**Topics, K**

**K-Causality** > see causality conditions.

**k-Essence** > s.a. causality;
quintessence;
time in gravity.

* __Idea__: (Kinetic-energy-driven quintessence)
A scalar field with a Lagrangian of a special form, that in cosmology 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]; Yang & Gao CQG(11)-a1006 [phase-space
analysis].

> __In cosmology__: see bianchi-I models; cosmological
acceleration; dark energy.

**K-System** > see under Kolmogorov
System.

**K-Theory** > s.a. bundle [gerbes];
KK-Theory; tiling.

* __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; Wegge-Olsen 93 [also C*-algebra];
Husemoller 94; Blackadar 98 [operator
algebras].

@ __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's Lemma** > see Recurrence.

**Kac-Moody Algebra**

* __Idea__: An infinite-dimensional
Lie algebra, with generators satisfying

[*T*_{n}^{i}, *T*_{m}^{j}]
= i *f*^{ ijk}* T*_{m+n}^{k} +
*K* δ^{ ij} *m* δ_{m+n,0} ,

where *K* is an operator such that [*K*, *T*_{n}^{i}]
= 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 \(\langle\)*A*, *B*\(\rangle\) =
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);
Zhe-Xian 91; Ray BAMS(01)
[generalized]; Wassermann a1004-ln; Gómez et al JGP(12) [geometric approach].

@ __And physics__: Dolan PRL(81)
[2D chiral models], PLB(82)
[4D self-dual Yang-Mills]; Goddard & Olive ed-88; Fuchs LNP(97)ht [and
conformal field theory]; > s.a. types
of spacetime singularities.

**Kac-van Moerbeke Lattice** > see toda lattice.

**Kadanoff-Baym Equations** > s.a. Boltzmann Equation;
early-universe cosmology.

* __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-conf [stochastic interpretation].

**Kadomtsev-Petviashvili Equation**

* __Idea__: A completely integrable partial differential equation
used to describe non-linear wave motion; It generalizes the 1D Korteweg–de Vries (KdV) equation.

@ __References__: in Xu a1205-ch [algebraic approach].

> __Online resources__: see MathWorld page; Scholarpedia page;
Wikipedia page.

**Kähler Metric, Structure** > see symplectic structures.

**Kalb-Ramond Field** > see types of gauge theories.

**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 |∂^{2}*H*_{0}
/ ∂*I*_{i }∂*I*_{j}| ≠ 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]; Pöschel a0908-ln [detailed];
Dumas 14 [friendly introduction, mathematical];
most books on chaos.

@ __Related topics__: Gallavotti & Gentile CMP(02)mp/01 [invariant
tori]; Broer BAMS(04)
[Kolmogorov's 1954 paper]; De Simone RVMP(07)
[renormalization proof]; Yuan CMP(07)
[nearly integrable Hamiltonian systems of infinite dimensions].

@ __Generalizations__: Evans CMP(04)
[quantum analog]; Jo & Jong a1505 [for generalized Hamiltonian systems without action-angle variables].

> __Online resources__: see MathWorld page; Wikipedia page.

**Kaniadakis Framework** > see non-extensive statistical mechanics.

**Kaons** > see hadrons.

**Kapitza-Dirac Effect / Diffraction** > see diffraction.

**Kardashev Civilizations** > see civilizations.

**Karlhede Classification** > see petrov classification;
types of lorentzian geometries [classification].

**Karlhede Invariant** > see riemann tensor.

**Kasner Solution** > see bianchi I models.

**Kauffman Bracket Polynomial** > see knot invariants.

**Kawai-Lewellen-Tye
Relations** > see covariant quantum gravity.

**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]; Lipa et al a1203 [prospects for an experiment in low Earth orbit].

**Kepler Conjecture** > see sphere [packings].

**Kepler Laws, Problem** > see orbits
in newtonian gravity; Runge-Lenz Vector.

**Kerr Solution** > s.a. particles in kerr spacetimes.

**Kerr State**

* __Idea__: A type of squeezed state.

@ __References__: Stobińska et al PRA(08)qp/06 [Wigner
function].

** Kerr-Bolt, Kerr-de Sitter, Kerr-NUT, Kerr-Sen Solutions** > see modified
kerr solutions.

**Kerr-CFT Correspondence** > see kerr solution.

**Kerr-Schild Metric / Solution** > s.a. generation
of solutions; kerr-newman [boosted]; lorentzian geometry [flat deformation theorem].

* __Idea__: A spacetime metric
written as
a linear superposition of the flat spacetime (or de Sitter / anti-de Sitter spacetime) metric *η*_{ab} and a squared null vector *k ^{a}*, multiplied by some scalar function

*g*_{ab} = *η*_{ab} + *H
k*_{a}*k*_{b} .

* __ Examples__: The class includes the Kerr and Kerr-(A)dS solutions.

@ __General references__: Kerr & Schild in(65); 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 in(09)-a0706
[historical]; Bini et al IJGMP(10)-a1408.

@ __Generalizations__:
Vaidya & Bhatt Pra(74); Málek & Pravda CQG(11)-a1009 [with (A)dS background]; Málek CQG(14)-a1401 [extended Kerr-Schild spacetimes]; Gürses et al CQG(17)-a1603 [Kerr-Schild-Kundt metrics as universal metrics].

@ __Higher-dimensional__: Ortaggio et al CQG(09)-a0808,
AIP(09)-a0901.

@ __In modified theories__: Macías & Camacho GRG(05)
[2+1, topologically massive]; Ett & Kastor JHEP(11)-a1103 [in Lovelock gravity].

**Kervaire Problem** > see differentiable manifolds [classification of exotic spheres].

**Killing Fields / Vectors** > s.a. killing
tensors / forms.

**Killing Horizon ** > see horizons.

**Kinematics** > s.a. Configuration
Space; special-relativistic kinematics.

* __Idea__: The study of the
possible configurations or states of a system and relationships between them,
including possible motions and transformation laws under changes of reference frame, independently
of the dynamics (e.g., of the Hamiltonian).

* __Rem__: One example in which
there have been different opinions on whether a certain phenomenon is kinematical
or dynamical is the interpretation of the length contraction and time dilation
[> see kinematics of special relativity]; There
are also situations in which structures that are normally considered part of
the dynamics are treated as part of the kinematics [> see covariant
symplectic structures], or structures that are normally considered part of the kinematics
are treated as dynamical [> see quantum gravity].

@ __References__: Martínez 09 [I]; in Janssen SHPMP(09); Spekkens a1209-FQXi [kinematics and dynamics must yield to causal structure]; Curiel a1603 [and the structure of a physical theory].

> __Related topics__: see Dynamics; Motion.

**Kinetic Energy** > see energy.

**Kinetic Focus** > see lagrangian dynamics.

**Kinetic Theory** > s.a. statistical
mechanics; thermodynamics.

* __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.

@ __Books__: Kennard 38; Jeans 40; Chapman & Cowley
91;
Brush 03; Loeb 04.

@ __General references__: Beck JSP(10) [deterministic approach].

@ __Conceptual__: Brush 76 [history]; de
Regt BJPS(96) [and philosophy].

@ __Relativistic__: García-Perciante et al JNT(12)-a1007 [and microscopic
description of dissipation–heat flow and viscosity]; Sarbach & Zannias AIP(13)-a1303 [intro], CQG(14)-a1309 [geometric perspective]; Ringström 13, [CQG+(15)].

@ __Related topics__: Latyshev & Yushkanov TMP(10)-a1001 [for degenerate
quantum gases]; > s.a. Boltzmann
Equation; gas; Maxwell-Boltzmann Distribution; Transport Phenomena.

> __Online resources__:
see Wikipedia page.

**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.

@ __Other examples__: Alonso-Izquierdo & Mateos-Guilarte AP(12)-a1205 [(1+1)-dimensional scalar field theory models].

**Kinnersley Black Hole** > see black-hole thermodynamics; generating solutions to einstein's equation.

**Kirby Calculus** > see 4D manifolds.

**Kirby-Siebenmann Invariant**

* __Idea__: An object in
H^{4}(*M*; \(\mathbb Z\)_{2}),
which equals (index *ω*)_{mod
8} when
the intersection form is even.

**Kitaev Chain** > see geometric phase.

**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, S^{2} =
SO(3)/SO(2), E^{2} = ISO(2)/SO(2),
and H^{2} = SO(2,1)/SO(2);

– With Lorentzian-signature
metrics, dS^{3,1} = SO(4,1)/SO(3,1), 4D Minkowski
= E^{3,1} =
ISO(3,1)/SO(3,1), and AdS^{3,1} = SO(3,2)/SO(3,1).

> __Online resources__: see Wikipedia
page.

**Klein Paradox** > s.a. dirac field theory.

* __Idea__: In relativistic
quantum mechanics, the surprising result obtained by Oskar Klein in 1929, applying
the Dirac equation to electron scattering by a potential barrier (or well),
that if its height *V*_{0} is of the
order of the electron mass the barrier is nearly transparent,
and as *V*_{0} approaches infinity the
reflection diminishes and the electron is always transmitted (the particle can effectively
continue on by transforming into its antiparticle); In quantum field theory, the phenomenon
by which if the potential is strong enough it becomes
supercritical and emits positrons or electrons spontaneously.

@ __General references__: Klein ZP(29);
Bongaarts & Ruijsenaars AP(76)
[as many-particle problem]; Bakke & Wergeland PS(82);
Su et al JPA(93); Holstein AJP(98)jun;
Calogeracos & Dombey IJMPA(99)qp/98,
CP(99)qp,
Dombey & Calogeracos PRP(99) [rev];
Nitta et al AJP(99)nov [simulations];
Bounames & Chetouani PLA(01)-a0712;
Krekora et al PRL(04) [numerical solutions];
Dragoman PS(09)-qp/07 [experiment
with graphene, phenomenon does not occur]; Alhaidari PS(11)-a0907 [resolution];
Kononets FP(10);
Gerritsma et al PRL(11)-a1007 [quantum
simulation using trapped ions]; Payandeh et al ChPC(13)-a1305 [Krein quantization approach]; Truebenbacher a1704 [new approach].

@ __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]; Cardoso et al CJP(09)-a0905 [not
for massive bosons with non-minimal interactions]; Wagner et al PRA(10)
[bosonic analog]; De Leo & Rotelli JPA(11)-a1202 [tests in graphene]; Ghosh IJTP(14)-a1202 [with generalized uncertainty principle]; Dodaro a1312 [in the pilot-wave interpretation]; > s.a. Refraction [classical
analog in metamaterials].

> __Online resources__: see Wikipedia page.

**Klein-Gordon Fields** >
s.a. klein-gordon fields in curved spacetime; quantum
klein-gordon fields.

**KLS Model (Katz-Lebowitz-Spohn)** > see non-equilibrium
thermodynamics.

**KLT Relations** (Kawai-Lewellen-Tye) > s.a. unimodular gravity.

@ __References__: Kawai et al NPB(86).

**Klyachko Inequality** > see Contextuality.

**KMS States** > see spin models.

**Knee** > see cosmic rays.

**Knot Theory** > s.a. knots
in physics; knot invariants.

**Knowledge** > s.a. Epistemology; Explanation; Understanding.

@ __General references__: Josephson in(03)-a1307 [are scientific theories the result of the particular mathematical and experimental tools we use?]; Cottey a1102 [knowledge-inquiry and wisdom-inquiry in nuclear-physics textbooks]; Alexanian a1506 [William Oliver Martin's *The Order and Integration of Knowledge*].

@ __Knowledge and beliefs__: Andrews a1205 [knowledge and justification of beliefs]; Martins a1508 [beliefs about the real world and probabilistic knowledge].

**Knudsen Number** > s.a. Maxwell-Boltzmann Distribution.

* __Idea__: A dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale; It is used to distinguish situations in which statistical mechanics or the continuum approximation are better descriptions for a fluid.

> __Online resources__: see Wikipedia page.

**Kobayashi-Maskawa Matrix** > see Cabibbo-Kobayashi-Maskawa
Matrix.

**Koch Curve** > see fractals.

**Kochen-Specker Experiment / Theorem** > s.a. experiments
in quantum mechanics / realism; Topos.

* __Idea__: Usually interpreted
to imply that predictions of non-contextual hidden variable theories cannot
agree with Copenhagen quantum mechanics.

@ __General references__: Kochen & Specker JMM(67);
Lenard in(74); Peres JPA(91);
Gill & Keane JPA(96)qp/03 [geometric];
Hamilton JPA(00)
[obstruction-based approach]; Cabello et al PLA(05)qp [proof
in any *D* > 3]; Nagata JMP(05)
[inequalities]; Malley PLA(06)qp [implication];
Rudolph qp/06 [and
ontological models]; Brunet PLA(07)
[and a priori knowledge]; Straumann a0801 [simple proof];
Lisoněk et al PRA(14)-a1308 [simplest set of contexts]; Calude et al TMMP-a1402 [two geometric proofs]; Malley & Fine PLA(14)-a1407 [simplified]; Loveridge & Dridi a1511 [mathematical aspects of Mermin's proof]; Rajan & Visser a1708 [simplified geometrical proof].

@ __Experimental precision / nullification__: Meyer PRL(99)qp +
Mermin qp/99,
Clifton & Kent PRS(00)qp/99 ["nullification"];
Appleby PRA(02)qp/00, qp/01 ["nullification"
of "nullification"]; Cabello PRA(02)qp/01;
Breuer PRL(02)qp;
Appleby SHPMP(05)qp/03;
Peres qp/03/PRL;
Barrett & Kent SHPMP(04).

@ __Single particle__: Simon et al PRL(00)qp;
Cabello PRL(03)
[qubit]; Huang et al PRL(03) [photons, test];
D'Ambrosio et al PRX(13) [single-photon experiment].

@ __Generalized__: Cabello et al PLA(96), IJMPA(00)qp/99;
Peres FP(96)qp/95;
Aravind PRA(03)qp;
Hrushovski & Pitowsky SHPMP(04)qp/03-conf
[and Gleason's theorem]; Döring IJTP(05)qp/04 [for
von Neumann algebras]; Dowker & Ghazi-Tabatabai JPA(08)-a0711 [for
quantum measure theory]; Lisoněk et al a1401 [generalized parity proofs]; de Ronde et al SHPMP-a1404 [modal Kochen-Specker theorem, physical interpretation].

**Kodama State** > see Chern-Simons
Function; loop quantum gravity; quantum
gauge theory.

**Kolmogorov Backward / Forward Equation** > see fokker-planck
equation.

**Kolmogorov Probability** > see probability
in physics.

**Kolmogorov System or K-System** >
s.a. ergodic theory [ergodic hierarchy]; lyapunov exponents; Mixing.

* __Idea__: A dynamical
system in which trajectories mix due to local instabilities.

$ __Def__: A dynamical system (*X*, *μ*, *φ*)
with positive Kolmogorov-Sinai entropy *h*.

* __Relationships__: It implies
mixing and local instability (positive Lyapunov
exponents), and *h* ~ *h*_{0} ~
1 / *τ*_{c}.

* __Examples__: Bernoulli
shift; Discretized Bianchi IX.

@ __References__: in Zaslavsky et al 91.

**Kolmogorov-Sinai Entropy**

* __Idea__: The growth rate *h* of
the phase-space volume of a phase drop with time; By Liouville's theorem, for
a Hamiltonian system we have *h* = 0 if there is no coarse-graining; If *V* is a coarse-grained phase-space
volume, we estimate *V*(*t*)
= *V*_{0} e^{ht}, and define

*h*:= lim_{V0 → 0} lim_{t → ∞} (1/*t*)
ln *V*(*t*) .

* __Remark__: Notice that *h* is
not actually an entropy but the time derivative
of the entropy *S* ~ ln *V*;
It is related to the stability or instability (and random behavior) of the
system, and characterizes the rate of entropy production in a classical dynamical system.

* __And Lyapunov exponents__: Related by *h* = ∑_{i=1}^{n} *λ*_{i}^{†} [@ Pesin UMN 77].

@ __General references__: see Klimek & Lesniewski AP(96)
[non-commutative Connes-Størmer entropy].

@ __And chaos__:
Frigg BJPS(04); Kamizawa et al JMP(14) [relationship with entropic chaos degree and Lyapunov exponents].

@ __Related topics__: Bianchi et al a1709 [and growth of entanglement entropy for a quantum system].

**Komar Integral** > see energy in general relativity.

**Kondo Effect**

* __Idea__: A cooperative
many-body phenomenon where electrons in a metal interact via spin-exchange
with magnetic impurity atoms; The impurity
increases the scattering of electrons at the Fermi level, causing an anomalous
increase in resistance below a certain temperature; First observed in the 1930s and understood only three decades later, the Kondo effect attracted renewed interest with its realization in quantum dots.

@ __References__: news pw(13)aug [ferromagnetic].

> __Online resources__: see Wikipedia page.

**Kondo Problem**

* __Idea__: A single
magnetic impurity in a non-magnetic material.

@ __References__: Rajeev AP(10) [Lie-algebraic approach].

**Kontsevich Integral** > see integration.

**Koopman-von Neumann Formalism** > s.a. approaches
to classical mechanics.

* __Idea__: A Hilbert space/operator
approach to classical mechanics proposed by Koopman and von Neumann in the 1930s;
It was later shown that this formulation could also be written in a path-integral form.

@ __References__: Abrikosov et al AP(05)qp/04 [path-integral version, from dequantization]; Gozzi FP(10)-a0910-proc
[and supermetrics in time]; Gozzi & Pagani PRL(10)-a1006; Bondar et al PRL(12)-a1105 [Ehrenfest quantization and unification of quantum and classical mechanics]; Klein QS:MF(17)-a1705 [proposal of new phase space function].

> __Types of systems__:
see dissipative systems [Koopman operator]; electrodynamics; yang-mills
theories.

> __And quantization__:
see approaches to quantum mechanics; canonical
quantum mechanics; geometric quantization.

**Korteweg-de Vries Equation** > see types of integrable
systems.

**Kottler Metric / Solution** > s.a. schwarzschild-de
sitter; solutions with symmetries.

* __Idea__: The static form of the Schwarzschild-de Sitter metric, when written using Schwarzschild type coordinates.

**Kovalevskaya Top** > see systems
in classical mechanics.

**KP Equation / Hierarchy** > see integrable
systems.

**Krajewski Diagrams**

@ __References__: Stephan JMP(09)-a0809 [and
the standard model].

**Kramers Degeneracy Theorem**

* __Idea__: The energy levels of systems with an odd total number of fermions remain at least doubly degenerate in the presence of purely electric fields.

@ __References__: Roberts PRA(12)-a1208 [without appealing to eigenvectors of the Hamiltonian].

> __Online resources__: see Wikipedia page.

**Kramers Equation**

* __Idea__: A partial
differential equation, arising as a special form of the Fokker-Planck equation
used to describe Browian motion in a potential.

@ __Methods__: Zhdanov & Zhalij JPA(99)mp [separation of variables].

> __Online resources__:
see MathWorld page.

**Kramers-Kronig Relations** > see dispersion.

**Krasnikov Tube** > see wormholes.

**Kraus Representation** > see quantum open systems.

**Krein Quantization** > see Klein Paradox.

**Krein Space**

* __Idea__: An indefinite inner product space (*K*,
\(\langle\cdot,\cdot\rangle\), *J*), in which the inner product (*x*, *y*):=
\(\langle x, Jy\rangle\) is positive-definite and *K* possesses a majorant topology.

* __Physical motivation__: Krein spaces
appear in the study of unitary irreducible representations of the de Sitter
group, which can be used to classify elementary particles when modeled by fields
propagating on a de Sitter background; In Krein-space quantization, the negative norm states are unphysical, and are used as mathematical tools for regularizing the theory.

@ __General references__: Gazeau et al Sigma(10)-a1001 [in de Sitter quantum theories].

@ __Krein-space quantization__: Sojasi & Mohsenzadeh IJTP(12)-a1202 [and ultraviolet divergences of Green functions]; Pejhan et al AP(14)-a1204 [and Casimir effect].

> __Applications__: see fock space [generalized]; approaches to quantum gravity; black-hole radiation; Klein Paradox; modified approaches to QED; non-commutative geometry; regularization schemes; Weyl Algebra.

> __Online resources__: see Encyclopedia of Mathematics page; Wikipedia page.

**Kretschmann Scalar Invariant** > s.a. riemann
tensor; schwarzschild geometry.

$ __Def__: The curvature scalar quantity *K*:= *R*_{abcd} *R*^{abcd}.

* __Motivation__: It is used as a convenient rough measure of how relativistic a system is, because it increases with curvature and does not automatically vanish for a vacuum solution of general relativity (the diagnostic power of the Ricci scalar is limited for this reason).

@ __References__: Gkigkitzis & Haranas PhInt(14)-a1406 [for black holes, and singularities, entropy and information].

> __Online resources__: see Wikipedia page.

**Kron Reduction** > see graphs.

**Kronecker Delta
***

**Kronecker Index** > see cohomology.

**Kronecker Power / Product** > see matrices.

**Kruskal Extension** > s.a. schwarzschild
spacetime.

* __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

d*s*^{2} = – (2*M* / *r*)
e^{–r/2M} d*u* d*v* +
*r*^{2}(d*θ*^{2} +
sin^{2}*θ* d*φ*^{2})
,

where *u*:= *t* – *r**, *v*:= *t* + *r**,
and *r**:= *r* + 2 *M* ln(*r*/2*M*–1)
is the tortoise coordinate.

@ __General references__: Kruskal PR(60);
in Birrell & Davies 82; Boersma PRD(97)
[identification]; Lake CQG(10), a1202 [Kruskal–Szekeres completion].

@ __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]; Augousti et al EJP(12) [use for infalling observers, pedagogical]; Sbierski a1507, a1711-proc [C^{0} inextendibility].

**Kuiper Belt** > see solar
system.

**Kullback-Leibler Distance** > see entropy [relative entropy].

**Kummer Tensor Density**

@ __References__: Baekler et al AP(14)-a1403 [introduction, in electrodynamics and gravity].

**Kundt Spacetimes / Waves** > s.a. chaotic
motion.

* __Idea__: Spacetimes
with a non-expanding, shear-free, twist-free, geodesic principal null congruence.

* __Result__: Degenerate
Kundt spacetimes (the ones in which the preferred kinematic and curvature null
frames are all aligned) are the only spacetimes in 4 dimensions that
are not \(\cal I\)-non-degenerate,
so that they are not determined by their scalar polynomial curvature invariants.

@ __General references__: Griffiths et al CQG(04)
[type III, non-zero cosmological constant, generalized]; Jezierski CQG(09)
[and degenerate Killing horizons]; Coley et al CQG(09)-a0901
[degenerate]; McNutt et al CQG(13) [invariant classification].

@ __Types of matter__: Fuster AIP(06)gq/05 [type
III, with null Yang-Mills field]; Tahamtan & Svítek a1505 [with minimally-coupled scalar field].

@ __In higher dimensions__:
Podolský & Žofka CQG(09)-a0812; Podolský & Švarc CQG(13)-a1303 [explicit algebraic classification], CQG(13)-a1306 [physical interpretation using geodesic deviation]; Podolský & Švarc CQG(15)-a1406 [Weyl tensor algebraic structure].

@ __In other theories__:
Brännlund et al CQG(08)-a0807 [and
supersymmetry]; Chow
et al CQG(10)-a0912 [topologically
massive gravity].

**Kunneth Formula / Theorem** > see homology.

**Kuratowski Lemma** > see axiom of
choice.

**Kustaanheimo-Stiefel Transformation**

* __Idea__: A transformation that maps the non-linear and singular equations of motion of the 3D Kepler problem to the linear and regular equations of a 4D harmonic oscillator; It is used extensively in studies of the perturbed Kepler problem in celestial mechanics and atomic physics.

@ __References__: Kustaanheimo & Stiefel JRAM(65); Bartsch JPA(03)-phy/03 [geometric Clifford algebra approach]; Saha MNRAS(09)-a0803 [interpretation, and quaternion form].

> __Online resources__: see Encyclopedia of Mathematics page.

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23 dec 2017