**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 / Statistics**
> 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 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-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.

**Kilonova**

* __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.

@ __Other examples__: Alonso-Izquierdo & Mateos-Guilarte AP(12)-a1205,
Alonso-Izquierdo PhyD-a1711 [(1+1)-dimensional scalar field models, and dynamics].

**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 EJP-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*];
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 a1806
[01-gadgets, subgraphs of a Kochen-Specker graph];
Elford & Lisoněk a1905 [analytical Kochen-Specker sets];
> 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* ~ *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 \(V(t)
= V_0^{~}\, {\rm e}^{ht}\), 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];
Bondar et al PRS-a1802
[and classical-quantum correlation dynamics].

> __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.

**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**

* __Expansion__: The
Kronecker delta δ_{nm},
where *n* and *m* vary over *N* possible values,
can be expanded as δ_{nm}
= *N*^{−1}
∑_{k = 1}^{N}
exp{2πi *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.

**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 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.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 29 jul 2019