Topics, L

@ 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 JPA(09)-a0906 [for continuous spectra, and coherent states]; Cardoso et al PRD-a1706, a1707 [mass ladder operators for scalar fields].

LAGEOS (LAser GEOdynamics Satellites) > s.a. phenomenology of torsion; tests of the equivalence principle; tests of general relativity with orbits.
> Online resources: see NASA Science Missions page.

Lagrange Bracket > see phase space.

Lagrange Points > see orbits in newtonian gravity [3-body problem].

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.

Lagrange-Poincaré Equations
@ References: Ellis et al JGP(11) [for field theory].

Lagrangian Formulation of a Theory > s.a. 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 = Ln(r)(x) of the equation xy'' + (r + 1 – x) y' + ny = 0.
@ References: Moya-Cessa a0809 [new expression].
> Online resources: see MathWorld page; PlanetMath page; Wikipedia page.

Lamb Shift > see QED phenomenology.

Lamb Waves
* Idea: Elastic waves propagating on solid plates, whose particle motion is in the plane of the direction of wave propagation and the plate normal.
> Online resources: see Wikipedia page.

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]; Arrighi & Dowek qp/06-proc [linear-algebraic].

Lambda Ring
* Idea: A commutative ring together with some operations λn on it behaving like the exterior powers of vector spaces; Introduced by Grothendieck.
@ References: Yau 10.
> Online resources: see Wikipedia page.

Lambda Symmetries
@ References: Cicogna a1102-proc [for dynamical systems, Hamiltonian and Lagrangian equations].

Lambert W Function > see quantum systems.

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-conf [eigenvalues]; Bena et al PS(06) [solitons, statistical mechanics].

Lanczos Algorithm > see operator theory.

Lanczos-Lovelock Lagrangian > see higher-order theories of gravity and types of higher-order theories; gravitational thermodynamics.

Lanczos Potential or Tensor

Landau Gauge > see gauge choices.

Landau Levels
* Idea: Quantum levels for charged particles in a magnetic field.
@ References: Onofri IJTP(01)qp/00 [on a torus]; Onorato PLA(12) [for 2D relativistic particles].
> Online resources: see Wikipedia page.

Landau Model / System > s.a. quantum particles.
* Idea: A charged quantum particle moving in a 2D plane in the presence of a background magnetic field perpendicular to the plane.
@ Generalizations: Ivanov TMP(08)-a0705-conf [superextension on a plane]; Dulat & Li EPJC(09)-a0802, Gangopadhyay et al PLA(15)-a1412 [in non-commutative phase space].

Landau Pole Problem > s.a. QCD; QED; vacuum.
* Idea: The fact pointed out by Lev Landau and collaborators in 1955 that, in QED, the effective running coupling constant α has a pole at a high, but finite value of the energy if renormalized; Or the energy scale at which the coupling constant of the theory becomes infinite.
* Approach: The original derivation of the singularity was based on the summation of one-loop diagrams for the vacuum polarization tensor for photons in perturbation theory, and the validity of the expansion was looked upon sceptically by many people; It was later realized, however, that this singularity appears at leading order in a 1/Nf expansion, where Nf is the number of species of electrons, or number of flavors, which implies that in the infinite-flavor limit this singularity is exact provided the perturbation series in the 1/Nf expansion converges.
@ References: in Azam ht/01, EJTP-a0806.
> Online resources: see Wikipedia page.

Landau-Ginzburg Model > see phase transitions; superconductivity.

Landau-Lifshitz Conjecture > see quantum measurements.

Landau-Lifshitz Pseudotensor > see stress-energy pseudotensor.

Landau-Zener Transition
* Idea: A transition between energy levels at an avoided crossing.
@ References: Shevchenko et al PRP(10) [Landau-Zener-Stückelberg interferometry].

Landauer's Erasure Principle > s.a. entropy; Erasure; information.
* Idea: The loosely formulated notion that erasure of information produces heat, and at least the equivalent amount of entropy; The erasure of each bit of information must always incur a cost of k ln 2 in thermodynamic entropy, or kT ln 2 in work; 2011, No demonstration of Landauer's Principle has succeeded so far; 2013, The principle has been used to explain the equality of the psychological and thermodynamic arrows of time.
@ General references: Landauer IBM(61); Plenio & Vitelli CP(01); Bennett SHPMP(03); Jacobs qp/05 [from statistical mechanics]; Daffertshofer & Plastino PLA(05) [derivation from entropy]; Maroney PRE(09)qp/07 [and thermodynamics of indeterministic operations]; Norton SHPMP(11) [on possible proofs]; news pw(12)mar [measurement].
@ Quantum: Hilt et al PRE(11)-a1004, Anders et al EPTCS(10)-a1006 [in the strong-coupling quantum regime]; Jakšić & Pillet JMP(14)-a1406; Peterson et al PRS(16)-a1412 [experimental demonstration of information to heat conversion]; Lorenzo et al PRL(15)-a1503 [in multipartite open quantum system dynamics].
@ Other variations: Daffertshofer & Plastino PLA(07) [in a gravitational field]; Curilef et al PLA(08) [extension to non-equilibrium systems]; Faist et al nComm(15)-a1211 [version with tighter, process-specific bound]; Reeb & Wolf NJP(14)-a1306 [improved version, and proof based on quantum statistical physics concepts]; Goold et al PRL(15)-a1402 [for generic non-equilibrium dynamics]; Hanson et al a1510 [for Repeated Interaction Systems].
> Online resources: see Wikipedia page.

Landé g-Factor >s.a. Gyromagnetic Ratio.
* Idea: The proportionality constant g appearing in the relationship μ = (e\(\hbar\)g/2mc) S, between the magnetic dipole moment μ of a particle and its spin S, where e is its charge and m its mass.
@ References: Pfister & Pfister AJP(02)nov.

Landen Transformation > see Elliptic Functions [Jacobi].

Landscape > see string-theory phenomenology; cosmological-constant problem.

Langevin Equation > s.a. brownian motion; lattice field theory; quantum measurement.
* Idea: A differential equation used to describe brownian motion in the Einstein-Smoluchowski theory,

dx/dt = F(x) + D1/2 ξ(t) ,

with F(x) = K(x)/mα = external force, D = kT/mα diffusion constant, α = friction coefficient, ξ(t) = stochastic source (e.g., gaussian white noise).
@ References: Beck & Roepstorff PhyA(87) [from deterministic dynamics]; Kleinert AP(01) [from the forward-backward path integral]; Frank JPA(04) [free electron gas]; Kleinhans et al PLA(05) [drift and diffusion coefficients]; Dunkel & Hänggi PRE(06)cm [from microscopic collisions, including relativistic]; Eliazar & Cohen PhyA(13) [degree of randomness]; Coffey & Kalmykov 12; Das et al IJMPA(15)-a1411 [path-integral approach]; > s.a. sub-quantum theories.
@ Variations: Ramshaw AJP(10)jan [discrete analog]; Satin & Gangal a1404 [on a fractal curve].
> Online resources: see Wikipedia page.

Langlands Conjecture > see conjectures.

Langlands Program > see mathematics.

Laniakea > The supercluster the Milky Way galaxy belongs to; > s.a. milky way.
@ References: Tully et al nat(14)sep; video yt(14)sep; > s.a. Wikipedia page.

Laplace Equation / Operator

Laplace Transform > s.a. bessel functions.
$ Def: Given a function f, its Laplace transform is defined by L f(k):= 0 dx f(x) ekx.
@ References: Corinthios PRS(07) [generalized]; Skarke AJP(13)jul-a1209 [as an involution]; Plastino & Rocca PhyA(13)-a1302 [q-Laplace (Tsallis-Laplace) transform].
> Online resources: see MathWorld page.

Laplace / Laplace-Runge-Lenz Vector > see under Runge-Lenz Vector.

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

Lapse Function > s.a. ADM formulation and initial-value formulation of general relativity.
* Idea: In a foliation of a spacetime by spacelike hypersurfaces labelled by a time function t of timelike gradient, the lapse N is the function that relates the infinitesimal change dt in t betwen nearby surfaces, and the elapsed proper time dτ along the direction normal to the hypersurfaces, by dτ = N dt.
@ References: Cederbaum proc(15)-a1309 [level sets].

LARES (LAser RElativity Satellite) > s.a. tests of general relativity with orbits.
* Idea: A satellite mission launched in February 2012; It will test some fundamental physics predictions and provide accurate measurements of the framedragging effect predicted by Einstein's theory of general relativity.
* Rem: 2013, The satellite is possibly the highest mean density orbiting body in the Solar System, and it provides therefore the best realization of a test particle ever reached experimentally and provides a unique possibility for testing the predictions of general relativity.
@ References: Ciufolini et al a1302-fs [and geodesic motion]; Ciufolini et al CQG(13)-a1310 [Monte Carlo simulations and systematic errors].

Large Deviations > see Extreme Value Statistics; statistical mechanics.

Large Inductive Dimension > see dimension.

Large-N Expansion Method in QCD ('t Hooft and Veneziano Limits) > s.a. QCD; QCD phenomenology; renormalization of gauge theories.
* Idea: A computational tool for QCD used for calculations of predictions of the theory at energies below 1 GeV, where the theory becomes strongly coupled; If one calculates the large-N limit of propagators and transition amplitudes for a non-Abelian gauge theory as one keeps the value of λ:= g2/N fixed, where g is the coupling constant, non-planar diagrams which contain higher-order quantum corrections drop out; The number N is the number of colors, whose physical value is 3, but expansions in 1/N often give good results even just keeping the leading terms.
@ 1/N expansion, large-N limit: 't Hooft NPB(74); Witten PT(80)jul; Das RMP(87); Veneziano ed; 't Hooft ht/02-conf, ht/04-conf; Moshe & Zinn-Justin PRP(03)ht [rev]; Makeenko ht/04-conf [rev]; Weinberg PRL(10) [large-N renormalizable effective theory]; Bouatta & Butterfield a1208 [conceptual understanding, beta-function behavior]; Sazdjian a1512-ln.
> Online resources: see Wikipedia page.

Large-Numbers Hypothesis > s.a. [not the same as Bernoulli's Law of Large Numers] / 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.
@ General references: Dirac PRS(38), PRS(74); Carter pr(67)-a0710; Dirac PRS(79) [and modified general relativity]; Brézin & Wadia ed-93; 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]; Jentschura AdP(14)-a1403.
@ Related numerologies: Page JCAP(11)-a1108 [age and size of the observable universe].
> Online resources: see Wikipedia page.

Larmor Formula > see radiation.

Larmor Frequency > see magnetism.

Lasers > s.a. optical technology; photons.
* Atom laser: A device in which millions of individual atoms propagate through space with minimal spreading, like photons in a coherent photon laser beam; 2014, Atom lasers are still in the early stages of research with much work to be done, but applications may include atom lithography, atom interferometry, and magnetometry.
* Laser cooling: Particles to be cooled are placed at the intersection of several laser beams, with their frequency set slightly below the energy of one of the particle's excited states; Particles will preferentially absorb photons from the laser beam propagating opposite to their initial velocity, and when the photon's momentum is transferred to the particle, the particle's velocity is reduced; When the particle decays back to the ground state it emits another photon in a random direction, and over many cycles of absorption and emission the net effect is a decrease in the particle's velocity; An apt analogy is slowing down a bowling ball by repeatedly hitting it with tennis balls.
@ Optical lasers: Focus Phy(14) [non-linear optics, ultraviolet lasers from optical harmonics].
@ Acoustic analog: news pw(10)feb; Mahhoob et al PRL(13) + Mendonça Phy(13); > s.a. sound.
@ Types, applications: news PhysOrg(14)may [brightest atom laser]; Jayich et al PRX(16) + Barry Phy(16) [laser cooling with ultrafast lasers]; > s.a. temperature [cooling].
> And gravity: see solutions of general relativity with matter.

Laser-Ranging Experiments > see tests of general relativity.

LATOR Mission > see light deflection.


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

Lattice Gas > s.a. gas [dipole gas]; mathematical conjectures [Riemann hypothesis].
* Idea: A class of models in which particles are located at sites in a lattice and their evolution consists in jumping between sites.
@ General references: Quastel & Valkó ARMA(13)-a1211 [diffusivity].
@ Quantum lattice gas: Meyer PRE(97) [physical processes].
@ Asymmetric simple exclusion process, ASEP: MacDonald et al Biopol(68); Gorissen et al PRL(12) + Parmeggiani Phy(12) [exact current statistics].
@ Related topics: Mirahmadi et al MPLA(15)-a1405 [attractive and repulsive forces].

Laves Lattices > see 2D ising model.

Law of Large Numbers
* Idea: When a large number of measurements are made of a quantity x with a probability dostribution f(x), the average of the measured values approaches the expected value.
@ References: Denker BAMS(13) [300 years].
> Online resources: see MathWorld page; Wikipedia page.

Laws (Physical Laws) > s.a. physics [nature of laws]; physical theories.
* Distinction: Physical laws describe the regularities in nature, while mechanisms explain them; > s.a. Explanations.
* Weinberg's laws of theoretical physics: First, The conservation of Information (you will get nowhere by churning equations); Second: Do not trust arguments based on the lowest order of perturbation; Third, You may use any degrees of freedom you like to describe a physical system, but if you use the wrong one, you'll be sorry [@ in Weinberg in(83)].
@ General references: Peres FP(80) [physicist's role]; Yamakawa & Kreinovich IJTP(99) [why second-order equations]; Frisch PhSc(04)dec [and initial conditions, electrodynamics]; Winsberg PhSc(04)dec [and statistical mechanics]; Crease pw(07)jul [laws expressing impossibilities]; Stenger 06 [comprehensibility]; Smolin a1201 [unification of laws and states]; Kak a1206 [observability and computability]; > s.a. probability and statistics in physics [laws from experimental data]; > s.a. Comprehensibility.
@ Nature of laws: Caticha AIP(02)gq/01 [as laws of inference]; Butterfield phy/04 [laws and models]; Davies in(07)qp [laws as software for universe and limitations]; Watson SHPSA(12) [Leibniz's account of the relation between laws and deductive systems]; Dorato & Esfeld a1411 [dispositionalism vs primitivism]; Inamori a1707 [observations and physical laws]; > s.a. mathematical physics.
@ Related topics: Lange PhSc(08)jan [changing laws]; dos Santos RG-a1405-proc + news tr(14)jun [using Second Life to simulate different laws of physics].
> Specific areas: see Dynamics; Newton's Laws / laws of black-hole dynamics; laws of thermodynamics.

Lax Tensor Equation / Pair > see integrable system.

LBNE (Long-Baseline Neutrino Experiment) > see neutrino mixing.

Leaf of a Foliation > see foliations.

Least-Squares Fit / Method > see statistics in physics.

Lebesgue Integral > see integration.

Lebesgue Number of a Cover > see cover.

Lee-Wick Finite Electrodynamics > s.a. modified QED.
* Idea: A U(1) gauge theory where a (gauge-invariant) dimension-6 operator containing higher-derivatives is added to the free Lagrangian of the U(1) sector..
@ References: Accioly et al a1012.

Lee-Wick Models
* Idea: Higher-derivative quantum field theories that are claimed to be unitary thanks to a peculiar cancellation mechanism.
@ References: Anselmi & Piva a1703 [new formulation].

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

Lee-Yang Theorem / Theory (Statistical Mechanics) > s.a. magnetism [imaginary magnetic fields].
* Idea: A theory of phase transitions for certain statistical mechanical models, based on the zeros of their partition functions.
@ References: Yang & Lee PR(52); Lee & Yang PR(52); Blythe & Evans BJP(03) [pedagogical account]; Bena et al IJMPB(05) [overview]; Iurato a1410 [historical roots].
> Online resources: see Wikipedia page.

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

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

Legendre Polynomials

Legendre Transform of a Sequence
* Idea: A linear, invertible transformation between infinite (or finite) sequences, where the transformation involves binomial coefficients.
> Online resources: see MathWorld page.

Legendre Transformation > s.a. lagrangian dynamics; schrödinger equation.
* In physics: A transformation between the space of configurations and generalized velocities and the phase space of a system, that maps the Lagrangian function on the former to the Hamiltonian function on the latter.
@ General references: Zia et al AJP(09)jul [in classical mechanics, statistical mechanics and thermodynamics]; Ornigotti & Aiello a1407/AJP [Faddeev-Popov method]; Jackson et al JPA(17)-a1612 [for qft].
@ In thermodynamics: Kalogeropoulos a1704 [non-extensive thermodynamics].
> Online resources: see MathWorld page.

Leggett Inequalities
@ References: Lapiedra & Socolovsky a0806 [and the arrow of time]; Bacciagaluppi AIP(09)-a0811 [Leggett's theorem without inequalities]; Socolovsky IJTP(09); Wechsler a0912 [general derivation, comment on assumptions]; Navascués PRA(14)-a1303 [violation by all entangled states].
> Online resources: see Wikipedia page.

Leggett-Garg Inequality > s.a. realism [macrorealism].
* Idea: A mathematical inequality satisfied by all macrorealistic physical theories (ones in which a macroscopic system is always in one or other of its macroscopically distinguishable states); Similarly to the Bell inequalities, in quantum mechanics the Leggett-Garg inequality is violated, meaning that the time evolution of a system cannot be understood classically.
@ References: Robens et al PRX(15)-a1404 [and the concept of classical trajectories]; Bacciagaluppi IJQF-a1409 [pilot waves and contextuality]; Maroney & Timpson a1412 [meaning]; Clemente & Kofler PRL(16)-a1509 [argument for retiring it]; Hari Dass a1509 [and weak vs strong measurements]; Formaggio et al PRL(16)-a1602 + news sa(16)jul [test with neutrino oscillations over hundreds of km]; Hess et al AMP(16)-a1605 [epistemology]; Gangopadhyay & Sinha Roy EPL(16)-a1608 [and the Page-Wootters mechanism].
> Online resources: see Wikipedia page.

Leibniz Algebroid
* Result: Each Nambu-Poisson manifold has associated a canonical Leibniz algebroid.
@ References: Ibáñez et al JPA(99)mp [proposal]; Jurčo & Vysoký JGP(15)-a1503 [connection, torsion and curvature].

Leibniz Bialgebra > see algebra.

Leibniz Principle > see Identity of Indiscernibles.

Leidenfrost Effect
* Idea: The phenomenon that makes water droplets skate across a frying pan; A thin layer of vapor forms beneath the water droplet and protects it from evaporating quickly, allowing it to levitate and skitter about freely; A similar effect is the one by which hot bodies sink faster in water.
@ References: news wired(11)jun; Celestini et al PRL(12) + news PhysOrg(12)jul.

Lemaître-Tolman-Bondi (LTB) Solutions > s.a. gravitational lensing; spherical symmetry; wormhole solutions.
* 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 Lemaître-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; Sussman a1001 [quasi-local integral scalar variables]; Giesel et al CQG(10) [in terms of Dirac observables]; Herrera et al PRD(10)-a1006 [symmetries, dissipative case]; Mattsson & Mattsson a1007 [role of shear in averaging].
@ 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 A&A(10)-a0906 [giant void not necessary]; Romano JCAP(10)-a0911, PRD(10)-a0912; Iorio JCAP(10)-a1005 [solar system constraints]; Marra & Pääkkönen JCAP(10)-a1009 [with cosmological constant, constraints]; Sussman CQG(11)-a1102 [back-reaction and effective acceleration]; Romano & Sasaki GRG(12); Célérier A&A(12)-a1108; Krasiński PRD(14) [mimicking acceleration]; Moffat a1608.
@ Thermodynamics: Chakraborty et al GRG(11)-a1006; Biswas et al a1106 [Hawking-like radiation from the dynamic horizon]; Sussman & Larena CQG(14)-a1310 and CQG+ [gravitational entropy proposals]; Mishra & Singh PRD(14) [canonical Weyl curvature entropy].
@ Phenomenology and fields: Garfinkle CQG(10)-a0908 [motion of galaxy clusters]; Bolejko et al JCAP(11)-a1102 [best fit to data sets]; Arakida JAA(12)-a1204 [local effects, not an explanation for the secular increase in the astronomical unit]; Zecca IJTP(14) [spin-0, 1/2 and 1 field equations]; > s.a. cosmology in general relativity; spin-3/2 field theories.
@ Perturbations: Zibin PRD(08) [scalar]; Waters & Nolan PRD(09)-a0903 [self-similar, gauge-invariant]; Clarkson et al JCAP(09)-a0903; Leithes & Malik CQG(15)-a1403 [gauge invariants]; Meyer et al JCAP(15)-a1412; > s.a. metric matching.
@ Generalization: Lasky & Lun PRD(06)gq [with pressure]; Fanizza & Tedesco PRD(15)-a1412 [inhomogeneous and anisotropic].
@ 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 CQG(10)-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; Lorentz-FitzGerald Contraction.

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.
@ General references: Khordad PhyA(08) [viscosity, integral equation method]; Papari et al PhyA(09) [thermal conductivity]; Čelebonović a0902-conf [coexistence of phases, and application to astronomy]; Yuhjtman a1501 [stability constant, estimate].
@ Related topics: Romero-Bastida & Braun JPA(08) [perturbations, Lyapunov modes]; Gómez & Sesma EPJD(12)-a1401 [scattering length].
> Online resources: see Wikipedia page.

Lens Spaces > see 3D manifolds.

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

Lensing > see gravitational lensing; lensing in specific model spacetimes; types of 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.

LETSGO (LEnse-Thirring Sun-Geo Orbiter)
* Idea: A proposed space-based mission involving the use of a spacecraft moving along a highly eccentric heliocentric orbit perpendicular to the ecliptic, whose mission is to accurately measure the solar angular momentum with frame-dragging.
@ References: Iorio AAstr(13)-a1104.

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-MGX, gq/04-proc [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.

Levitation > see acoustics; Leidenfrost Effect.

Lévy Flight > s.a. fractional calculus [in quantum mechanics].
* Idea: A random walk in which the step-lengths have a probability distribution that is heavy-tailed; The term was coined by Benoît Mandelbrot.
> Online resources: see MathWorld page; Wikipedia page.

Lévy Process / Walk
* Idea: A continuous-time analog of a random walk.
@ General references: Barndorff-Nielsen et al 01; Eliazar & Shlesinger PRP(13) [and other fractional motions].
@ Related topics: Salari et al a1310 [trivial non-chaotic map lattice asymptotically indistiguishable from a Lévy walk].
> Online resources: see Wikipedia page.

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]; Holst & Tsogtgerel CQG(13) [on compact manifolds with boundary].

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; The result extends to electrovac Kaluza-Klein theory.
$ Def: If (M, g) is topologically Euclidean, asymptotically flat and empty, and stationary, with the electromagnetic and matter current staticity conditions j[a kb] = 0 and u[a kb] = 0, where k is the stationary Killing vector field, j is the electric current vector and u the matter current vector appearing in Jmatterab = ρ ua ub + pab, then the staticity conditions F[ab kc] = 0 and k[a;b kc] = 0 will also hold.
@ References: Lichnerowicz 55; Carter in(73); Nelson PRD(10)-a1010 [in 4th-order gravity].

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.

Lieb-Robinson Bounds
* Idea: Bounds on the speed of signal propagation in discrete quantum-mechanical systems with local interactions, originally introduced in the 1970s to describe solid-state spin systems, which generalize the concept of relativistic causality beyond field theory.
@ General references: Lieb & Robinson CMP(72); Prémont-Schwarz & Hnybida PRA(10)-a1002 [new bounds]; Poulin PRL(10) [for general Markovian quantum evolution]; Schuch et al PRA(11)-a1010; Nachtergaele & Sims IAMP(10)oct-a1102 [review of applications]; Them PRA(14)-a1308 [tests]; Kliesch et al in(14)-a1408 [and the simulation of time evolution of local observables in lattice systems].
@ Specific systems: Cheneau et al nat(12)jan + news at(12)jan [1D quantum gas in an optical lattice, experiment]; Islambekov et al JSP(12)-a1204 [for the Toda lattice]; Barmettler et al PRA(12) [1D interacting Bose gas, model].
> Online resources: see Tobias Osborne's blog post.

Lieb-Thirring Inequalities > see generalized particle statistics.

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

\[ \phi = {e\over4\pi\epsilon_{_0}}\left[{1\over r-({\bf r}\cdot{\bf u})/c}\right]\quad {\rm and}\quad
{\bf A} = {\mu_{_0}e\over4\pi} \left[{{\bf u}\over r-({\bf r}\cdot{\bf u})/c}\right],\quad
{\rm or}\quad A_a = {e\over4\pi\epsilon_{_0}}\left({1\over cs},{{\bf u}\over s}\right), \]

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. hořava 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); He et al PRD(11)-a1107 [in higher dimensions, and 4D Lorentz invariance]; Alexandre IJMPA(11)-a1109 [intro, and particle physics]; Kikuchi PTP(12)-a1111 [scalar theory, restoration of Lorentz symmetry]; Faizal & Majumder AP(15)-a1408 [and the GUP]; Chapman et al a1508 [supersymmetric].

Lifting of a Map
$ Def: Given two maps f : YX and p: EX, a lifting of f is a map g: YE, such that pg = f.

Light > s.a. photon.

Light Cone > s.a. Paneitz Operator [volume of past light cone].
@ General references: in Klainerman & Nicolò 02 [properties of intersections].
@ Light-cone slices: Choquet-Bruhat et al CQG(09)-a0905 [area theorem]; Grant AHP(11)-a1008 [area monotonicity properties and comparison results].
> Fluctuations: see origin of black-hole entropy; semiclassical general relativity.
> In analog spacetime systems: see Luttinger Liquid.

Light Deflection

Lilienfeld Transition Radiation > see radiation.

Limit > s.a. lorentzian geometry [limits of spacetimes].

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 / A theory of open (damped) quantum systems.
@ Solutions: Nakazato et al PRA(06)qp [in Kraus representation]; Brodier & Ozorio de Almeida PLA(10)-a0808 [semiclassical, approximate]; Honda et al JMP(10)-a1004 [for a harmonic oscillator]; Brasil et al RBEF(13)-a1110 [derivation in a simple model].
> Generalizations: see Master Equation.

Lindelöf Property > see types of topologies.

Lindley Paradox > see probability theory.

Line, Line Bundle > see line.

Linear Algebra > see elementary algebra.

Linear Independence > see vectors.

Linear Space > see vector space.

Linearity in Physics > s.a. electromagnetism; field theory; formulations of quantum mechanics; Non-Linear Quantum Mechanics; quantum field theory formalism and types; sigma models; Superposition Principle.
@ References: Müller et al a1608 [as a consequence of the linearity of probabilities].

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; types of orders.
* Idea: May be algebraically described as certain morphisms in the category of tangles.
* Linking number: Has the properties lk(γ1 \(\circ\) γ2, γ3) = lk(γ1, γ3) + lk(γ2, γ3), and lk(γ–1, η) = –lk(γ, η).
* 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: 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]; Hillman 12.
@ 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: Buniy et al a1010-proc [examples]; > s.a. gravitational-wave solutions; quantum systems.

@ 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; Jackiw TMP(06)ht/05 [2D, Weyl symmetry]; Banerjee et al EPL(10)-a0807 [diffeomorphism anomaly]; Abadi et al IJTP(11)-a0904 [rev]; Menotti JPA(11)-a1106 [on the torus].
@ Quantum: Bershadsky & Klebanov PRL(90) [path-integral quantization]; Jorjadze & Weigt NPB(01)ht [Moyal quantization]; Menotti JPCS(06)ht/05 [and semiclassical]; Ambjørn & Budd NPB(14)-a1405 [geodesic distances].
@ 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 & Urbański a0806.

Liouville Measure
* Idea: The measure on the phase space for a dynamical theory induced by the symplectic structure.
@ References: used in Carroll & Tam a1008.

Liouville Theorem > see phase space.

Liouvillian Operator
* Idea: The operator governing the evolution of a density matrix in quantum theory.
@ References: Kakofengitis & Steuernagel a1410.
> Online resources: in the Wikipedia page on Liouville's Theorem.

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

Lipschitz Condition > s.a. analysis; Contraction Mapping; distance; distances between metrics; types of lorentzian geometries [Lipschitz metrics].
$ Locally Lipschitz real function: A function f on an open set O ⊆ \(\mathbb R\) is said to be locally Lipschitz if, for each open set UO with compact closure, there is some constant K such that (we use the Euclidean norm in \(\mathbb R\)n)

for all x, yU,   |f(x)–f(y)| ≤ K |xy| .

* 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 C1– if the coordinates of f(p) are locally Lipschitz functions of those of p.
$ For a map between metric spaces: If (X, dX) and (Y, dY) are metric spaces, f : XY is Lipschitz if there exists a constant K such that

for all x, yX,   dY(f(x), f(y)) ≤ K dX(x,y) .

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

for all x, yX,   K' dX(x,y) ≤ dY(f(x), f(y)) ≤ K dX(x,y) .

Lipschitz Distance > see distance between metrics.

Liquid Matter > see condensed-matter physics; crystals [liquid crystals]; fluids; statistical-mechanical systems.

LISA > see space-based gravitational-wave interferometers.

Lithium > see elements; standard cosmological model [7Li problem]; matter near black holes [lithium synthesis].

Little Group
@ General references: Başkal et al Symm(17)-a1707 [loop representation].
@ Applications: Kim cm/96-in, NPPS(01)ht [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].

Lobachevsky Geometry / Plane > see 2D geometry [hyperbolic] / 2D ising model.

Local Bubble > see solar system.

Local Group of Galaxies

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

Local Sheet > see milky way galaxy.

Local Void > see galaxy distribution.

@ References: Pultr in(88).

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

Localization / Localizability

Locally Convex Topological Vector Space > see vector spaces.

Locally Homogeneous Space > see Homogeneous Manifold.

Locally Spherically Symmetric Spacetime > see spherical symmetry in general relativity.

Locally Symmetric Spacetime > see types of lorentzian geometries.

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).

Log-Spectral Distance > see types of metric spaces.

Logarithm > s.a. asymptotic flatness [logarithmic transformations]; entropy [deformed logs]; operator theory [log of an operator].
* Change of base: In general, logb x = (loga x)/(loga b); In particular, Log x = (ln x)/(ln 10) = (Log e) (ln x).


Logotropic Fluid
* Idea: One with an equation of state of the form p = A ln(ρ/ρ*), where ρ is the mass density, ρ* a reference density and A the logotropic temperature.
@ References: Chavanis a1504/EPJP [and cosmology].

London's Equations > see electricity.

Long Line > see paracompact.

Longevity > see civilizations [Doomsday Argument].

Loop, Loop Group, Loop Space

Loop Diagram > see quantum field theory formalism.

Loop Quantum Cosmology > s.a. models and phenomenology.

Loop Quantum Gravity > s.a. connection representation and loop representation of quantum gravity.
* 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 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 Fem = q (E + v × B) .
* Relativistic version: If dτ = dt (1–v2)1/2 is the proper-time interval,

m d2xa/dτ2 = e Fab dxb/dτ .

@ General references: Buitrago EJP(95)-a0901 [and geometry of Minkowski space]; van Drie mp/00 [geometric, in terms of connections]; Garat gq/06 [as Fermi-Walker transport, in geometrodynamics]; Buitrago & Hajjawi JMP(07)-a0901 [extended to spinors].
@ Claimed contradiction with special relativity: Mansuripur PRL(12) + news pt(12)apr + comment Boyer AJP(12)nov, Griffiths & Hnizdo AJP(13)aug; news sci(13)feb [paradox resolved].

Lorentz Gas > see gas.

Lorentz Gauge > the correct spelling is Lorenz Gauge.

Lorentz Group > s.a. lorentz-symmetry modifications and violation; lorentz violation models, 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 / γ; In Lorentz's view this was a dynamical contraction, while in the currently accepted special-relativistic view it is a purely kinematical effect.
* And experiment: It predicts a fringe shift in a Michelson interferometer upon a change in speed with respect to the ether (although not upon rotating the apparatus); This is not observed.
@ General references: Lorentz Nat(21)feb; in Panofsky & Phillips 62, 278–279; Martínez SHPMP(07); Barceló & Jannes FP(08)-a0705 [in condensed-matter analogs of Lorentzian geometry]; Franklin EJP(10) [Bell's spaceships and rigid-body motion]; Rafelski a1708 [measurement].
@ History: Brown AJP(01)oct-gq; Moylan a1006 [arguments from moving charge distributions].

Lorentzian Geometry / Metric / Structure > s.a. types of lorentzian geometries.

Lorenz Gauge > see gauge choices.

Lorenz System / Attractor > s.a. Attractors; quantum computing [simulations].
* 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].
> Online resources: see MathWorld page; Wikipedia page.

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 PhD(04)qp [chaotic systems]; Combescure & Robert AHP(07)qp/05 [semiclassical limit]; Gorin et al PRP(06) [and fidelity decay]; Ares & Wisniacki PRE(09)-a0908 [and the local density of states]; Haikka et al PRA(12) [and measures of non-Markovianity, and criticality].
> Online resources: see Scholarpedia page.

Lounesto Classification > see spinors.

Lovász Local Lemma
@ References: Ying a1010 [generalization].

Lovász Number > see graph.

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

Lovelock Gravity

Lovelock Identity > see tensors.

Lovelock Tensor
@ And curvature: Farhoudi GRG(09)gq/95 [as generalized Einstein tensor]; Briggs gq/96 [quintic], gq/97 [quartic], gq/98 [more general]; > s.a. tensors [identities].

Lovelock Theorem
* Idea: A result giving an explicit and complete list of all symmetric, natural (0,2)-tensors on an arbitrary pseudo-riemannian manifold (X, g) that are divergence-free and whose coefficients depend on second derivatives of the metric; The Einstein tensor is the simplest example.
@ References: Navarro & Navarro JGP(11)-a1005 [simplified proof].

Lowering Operator > s.a. Ladder Operators; spherical harmonics.
* Idea: An operator acting on a set of states or functions labelled by a discrete parameter; > Similar to an annihilation operator, but without the particle interpretation, and the opposite of a Raising Operator.

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.

Lubricants > see Friction.

Lukash Metric > see bianchi models [VIIh].

Luminosity Distance in Cosmology > see large-scale geometry of the universe [including redshift-luminosity relation]; 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]; particle models.

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).
@ General references: Giamarchi Phy(09) [viewpoint on electron transport results]; Langmann LMP(10) [2D analog].
@ Related topics: Dubail et al a1705 [emergence of curved light cones].

Luttinger's Theorem > s.a. types of superconductors [cuprates].
* Idea: The volume enclosed by a material's Fermi surface is directly proportional to the particle density; Or, The number of electrons in a material is the same as the number of electrons in all of its atoms added together.
> Online resources: see Wikipedia page.

Lutzky's Theorem > see symmetries.

* Idea: A zero-mass particle, that moves at the speed of light; For example, the photon.

Lyapunov Exponents

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]; Chaubey & Shukla IJTP(13) [Bianchi models].

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