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
/ ∂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];
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.
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-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.
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.
@ 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 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
ds2 = − (2M / r) e−r/2M du dv + r2(dθ2 + sin2θ dφ2) ,
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];
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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13 mar 2021