* Idea: The chaotic map u
u –
1 if u 
1
and u–1 – 1
if
u < 1, which appears in mixmaster dynamics.Topics, F
F-Theory > see M-theory.
Factorial Function > s.a. Stirling's
Formula.
$ Def: The factorial
of a positive integer n is n! := 1 · 2 · 3 · ... · n.
* Extensions: 0! = 1,
and for other numbers use the gamma function.
* Derived functions: The
left factorial function, !n =
0! + 1! + 2! + ... + (n–1)! There is a conjecture that n does
not divide !n for any n > 2 [> MathPages page].
@ Approximations: Memin AJP(83),
Fletcher AJP(86).
Factor > see observable.
Factorization
@ References: Mielnik & Rosas-Ortiz JPA(04) [rev].
Faddeev Model > see QCD phenomenology.
Faddeev-Jackiw Method > see hamiltonian dynamics; quantization of constrained systems; symmetries.
Faddeev-Popov Procedure > see path integral quantization of gauge theories.
Fantappiè Group > see special relativity.
Faraday Lines of Force > see gauge theories.
Faraday Effect / Rotation > see polarization.
Faraday Tensor > see electromagnetic field equations.
Faraday's Law of Induction > see electricity; electromagnetic field equations.
Farey Map > s.a. chaotic
systems.
* Idea: The chaotic map
u u –
1 if u 1
and u–1 – 1
if
u < 1, which appears in mixmaster dynamics.
Fedosov Formalism, Quantization > see deformation quantization.
Feichtinger Algebra > see quantum mechanics in phase space.
Feigenbaum Number
* Value: The number c
= 3.5699456718...
Fermat Geometry > see optics [optical geometry].
Fermat's Last Theorem > s.a. conjectures [Beal conjecture].
$ Def: The conjecture
that xn + yn = zn has
no integer solutions with n > 2 and xyz 
0.
* History: Proved by
Euler for n = 3 and 4 (Gauss corrected the n = 4 proof),
Dirichlet for n = 5 and 14, Legendre (independently) for n =
5, Lamé for n = 7 (his claim of a general proof turned out
to be false), Kummer for n < 100; later shown (also by computer)
to be true for n < 125,000; Now related to elliptic curves and
cusp forms; Was finally proved in 1993–1995 by A Wiles.
* Remark: It is clearly
sufficient to prove it for n = 4 or odd prime.
* Solutions for n =
2: 32 + 42 =
52; 52 +
122 = 132;
... (infinitely many).
@ General references: Ribenboim 79; Goldfeld ThSc(96)mar
[history]; Singh & Ribet SA(97)nov; Ribenboim
99
[advanced];
Mozzochi 00.
@ Wiles' approach: Rubin & Silverberg BAMS(94); Wiles AM(95); Cornell
et al 97.
@ Related topics: da Costa et al IJTP(93)
[and dynamical systems]; AS 82(94)144
[and
arithmetic].
Fermat's Principle > s.a. finsler
geometry.
* Idea: The path followed
by light between two points is the one that minimizes the time.
* Kovner's version: A
variational principle (as opposed to a differential equation) whose solutions
are the past-oriented null geodesics from an observation event p0
to a timelike curve 
S (worldline
of light source), in an arbitrary spacetime.
@ General references: Helfgott & Helfgott AJP(02)
[and law of refraction]; Ogawa qp/02 [and
wave equation].
@ In general relativity: Kovner ApJ(90);
Perlick CQG(90),
CQG(90);
Perlick JMP(95),
Giannoni & Masiello GRG(96)
[Morse theory for light rays].
@
In general relativity, application to lensing: Nandor & Helliwell AJP(96);
Giannoni et al JMP(02).
> Online resources: see Wikipedia page.
Fermi Energy > see Chemical Potential.
Fermi Paradox > see Civilizations.
Fermi Surface
* Idea: The surface in
momentum space that separates occupied and non-occupied states in the ground
state (for an ideal gas of fermions).
Fermi Theory > see Weak Interactions.
Fermi Transport, Normal Coordinates > see coordinates.
Fermi's Golden Rule > see Golden Rule.
Fermi-Pasta-Ulam-Tsingou Model / Paradox > see energy [time
to equipartition].
* Idea: A problem of
central importance in the theories of solitons and chaos, a model that gave
rise to the
first numerical
simulation for scientific problems and marked
the beginning of non-linear physics; The idea was to simulate the one-dimensional
analogue of atoms in a crystal as a long chain of masses linked by springs
that obey Hooke's law, plus a weak non-linear term;
A
purely linear interaction would ensure that energy introduced into a single
Fourier vibrational mode always remains in that mode, while the non-linear
term allows the transfer of energy between modes; They found that, under certain
conditions, the energy does not drift toward
the equipartition predicted by statistical physics, but periodically returns
to the original mode; That highly remarkable result, known as the FPU paradox,
shows that non-linearity is not enough to guarantee the equipartition of energy;
In the 1960s, pursuing the solution of the FPU paradox, Norman Zabusky and
Martin Kruskal were able to explain the periodic behavior in terms of the
dynamics of solitons; Another line of thought, developed in parallel and based
on Fourier-mode analysis, proved that in general orbits of slightly perturbed
integrable Hamiltonian systems remain
quasi-periodic (KAM theorem); If the perturbation is too strong, the recurrence
is destroyed and the equipartition of energy is quickly established.
@ General references: Fermi et al rp(55); Tuck & Menzel AiM(72);
Dauxois et al EJP(05);
Dauxois PT(08)jan-a0801 [history].
@ FPU paradox: Berman & Izrailev Chaos(05)n.CD/04 [rev].
Fermi-Walker Transport and Coordinates
* Idea: A generalization
of Fermi transport along a non-geodesic curve; For a vector perpendicular
to
that curve, it is parallel transport with an additional term to keep the vector
perpendicular to the curve.
* For a geodesic: It coincides
with parallel or Fermi transport.
@ References: Manoff IJMPA(00)gq [generalization,
conformal transport]; Bini & Jantzen
NCB(02)gq-in
[and gravitomagnetism]; Klein & Collas a0712 [coordinate
transformations]; Maluf & Faria AdP(08)-a0804 [Fermi-Walker
transported frames].
Fermion > see particle statistics [including fermion number]; gas; kaluza-klein phenomenology; lattice field theory [including doubling]; particle types; supersymmetry.
Fermionic Projector
@ References: Finster ht/06.
Fermionization > s.a. particle
statistics.
* Idea: A phenomenon
in which strongly correlated bosonic particles exhibit fermionic behavior,
for example mimicking the exclusion principle.
@ Related topics: Paredes et al Nat(04)may
+ pw(04)may
[in
Tonks-Girardeau gas].
Ferromagnetism > see magnetism.
Feshbach-Villars Equation > see spin-1/2 fields.
Feynman Diagram / Graph > s.a. quantum
field theory techniques [perturbative]; history
of physics.
@ General references: 't Hooft & Veltman yr(73); Bilenky 74; Veltman
94; Frankel AS(03);
Kaiser
AS(05);
Scadron
06.
@ In other areas: Mattuck 76 [many-body problem]; Penco & Mauro
EJP(06)ht [in
classical mechanics].
@ Related topics: Ohl CPC(95)hp [LaTeX
drawings]; Suzuki & Schmidt
JHEP(97)
[negative-dimensional technique]; Easther et al PRD(00)hp/99 [numerical];
Kreimer LMP(00)ht/99 [shuffling
identities]; De Pietri & Petronio
JMP(00)gq [and
manifolds]; Ogreid & Osland JCAM(02)mp/00 [and
infinite series]; Connes & Kreimer
AIHP(02)ht [as
Lie algebra]; Larin PLB(99)
[expansion method]; Djah et al mp/05 [for
non-Gaussian
measures]; Davydychev ht/05-in
[N-point, geometrical method]; Baikov PLB(06)
[irreducibility criterion]; Baratin & Freidel CQG(07)gq/06,
CQG(07)
[and
3D-4D topological BF spin foam model]; Bender et al qp/06 [theories
and combinatorics]; Argeri & Mastrolia IJMPA(07)
[differential equation method]; Casadio a0806 [gravitationally
modified]; Marcolli & Rej a0807 [and supermanifolds].
Feynman Gauge > see gauge.
Feynman's Operator Calculus > see modified quantum mechanics.
Feynman's Reverse Sprinkler > see physics teaching.
Feynman-Fields Model
* Idea: A model for quark fragmentation.
Feynman-Kac Formula
* Idea: A relationship
between partial differential equations and stochastic processes.
@ References: DeWitt-Morette & Zhang PRD(83) [in phase space, and coherent state
transitions].
> Online resources:
Wikipedia page;
Answers.com page.
Fiber > s.a. fiber
bundle; principal
fiber bundle.
$ Def: The image of a
curve : [0,1] → R2 which is C1, has nonzero tangent
vector, and does not self-intersect.
$ Fiber system: A closed
subset of R2 which can
be represented as a countable (at most) union of fibers i which
intersect only at endpoints, and such that any compact K 
R2 intersects
only finitely many i's.
$ Fiber process:
A random variable with values in the set of fiber systems as a 
-algebra.
@ References: Parkhouse & Kelly PRS(95)
[random 3D packing of straight
fibers].
Fiber Derivative > same as Legendre Transformation.
Fibonacci Numbers > see sequences; integration; / For Fibonacci Operator, see quantum effects.
Fibration > s.a. bundle.
$ Def: A map 
: X
→ B of a manifold X to a manifold B, such that
(X,
B, ) is a bundle.
@ References: Daverman T&A(05) [approximate fibrations, fibrator properties].
Fick's Law > see diffusion.
Field > s.a. ring.
$ Def: A commutative ring
with inverses.
@ References: Kaplansky 72; Lidl & Niederreiter 97 [finite]; Roman 06.
Field Lines > s.a. electromagnetism.
@ References: Belcher & Olbert AJP(03) [motion].
Field Strength Measurement > see measurements in quantum mechanics.
Field Theory > s.a. boundaries in field theory; higher-spin, scalar, topological, types of field theory.
Fierz Identities
@ References: Nishi AJP(05) [simple derivation].
Fierz-Pauli Lagrangian > see under Pauli-Fierz.
Fifth Order Algebraic Equation > see elementary algebra [quintic].
Filter on a Set X {see B Davis'
dissertation.}
* Idea: A generalization
of the notion of family of neighborhoods of a point x, used to define
limit of a function on a non-topological space; Can be obtained from a filter
base
by adding all subsets of X containing one of the sets.
$ Def: A filter on X is
a set F of subsets of X such that (1) If A X
and A contains an element of F, then A 
F;
(2) F is
closed under finite intersections; (3) The empty set is not in F.
* Examples: The set of
all neighborhoods of a given non-empty subset of a topological space, e.g.,
the neighborhood filter N(x) of x X; A Fréchet
filter is the set of complements of
finite subsets of N.
* Meshing: Two filters
F and G mesh, F # G, iff for all f F and
g G, the intersection
f 
g Ø.
* Fineness: A filter F is
finer than G, F G,
iff for each g G there is
a f F with f g.
* Convergence: A filter
F converges to x X iff F N(x).
@ References: Bourbaki, sec9.1 [duality with covers].
Filter Base for a Set X
$ Def: A family 
of
subsets of X, such that (1) For all A and B in
, there exists C in with C A B;
(2) Ø is not in .
* Example: Given any net
= {xk}k in K
on
a topological space X, the family k =
{F} of subsets of X, with Fk =
{xl |
l k}
is a
filter base. (Viceversa, given any filter base on X one can get many nets associated
with it.)
Filtered Algebra > see algebra.
Fine-Tuning > s.a. Hierarchy
Problem; inflation, phenomenology and versions.
@ References: Koperski BJPS(05)
[and probabilistic arguments].
Finite Geometry > s.a. combinatorics,
geometry.
$ Def: A collection of n objects
(points) and a choice of subsets of these (lines).
* Motivation: Combinatorial
design, coding theory.
* Example: Projective
plane, a special finite geometry of order n, i.e., each line contains n +
1 points.
* Results: There are
no finite geometries of order n if remainder(n/4) = 1 or
2, and n p2 + q2;
There is none of order 10 (computer proof).
* Conjecture: There
are finite geometries of order n only if n is a power of
a single prime.
@ References: Batten 86; Ikeda AdP(90)
[and
general relativity].
Finite Part Distribution > see distribution.
Finite-Temperature Field Theory > see types of field theories and quantum field theories.
Finkelstein Extension > see coordinates for schwarzschild.
First Countable Topological Space > s.a. types of topological spaces.
First Fundamental Form
* Idea: A symmetric rank-2
covariant tensor defined on a hypersurface of a manifold, that gives the metric
on it induced by the metric in the whole manifold; In gravitation, often a spatial
metric on a spacelike hypersurface.
Fisher Information > see information.
Fisher Metric > see distances; metrics; riemannian geometry.
Fitting
@ Curve fitting: Sorkin pr(80); Turney BJPS(90)
[balancing stability and accuracy].
Fitting Problem in Cosmology > see cosmological models; general relativistic cosmology.
Fitzgerald-Lorentz Contraction > see under Lorentz-Fitzgerald.
Five Lemma > see exact sequence.
Flag
$ In a vector space V:
A sequence W1 W2 ···
Wk of
subspaces of V.
$ In a topological space X:
A sequence A1 A2 ···
Ak of
subsets of X with dim(Ai) < dim(Ai+1).
Flag Manifold
$ Def: A homogeneous
space M = G/K, where G is a compact semisimple
Lie group, and K the centralizer
of
a torus in G; Equivalently, M can be identified with the
adjoint orbit Ad(G)w of an element w in the Lie
algebra of G.
@ References: Boya et al RPMP(03)mp/02 [volumes];
Arvanitoyeorgos IJGMP(06) [geometry].
Flat
* Idea: A subset of
a set which includes all points that depend on it; Used in Matroid
Theory.
Flat Manifolds > see types of spacetimes.
Floating
@ References: Vella & Mahadevan AJP(05) ["Cheerios effect"].
Floer Homology > see homology types.
Floquet Spectrum
@ References: e.g., in Graffi & Yajima mp/00 [forced
harmonic oscillator].
Flow of a Vector Field > see vector fields.
Fluid > s.a. perfect fluid.
Flux > see gauge theories [flux tubes]; QED [flux quantization].
Flyby Anomalies > s.a. anomalous
acceleration.
* Idea: Unexplained velocity
jumps of 3.9, –4.6, 13.5, –2, 1.8 and 0.02 mm/s observed near closest approach
during the Earth flybys of six spacecraft.
@ References: McCulloch a0806 [and modification of inertia].
Foam > s.a. matter [meta-materials]; spacetime
foam.
* Applications: Beer;
Galaxy distribution; Soap.
@ References: Klarreich AS(00)mar
[soap foam]; Perkowitz ThSc(00);
Weaire & Hutzler 01;
Diebels & Steeb
PRS(02)
[elastic
moduli].
> Online resources: Trinity College Dublin group.
Focusing of Geodesics > see geodesic.
Fold
* Idea: One of two
generic singularities in mappings from 2-surfaces to a plane.
@ References: Akhmetiev & Sadykov T&A(03),
Sadykov T&A(04)
[for maps 4M →
3N].
Foldy-Wouthuysen Representation, Transformation > see quantum dirac fields and klein-gordon fields.
Foliation > s.a. embeddings [including webs]; extrinsic curvature [extremal surface].
Forgetful Functor > see functors.
Form (Differential form)
Formal Groups > see lie groups.
Foucault's Pendulum > see Pendulum.
Four-Color Theorem > s.a. Coloring.
* Idea: In coloring a
2D map on a surface [homeomorphic to R3],
it is never necessary to use more than 4 colors.
* History: It was proved
in 1976 by K Appel and W Haken (University of Illinois), with extensive computer
use, since the proof is too long to be checked by hand;
After the work was published, mathematicians began finding mistakes
in it; In each case, Haken and Appel quickly fixed the error, but
to many mathematicians
this left a very bad taste for computer proofs.
@ References: Ill Journal of Math, vol 21; Olivastro ThSc(92)may;
Wilson 02 [r pw(03)apr].
Fourier Analysis / Series / Transform
Fourier's Law > see heat.
Fourth Order Algebraic Equation > see elementary algebra.
FPU Paradox > see under Fermi-Pasta-Ulam Model.
Fractal > s.a. fractals in physics.
Fractal Dimension > see dimension.
Fractional Derivatives > see analysis.
Fragility in Cosmology > see cosmology in general relativity; cosmic topology.
Frame > s.a. coordinates;
Reference Frame [more physical point of
view]; tetrad.
@ Causal types: Morales gq/06-in; > s.a. special relativity.
Frame Bundle > s.a. principal
fiber bundles.
* Idea: A principal fiber
bundle
whose
base manifold is an n-manifold M, fiber
the frames (sets of n linearly independent vectors) at p M,
structure
group G = GL(n, R).
@ References: Ståhl gq/00/JMP
[over spacetime, geometry].
> Related topics: see approaches
to quantum gravity [quantum frame bundles].
Frame Dragging > see tests of general relativity with orbits.
Frame Theory (Ehlers)
* Idea: A general theory
of gravitation including the Newton-Cartan theory and general relativity
as special cases.
@ References: Ehlers CQG(97).
Franck-Hertz experiment > see experiments in physics.
Fréchet Algebra
* Idea: A complete,
metrizable, topological algebra whose topology is defined by an increasing
family {qn} of multiplicative seminorms.
Fréchet Derivative > see Banach Space.
Fréchet Space
$ Def: A complete metric
vector space.
* Motivation:
Some different notions of differentiability coincide.
Fredholm Alternative > s.a. integral equations.
Free Action of a Group on a Manifold > see group action.
Free Energy > s.a. thermodynamics.
$ Statistical definition: The
quantity F:= –kT log Z, where Z is
the partition function for the system.
$ Helmholtz: The thermodynamic
quantity F:= E – TS, the amount
of energy that can be converted into work in a T = constant reversible
(S = const) transformation.
$ Gibbs: The thermodynamic
quantity F:= E – TS + pV.
@ General references:
Coffey HSPBS(06)
[historical, and the third law of thermodynamics].
@ Geometric: Pollicott & Weiss CMP(05)
[for surfaces with variable
negative curvature].
Free Fall > see equivalence principle; Projectile Motion.
Free Group > see types of groups.
Free Product of Goups > see group.
Free Will > see philosophy of
physics.
@ "Free will theorem":
Conway & Kochen a0807 [strengthening].
Frenet-Serrat Curvature > see classical particle models.
Frequency > see wave equations.
@ Frequency operator: Caves & Schack AP(05) [and quantum probability
postulate].
Fresnel Integrals
* Idea: Integrals of
the type 
–inftyinfty dx exp{
ix2}
= 1/2;
Can be proved taking the a → 0 limit of generalized Gaussian integrals.
Freud Pseudotensor / Superpotential > see stress-energy pseudotensors.
Friction > s.a. Stokes'
Law.
* History: In the 1700's,
Amontons and Coulomb found that the force needed to push an object across
a surface depends on its mas, but not on the area of the contact surface;
Microscopic 'asperities' on the surfaces have traditionally been blamed for
this; In 2001, Gerde & Marder proposed a 'micro-crack' theory.
@ General references: Krim SA(96)oct; Persson 98 [sliding, r PT(99)jan];
Hähner & Spencer
PT(98)sep;
Gerde & Marder
Nat(01)sep
+ pw(01)sep
[new theory]; Scherge & Gorb 01 [biology]; Krim AJP(02)RL
[microscopic and macroscopic]; Peters CP(04)
[mesoscale]; Barnett & Cresser PRA(05)
[Markovian quantum theory]; Krim pw(05)feb
[nanoscale]; Besson et al AJP(07)
[experiments and models].
@ Examples: Deakin & Troup AJP(98) [air resistance, projectile
motion]; Cross AJP(02) [bouncing ball], AJP(05) [increase with speed].
@ Related topics: Salazar et al
PhysEd(90)
[in direction of center of mass motion]; Barrena et al PRL(99)
+ pn(99)mar
[origin];
Johansen & Sornette
PRL(99)
+ pn(99)jun
[and
sound]; Ringlein & Robbins AJP(04)
[atomic origins]; Raine EJP(05)
[fluctuations and dissipation]; news pw(06)jul
[overcoming, in nanosized mechanical
devices].
Friedmann Equation and Solutions
Friedmann-(Lemaître)-Robertson-Walker Spacetime > s.a. perturbations and quantum cosmology.
Friedrichs Model > s.a. quantum
systems.
* Idea: A model with
a small system with a finite number of states ("atom") coupled to a reservoir
with an infinite number of states ("radiation"); Used as a model for an unstable
system.
@ References: Antoniou et al PRA(01)qp/00 [(anti)-Zeno
effect], qp/01 [N-level,
decay]; Baumgärtel RVMP(06)mp/05,
addendum RVMP(07)
[resonances
and Gamov vectors]; Derezinski & De Roeck qp/06 [stochastic
limit]; Courbage et al PLA(07)
[and kaon phenomenology].
Frobenius Algebra
@ References: Kaufmann CMP(04)
[second quantization].
Frobenius Manifold
* History: Introduced
by Dubrovin as a coordinate-free approach to the Witten-Dijkgraaf-E Verlinde-H
Verlinde (WDVV) differential equations obtained in topological field theory
in the 1990's; They play a fundamental role in apparently unrelated areas
of
mathematics;
Besides the theory of Gromov-Witten invariants of symplectic manifolds, they
also
come up in singularity theory, the theory of isomonodromic deformations
of linear differential equations, the theory of Coxeter groups and their
extensions and the theory of integrable systems of KdV-type.
@ References: Manin 99; Strachan JGP(01)m.DG/99,
DG&A(04)m.DG/02 [submanifolds];
Mironov & Taimanov TMP(07)mp/06 [algebraic
examples]; Mokhov a0710 [as submanifolds of pseudo-euclidean
spaces].
Frobenius Theorem [@ MathWorld page]
Frobenius-Perron Operator
$ Def: The operator U generating
discrete time evolution for a classical distribution function 
, i.e.
n+1(x) = U n(x).
Froude Number
* Idea: A concept related
to how an object moves in a fluid (tumble and flutter), the ratio of the
time it takes for it to fall its own length to the time it takes for it to
move from side to side.
Frozen Formalism > see time in gravity.
Frustration > see spin models.
Fubini-Study Metric > see types of metric.
Fuchsian Analysis / Equations > see gowdy spacetime.
Fugacity
$ Def: The quantity z:=
exp{
},
in terms of which the partition function for the grand canonical ensemble can
be written as Z = exp{zV/
3}.
Full Chronological Space > see Chronological Space.
Functional Analysis, Derivative, Equations
Functions > s.a. functions and maps on differentiable manifolds.
Fundamental Group > s.a. homotopy.
Fundamental Homology Class
$ Def: For a compact
oriented n-manifold M, the unique in
H n(M; Z)
such that x()
= x, where ...
Fundamental Theorem of Algebra > see elementary algebra.
Fundamental Theorem of Calculus
$ Def: The statement
that ab dx = b – a.
Fusion > see nuclear physics and technology.
Fusion Bases, Rules
* Fusion bases: The sets of inequalities governing fusion rules.
* Fusion rules: Also
known as representation rings for groups.
@ Fusion bases: Bégin et al mp/00-in,
JMP(00), JMP(00)
[for affine Lie
algebras], JMP(02)ht/01 [as
facets of
polytopes].
@ Fusion rules: Zimboras m.GR/05-in
[compact
groups, information contained].
Fuzzy Manifolds > see differential geometry.
Fuzzy Set Theory > see set theory.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
21 jul 2008