$ Def: A subspace I
A is
a left (right) ideal if for all i in I and a in A, ai belongs
to I (resp, ia is in I); AI I or
IA I.* For a ring R: A submodule of R considered as an R-module.
Topics, I
i, Imaginary Unit
@ References: Nahin 98 [I, history].
Ice > see Water.
Icosahedral Group
@
References: Cesare & Del Duca RNC(87).
Ideal Elements of a Physical Theory > see physical theories.
Ideal Gas > see gas; thermodynamics.
Ideal of an Algebra
$
Def: A subspace I A is
a left (right) ideal if for all i in I and a in A, ai belongs
to I (resp, ia is in I); AI I or
IA I.
* For a ring R: A submodule of R considered as an R-module.
Idealizations > see Models.
Identical Particles > see particle statistics.
Identity of Indiscernibles > s.a. particle
statistics.
* Idea: If two systems
are qualitatively identical then they are logically identical; It is violated
by indistinguishable quantum particles.
Image Charge > s.a. schrödinger
equation.
* Idea: A fictitious
charge used to solve boundary value problems; It is placed behind the boundary
and acts as a source which, together with the physical charges in the problem,
produces the same field on the boundary as the prescribed boundary conditions in
a new, simpler problem.
@ References: Roulet & Saint Jean AJP(00).
Imbedding, Immersion > see embedding.
Immirzi Parameter > see black hole entropy; connection formulation of generl relativity and quantum gravity; yang-mills gauge theory.
Implicit Function Theorem
Incidence Algebra / Structure > s.a. posets.
@ Incidence geometry: Buekenhout ed-95 [handbook].
Incidence Matrix
@
References: Hodge 52.
Incompleteness Theorem > see under Gödel's Theorem.
Independence > s.a. matroids.
* Remark: The abstract notion
has been formalized in the theory of matroids.
* Specific notions: Geometrically
independent points (> see affine structures);
Linearly independent vectors.
Index of a Critical Point of a Function f
$ Def: If a is
a non-degenerate (i.e., non-singular Hessian) critical point of f,
its index is the number of negative terms in
the
quadratic expansion of f in a neighborhood of a.
Index of an Elliptic Operator D
$ Def: The difference dim(ker D) – dim(coker D).
Index of Refraction > see Refraction.
Index of a Vector Field
$ Def: If x is
a zero of a vector field on an n-dimensional manifold M,
the index of v at x is
the degree of the map Sn–1 → Sn–1 defined
by v (normalized
with some flat metric) on a small sphere surrounding x.
* Properties: indx(–v) = (–1)n indx(v).
Index Theorem (Atiyah-Singer) >
s.a. anomaly.
* Idea: It relates
the number of zero modes (in general the index) of an elliptic differential
operator D on a closed manifold M to
characteristic classes of the tangent bundle of M and of the vector
bundles on which D acts; "Basically
a formula that counts the number of solutions to another
equation" (M
Atiyah 2004, on receiving the Abel Prize), "A cornerstone of maths; it
is one of the most fundamental results of the last 50 years" (Elmer
Rees).
* Special cases: The Hirzebruch signature theorem, the Riemann-Roch
theorem.
@ General references: Shanahan 77; Gilkey 84; Booss & Bleecker 85;
Esposito gq/95-ln.
@ Special manifolds: Niemi & Semenoff pr(85) [infinite]; Peeters & Waldron
JHEP(99)ht [with
boundary].
@ Generalizations: Longo
CMP(01)
[quantum]; Harikumar et al JPA(07)ht/06 [q-deformed
fuzzy sphere]; > s.a. quantum graphs.
Individuality > s.a. foundations
of quantum mechanics [individual particles].
@ In quantum mechanics: Pesic 02; Sant'Anna qp/04 [particles].
Induced Gravity > s.a. bianchi
models, and bianchi IX; gravity.
* Idea: Gravity is not
fundamental, but becomes dynamical as a result of quantum effects in the system
of heavy constituents of an
underlying theory, electromagnetic or other.
* Example: Sakharov's
theory of gravity as a long-range Casimir force [@ NS(81)apr,
NS(90)jul28].
* Drawbacks: The biggest
problem is that it was shown that G is not calculable.
@ General references: Sakharov (68); Adler PLB(80)
[formula for G], RMP(82);
Puthoff PRA(89)
[stochastic electrodynamics]; Haisch et al phy/98-in;
Barceló et al IJMPD(01)gq [based
on general relativity analogs]; Visser MPLA(02)gq [status];
Chernitskii gq/02-in
[from non-linear electrodynamics]; Davidson & Gurwich gq/06-in
[and dark matter].
@ Other dynamical origin: Dhar NPB(97)
[c =
1 matrix model]; Laughlin IJMPA(03)gq-in
[emergent]; Wetterich
PRD(04)ht/03 [from
spinors]; Kan & Shiraishi PTP(04)gq/03;
Makhlin hp/04/PRL
[Dirac
field]; > s.a. Higgs Mechanism [gravitational],
Stochastic Gravity.
Induction, Electromagnetic > see Faraday's Law.
Induction, Mathematical
* Analogy: When subscribing
to a newspaper, say (i) Deliver it tomorrow; (ii) If you deliver it one day,
make sure you deliver it the following day (R Smullyan).
Inductive Family or System > see sequences.
Inductive Limit > see limit.
Inequalities > for physics ones, see effects in quantum field theory, states in quantum mechanics.
Inertia (including inertial frame, observer) > s.a. Moment of Inertia.
Infeld-van der Waerden Symbols > see Soldering Form.
Inference > see probability; statistics.
Infinite > s.a. Denumerability.
* History: The actual infinite was introduced by Cantor around 1871 when
studying uniqueness of trigonometric series for cases with complicated sets
of exceptional
points.
* Different infinities: 
=
card Z, +
1 (notice: 
1 + = ),
..., · 2 (notice:
2 · =
);
Surreal numbers lie somewhere between and +
1.
@ General references: Cantor 15; Zippin 62; Maor 87; Berry & Howls PW(93)jun;
Rucker 95; Barrow 03 [play]; Clegg 03; Wallace 03 [history, r pw(04)apr];
Sergeyev 04 [arithmetic]; Barrow 05 [math and physics].
@ Related topics: Donald qp/03 [many-minds
and mathematical
vs physical "existence"].
Infinitesimal > s.a. differential
geometry; non-standard
analysis.
* Idea: Numbers that
lie between zero and every positive standard number.
*
History: Introduced by Leibniz and Newton, opposed from the beginning
by Berkeley; Always very controversial; Replaced by limits, they came back
in the 1960's
with non-standard analysis; New approach by Nelson.
@ References: Bell MI(95) [and the continuum].
Inflation > s.a. phenomenology; types of inflation.
Inflaton > see scalar-tensor theories.
Influence Functional > see quantum systems [dissipative].
Information Theory > s.a. quantum information.
Infraparticles > see Unparticles.
Initial Conditions > s.a. cosmology and
cosmological models; quantum
cosmology boundary conditions; singularities; theory.
@ In thermodynamics: Callender BJPS(04)
["special" initial conditions].
Initial Value Formulation > see for general relativity.
Injective Module > see types of modules.
Injectivity Radius > see lorentzian geometry.
Inner Product > see vector.
Instabilities
@ References: in Arnold 78 [beautiful fluid example];
Sorkin ApJ(81), ApJ(82); Price AJP(82);
Kiessling AAM(03)ap/99 [Jeans
swindle].
@ In statistical mechanics:
Simon & Sokal JSP(81)
[balance of energy vs entropy].
> In gravitation: see
astrophysics; black
holes [black strings and Gregory-Laflamme instability]; fluid; gravitational
radiation.
> In other systems:
see classical systems [unstable]; dissipation; fluid [incuding
astrophysics]; geons.
Instantons > s.a. gravitational instantons.
Insulators
* Types: In ordinary insulators
every possible electron state is filled (with two electrons of opposite spin
orientation); No electric current is
then possible and the material is insulating; In a Mott insulator only half the
electronic states are occupied, but still no electric current flows because strong
electron repulsions prevent any electron motion; Properties of Mott insulators
are believed to be important for understanding hight-Tc superconductivity in cuprates.
Integrability Conditions > see partial differential equations.
Integrable System > s.a. quantum systems.
Integral Curve of a Vector Field > see vector field.
Integral Domain
$ Def: A commutative
ring with identity and no proper zero divisors.
Integral Geometry > see geometry.
Integral Submanifold > see manifolds.
Integral Transforms > s.a.
[integration]; fourier; Laplace
Transform.
@ General references: Davies 85; Bateman 54.
@ Other types: Gasaneo & Colavecchia
JPA(03)
[using 2-body Coulomb wave functions].
Integration Theory > s.a. integration on manifolds.
Interaction Representation > see representations of quantum theory.
Intergalactic Matter > s.a. contents
of the universe; dark
matter.
@ References: Scannapieco et al SA(02)oct; Mörtsell & Goobar JCAP(03)
[dust, constraints]; Simcoe AS(04)#1;
Kim et al ApJ(05)ap
[dynamics and velocity field]; Barkana & Loeb RPP(07)ap/06
[physics and early history]; Meiksin a0711-RMP [rev].
Interior Product
* Idea: A product between
a vector field X and a 1-form 
,
denoted by iX := (X).
Intermediate Vector Bosons > see electroweak; particle types.
Interpolation Spaces
@ References: Bergh & Löfström 76.
Interpolation Techniques > see observational cosmology.
Interpretation of a Theory > s.a. interpretations
of quantum mechanics; quantum field theory.
* Idea: "A
description in ordinary language of what an observer would see or experience
when the mathematical quantities used by the theory to describe the state of
the system take on any of their allowed values".
Intersection of Sets > see sets.
Intertwiner, Intertwining Operator > s.a. group
representations; spin
networks.
* Idea: A Contraction
matrix for a set of representations of a group.
@ References: Huang et al m.QA/04 [modules for vertex operator algebras].
Interval > see posets.
Invariance of a Theory > see symmetry.
Invariant in Dynamics > see observables.
Invariant Vector Field on a Lie Group, on a Manifold > see vector fields.
Inverse > see group theory; matrix.
Inverse Function Theorem
Inverse Limit > see projective limit.
Inverse Scattering > see scattering.
Inverse System of Spaces and Maps > see projective system.
Invisibility Cloak > see metamaterials.
Involution > see algebra.
Ionization > see atomic physics.
Irreducible Mass of a Black Hole [> s.a. black
holes.]
* Idea: The energy contained
in the black hole that cannot be extracted by Penrose
processes, i.e., degraded
energy that is not stored in rotation; Classically it can never decrease.
* Expression: Given by
Mirr2 =
A / (16
G2)
,
or (1/2) [M2 + (M4 – J2)1/2]
for a Kerr black hole.
* Special cases: For
a Schwarzschild black hole it coincides with the total
mass;
For a system of several black holes, it is Edegr =
(
Mirr2)1/2 < Mirr,
so, by combining "dead" Schwarzschild
black holes one can still obtain energy.
@ Introduction of concept: Christodoulou PRL(70).
Irreversibility > see arrow of time.
Island Universes > see history of cosmology [galaxies]; cosmological models [cosmological constant sea].
Isolated Horizon > see horizons.
Isolated Object / System [> s.a. systems].
* Idea: One
which does not interact with any other system; As pointed out by D Zeh (1970),
there can never be a truly isolated object; Is this a conceptual difficulty
for the universal
validity
of laws?
* Quasi-isolated system:
One subject to small random uncontrollable perturbations; In general
stochastically unstable; > s.a. arrow of time.
@ References: Cox GRG(07)
[in practice].
> Various contexts:
see asymptotic flatness; quantum mechanical systems; relativistic
cosmology [local effects].
Isometry of a Manifold with Metric > see differential geometry.
Isomorphism
* Idea: A mapping
that preserves all the relevant structures of objects in a category.
$ General def: An f 
Hom(X, Y)
is an isomorphism if 
g Hom(Y, X)
such that f 
g =
idX and g f =
idX.
$ For sets: A one-to-one and onto map.
$ For linear spaces:
A bijection f : X → Y, where X and
Y are linear spaces, preserving the linear structure.
* Isomorphism problem: The problem of deciding whether any two finite presentations
represent isomorphic groups; It was proved unsolvable in the late 1950's.
@ Isomorphism problem: Stillwell BAMS(82).
Isospin > s.a. modified
quantum mechanics [quaternions and (iso)spin]; particle
types; QCD; standard
model.
@ References: Fernandes & Letelier PLA(05)
[motion of particle with isospin].
Isotopy Theory > see algebraic topology.
Isotropic Coordinates > s.a.
[coordinates]; coordinates
for schwarzschild.
$ Def: Coordinates in
which a metric (e.g., spatially spherically symmetric) takes the form
ds2 = –exp{2
}dt2 +
exp{2
} (dr2 + r2 d
2)
.
@ References: in Misner et al 73, ex 23.1 (p595), ex 31.7 (p840).
Isotropic Metric on a Manifold > s.a.
bianchi models; cosmological
models in general relativity; observational
cosmology.
* Idea: An n-dimensional
Riemannian manifold M is isotropic at p in M if
there is an action of SO(n–1) on M such that p is
a fixed point (the only one in some neighborhood of p) and all group elements
act
as isometries.
* Result: Isotropy
about every point implies homogeneity.
Isotropic Submanifold > see symplectic structures.
Isotropy Group of a Point
* Idea: The group of
spatial isometries of an asymptotically flat spacetime which leaves a given
point p in M fixed.
Israel's Theorem > s.a. black
hole hair and uniqueness.
* Idea: The only static
and asymptotically-flat vacuum space-time possessing a regular horizon is the
Schwarzschild solution (or Reissner-Nordström
in the electrovac case).
@ References: Israel PR(67),
CMP(68);
Herrera a0711-IJMPD
[physical consequences].
Ito Calculus > see analysis.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
22 jul 2008