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; unimodular gravity.

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

[Tni, Tmj] = i f ijk Tm+nk + K δ ij m δm+n,0 ,

where K is an operator 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 \(\langle\)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.

KAGRA > see gravitational-wave interferometers.

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 |∂2H0 / ∂IiIj| ≠ 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 / Statistics > see non-extensive statistical mechanics.

Kantowski-Sachs Models

Kaons > see hadrons.

Kapitza-Dirac Effect / Diffraction > see diffraction.

Kardar, Parisi, and Zhang Equation > see KPZ Equation.

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 types of 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].

Kent's Formulation of Relativistic Quantum Mechanics > see realism.

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 fields in kerr spacetimes.

Kerr-Newman 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 ηab and a squared null vector ka, multiplied by some scalar function H, or

gab = ηab + H kakb .

* 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]; > s.a. action for general relativity [distributional].
@ 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 / spinors / forms.

Killing Horizon > see horizons.

* Idea: The result of the mergers of binary compact objects; So called because their transients peak at a luminosity that is a factor approximately \(10^3\) higher than a typical nova; The observation of gravitational waves from a binary neutron star merger in 2017 proved the theory that the nuclear reactions happening within the kilonova, called r-process (rapid neutron capture process), were the source of most, if not all, of the universe's heavy metals such as gold, platinum and silver.
@ References: Metzger et al MNRAS(10)-a1001 [proposal].

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.
@ Related topics: Alonso-Izquierdo & Mateos-Guilarte AP(12)-a1205, Alonso-Izquierdo PhyD(18)-a1711 [(1+1)-dimensional scalar field models, and dynamics]; Bazeia et al a2009 [fermions interacting with kinklike structures in 2D].

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 H4(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 AdS3,1 = 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 EJP-a1704 [new approach]; Wang a2010 [resolution].
@ 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]; Wolpert a1711 [constraints on physical reality]; Leifer a1810-FQXi [knowledge as a scale-free network].
@ 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]; Ramanathan et al Quant(20)-a1807 [01-gadgets, subgraphs of a Kochen-Specker graph]; Elford & Lisoněk a1905 [analytical Kochen-Specker sets]; Acacio de Barros et al a2103 [assumptions]; > s.a. generalized bell inequalities.
@ 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 ~ h0 ~ 1 / τc.
* 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 \(V(t) = V_0^{~}\, {\rm e}^{ht}\), and define

h:= limV0 → 0 limt → ∞ (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=1n λi [@ 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 JHEP(18)-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]; Bondar et al PRS(18)-a1802 [and classical-quantum correlation dynamics]; Morgan AP(20)-a1902 [algebraic approach].
> Types of systems: see dissipative systems [Koopman operator]; electrodynamics; macroscopic quantum systems [hybrid]; 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.

KPZ Equation
* Idea: A non-linear stochastic partial differential equation, arguably the simplest possible equation of motion for the dynamics of an interface including all the ingredients for non-trivial growth: irreversibility, non-linearity, stochasticity, and locality; 2003, It was proposed in 1986 Kardar, Parisi, and Zhang, and is still the subject of intensive research by means of simulations, field theoretic and approximation methods.
@ References: in Prähofer PhD(03); Halpin-Healy & Takeuchi JSP(15)-a1505.

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:= Rabcd Rabcd.
* 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
* Expansion: The Kronecker delta δnm, where n and m vary over N possible values, can be expanded as δnm = N−1 k = 1N exp{2πi k(n>−m)/N}; Proof: For n = m, the exponential is 1 and the sum equals N; For nm, the sum is equal to (sum of all N-th roots of unity)n>−m = 0.

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

ds2 = − (2M / r) er/2M du dv + r2(dθ2 + sin2θ dφ2) ,

where u:= tr*, 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]; 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, JPCS(18)-a1711 [C0 inextendibility]; Ashtekar & Olmedo a2005 [quantum extension, properties].

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 EPJC(17)-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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