ij m m+n,0 ,Topics, K
K-Causality > see causality conditions.
K-Essence > s.a. causality; cosmological
acceleration; dark energy; quintessence;
time in gravity.
* Idea: (Kinetic-energy-driven
quintessence) A scalar field
with a Lagrangian of a special form, that causes its energy density to track
that of radiation when the universe is
radiation-dominated,
and to follow its own evolution (first around a cosmological-constant-like
value, then a different attracting behavior, when
the
universe becomes matter-dominated; The motivation is to solve the coincidence
problem; & Mukhanov.
@ References: Rendall CQG(06)gq/05 [dynamics];
Bonvin et al PRL(06)
[no-go theorem].
K-System > see under Kolmogorov System.
K-Theory > s.a. bundle [gerbes]; KK-Theory.
* Idea: The Abelian group
constructed from the space Vect(M) of equivalence classes of vector bundles
over M, using the Grothendieck construction; A generalized cohomology theory
(does not satisfy the dimension axiom for cohomology, and
the K-theory of a point is not trivial), used to classify vector bundles; Notice
however that it does not fully classify them, but only up to stable equivalence.
* Remark: Its dual homology theory does not seem to be useful.
@ General references: Atiyah 67; Milnor 74; Bak 81; Blackadar 86 [operator
algebras]; Wegge-Olsen 93 [also C*-algebra]; Husemoller 94.
@ And physics: Witten JHEP(98)ht [D-branes], IJMPA(01)ht/00 [strings];
Braun ht/00 [K-torsion];
Freed mp/02-ln,
Woit ht/02 [and
quantum field theory].
@ Generalizations: Mickelsson LMP(05) [twisted, invariants].
Kac-Moody Algebra
* Idea: An infinite-dimensional
Lie algebra, with generators satisfying
[Tni, Tmj]
= i f ijk Tm+nk +
K ij m m+n,0 ,
where K is an op such that [K, Tni] = 0 (it is
effectively a c-number for the
algebra).
* Example: If K = 0, we get a loop algebra.
* Remark: It is based on some compact simple Lie algebra.
* Scalar product: One with Lorentzian signature can be defined, improperly
denoted by 
A,B
=
tr AB, requiring that: tr AB = tr BA,
and tr[A,B]C + tr B[A,C]
= tr[A,BC] = 0 (to
guarantee group invariance).
@ General references: Kac Izv(68); Moody JA(68);
Ray BAMS(01)
[generalized].
@ And physics: Dolan PRL(81)
[2D chiral models], PLB(82)
[4D self-dual Yang-Mills]; Goddard & Olive ed-88; Fuchs
ht/97-ln
[and conformal field theory].
Kac-van Moerbeke Lattice > see toda lattice.
Kadanoff-Baym Equations > s.a.
early universe.
* Idea: The (non-)equilibrium
real time Green's function description (or "closed time path Green's
function" – CTPGF)
of transport equations.
@ References: Greiner & Leupold AP(98)hp,
hp/98-in [stochastic
interpretation].
Kähler Metric, Structure > see complex structure and symplectic structure.
Kalb-Ramond Field > see types of gauge theory.
Kalman Filter
@ References: in Casti 00.
Kaluza-Klein Theories > s.a. models and phenomenology.
KAM Theorem (Kolmogorov, Arnold, Moser) > s.a. Arnold
Diffusion.
* Idea: When perturbing
a completely integrable, non-degenerate (det |
2H0
/ Ii Ij| 
0)
Hamiltonian system, "most" of
the invariant tori, on which motion is quasi-periodic, persist for small perturbations;
The Lebesgue measure of the complement of their union is small.
* Remark: The set of invariant
tori is Lebesgue-measurable, although probably not Riemann-measurable, but
it may be that there is an R-measurable set of
points that move close to the unperturbed tori (but not quasi-periodically)
– true for 2D autonomous systems.
@ General references: in Gallavotti 83, p466; in Arnold 89; Bricmont
et al CMP(99)cd/98 [and
quantum field theory]; most books on chaos.
@ Related topics: Gallavotti & Gentile CMP(02)mp/01 [invariant
tori]; Evans CMP(04)
[quantum analog]; Broer BAMS(04)
[Kolmogorov's 1954 paper]; De Simone RVMP(07)
[renormalization proof]; Yuan CMP(07)
[nearly integrable Hamiltonian systems of infinite dimensions].
Kaons > see hadrons.
Kapitza-Dirac Effect > see diffraction.
Karlhede Classification > see lorentzian geometries; petrov classification.
Kasner Solution > see bianchi I models.
Kauffman Bracket Polynomial > see knot invariants.
KdV Equation / System > see integrable systems.
Kelvin's Circulation Theorem > see under Circulation.
Kemmer Equation
* Idea: A relativistic
(first-order, Dirac-like) field equation describing spin-0 and spin-1 particles.
@ References: Struyve et al PLA(04)qp/03 [paths
and Bohm interpretation].
Kennedy-Thorndike Test > s.a. special
relativity.
* Idea: A test of
the velocity independence of the speed of light.
@ References: Hils & Hall PRL(90)
[improved experiment].
Kepler Conjecture > ses sphere [packings].
Kepler Laws, Problem > see orbits in newtonian gravity; Runge-Lenz Vector.
Kerr State
* Idea: A type
of squeezed state.
@ References: Stobinska et al qp/06 [Wigner
function].
Kerr-Schild Solution > s.a. generation
of solutions; kerr-newman [boosted].
@ General references: Gergely & Perjés PLA(93)gq/02,
JMP(94)gq/02,
JMP(94)gq/02,
AdP(94)gq/02 [vacuum];
Sopuerta JMP(98)
[generalized stationary]; Coll et al GRG(01)
[generalized transformations]; Hildebrandt GRG(02)gq,
GRG(02)gq;
Ivanov PRD(05)gq/04,
Natorf GRG(05)gq/04
[and news, gravitational radiation]; Burinskii G&C(05)
[multi-particle]; Kerr a0706-in
[historical].
@ In modified theories: Macías & Camacho GRG(05)
[2+1, topologically massive].
Killing Form, Horizon, Spinor, Tensor, Vector > see killing fields.
Kinematics > s.a. Configuration
Space; special
relativistic.
* Idea: The study of the
possible configurations or states of a system, regardless of the dynamics (e.g.,
of the Hamiltonian).
Kinetic Theory > s.a. [statistical
mechanics; thermodynamics]; Boltzmann
Equation; gas; Maxwell-Boltzmann.
* Idea: The interpretation
of thermodynamics in terms of which T corresponds to the average kinetic
energy of molecules; Gave rise to statistical mechanics, and allows to derive
properties such as viscosity, thermal conduction, and diffusion in non-uniform
gases based on the solution
of the Maxwell-Boltzmann equations.
@ References: Kennard 38; Jeans 40; Brush 76 [history]; Chapman & Cowley
91;
de
Regt BJPS(96)
[and philosophy]; Brush 03; Loeb 04.
Kink > s.a. geons;
topological defects.
* Idea: A solution of a
field theory (with non-simply-connected target space) which cannot be smoothly
deformed to a constant field.
@ Gravitational: Shastri & Zvengrowski RVMP(91).
@ Topological fermions: Williams & Zvengrowski IJTP(77),
Faber FBS(01)ht/99; > s.a. particle
statistics; spinors in field theory.
Kirby Calculus > see 4D manifolds.
Kirby-Siebenmann Invariant
* Idea: An object in
H4(M; Z2),
which equals (index 
)mod
8 when
the intersection form is even.
KK-Theory > s.a. K-Theory.
* Idea: A bivariant
version of topological K-theory, useful in the index theory for elliptic pseudo-differential
operators.
@ References: Jensen & Thomsen 91.
Klein Bottle > see 2D manifolds.
Klein Geometry > s.a. geometry [history,
relationships]; Cartan Geometry.
* Idea: A conception
of geometry proposed in Felix Klein in 1872 with his Erlangen Programme,
in which a geometry is characterized by an
underlying set X and a group G of transformations acting
on it, that are to be considered
as equivalences; In modern terminology, if Euclidean geometry describes flat
Euclidean space, Klein geometry describes general homogeneous manifolds.
* Examples: If the
stabilizer group of an (arbitrary) element of X is denoted by H,
one can express X = G/H, and some examples are
– With positive-definite
metrics, S2 =
SO(3)/SO(2), E2 = ISO(2)/SO(2), and H2 =
SO(2,1)/SO(2);
– With Lorentzian-signature
metrics, dS3,1 = SO(4,1)/SO(3,1), 4D Minkowski
= E3,1 =
ISO(3,1)/SO(3,1), and AdS3,1 = SO(3,2)/SO(3,1).
> Online resources: Wikipedia page.
Klein Paradox > s.a. dirac
field theory.
* Idea: The
phenomenon by which a potential well or barrier in the Dirac equation can become
supercritical and emit positrons or electrons spontaneously if the potential
is strong enough.
@ General references: Bongaarts & Ruijsenaars AP(76)
[as many-particle problem]; Bakke & Wergeland PS(82);
Holstein AJP(98);
Calogeracos & Dombey IJMPA(99)qp/98,
CP(99)qp [rev];
Nitta et
al
AJP(99)
[simulations]; Bounames & Chetouani PLA(01)-a0712;
Krekora et al PRL(04)
[numerical
solutions]; Dragoman qp/07 [experiment
with graphene, phenomenon does not occur].
@ Variations: Grübl et al JPA(01)qp/02 [and
Bohmian trajectories]; Ghose et al PLA(03)qp [not
found for bosons]; De Leo & Rotelli PRA(06) [and potential barrier].
Klein-Gordon Fields > s.a. klein-gordon fields in curved spacetime; quantum klein-gordon fields.
KMS States > see spin models.
Knot Theory > s.a. knots in physics; knot invariants.
Kobayashi-Maskawa Matrix > see Cabibbo-Kobayashi-Maskawa.
Koch Curve > see fractals.
Kochen-Specker Experiment / Theorem > see experiments in quantum mechanics.
Kodama State > see Chern-Simons Function; loop quantum gravity; quantum gauge theory.
Kolmogorov Probabiity > see probability in physics.
Kolmogorov System or K-System >
s.a. lyapunov exponents; Mixing.
* Idea: A dynamical
system in which trajectories mix due to local instabilities.
$ Def: A dynamical system (X, 
, 
)
with positive Kolmogorov-Sinai entropy h.
* Relationships: It implies
mixing and local instability (positive lyapunov
exponents), and h 
h0 1
/ 
c.
* Examples: Bernoulli
shift; Discretized Bianchi IX.
@ References: in Zaslavsky et al 91.
Komar Integral > see energy in general relativity.
Kondo Problem
* Idea: A single
magnetic impurity in a non-magnetic material.
Kontsevich Integral > see integration.
Koopman-von Neumann Formalism > s.a. dissipative
systems [Koopman operator]; Yang-Mills
theories.
* Idea: A Hilbert space/operator
approach to classical mechanics; > see approaches to
classical mechanics.
> And quantization:
see approaches to quantum mechanics; canonical
quantum mechanics; geometric quantization.
Korteweg-de Vries Equation > see integrable systems.
Kottler Metric / Solution > see schwarzschild-de sitter; solutions with symmetries.
Kovalevskaya Top > see systems in classical mechanics.
KP Equation / Hierarchy > see integrable systems.
Kramers Equation
@ Methods: Zhdanov & Zhalij JPA(99)mp [separation of variables].
Kramers-Kronig Relations > see dispersion.
Krasnikov Tube > see wormholes.
Krein Space > see fock space [generalized]; quantum field theory techniques.
Kretschman Invariant > see riemann tensor.
Kronecker Delta
* Expansion:
The Kronecker delta nm,
where
n and m vary over N possible values, can be expanded
as nm =
N–1 
k=1N exp{2
i k(n–m)/N};
Proof:
For n = m, the exponential is 1 and the sum equals N;
For n m,
the sum is equal to (sum of all N-th roots of unity)n–m =
0.
Kronecker Index > see cohomology.
Kruskal Extension > s.a. schwarzschild.
* Idea: The maximally
extended Schwarzschild solution, obtained by introducing coordinates that extend
across the horizon.
$ Def: The Schwarzschild
metric, with line
element written in the form
ds2 = – (2M / r) e–r/2M du dv +
r2(d
2 +
sin2 d2)
.
where u:= t – r*, v:= t + r*,
and r*:= r + 2 M ln(r/2M–1)
is the tortoise coordinate.
@ General references: Kruskal PR(60);
in Birrell & Davies 82; Boersma PRD(97)
[identification].
@ Related topics: Gibbons NPB(86)
[elliptic interpretation, and quantum mechanics]; Gautreau IJMPA(99)
[Kruskal-Szekeres incompleteness??]; Qin gq/00 [causal
structure]; Varadarajan PRD(01)gq/00 [as
canonical variables].
Kuiper Belt > see solar system.
Kundt Spacetimes / Waves > s.a. chaotic
motion.
* Idea: Spacetimes
with a non-expanding, shear-free, twist-free, geodesic principal null congruence.
@ References: Griffiths et al CQG(04)
[type III, non-zero cosmological constant, generalized]; Fuster gq/05-in
[type III, with null Yang-Mills field].
Kunneth Formula / Theorem > see homology.
Kuratowski Lemma > see axiom of choice.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008