phase space].

**Goldman Bracket**

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

**Goldstone Boson / Theorem** > see symmetry
breaking.

**Gonihedric Action / System** > see path integrals.

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

**Googol**

* __Idea__: A number equal
to 10^{100}, a 1 followed by 100 zeros.

**Googolplex**

* __Idea__: A number equal
to 10^{10^{100}}, a 1 followed by a googol of zeros, and the second largest number with a name.

> __Online resources__: see googolplex.com; Wikipedia page.

**Googolplexian**

* __Idea__: A number equal
to 10^{10^{10^{100}}}, a 1 followed by a googolplex of zeros, and the largest number with a name.

> __Online resources__: see googolplexian.com.

**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].

**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:= *U* – *TS* – *μN*.

> __Online resources__: see Wikipedia page.

**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]

**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].

@ __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].

@ __Related topics__: Lobo & Garattini a1004 [in non-commutative geometry, and
stability].

**Gravimeter**

* __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 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 a1612 [and supertranslation, in four and higher dimensions].

@ __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].

@ __In cosmological spacetimes__: Bieri et al PRD(16)-a1509 [de Sitter spacetime]; Tolish & Wald PRD(16)-a1606, Chu a1611 [spatially flat FLRW cosmology]; Bieri et al a1706 [\(\Lambda\)CDM cosmology];

@ __Other effects and examples__: Oikonomou ASS(14)-a1403 [effect on primordial black holes in *f*(*R*) theories and scalar-tensor theories]; Tolish et al PRD(14)-a1405 [simple example]; Lasky et al PRL(16)-a1605 [and GW150914]; Compère IJMPD(16)-a1606-GRF [and supertranslations]; Kolekar & Louko a1703 [for uniformly accelerated observers]; Zhang et al a1704, a1705 [for an exact plane wave solution, and soft gravitons]; > s.a. angular momentum.

@ __Variations__: Pasterski et al JHEP(15)-a1502 [gravitational spin memory]; Du & Nishizawa PRD(16)-a1609 [in scalar-tensor theories, new types of 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 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 a1701; Dudas et al 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 a1706 [constaints on gravitinos and reheating].

> __Online resources__: see Wikipedia page.

**Gravity's Rainbow** > see 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__: 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 [S^{2}];
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 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__: 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 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
transformations]; group; Gyrogroup;
lie group; principal
fiber bundle; Semigroup.

$ __Def__: A non-empty set *S* with a binary operation +.

* __And other structure__:
One can always obtain a group by considering Aut(*S*,+).

@ __References__: Renault 80; 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].

> __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,
*A*_{a}
= *A*_{a}^{(+)}
+ *A*_{a}^{(–)};
Require that quantum states satisfy *∂*_{a} *A*^{a (+)} |*ψ*\(\rangle\) =
0; This is equivalent to \(\langle\)*ψ*| *∂ _{a} A*

*

@

**Gyraton** > see gravitational wave solutions.

**Gyrogroup**

$ __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*/2*m* appearing in
*μ* = (*e\(\hbar\)g*/2*mc*) *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.

* __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.

@

**Gyroscope** > s.a. Geodetic Precession.

* __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].

**GZK Cutoff / Effect / Puzzle**
> s.a. ultra-high-energy cosmic rays.

* __GZK effect__: Cosmic
rays of energies above *E* > 5 × 10^{19} 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* + *ν* → *Z*
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)
[*n*s rather than *p*s]; > 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].

main page – abbreviations – journals – comments – other
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send feedback and suggestions to bombelli at olemiss.edu – modified
13 jun 2017