$ Def: A transformation consisting in a rotation by
around
the z-axis
in isospace, followed by C conjugation.Topics, G
g-Factor > see under Gyromagnetic Ratio.
G-Parity
$ Def: A transformation consisting in a rotation by 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].
> Online resources:
Wikipedia page;
Mauro Cherubini's page; Takasho Ohyama's page.
Galaxies > s.a. distances and distribution; formation and evolution.
Galerkin Approximation
Galerkin Duality > see lattice gauge theory.
Galilean Group / 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(
, 
, 
),
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].
@ In quantum mechanics:
Dieks FPL(90)
[Galilean boosts and Sagnac's phase]; Giulini AP(96);
Greenberger PRL(01).
Galileon Field
* Idea: A scalar field
whose field equations in flat spacetime with the are strictly of second order
(they do not contain the undifferentiated or once differentiated field, nor
derivatives of order higher than 2); 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: One could look
for generalizations to higher-spin field theories; Skyrmion theory could be
a group-valued analog.
@ References: Nicolis et al PRD(09)-a0811;
Deffayet et al PRD(09)-a0901 [special
non-minimal coupling and second-order equations], PRD(09)-a0906 [extension
to curved backgrounds]; Chow & Khoury PRD(09)-a0905 [cosmology];
Silva & Koyama PRD(09)-a0909 [cosmology].
Gallai's Conjecture > see graph theory.
Galois Group
Galois Theory, Field (Extension) > s.a. modified
quantum mechanics; modified quantum field theories.
@
Galois theory: Artin 44; Bastida 84; Garling 87; Weintraub 06 [intro].
@ Related subjects: Snaith AMS(94) [Galois
modules]; Denecke et al 04 [Galois
connections]; Stachowiak a0810-PhD [differential Galois group theory, integrability
and chaos].
Gamma Distribution > s.a. measure;
[probability].
$ Def: The distribution
_(
,
)
generalizing the Poisson distribution, defined by the probability density function
f(x) = ^{/}
x^{–1} exp{– x^}/(/)
, for
x 
0 .
@ References: in Santaló 76.
Gamma Function > s.a. Beta
Function.
$ Def: The function
* Properties: (n+1)
= n (n),
and, for integer argument, (n+1)
= n!; For half-integers, use (1/2)
= 1/2.
@ Generalized: Jurzak LMP(05).
Gamma Matrices > s.a. clifford
algebra; spinors
in field theory.
$ Def: Four matrices 
a, a =
0, 1, 2, 3 (a.k.a. Dirac matrices), defined by {a, b}
= 2 gab I.
* Form: They
generate a representation of the clifford algebra;
In the "real" representation, 1, 2, 3 are
real, 4 =
i 0 pure
imaginary, and they are all hermitian; 5:= 1 2 3 4 is
imaginary hermitian.
* Properties: They satisfy
the identities
[m, n]
= 4 mnab 5 – 4
(gmagnb – gmbgna)
+ 2 (–gma[n,b]
+ gmb[n,a]
+ gna[m,b] – gnb [m,a])
,
and, if we define ab...
d:= [ab... d ],
abc mn =
2 gc[m gn]ba +
2 gm[bgn]ac +
2 ga[mgn]cb +
2 gc[mn]ab +
2 gb[mn]ca +
2 ga[mn]bc + abcmn .
@ References: Deloburgo & Prasad NC(74) [n-dimensional]; Veltman NPB(89); Gran ht/01 [Mathematica package]; Bondarev NPB(06) [trace calculation].
Gamma Metric > see axisymmetric spacetimes.
Gamma Rays > see gamma-ray astronomy.
Gamow Functional / Vector > see resonance.
Gannon's Theorem > see singularities.
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 Theory > s.a. lattice gauge theory; solutions; types of gauge theories; yang-mills theories.
Gauge Theory of Gravitation > s.a.
[gauge
theory; gravity]; higher-order
theories; linearized gravity;
supersymmetric field theories.
* Idea: The variables
are gauge fields in flat spacetime, a tetrad (translational potential) and
a connection (rotational potential).
* In terms of symmetry
breaking: The n-dimensional general relativity symmetry group
GL(n, R) is broken to the Lorentz group.
@ General references: Utiyama PR(56);
Thirring AP(60), APA(78);
Kibble JMP(61);
Yang PRL(74)
[and Guilfoyle & Nolan GRG(98)gq];
Asorey & Boya IJTP(79) [observational difference wrt electromagnetic case];
Lenzen GRG(85)
[quadratic S]; Dehnen & Ghaboussi PRD(86);
Ghaboussi et al PRD(87);
Frønsdal JGP(90);
Hecht et al PRD(91);
Hehl et al PRP(95)gq/94;
Wu et al ht/02;
Sardanashvily TMP(02)gq;
Francis & Kosowsky
gq/03-wd
[solution]; Chamseddine IJGMP(06)ht/05
[rev]; Vignolo et al IJGMP(06)
[general relativity as constrained gauge theory].
@ Books, reviews: Ivanenko & Sardanashvily PRP(83);
Gronwald & Hehl gq/96-in
[rev]; Blagojevic 01; Sardanashvily gq/02;
Blagojevic gq/03-in;
Tiemblo & Tresguerres gq/05-in
[non-linear framework]; Sardanashvily IJGMP(06)gq/05
[geometric].
@ Lorentz group: Carlevaro & Montani a0903; Cho IJMPA(09)
[abelian decomposition].
@ Poincaré group: Edelen IJTP(85),
IJTP(85),
IJTP(85),
IJTP(85),
IJTP(86);
Edelen IJTP(89);
López-Pinto et al CQG(97)gq/96 [Hamiltonian];
Batakis gq/97;
Blagojevic AdP(01)ht/00 [and
teleparallel]; Yo & Nester IJMPD(02)gq/01 [Hamiltonian];
Leclerc PRD(05)gq
[teleparallel limit];
Frolov G&C(04)gq/05 [foundations];
Obukhov IJGMP(06)gq;
Leclerc IJMPD(07)
[second-order formalism]; Minkevich APPB(09)-a0808;
Ali et al IJTP(09)-a0907 [rev].
@ SL(2, C):
Nissani PRP(84);
Carmeli et al 90; Carmeli & Malin IJTP(98)
[quantum].
@ SO(3):
Mattes gq/03-PhD;
Kaul PRD(06)gq [complex
SU(2), and
supergravity].
@ SO(D+1):
Botta Cantcheff GRG(02)gq/00 [with
cosmological constant and torsion].
@ SO(4,1): MacDowell & Mansouri
PRL(77)
[with 
> 0].
@ Other: Wallner PRD(90),
Mielke PLA(90)
[new variables]; Gaitan gq/01;
Bertolini IJMPA(03)ht-ln;
Wu CTP(04)ht/03 [shielding
effect]; Wu & Zhang gq/05 [solutions
and tests]; Hestenes FP(05)
[with geometric calculus]; Cuzinatto et al a0712 [second-order,
and acceleration]; Anabalón et al JPA(08)
[SO(4,2)].
> Other versions: see
formulations of general relativity [as gauge theory
of
the
diffeomorphism
group].
> Related topics: see FRW spacetimes; Higgs
Mechanism; phenomenology; torsion.
Gauss' Law > s.a. electromagnetic
field equations; lattice gauge theories; solutions
of gauge theories.
@ References: Donohoe AJP(08)oct
[for the electric field].
Gauss Map
*
Idea: The chaotic "return" map on [0,1], defined
by v 
1/v –
[1/v], which appears in mixmaster dynamics.
@ References: in Barrow PRP(82); in Motter PRL(03)gq.
Gauss' Theorem > see integration on manifolds.
Gauss-Bonnet Gravity > s.a.
black holes and thermodynamics;
holography; wormhole
solutions.
* Idea: A higher-order
gravity theory with 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.
@ General references: Neupane & Dadhich CQG(09)-a0808 [as
incorporating features of quantum gravity in classical theory].
@
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];
Maeda CQG(06)gq/05
[effect on collapse]; Dadhich ht/06-in;
Gürses GRG(08) [with scalar field, solutions].
@ Solar-system tests: Sotiriou & Barausse PRD(07)gq/06 [+
scalar field, post-newtonian]; Davis a0709 [f(G)
theories].
@ 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-in
[constraints]; Andrew et al GRG(07)-a0708;
De Felice et al PRD-a0911 [small-scale
matter instability]; > s.a. cosmological
acceleration; friedmann
equation.
@ 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.
@ Quantum: Boernsen et al a0709 [dimensional
regularization]; Niu & Pak a0709 [coupled
to torsion]; Charmousis & Padilla JHEP(08)-a0807 [vacuum
instability].
> Related theories: see brans-dicke theory; higher-order
gravity.
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)Rabcd =
Rabcd +
(Kad Kbc –
Kac Kbd)
, (d–1)Rabcd nd
= Db
Kac – Da 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].
Gaussian Curvature > see riemann tensor.
Gaussian Functions [including Gaussian integrals]
Gaussian Integers > see numbers.
Gaussian Normal Coordinates > see coordinates.
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]; Cucic a0812.
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 
Z;
If Cnr
are the Gegenbauer polynomials,
F T{F(t)}
fnr
:= 
–1+1 (1–t2)r–1/2 Cnr(t) F(t)
dt .
* Inversion formula: For –1 < t <1,
* 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 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 {homos: A → 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 Covariance > see under Covariance.
General Relativity > s.a. 3D general relativity; formulations; modifications; tests.
Generalized Functions > see under distributions.
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 a0910 [exponential formula].
Generator of an R-Module > see module.
Generic Spacetime > see types of spacetimes.
Genus of a 2-Surface > see 2D manifold.
Geodesic Completeness > see differential geometry.
Geodetic Set, Number > see graph theory.
Geometric Algebra > s.a. special
relativity.
@ References: Doran et al AIEP(96)qp/05 [spacetime
algebra and electron physics]; Doran & Lasenby
03; Hestenes AJP(03)feb,
AJP(03)jul;
Henselder
et al mp/04.
Geometric Number Theory > see number theory.
Geometric Optics Approximation > see electromagnetism; gravitational phenomenology; wave phenomena.
Geometric Series > see series.
Geometric Topology > s.a.
combinatorics.
@ References: Moise 77 [2D and 3D].
Geometrically Independent Points > see affine structures.
Geometrodynamics > see canonical formulation of general relativity [classical]; gravitating matter; quantum geometrodynamics.
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 or sheaves
of groupoids provide a geometric realisation of three-dimensional integral
cohomology through their Dixmier-Douady class.
@ General references: Lupercio & Uribe m.AT/01, 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 mp/06-in
[quantum field theory].
Germ of a Function at a Point
$ Def: 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).
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, path-integral
quantization of general relativity.
* Idea: States of a
quantum field theory with negative norm; Classically, they correspond
to instabilities in interacting theories with higher derivatives, and theories
that have them are considered physically unacceptable.
@ References: Krause & Ng IJMPA(06)ht/04 [ghost
modification of general relativity and
cosmology]; Smilga NPB(05)
[benign and malicious]; Slavnov JHEP(08)-a0807 [Yang-Mills
theory with gauge-invariant
ghost field Lagrangian].
> Related topics: see
types of quantum field theories [higher-derivative].
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 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; Pesic AJP(91)nov
[and quantum mechanics].
@
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.
@ Theories and topics: Kiefer & Kolland GRG(08)-a0707 [for
black-hole entropy].
Gibbs-Duhem Relation
* Idea: A relationships
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 + 
N;
Other relations, such as G = N, 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 dT – V dP + N d =
0.
Ginzburg-Landau Model > see under Landau-Ginzburg model.
Girth > see graph invariants.
Gisin's Theorem > see bell inequalities.
Glass > see condensed matter; Disordered Systems; fluctuations [FD theorem for glassy systems]; phase transitions.
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 
is
a (real or complex) Hilbert space of dimension > 2 and a
probability measure on the subspace lattice L(),
then there exists a density operator W on such
that for all E in L(),
(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].
> 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 > mprot,
but hard to recognize.
* 1995: Not confirmed
yet; Evidence from lattice calculations that 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).
@ 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].
@ Spectrum: Morningstar & Peardon PRD(99);
Bugg et al PLB(00);
Frasca a0704 [strong
coupling and lattice calculations].
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
* 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;
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];
Durkee & Reall CQG(09)-a0908 [higher-dimensional
generalization].
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: Livio 02; Kak Foarm(06)phy/04 [physics
of aesthetics]; Moorman & Goff EJP(07) [in coupled-oscillator problem].
(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-in
[and 3D quantum gravity].
Goldstone Boson / Theorem > see symmetry breaking.
Googol
* Idea: A number equal
to 10100.
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].
GPS (Global Positioning System) > s.a.
coordinates.
@ References: Parkinson & Spilker ed-96; in Hartle 03; Coll et al
a0906-rp [relativistic positioning systems, status].
Gradient > see vector calculus.
Gram-Schmidt Orthogonalization Procedure > see Orthogonalization.
Grand Canonical Ensemble > see states in statistical mechanics.
Granular Materials > see gas; metamaterials.
Graph Theory > s.a. graphs in physics; graph invariants, and types and embeddings.
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].
@ 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 [gravitational-wave
signatures].
Graviscalar > s.a. scalar-tensor
theories.
* Idea: Scalar components
of the gravitational field, e.g., graviscalar Kaluza-Klein excitations.
Gravitation Theories > s.a. 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 [@ Einstein], even changing the
field equations, but failed; The modern version of the concept has become
that of geon.
@ References: Davidson & Guendelman PRD(86).
Gravitational Collapse > s.a. critical collapse.
Gravitational Phenomenology > s.a. gravitating objects; lensing; thermodynamics.
Gravitational Radiation / Waves > s.a. detection; interferometers; propagation; sources.
Gravitational Slip > see phenomenology of gravity [cosmological].
Gravitino > s.a. supergravity.
* Idea: The spin-3/2
supersymmetric partner of the graviton.
@ Mass: Tkach et al MPLA(99);
Takahashi et al PLB(08)-a0803 [from
gravitational wave background].
@ Other phenomenology: Brignole et al JHEP(97)ht [interactions],
NPB(98)ht [at e+e– colliders],
JHEP(99)
[muon anomalous magnetic moment];
Feruglio APPB(97)hp-in;
Lemoine PRD(99)hp [and
inflation]; Kirchbach & Ahluwalia PLB(02)ht, ht/02-in;
Gorbunov et al JHEP(08)-a0805 [as
warm dark matter candidate].
> Online resources: see Wikipedia page.
Greechie Diagram > s.a. non-commutative
geometry [Greechie logic].
@ References: Mckay et al IJTP(00) [algorithms].
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;
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;
Implies that one can't make a global gauge choice.
@ General references: Schön & Hájícek CQG(90)
[quadratic constraints]; Langmann & Semenoff PLB(93)
[U(N) and U(N) gauge theory on a circle]; McMullan CMP(94);
Langfeld hl/03 [toy
model]; Fleischhack CMP(03)
[generalized
connections]; Esposito et al IJGMP(04)ht [intro].
@ And phenomenology:
Ilderton a0709-in
[confinement]; Holdom PRD(09)-a0901 [infrared
behavior, confinement].
@ In euclidean Yang-Mills theory: Killingback PLB(84)
[on the 4-torus];
Sobreiro & Sorella ht/05-in.
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-Lawson-Rosenberg Conjecture
@ References: Schick Top(98) [counterexample].
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].
@ 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].
Gross-Pitaevskii Equation >
s.a. composite quantum systems.
* Idea: A non-linear
model equation for the order parameter or wavefunction of a Bose-Einstein condensate.
@ References: Gravejat CMP(03)
[no superluminal traveling waves]; Konotop & Kevrekidis
PRL(03)
[Bohr-Sommerfeld quantization]; Erdös et al PRL(07)mp/06 [derivation];
Béthuel et al CMP(09)
[traveling wave solutions]; Pickl a0907 [time-dependent, new derivation].
> Online resources: 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, p 63 ff].
* Examples: Used to
construct K-theory, Burnside rings,...
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)
[grupoid over frame bundle of spacetime].
Grover's Algorithm > see quantum computing [search algorithm].
Growth Process > see stochastic processes.
GUP > the generalized uncertainty principle.
Gupta-Bleuler Formalism
* 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 (+) |
=
0; This is equivalent to 
| a Aa (–) =
0, and implies |a Aa | =
0.
* Limitations: It is
a gauge-fixed procedure; 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].
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. higher-spin
field theory; Landé g-Factor; particle
types.
$ Def: The combination
of
constants = eg/2m appearing
in = (eg/2m)
S,
where is
the magnetic
dipole moment of a particle, e its charge, g its Landé g-factor, m its
mass, and S its
spin.
* For spin-1/2: 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).
* Electron: As of 2006, g/2
= 1.001 159 652 180 85 (76); As of 2008, 1.001 159 652 180 73 (28).
* Baryons: For
the
proton, g =
5.58,
and
for the
neutron, g = –3.82, because of large QCD corrections.
@ Electron:
Welton PR(48)
[and electromagnetic field fluctuations]; Odom et al PRL(06)
+ pn(06)jul
[best
measurement]; Aoyama et al PRL(07)
[theory, 8th-order contribution]; Hanneke et al PRL(08); > s.a. modified
QED.
@ Other objects: Belinfante PR(53)
[spin-3/2 particle].
@ 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
* Directionality claim:
According to results reported in 1989, gyroscopes spinning in one direction
change weight; Proved to be wrong.
@ General references: Jonsson AJP(07)may-a0708 [precession,
in special and general relativity]; Slawianowski et al a0802 [in
curved manifolds].
@ 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.
GZK Cutoff / Effect / Puzzle > see cosmic rays.
main page – abbreviations – journals – comments – other
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send feedback and suggestions to bombelli at olemiss.edu – modified
9 jan 2010