Topics, L

L-Functions
@ References: Gelbart & Miller BAMS(04).

Lace Expansion > see ising model.

Ladder Operators > s.a. annihiliation and creation operators.
* Idea: A set of generalized raising/creation and lowering/annihilation operators.
@ References: Ben Geloun & Klauder a0906 [for continuous spectra, and coherent states].

Lagrange Bracket > see phase space.

Lagrange Spaces
* Idea: Spaces with non-linear connections.
@ References: Miron & Anastasiei 93.

Lagrange Structure > s.a. path integrals.
* Idea: A structure that is more general than the Lagrangian formalism in the same sense as Poisson geometry is more general than the symplectic one.

Lagrangian Formulation of a Theory > s.a. examples of lagrangian systems; higher-order lagrangians.

Lagrangian Observers > see Observers.

Lagrangian Submanifold > see symplectic structure.

Laguerre Polynomials > s.a. coherent states [generalized Laguerre functions].
* Generalized: Solutions y = L_n^(r)(x) of the equation xy'' + (r + 1 – x) y' + ny = 0.
@ References: Moya-Cessa a0809 [new expression].

Lamb Shift > see QED phenomenology.

Lambda Calculus
* Idea: May be regarded both as a programming language and as a formal algebraic system for reasoning about computation; Applied in the classical theory of computation, provides a computational model equivalent to the Turing machine.
@ References: van Tonder SICOMP(04)qp/03 [for quantum computation].

Lamé Equation, Functions, Operator
@ References: Ruijsenaars JMP(99), JMP(99) [generalized]; Maier JDE(04)m.CA/02 [algebraic solutions], PTRS(08)mp/03; Takemura m.CA/03, m.CA/04-in [eigenvalues]; Bena et al PS(06) [solitons, statistical mechanics].

Lanczos Lagrangian > see higher-order theories of gravity.

Lanczos Potential or Tensor

Landau Gauge > see gauge choices.

Landau Levels
@ References: Onofri IJTP(01)qp/00 [on a torus].

Landau Model > see quantum particles.

Landau Pole Problem > s.a. QCD; QED; vacuum.
* Idea: The fact that, in QED, the effective running coupling constant has a pole at a high, but finite value of the energy if renormalized.
* Status: In a non-Abelian theory, solved by confinement and asymptotic freedom. <--??--> The problem remains in the standard model of particle physics, because the coupling constants for the three interactions don't seem to join at the available energies, so we don't see that the behavior is taken over by the non-Abelian nature of the theory.
@ References: in Azam EJTP-a0806.

Landau-Ginzburg Model > see phase transitions; superconductivity.

Landau-Lifshitz Pseudotensor > see stress-energy pseudotensor.

Landauer's Erasure Principle > s.a. entropy; information.
* Idea: The loosely formulated notion that erasure of information produces heat, and at least the equivalent amount of entropy; The erasure of n bits of information must always incur a cost of k ln n in thermodynamic entropy.
@ References: Plenio & Vitelli CP(01); Bennett SHPMP(03); Jacobs qp/05 [from statistical mechanics]; Daffertshofer & Plastino PLA(05) [derivation from entropy], PLA(07) [in a gravitational field]; Maroney qp/07 [and thermodynamics of indeterministic operations]; Curilef et al PLA(08) [extension to non-equilibrium systems].

Landé g-Factor > s.a. Gyromagnetic Ratio.
@ References: Pfister & Pfister AJP(02)nov.

Landen Transformation > see Elliptic Functions [Jacobi].

Langevin Equation > see brownian motion; lattice field theory; quantum measurement.

Langlands Conjecture > see conjectures.

Langlands Program > see mathematics.

Laplace Equation / Operator

Laplace Transform > s.a. bessel functions.
$ Def: Given a function f, its Laplace transform is defined by Lf(k):= dx f(x) e^–kx.
@ References: Corinthios PRS(07) [generalized].
> Online resources: see MathWorld page.

Laplace Vector > see Runge-Lenz Vector.

Laplacian (Laplace-Beltrami) Operator > see laplace equation.

Lapse Function > see initial-value formulation of general relativity.

Large Deviations > see statistical mechanics.

Large Inductive Dimension > see dimension.

Large-N Expansion Method > see QCD.

Large-Number Hypothesis > s.a. anthropic principle; gravitational constant.
* Idea: In general, the claim that, if two combinations of parameters of fundamental physical theories and fundamental constants agree, it can't be a coincidence; Leads to predicting relationships between parameters; In Dirac's words, "The Large Numbers hypothesis asserts that all the large dimensionless numbers occurring in Nature are connected with the present epoch, expressed in atomic units, and thus vary with time. It requires that the gravitational constant G shall vary, and also that there shall be continuous creation of matter"; Related to what became later known as the anthropic principle
* Examples: The main ones involve the hierarchy problem for the fundamental interactions.
@ References: Dirac PRS(38), PRS(74); Carter pr(67)-a0710; Dirac PRS(79) [and modified general relativity]; Beesham IJTP(94), IJTP(94); Mena & Carneiro PRD(02)gq/01 [and holography]; Ray et al a0705/GRG [rev]; Funkhouser PRS(08)phy/06 [and cosmological constant scaling].

Larmor Formula > see radiation.

Larmor Frequency > see magnetism.

Laser > see optical technology; photons.

Laser-Ranging Experiments > see tests of general relativity.

LATOR Mission > see tests of general relativity with light.

Lattice

Lattice Field Theory > s.a. lattice gauge theories.

Lax Tensor Equation / Pair > see integrable system.

Lebesgue Integral > see integration.

Lebesgue Number of a Cover > see cover.

Lee-Yang Circle Theorem > see functions [polynomials].

Lefshetz Fixed-Point Theorem > see fixed-point theorems.

Left Translation in a Lie Group
$ Def: The left action of G on itself by L: G → G by L_g(h):= gh.

Legendre Polynomials

Legendre Transformation > s.a. lagrangian dynamics.
@ References: Zia et al AJP(09)jul [in classical mechanics, statistical mechanics and thermodynamics].

Leibniz Principle
* Idea: The principle of the identity of indiscernibles.
@ References: Castellani & Mittelstaedt FP(00) [in classical and quantum physics].

Lemaître-Tolman-Bondi Solutions > s.a. lensing.
* Idea: Spherically symmetric dust solutions used to model matter collapse.
* And cosmological acceleration: The solutions have also been used as models for an alternative explanation of the dimming of distant supernovae based on the idea that the acceleration may be due to the effect of inhomogeneities on the global expansion; According to this proposal we live in a special place in the universe, near the center of a large spherical void described by a Lemaitre-Tolman-Bondi metric.
@ General references: Sussman & Trujillo CQG(02)gq/01 [dust]; Ribeiro CPC(02)gq [code for null geodesics]; Sussman CQG(08)-a0709 [as dynamical system]; Van Acoleyen JCAP(08)-a0808.
@ Cosmological acceleration: Alnes et al JCAP(07)ap/05 [+ dust, do not account for acceleration]; Chuang et al CQG(08)ap/05; Paranjape & Singh CQG(06)ap [and averaging]; Garfinkle CQG(06)gq [dark-energy model]; Romano PRD(07); Enqvist GRG(08); García-Bellido & Haugboelle JCAP(08)-a0802 [local void]; Vanderveld et al a0904 [and smoothness]; Célérier et al a0906 [giant void not necessary].
@ Other phenomenology: Garfinkle a0908 [motion of galaxy clusters].
@ Perturbations: Zibin PRD(08) [scalar]; Waters & Nolan a0903 [self-similar, gauge-invariant]; Clarkson et al JCAP(09)-a0903; > s.a. metric matching.
@ Generalization: Lasky & Lun PRD(06)gq [with pressure].
@ Quantum: Kiefer et al PRD(07)gq [solution of Wheeler-DeWitt equation, and Hawking radiation]; Bojowald et al PRD(08)-a0806, PRD(09)-a0906 [in lqg]; Franzen et al a0908 [with positive cosmological constant].

Length, Length Space > see distances; geometrical operators in quantum gravity; riemannian geometry [length scales].

Length Contraction > see kinematics of special relativity.

Length of a Map
@ References: in Gromov 81.

Lennard-Jones Fluid / Potential > s.a. van der Waals Equation of State.
* Idea: A semi-empirical potential used to represent the interaction between two molecules; It is strongly repulsive for short separations, and weakly attractive for larger separations, with an overall form u(r) = u₀ [(r₀/r)¹² – 2(r₀/r)⁶]; Can be used to obtain a van-der-Waals-type equation of state for a fluid.
@ References: Khordad PhyA(08) [viscosity, integral equation method]; Romero-Bastida & Braun JPA(08) [perturbations, Lyapunov modes]; Papari et al PhyA(09) [thermal conductivity]; Celebonovic a0902-in [coexistence of phases, and application to astronomy].

Lens Spaces > see 3D manifolds.

Lense-Thirring Effect > see tests of general relativity with orbits.

Lensing > see gravitational lensing.

Lenz's Law > see electromagnetic field equations; electricity.

Lepage Form > see hamiltonian dynamics.

Lepage-Dedecker Formalism > see symplectic structures.

Leptogenesis > see early-universe cosmology.

Leptons > see particle types.

Leptoquark > see GUTs.

Letelier-Gal'tsov Metric > see cosmic strings.

Levi-Civita Connection > see affine connection.

Levi-Civita Spacetime > s.a. types of spacetimes.
@ General references: Magnon JMP(81) [euclidean version from Wick rotation]; Konkowski et al CQG(04)gq, gq/04-in, gq/04-in [quantum singularities]; da Silva et al JMP(06).
@ Higher-dimensional: Sarioglu & Tekin PRD(09)-a0901; Ponce de León MPLA(09)-a0904.

Levi-Civita Tensor > see under Alternating Tensor.

Levinson's Theorem > see scattering.

Lewis Metric
* Idea: A stationary cylindrically symmetric vacuum solution of Einstein's equation.
@ References: Herrera & Santos JMP(98)gq/97 [geodesics]; Gariel et al JMP(00) [mechanical interpretation of equations]; Davidson CQG(01) [as exterior for rotating cylinder]; da Silva et al CQG(02)gq [rotating shell source]; Ali NCB(04) [rotating cylindrical sources]; Gariel et al JMP(06) [and Painlevé transcendent III].

Lewis Phase > see geometric phase.

Lewis-Papapetrou Metric > see types of spacetimes.

Lichnerowicz Conditions > see metric matching.

Lichnerowicz Equation
* Idea: An equation which arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity.
@ References: Hebey et al CMP(08)gq/07 [results for compact manifolds].

Lichnerowicz Theorem > s.a. Hawking-Lichnerowicz Theorem.
* Idea: A smooth, stationary, topologically trivial and asymptotically flat vacuum solution of the Einstein equation is just flat; Extends to electrovac Kaluza-Klein.
$ Def: If (M, g) is topologically Euclidean, asymptotically flat and empty, and stationary, with the electromagnetic and matter current staticity conditions j_[a k_b] = 0 and u_[a k_b] = 0, where k is the stationary Killing vector field, j is the electric current vector and u the matter current vector appearing in J_matter^ab = u^a u^b + p^ab, then the staticity conditions F_[ab k_c] = 0 and k_[a;b k_c] = 0 will also hold.
@ References: Lichnerowicz 55; Carter in(73).

Lichnerowicz-Obata Conjecture > see group action.

Lie Algebra [including generalizations]

Lie Bracket, Lie Derivative

Lie Group > s.a. examples and representations.

Lie-Poisson Manifold
@ Quantization: Racanière m.DG/04.

Liénard-Wiechert Potentials > s.a. Robinson-Trautman Spacetimes [gravitational analog].
$ Def: The electromagnetic potentials for a charge moving with velocity u, normally written in the form

where s:= r – (r · u)/c, and [  ] stands for "retarded".
@ References: Panofsky & Phillips 62; Jackson 75; Heras AJP(96)apr [derivation]; Jackson IJMPA(02)hp; Gsponer phy/06, phy/06 [arbitrary acceleration, complete current density].

Lifshitz-Type Theories > s.a. Horava Gravity; renormalization.
* Idea: Field theories with different scaling in the time and space directions.
Original theory (Lifshitz 41): A simple scalar theory, proposed in a study of the behavior of critical points (such as the triple point) in condensed matter systems, with the usual Galilean invariance of Newtonian mechanics but without Lorentz invariance, in which Lorentz symmetry appears as an accidental symmetry at large distances; Quantum corrections, which become larger at larger distances, give rise to a Lorentz-invariant theory at distances much larger than the scale at which the theory is defined.
@ References: Lifshitz ZETP(41).

Lifting of a Map
$ Def: Given two maps f : Y → X and p: E → X, a lifting of f is a map g: Y → E, such that pg = f.

Light > s.a. photon.

Light Cone
@ References: in Klainerman & Nicolò book [properties of intersections]; Choquet-Bruhat et al CQG(09)-a0905 [light-cone (cross-section area) theorem].

Limit

Limit Cycle
* Idea: A closed phase curve, which represents oscillatory motion.
@ Method for finding: Delamotte PRL(93).

Lindblad Equation / Theory > s.a. dissipation.
* Idea: A master equation describing dissipative quantum dynamics / Theory of open/damped quantum systems.
@ Solutions: Nakazato et al PRA(06)qp [in Kraus representation]; Brodier & Ozorio de Almeida a0808/PRA [semiclassical, approximate].
> Generalizations: see Master Equation.

Lindelöf Property > see types of topologies.

Line, Line Bundle > see line.

Linear Space > see vector space.

Linearity in Physics > see electromagnetism; field theory; formulations of quantum mechanics; Non-Linear Quantum Mechanics; quantum field theory formalism and types; sigma models.

Link of a Vertex in a Simplicial Complex
$ Def: The union of all subsets not containing v of the simplices that intersect v.

Link Theory > s.a. knots (+ invariants + in physics); types of distances.
* Idea: May be algebraically described as certain morphisms in the category of tangles.
* Linking number: Has the properties lk(_{1 2}, ₃) = lk(₁, ₃) + lk(₂, ₃), and lk(^–1, ) = –lk(, ).
* Borromean rings: A set of three loops such that each pair is unlinked but the three are linked together.
* Whitehead link: Has zero helicity.
@ Invariants: Hillman 02; Jin & Zhang PhyA(04) [Jones polynomials for some links]; Akhmetiev JGP(05) [3-component links]; Chernov & Rudyak G&T(05)m.GT/03 [general theory]; Buck & Flapan JPA(07) [topological characterization].
@ In 4D: Bartel & Teichner G&T(99) [S²-links are null-homotopic]; Hirose T&A(03) [4D analog of torus links]; Bodecker & Hornig PRL(04) [for triples of closed 2-forms].
@ Related topics: Jacobsen & Zinn-Justin mp/01 [enumeration]; Chernov & Rudyak CMP(08) [of spacetime manifolds, causality and linking]; > s.a. Ribbons.
> In physics: see quantum systems.

Linkages
@ References: Tamburino & Winicour PR(66); Winicour JMP(68); Bramson PRS(75); Geroch & Winicour JMP(81); Ashtekar & Winicour JMP(82).

Linking Numbers > see Link Theory; topological field theories.

Liouville Equation / Theory > s.a. 2-dimensional gravity.
@ General references: Chambré JCP(52); Matsuno JMP(87); Teschner CQG(01) [rev]; Nakayama IJMPA(04)ht-MS & IJMPA(04); Jackiw TMP(06)ht/05 [2D, Weyl symmetry]; Banerjee et al a0807 [diffeomorphism anomaly]; Momeni & Afrazeh IJMPB-a0904 [rev].
@ Quantum: Bershadsky & Klebanov PRL(90) [path-integral quantization]; Jorjadze & Weigt NPB(01)ht [Moyal quantization]; Menotti JPCS(06)ht/05 [and semiclassical].
@ Generalizations: Tarasov Chaos(04)nl.CD/03 [fractional].
@ Related topics: Ferrari & Paturej PLB(08)mp/06 [and Brownian motion].

Liouville Structure
* Idea: A structure isomorphic to a cotangent vector fibration; An essential ingredient of every variational formulation of a physical theory.
@ References: Tulczyjew & Urbanski a0806.

Liouville Theorem > see phase space.

Lippmann-Schwinger Equation
@ References: de la Madrid JPA(06)qp, JPA(06)qp [rigged Hilbert space approach].

Lipschitz Condition > s.a. analysis; distance; distances between metrics.
$ Locally Lipschitz real function: A function f on an open set O R is said to be locally Lipschitz if, for each open set U O with compact closure, there is some constant K such that (we use the Euclidean norm in Rⁿ)

for all x, y U,   |f(x)–f(y)| K |x–y| .

* Remark: A "spacetime interpretation" is that the graph of the function is a spacelike hypersurface in flat 2D space, with speed of light 1/K.
$ For a map between manifods: Similarly, a map f of two manifolds is locally Lipschitz or C^1– if the coordinates of f(p) are locally Lipschitz functions of those of p.
$ For a map between metric spaces: If (X, d_X) and (Y, d_Y) are metric spaces, f : X → Y is Lipschitz if K such that

for all x, y X,   d_Y(f(x), f(y)) K d_X(x,y) .

$ Bi-Lipschitz map between metric spaces: There exist two constants K, K' such that

for all x, y X,   K' d_X(x,y) d_Y(f(x), f(y)) K d_X(x,y) .

Lipschitz Distance > see distance between metrics.

Liquid Matter > see condensed-matter physics; crystals [liquid crystals].

Little Group
@ Applications: Kim cm/96-in, ht/01-in [particle symmetries].
@ Special cases: Banerjee et al MPLA(01) [Maxwell-Chern-Simons, as gauge generator]; Scaria & Chakraborty CQG(02)ht [linearized gravity, as gauge generator]; Lindner et al JPA(03)ht [for massless particles].

Local Group, Local Void > see milky way galaxy

Local Pseudogroup of Transformations of a Manifold > see differentiable maps.

Locale
@ References: Pultr in(88).

Locality, Localizability > s.a. locality in quantum mechanics and in quantum field theory.

Locally Convex Topological Vector Space > see vector spaces.

Lock and Key Paradox > see special relativistic kinematics.

Loewner Equation > s.a. Scaling (relevant for scale-invariant problems).
@ References: Kager et al JSP(04)mp/03 [solutions]; Kager & Nienhuis JSP(04)mp/03 [stochastic evolution]; Gruzberg & Kadanoff JSP(04).

Logarithm > see asymptotic flatness [logarithmic transformations]; entropy [deformed log's].

Logic

London's Equations > see electricity.

Long Line > see paracompact.

Longevity > see Doomsday Argument.

Loop, Loop Group, Loop Space

Loop Diagram > see quantum field theory formalism.

Loop Quantum Gravity > s.a. quantum gravity in the connection representation
* Idea: The name given to an approach to quantum gravity that originated with loop-based solutions of the connection representation of canonical quantum gravity and the of loop representation of quantum gravity; The name has stuck despite the fact that states of quantum gravity in this approach are now based on spin networks or spin-foam models.

Loop Transform

Lorentz Equations
* Diffusionless: They correspond physically to diffusionless convection.
@ Diffusionless: Huang PLA(03) [periodic and homoclinic orbits].

Lorentz Force > s.a. classical particles; the gravitational one arises in gravitomagnetism.
* Idea: The force felt by a charged particle moving in an electromagnetic field.
* Non-relativistic version: Given by F_em = q (E + v B) .
* Relativistic version: If d = dt (1–v²)^1/2 is the proper time interval,

m d²x^a/d² = e F^a_b dx^b/d .

@ References: Buitrago EJP(95)-a0901 [and geometry of Minkowski space]; van Drie mp/00 [geometric, in terms of connections]; Buitrago & Hajjawi JMP(07)-a0901 [extended to spinors].

Lorentz Gauge > the correct spelling is Lorenz Gauge.

Lorentz Group > s.a. modifications and violation [theory]; phenomenology and tests.

Lorentz-Abraham Model > see particle models.

Lorentz-Dirac Equation / Force > see self-force.

Lorentz-FitzGerald Contraction > s.a. kinematics of special relativity.
* Idea: Objects moving with velocity v are contracted in the direction of motion by L(v) = L₀ / .
* And experiment: It predicts a fringe shift in a Michelson interferometer upon a change in speed wrt the ether (although not upon rotating the apparatus); this is not observed.
@ References: Lorentz Nat(21); in Panofsky & Phillips 62, p278–279; Brown AJP(01)oct-gq [history]; Martínez SHPMP(07); Barceló & Jannes FP(08)-a0705 [in condensed matter analogs of Lorentzian geometry].

Lorentzian Geometry / Metric / Structure

Lorenz Gauge > see gauge choices.

Lorenz System / Attractor
* Chaotic nature: The mechanism leading to chaotic behavior in the Lorenz system is well understood; Homoclinic connections induce a strange invariant set around the zero fluid motion stationary point, associated with a Smale horseshoe.
@ References: Luzzatto et al CMP(05)m.DS/04 [attractor, mixing nature]; Álvarez-Ramírez et al PLA(05) [control by destruction of homoclinic connections].

Loschmidt Echo
* Idea: A quantity M(t) that measures the sensitivity of a quantum system to a perturbation; Defined as the (squared) overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians; Thus, it also serves as a quantification of irreversibility in quantum mechanics, and has been extensively studied in connection with the problems of quantum chaos, quantum computation and decoherence.
@ References: Wisniacki PRE(03)nl/02; Cucchietti et al PRL(03) [and decoherence/purity]; Cucchietti qp/04-PhD [chaotic systems]; Combescure & Robert AHP(07)qp/05 [semiclassical limit]; Gorin et al PRP(06) [and fidelity decay]; Ares & Wisniacki a0908.

Lounesto Classification > see spinors.

Lovász Number > see graph.

Love Number > s.a. multipoles [black-hole polarizability].
* Idea: A tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field.
@ References: Binnington & Poisson PRD(09)-a0906 [relativistic theory].

Lovelock Gravity > s.a. higher-order gravity.
* Idea: A theory with higher-order curvature terms, which are considered important for the low-energy action of string theories; Lovelock terms in the action consist of the dimensionally-extended Euler densities, polynomial scalar densities in the Riemann curvature tensor with the property that their Euler-Lagrange derivatives contain derivatives of the metric only up to second order (generic polynomial scalar densities lead to Euler-Lagrange equations with derivatives of the metric of order four); The theory is free from ghosts.
@ General references: Deruelle & Fariña-Busto PRD(90); Müller-Hoissen NPB(90); Cnockaert & Henneaux CQG(05)ht [and BRST cohomology]; Charmousis LNP(09)-a0805 [and brane world and black holes]; Canfora et al PRD(09)-a0812 [spontaneous compactification, 4D general relativity with small cosmological constant]; Bogdanos PRD-a0902 [scalar densities, diagrammatic techniques]; Willison PRD-a0904 [and Weyl's tube formula].
@ Black holes: Dehghani & Pourhasan PRD(09)-a0903 [instability]; Cai et al a0911 [vanishing mass and entropy]; > s.a. Birkhoff's Theorem; black holes in modified theories [including Lovelock-Born-Infeld].
@ Black-hole thermodynamics: Cai PLB(04)ht/03; Aiello et al CQG(05)gq [+ Hoffmann-Infeld electromagnetism]; Correa-Borbonet BJP(05)ht [entropy]; Cai & Ohta PRD(06)ht [pure Lovelock gravity].
@ Other solutions: Crisóstomo et al PRD(04)ht [collapsing thin shells, Hamiltonian]; Willison gq/05-PhD [hypersurfaces and branes]; Nozawa & Maeda CQG(06)gq/05 [collapse, final fate]; Dehghani & Dayyani PRD(09)-a0903 [wormholes]; Dehgani & Farhangkhah PLB(09)-a0904; > s.a. bianchi I spacetimes.

Lovelock Identity > see tensor.

Lovelock Tensor
@ And curvature: Briggs gq/96 [quintic], gq/97 [quartic], gq/98 [more general]; > s.a. tensor [identities].

Lowering Operator > essentially the same as annihilation operator; s.a. Ladder Operators.

LSZ Formalism
* Idea: A method of reducing S-matrix elements to expressions given in terms of Green functions.
@ References: in Bjorken & Drell 65, #16.7.

Lukash Metric > see bianchi models [VII_h].

Luminosity Distance in Astronomy > see cosmological expansion.

Luminosity in High-Energy Physics
* Idea: The number of interactions per unit cross section and unit time; A measure of beam intensity and collimation.

Lump > see field theory [localized configuration].

Lunar Ranging > see tests of general relativity with orbits.

Luttinger-Liquid Theory
* Idea: The cornerstone for the description of one-dimensional materials; In it, an electron that carries both spin and charge cannot exist as a well-defined particle: It splits into two collective excitations, one carrying spin but not charge (the spinon), the other carrying charge but not spin (the holon).
@ References: Giamarchi Phy(09) [viewpoint on electron transport results].

Lutzky's Theorem > see symmetries.

Lyapunov Exponent

Lyra Geometry > s.a. bianchi I models; other bianchi models; black holes in modified theories.
@ Cosmology: Rahaman et al IJMPD(02)gq/07 [inhomogeneous models].

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