Topics, L
L-Functions
@ References: Gelbart & Miller BAMS(04).
Lace Expansion > see ising model.
Lagrange Bracket > see phase
space.
Lagrange Spaces
* Idea: Spaces with non-linear
connections.
@ References: Miron & Anastasiei 93.
Lagrange Structure > s.a. path
integrals.
* Idea: A structure that
is more general than the Lagrangian formalism in the same sense as Poisson geometry
is more general than the symplectic one.
Lagrangian Formulation of a Theory > s.a.
examples of lagrangian systems;
higher-order lagrangians.
Lagrangian Observers > see Observers.
Lagrangian Submanifold > see symplectic
structure.
Laguerre Polynomials > s.a. coherent
states [generalized Laguerre
functions].
* Generalized: Solutions
y = Ln(r)(x)
of the equation xy'' + (r + 1 – x) y' + ny = 0.
Lamb Shift > see QED phenomenology.
Lambda Calculus
* Idea: May be regarded
both as a programming language and as a formal algebraic system for reasoning
about computation; Applied in the classical theory of computation, provides
a computational model equivalent to the Turing machine.
@ References: van Tonder SICOMP(04)qp/03 [for
quantum computation].
Lamé Equation, Functions, Operator
@ References: Ruijsenaars JMP(99), JMP(99)
[generalized]; Maier JDE(04)m.CA/02 [algebraic
solutions], mp/03;
Takemura
m.CA/03, m.CA/04-in
[eigenvalues]; Bena et al PS(06)
[solitons, statistical mechanics].
Lanczos Lagrangian > see higher-order
theories of gravity.
Lanczos Potential or Tensor
Landau Gauge > see gauge choices.
Landau Levels
@ References: Onofri IJTP(01)qp/00 [on
a torus].
Landau Model > see quantum particles.
Landau Pole Problem > s.a. QCD; QED; vacuum.
* Idea: The fact that,
in QED, the effective running coupling constant 
has
a pole at a high, but finite value of the energy if renormalized.
* Status: In a non-Abelian
theory, solved by confinement and asymptotic freedom. <--??--> The problem
remains in the standard model of particle physics, because the coupling constants
for
the three interactions don't seem to join at the available energies, so we
don't see that the behavior is taken over by the non-Abelian nature of the
theory.
@ References: in Azam a0806.
Landau-Ginzburg Model > see superconductivity.
Landau-Lifshitz Pseudotensor > see stress-energy
pseudotensor.
Landauer's Erasure Principle > s.a. entropy;
information.
* Idea: The
loosely formulated notion that erasure of information produces heat, and at
least the equivalent amount of entropy; The erasure of n bits of
information must always incur a cost of k ln n in thermodynamic
entropy.
@ References: Plenio & Vitelli CP(01);
Bennett SHPMP(03);
Jacobs qp/05 [from
statistical mechanics]; Daffertshofer & Plastino PLA(05)
[derivation from entropy], PLA(07)
[in a gravitational field]; Maroney qp/07 [and
thermodynamics of indeterministic
operations]; Curilef et al PLA(08)
[extension to non-equilibrium systems].
Landé g-Factor > s.a. Gyromagnetic
Ratio.
@ References: Pfister & Pfister AJP(02).
Landen Transformation > see Elliptic
Functions [Jacobi].
Langevin Equation > see brownian
motion; lattice
field theory; quantum measurement.
Langlands Conjecture > see conjectures.
Langlands Program > see mathematics.
Laplace Equation / Operator
Laplace Transform > s.a.
bessel functions.
$ Def: Given a function
f, its Laplace transform is defined by Lf(k):= 
dx f(x)
e–kx.
@ References: Corinthios PRS(07) [generalized].
> Online resources:
see MathWorld page.
Laplace Vector > see Runge-Lenz
Vector.
Laplacian (Laplace-Beltrami) Operator > see laplace.
Lapse Function > see initial
value formulation of general relativity.
Large Deviations > see statistical
mechanics.
Large Inductive Dimension > see dimension.
Large N Method > see QCD.
Large Number Hypothesis > s.a.
anthropic;
gravitational constant.
* Idea: In general, the
claim that, if two combinations of parameters of fundamental physical theories
and fundamental constants agree, it can't be a coincidence; Leads to predicting
relationships between parameters; In Dirac's words, "The Large Numbers
hypothesis asserts that all the large dimensionless numbers occurring in Nature
are connected
with the present epoch, expressed in atomic units, and thus vary with time.
It requires that the gravitational constant G shall vary, and also
that there shall be continuous creation of matter"; Related to what became
later known as the anthropic principle
* Examples: The main
ones involve the hierarchy problem for the fundamental interactions.
@ References: Dirac PRS(38),
PRS(74);
Carter pr(67)-a0710;
Dirac PRS(79)
[and modified general relativity]; Beesham IJTP(94), IJTP(94);
Mena & Carneiro PRD(02)gq/01 [and
holography]; Ray et al a0705-GRG
[rev]; Funkhouser PRS(08)phy/06 [and
cosmological constant scaling].
Larmor Formula > see radiation.
Larmor Frequency > see magnetism.
Laser > see optical technology; photons.
Laser Ranging Experiments > see tests
of general relativity.
LATOR Mission > see tests
of general relativity with light.
Lattice
Lattice Field Theory > s.a.
lattice gauge theories.
Lax Tensor Equation / Pair > see integrable
system.
Lebesgue Integral > see integration.
Lebesgue Number of a Cover > see cover.
Lee-Yang Circle Theorem > see functions [polynomials].
Lefshetz Fixed Point Theorem > see fixed
point theorems.
Left Translation in a Lie Group
$ Def: The left action
of G on itself by L: G → G by Lg(h):=
gh.
Legendre Polynomials
Legendre Transformation > see
lagrangian dynamics.
Leibniz Principle
* Idea: The principle
of the identity of indiscernibles.
@ References: Castellani & Mittelstaedt FP(00)
[in classical and quantum physics].
Lemaître-Tolman-Bondi Solutions
* Idea: Spherically
symmetric dust solutions used to model matter collapse.
* And cosmological acceleration:
The solutions have also been used as models with which to test the idea that
the acceleration
may be due to the effect of inhomogeneities on the global, averaged expansion.
@ General references: Sussman & Trujillo CQG(02)gq/01 [dust];
Ribeiro CPC(02)gq [code
for null geodesics]; Sussman CQG(08)-a0709 [as
dynamical system].
@ Cosmological acceleration:
Alnes et al JCAP(07)ap/05 [+
dust, do not account for acceleration]; Chuang et al ap/05;
Paranjape & Singh CQG(06)ap [and
averaging]; Garfinkle CQG(06)gq [dark
energy model]; Romano PRD(07);
Enqvist GRG(08); Garcia-Bellido & Haugboelle a0802 [local
void].
@ Generalization:
Lasky & Lun PRD(06)gq
[with pressure].
@ Quantum: Kiefer et al PRD(07)gq [solution
of Wheeler-DeWitt equation, and Hawking radiation].
Length, Length Space > see distances; geometrical
operators in quantum gravity; riemannian geometry [length
scales].
Length Contraction > see kinematics of
special relativity.
Length of a Map
@ References: in Gromov 81.
Lennard-Jones Potential > s.a.
van der Waals Equation of State.
* Idea: A semi-empirical
potential used to represent the interaction between two molecules; It is strongly
repulsive for short separations, and weakly attractive for larger separations,
with an overall form u(r) = u0 [(r0/r)12 – 2(r0/r)6];
Can be used to obtain a van-der-Waals-type equation of state for a fluid.
@ References: Khordad PhyA(08) [viscosity, integral equation method].
Lens Spaces > see 3D manifolds.
Lense-Thirring Effect > see tests
of general relativity with orbits.
Lensing > see gravitational lensing.
Lenz's Law > see electromagnetic
field equations; electricity.
Lepage Form > see hamiltonian dynamics.
Lepage-Dedecker Formalism > see symplectic
structures.
Leptogenesis > see early universe cosmology.
Leptons > see particle types.
Leptoquark > see GUTs.
Letelier-Gal'tsov Metric > see cosmic strings.
Levi-Civita Connection > see affine
connection.
Levi-Civita Spacetime > s.a. types
of spacetimes.
@ References: Magnon JMP(81) [euclidean version from Wick rotation]; Konkowski et al CQG(04)gq, gq/04-in,
gq/04-in [quantum
singularities]; da Silva et al JMP(06).
Levi-Civita Tensor > see under Alternating
Tensor.
Levinson's Theorem > see scattering.
Lewis Metric
* Idea: A stationary cylindrically symmetric vacuum solution of Einstein's equation.
@ References: Herrera & Santos JMP(98)gq/97 [geodesics];
Gariel et al JMP(00)
[mechanical interpretation of equations]; Davidson CQG(01)
[as exterior for rotating cylinder]; da Silva et al CQG(02)gq [rotating
shell source]; Ali NCB(04)
[rotating cylindrical sources]; Gariel et al JMP(06)
[and Painlevé transcendent
III].
Lewis Phase > see geometric phase.
Lewis-Papapetrou Metric > see types
of spacetimes.
Lichnerowicz Conditions > see metric
matching.
Lichnerowicz Equation
* Idea: An equation which
arises from the Hamiltonian constraint equation for the Einstein-scalar field
system in general relativity.
@ References: Hebey et al gq/07 [results for compact manifolds].
Lichnerowicz Theorem > s.a. Hawking-Lichnerowicz
Theorem.
* Idea: A smooth, stationary,
topologically trivial and asymptotically flat vacuum solution of the Einstein
equation is just flat; Extends to electrovac Kaluza-Klein.
$ Def: If (M, g)
is topologically Euclidean, asymptotically flat and empty, and stationary,
with the electromagnetic and matter
current staticity conditions j[a 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 = 
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).
Lichnerowicz-Obata Conjecture > see group
action.
Lie Algebra [including generalizations]
Lie Bracket, Lie Derivative
Lie Group > s.a. examples and representations.
Lie-Poisson Manifold
@ Quantization: Racanière m.DG/04.
Liénard-Wiechert Potentials >
s.a. Robinson-Trautman Spacetimes [gravitational
analog].
$ Def: The electromagnetic
potentials for a charge moving with velocity u, normally written
in the form
where
s:= r – (r · u)/c,
and [ ] stands for "retarded".
@ References: Panofsky & Phillips 62; Jackson 75; Heras AJP(96)
[derivation]; Jackson IJMPA(02)hp;
Gsponer phy/06,
phy/06 [arbitrary acceleration, complete current density].
Lifting of a Map
$ Def: Given two maps f : Y → X and p: E → X,
a lifting of f is a map g: Y → E,
such that pg = f.
Light > s.a. photon.
Light Cone
@ References: in Klainerman & Nicolò book [properties of intersections].
Limit
Limit Cycle
* Idea: A closed phase
curve, which represents oscillatory motion.
@ Method for finding: Delamotte PRL(93).
Lindblad Equation / Theory
* Idea: A master equation
describing dissipative quantum dynamics / Theory of open/damped quantum systems.
@ References: Nakazato et al PRA(06)qp [solution
in Kraus representation].
Lindelöf Property > see types
of topologies.
Line, Line Bundle > see line.
Linear Space > see vector space.
Linearity in Physics > see electromagnetism;
field theory; formulations
of quantum mechanics; Non-Linear
Quantum Mechanics; quantum field theory formalism and types;
sigma models.
Link of a Vertex in a Simplicial Complex
$ Def: The union of
all subsets not containing v of the simplices that intersect v.
Link Theory > s.a. knots (+ invariants + in
physics); types of distances.
* Idea: May be algebraically described as certain morphisms in the
category of tangles.
* Linking number: Has
the properties lk(
1 
2, 3)
= lk(1,
3)
+ lk(2,
3),
and lk(–1,
) = –lk(,
).
* Borromean rings: A set of three loops such that each pair is unlinked
but the three are linked together.
* Whitehead link: Has zero helicity.
@ Invariants:
Hillman 02; Jin & Zhang PhyA(04)
[Jones polynomials for some links]; Akhmetiev JGP(05)
[3-component links]; Chernov & Rudyak G&T(05)m.GT/03 [general
theory]; Buck & Flapan JPA(07)
[topological characterization].
@ In 4D: Bartel & Teichner G&T(99) [S2-links
are null-homotopic];
Hirose T&A(03) [4D analog of torus links]; Bodecker & Hornig PRL(04)
[for
triples of closed 2-forms].
@ Related topics: Jacobsen & Zinn-Justin mp/01 [enumeration];
Chernov & Rudyak CMP(08) [of spacetime manifolds, causality and linking]; > s.a. Ribbons.
> In physics:
see quantum systems.
Linkages
@ References: Tamburino & Winicour PR(66); Winicour JMP(68); Bramson PRS(75); Geroch & Winicour
JMP(81); Ashtekar & Winicour JMP(82).
Linking Numbers > see Link Theory; topological
field theories.
Liouville Equation / Theory > s.a. 2-dimensional
gravity.
@ General references: Chambré JCP(52);
Matsuno JMP(87);
Teschner
CQG(01)
[rev]; Nakayama IJMPA(04)ht-MS & IJMPA(04);
Jackiw TMP(06)ht/05 [2D,
Weyl symmetry].
@ Quantum: Bershadsky & Klebanov PRL(90)
[path integral quantization]; Jorjadze & Weigt
NPB(01)ht [Moyal
quantization]; Menotti ht/05-in
[and semiclassical].
@ Generalizations: Tarasov Chaos(04)nl.CD/03 [fractional].
@ Related topics: Ferrari & Paturej mp/06 [and
Brownian motion].
Liouville Structure
* Idea: A structure
isomorphic to a cotangent vector fibration; An essential
ingredient of every variational formulation of a physical theory.
@ References: Tulczyjew & Urbanski a0806.
Liouville Theorem > see phase
space.
Lippmann-Schwinger Equation
@ References: de la Madrid JPA(06)qp,
JPA(06)qp [rigged
Hilbert space approach].
Lipschitz Condition > s.a. analysis;
distance; distances
between metrics.
$ Locally Lipschitz real
function:
A function f on an open set O 
R is
said to be locally Lipschitz if, for each open set U O with
compact closure, there is some constant K such that (we use the Euclidean
norm in Rn)
for all x, y 
U, |f(x)–f(y)|
K |x–y|
.
* Remark: A
"spacetime interpretation" is that the graph of the function is
a spacelike hypersurface in flat 2D space, with speed of light 1/K.
$ For a map between manifods:
Similarly, a map f of two manifolds
is
locally Lipschitz or 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 : X → Y is
Lipschitz if 
K such
that
for all x, y X, 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, y X, 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].
Little Group
@ Applications: Kim cm/96-in, ht/01-in
[particle symmetries].
@ Special cases: Banerjee et al MPLA(01)
[Maxwell-Chern-Simons, as gauge generator]; Scaria & Chakraborty CQG(02)ht [linearized
gravity, as gauge generator]; Lindner et al JPA(03)ht [for
massless particles].
Local Pseudogroup of Transformations of a Manifold > see
differentiable maps.
Locale
@ References: Pultr in(88).
Locality, Localizability > s.a. locality
in quantum mechanics and in quantum field theory.
Locally Convex Topological Vector Space > see vector
spaces.
Lock and Key Paradox > see special relativistic
kinematics.
Loewner Equation > s.a. Scaling (relevant for scale-invariant
problems).
@ References: Kager et al JSP(04)mp/03 [solutions];
Kager & Nienhuis JSP(04)mp/03 [stochastic
evolution]; Gruzberg & Kadanoff
JSP(04).
Logarithm > see asymptotic
flatness [logarithmic transformations]; entropy [deformed
log's].
Logic
London's Equations > see electricity.
Long Line > see paracompact.
Loop, Loop Group, Loop Space
Loop Diagram > see quantum field
theory formalism.
Loop Quantum Gravity > s.a. quantum
gravity in the connection representation
* Idea: The name given
to an approach to quantum gravity that originated with loop-based solutions of
the connection representation of canonical quantum
gravity and the of loop representation of quantum gravity; The
name has stuck despite the fact that states of quantum gravity in this approach
are now based on spin networks or spin
foam models.
Loop Transform
Lorentz Equations
* Diffusionless: They
correspond physically to diffusionless convection.
@ Diffusionless: Huang PLA(03) [periodic and homoclinic orbits].
Lorentz Force > 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 d
=
dt (1–v2)1/2 is
the proper
time interval,
m d2xa/d2
= e Fab dxb/d .
@ References: van Drie mp/00 [geometric,
ito connections].
Lorentz Gauge > the correct spelling is Lorenz
Gauge.
Lorentz Group > s.a. modifications
and violation [theory]; phenomenology
and tests.
Lorentz-Abraham Model > see particle models.
Lorentz-Dirac Equation / Force > see self-force.
Lorentz-FitzGerald
Contraction > s.a. kinematics of special
relativity.
* Idea: Objects moving
with velocity v are contracted in the direction of motion
by L(v) = L0 / .
* And experiment:
It predicts a fringe shift in a Michelson interferometer upon a change
in speed wrt the ether (although not upon rotating the apparatus); this
is not observed.
@ References: Lorentz Nat(21); in Panofsky & Phillips
62, p278–279;
Brown AJP(01)gq [history];
Martínez SHPMP(07);
Barceló & Jannes FP(08)-a0705 [in
condensed matter analogs of Lorentzian geometry].
Lorentzian Metric / Structure
Lorenz Gauge > see gauge
choices.
Lorenz System / Attractor
* Chaotic nature: The mechanism leading to chaotic behavior in the Lorenz system
is well understood; Homoclinic connections induce a strange invariant
set around the zero fluid motion stationary point, associated with
a Smale horseshoe.
@ References: Luzzatto et al CMP(05)m.DS/04 [attractor,
mixing nature]; Álvarez-Ramírez et al PLA(05)
[control by destruction of homoclinic
connections].
Loschmidt Echo
* Idea: A quantity M(t)
that measures the sensitivity of a quantum system to a perturbation; Defined
as the (squared) overlap of two wave functions evolved from the same initial
state but with slightly different
Hamiltonians;
Thus, it also serves as a quantification of irreversibility in quantum mechanics,
and has been extensively studied in connection
with the problems of quantum chaos, quantum computation and decoherence.
@ References: Wisniacki PRE(03)nl/02;
Cucchietti et al PRL(03)
[and decoherence/purity];
Cucchietti qp/04-PhD
[chaotic systems]; Combescure & Robert AHP(07)qp/05 [semiclassical
limit]; Gorin et al PRP(06)
[and fidelity decay].
Lounesto Classification > see spinors.
Lovász Number > see graph.
Lovelock Gravity > s.a. Birkhoff's
Theorem; [higher-order gravity].
* Idea: A theory with
higher-order curvature terms, which are
considered important for the low-energy action of string theories; Lovelock
terms in the action consist of the dimensionally extended Euler densities,
polynomial scalar densities in the Riemann curvature tensor with the property
that their Euler-Lagrange derivatives contain derivatives
of the metric only up to second order (generic polynomial
scalar densities lead to Euler-Lagrange equations with derivatives of the
metric of order four); The theory is free from ghosts.
@ General references: Deruelle & Fariña-Busto PRD(90);
Müller-Hoissen
NPB(90);
Cnockaert & Henneaux CQG(05)ht [and
BRST cohomology]; Charmousis a0805-ln
[and brane world and black holes].
@
Black hole thermodynamics: Cai PLB(04)ht/03;
Aiello et al CQG(05)gq [+
Hoffmann-Infeld electromagnetism]; Correa-Borbonet BJP(05)ht [entropy];
Cai & Ohta PRD(06)ht [pure
Lovelock gravity].
@ Phenomenology: Crisóstomo et al PRD(04)ht [collapsing
thin shells, Hamiltonian]; Willison gq/05-PhD
[hypersurfaces and branes]; Nozawa & Maeda CQG(06)gq/05 [collapse,
final fate]; > s.a. black holes in modified theories [including
Lovelock-Born-Infeld].
Lovelock Identity > see tensor.
Lovelock Tensor
@ And curvature: Briggs gq/96 [quintic],
gq/97 [quartic],
gq/98 [more
general]; > s.a. tensor [identities].
Lowering Operator > essentially the same as annihilation
operator.
LSZ Formalism
* Idea: A method of reducing
S-matrix elements to expressions given in terms of Green functions.
@ References: in Bjorken & Drell 65, #16.7.
Lukash Metric > see bianchi models [VIIh].
Lump > see field theory [localized configuration].
Lunar Ranging > see tests
of general relativity with orbits.
Lutzky's Theorem > see symmetries.
Luminosity in High Energy Physics
* Idea: The number
of interactions per unit cross section and unit time; A measure of beam intensity
and collimation.
Lyapunov Exponent
Lyra Geometry > s.a. bianchi
I models; black holes in modified theories.
@ Cosmology: Rahaman et al IJMPD(02)gq/07 [inhomogeneous
models].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008