**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 JPA(09)-a0906 [for continuous spectra, and coherent states];
Cardoso et al PRD(17)-a1706,
CQG(18)-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* = L_{n}^{(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 *λ*^{n}
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**
> s.a. quantum systems with special potenmtials.

@

**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.

**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];
Landry & Hammad a1909 [effect of a gravitational field].

> __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/*N*_{f}
expansion, where *N*_{f} 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/*N*_{f} 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; 2012, Classical version confirmed using
trapped micrometer-sized colloidal particles; 2013, The principle has been
used to explain the equality of the psychological and thermodynamic arrows
of time; 2018, Quantum version confirmed.

@ __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];
Wolf LNCS-a1810 [and lossless data compression];
Frank LNCS(18)-a1901 [physical foundations];
Myrvold a2007.

@ __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];
Yan et al PRL(18)
+ Esposito Phy(18) [single-atom demonstration];
Riechers & Gu a2103
[the lower bound is unachievable for nearly every initial state, for any reliable reset mechanism].

@ __And gravity__:
Daffertshofer & Plastino PLA(07) [in a gravitational field];
Herrera Ent-a2003 [and gravitational radiation].

@ __Other variations__:
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 CMP(17)-a1510 [for Repeated Interaction Systems].

> __Online resources__:
see Wikipedia page.

**Landé g-Factor**
>s.a. Gyromagnetic Ratio.

*

@

**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,

d*x*/d*t* = *F*(*x*)
+ *D*^{1/2}
*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); Used to describe the dynamics of how equilibrium is reached.

@ __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];
de Oliveira a1912 [quantum version];
> 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 Transform**
> s.a. bessel functions.

$ __Def__: Given a function
*f*, its Laplace transform is defined by L *f*(*k*):=
∫_{0}^{∞}
d*x f*(*x*) e^{−kx}.

* __Inverse__: The inverse
of the Laplace transform is also known as the Bromwich Integral;
> see MathWorld page.

@ __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 d*t*
in *t* betwen nearby surfaces, and the elapsed proper time d*τ* along
the direction normal to the hypersurfaces, by d*τ* = *N* d*t*.

@ __References__: Cederbaum proc(15)-a1309 [level sets];
Volovik a1806
[negative lapse function and negative temperature with fermions].

**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.

*

@

>

**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];
focus Phy(18)
[2018 Nobel prize, optical tweezers and high-intensity, ultrashort optical pulses];
> 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 distribution
\(f(x)\), the average of the measured values approaches the expected value.

@ __References__:
Denker BAMS(13) [300 years].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Law of the Excluded Middle**
> s.a. logic [quantum].

$ __Def__: For any proposition,
either that proposition is true or its negation is true; It is the third
of the three classic laws of thought.

**Laws (Physical Laws)**
> s.a. physical theories [nature of laws];
structure of 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];
Levinson a1709 [definition of law];
> s.a. probability and statistics
in physics [laws from experimental data]; 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];
Chen a2006 [some fundamental laws are vague];
> 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 JHEP(17)-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: *G* → *G* by
L_{g}(*h*):= *gh*.

**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 Ent(17)-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.

@ __General 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];
Hess et al AMP(16)-a1605 [epistemology];
Gangopadhyay & Sinha Roy EPL(16)-a1608 [and the Page-Wootters mechanism];
Adami a1908,
a1908 [cannot be violated].

@ __And neutrino oscillations__: Formaggio et al PRL(16)-a1602
+ news sa(16)jul [over hundreds of km];
Naikoo et al PRD(19)-a1901 [three-flavoured oscillations];
Naikoo et al a1906 [in subatomic systems].

@ __Other applications__:
Hari Dass a1509 [weak vs strong measurements];
Chevalier et al PRA(21)-a2009 [violation with classical light];
Giacosa & Pagliara a2104 [decays of unstable systems].

> __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*) = *u*_{0}
[(*r*_{0}/*r*)^{12}
− 2(*r*_{0}/*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.

**Lense-Thirring Metric**

* __Idea__: A metric representing the slow-rotation,
large-distance approximation to the gravitational field outside a rotating massive body.

@ __References__: Baines et al a2006 [Painlevé-Gullstrand form].

**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];
Gicquaud a1911
[existence and uniqueness of solutions].

**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} *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} = *ρ*
*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);
Nelson PRD(10)-a1010 [in 4th-order gravity].

**Lichnerowicz-Obata Conjecture**
> see group action.

**Lie Algebra** [including
generalizations—Lie algebroids, superalgebras, Lie *n*-algebras, etc]

**Lie Group**
> s.a. examples and representations.

**Lie-Poisson Manifold**

@ __Quantization__: Racanière m.DG/04.

**Lie-Trotter Formula**
> see Exponential Function.

**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];
Wang & Hazzard PRX(20) [tighter bounds].

@ __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];
Cheneau & Sanchez-Palencia Phys(20) [viewpoint, for long-range interactions].

> __Online resources__: see
Tobias Osborne's blog post.

**Lieb-Thirring Inequalities**
> s.a. generalized particle statistics.

* __Idea__: Inequalities providing
upper bounds on sums of powers of the negative eigenvalues of a Schrödinger
operator in terms of integrals of the potential.

@ __References__: Lieb mp/00-en [rev];
Frank et al a1808 [improvement on the bounds].

> __Online resources__:
see Wikipedia page.

**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 JHEP(15)-a1508 [supersymmetric];
Bonanno et al a2010 [regularization].

**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 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 [areatheorem];
Grant AHP(11)-a1008 [area monotonicity properties and comparison results].

@ __Properties__: De Lorenzo & Perez PRD(18)-a1707 [in Minkowski spacetime, thermodynamics].

> __Fluctuations__: see origin
of black-hole entropy; semiclassical general relativity.

> __In analog spacetime systems__:
see Luttinger Liquid.

**Light-Front Quantization**
> see approaches to quantum field theory.

**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; arrow
of time in physical theories.

* __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 JPA(17)-a1608 [as a consequence of the linearity of probabilities];
Brezhnev a1807 [from quantum empiricism].

**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(*γ*_{1} \(\circ\)
*γ*_{2},
*γ*_{3})
= lk(*γ*_{1},
*γ*_{3})
+ lk(*γ*_{2},
*γ*_{3}),
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;
Lescop a2001
[framed links in rational homology 3-spheres, *Z* invariant].

@ __In 4D__: Bartel & Teichner G&T(99) [S^{2}-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.

**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;
topology in physics.

**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];
Barkley & Budd a1908 [Hausdorff dimension].

@ __Generalizations__: Tarasov Chaos(04)nl.CD/03 [fractional].

@ __Related topics__: Ferrari & Paturej PLB(08)mp/06 [and Brownian motion].

**Liouville Space**
> see mixed states in quantum theory.

**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 *U* ⊂ *O* with compact
closure, there is some constant *K* such that (we use the Euclidean norm
in \(\mathbb R\)^{n})

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 there exists a constant *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]; fluids;
statistical-mechanical systems.

**LISA**
> see space-based gravitational-wave interferometers.

**Lithium**
> see elements; standard cosmological model
[^{7}Li 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 Pseudogroup of Transformations of a Manifold**
> see differentiable maps.

**Local Sheet**
> see milky way galaxy.

**Local Void**
> see galaxy distribution.

**Locale**

@ __References__: Pultr in(88).

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

**Locally Convex Topological Vector Space**
> see vector spaces.

**Locally Homogeneous Space**
> see Homogeneous Manifold.

**Locally Rotationally/Spherically Symmetric (LRS) 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,
log_{b} *x*
= (log_{a} *x*)/(log_{a} *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 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.

**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*τ* = d*t*
(1−*v*^{2})^{1/2}
is the proper-time interval,

*m* d^{2}*x*^{a}
/ d*τ*^{2} = *e*
*F*^{a}_{b}
d*x*^{b}/d*τ* .

@ __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].

@ __History, derivations__: Yaghjian a1911 [Maxwell's derivation from Faraday's law];
Natiello & Solari a2102 [argument based on relational electromagnetism];
> s.a. history of physics.

@ __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*) = *L*_{0} /
*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 EPJA(18)-a1708 [measurement].

@ __History__: Brown AJP(01)oct-gq;
Moylan a1006
[arguments from moving charge distributions];
Johansson a2012 [consistent with compression of aether].

**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 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.

**LRS Spacetime** > see under Locally Spherically Symmetric Spacetime.

**LSZ Formalism / Reduction Formula**

* __Idea__: A method of reducing
S-matrix elements to expressions given in terms of Green functions.

@ __References__: in Bjorken & Drell 65, #16.7;
Collins a1904 [new approach].

**Lubricants**
> see Friction.

**Lukash Metric**
> see bianchi models [VII_{h}].

**Luminosity Distance**
> s.a. large-scale geometry of the universe [including
redshift-luminosity relation]; cosmological expansion.

@ __References__: Ivanov et al JCAP(18)-a1802 [general non-perturbative results];
Calcagni et al PLB(19)-a1904 [gravitational waves].

**Luminosity**

@ __References__: Cardoso et al PRD(18)-a1803 [maximum luminosity].

**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 SciPP(17)-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.

**Luxon**

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

**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].

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

send feedback and suggestions to bombelli at olemiss.edu – modified 5 may 2021