Topics, F

F-Theory > see M-theory.

Fab Four Theories > see Horndeski Theory.

Factorial Function > s.a. Stirling's Formula.
$ Def: The factorial of a positive integer n is \(n! := 1 \cdot 2 \cdot 3 \cdot\, ... \cdot n\).
* 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.
@ Approximations: Memin AJP(83)sep, Fletcher AJP(86)feb.
> Online resources: see MathPages page; Wikipedia page.

Factor > see observable algebras.

Factorization > s.a. number theory [factoring numbers].
@ References: Mielnik & Rosas-Ortiz JPA(04) [of wave equations, rev].

Faddeev Formulation of Gravity
* Idea: A formulation in which the metric is obtained as a bilinear combination of 10 vector fields f_μ^A, or a 4 × 10 tetrad; A unique feature of this formulation is that the action remains finite for discontinuous fields (although continuity is recovered on the equations of motion), which makes it convenient for use with piecewise flat (simplicial) manifolds.
@ References: Faddeev TMP(11); Khatsymovsky CQG(13)-a1201 [first-order representation and Barbero-Immirzi parameter], a1201 [discrete form]; Khatsymovsky a1206; Khatsymovsky a1312 [on a piecewise-flat manifold]; Khatsymovsky MPLA(14)-a1408 [minisuperspace model].

Faddeev Model > s.a. Glueballs.
@ References: Shi & Hirayama JMP(12)-a1205 [solitonic solutions].

Faddeev-Jackiw Method > s.a. hamiltonian systems; quantization of constrained systems; symmetries in quantum physics.
* Idea: A quantization procedure for constrained gauge systems.
@ References: Faddeev & Jackiw PRL(88); Barcelos-Neto & Wotzasek MPLA(92), IJMPA(92); Montani & Wotzasek MPLA(93) [non-abelian systems]; Jackiw ht/93; Barcelos-Neto & Silva IJMPA(95) [reducible theory]; Müller-Kirsten & Zhang PLA(95), PLA(95); García & Pons IJMPA(98); Huang AP(10) [and superconductivity]; Prescod-Weinstein & Bertschinger CQG(15)-a1404 [extension to fields in curved spacetimes]; Toms PRD(15)-a1508 [and path integrals]; > s.a. dirac approach [comparison].
@ Specific theories: Foussats et al CQG(97) [2D supergravity theories]; Escalante & Rodríguez-Tzompantzi EPJC(16)-a1601 [topologically massive gravity]; Rodríguez-Tzompantzi EPJP(18)-a1804 [two interacting massive relativistic particles]; Rodrigues et al PRD(18)-a1808 [general relativity]; > s.a. BF theory; Proca Theory.

Faddeev-Niemi Equations
* Idea: Equations that correspond to the Yang-Mills equations of motion for a decomposed gauge field.
@ References: Niemi & Wereszczyński JMP(11)-a1011 [solutions].

Faddeev-Popov Ghosts > see Ghost Fields.

Faddeev-Popov Procedure > see path-integral quantization of gauge theories.

Fakeon (Fake Particle) > see types of particles.

False Vacuum > see vacuum.

Falsifiability > see theory of physical theories.

Fantappiè Group > see special relativity.

Fantappiè-Arcidiacono Relativity Theory / Spacetime > see Projective Relativity.

Faraday Lines of Force > see gauge theories.

Faraday Effect / Rotation > see polarization.

Faraday Tensor > see electromagnetic field equations.

Faraday Waves
* Idea: Finite-amplitude surface waves arising from an instability of an oscillating free surface.

Faraday's Law of Induction > s.a. electromagnetic field equations / Circulation; Eddy Currents.
* Idea: The law of electromagnetism stating how a time-dependent magnetic flux through a surface produces an electric field around its boundary.
@ Generalizations: Mannheim & Poveromo GRG(14) [gravitational analog, with torsion and non-symmetric metric].
> Online resources: see HyperPhysics page; Wikipedia page.

Farey Map > s.a. chaotic systems.
* Idea: The chaotic map u \(\mapsto\) u − 1 if u ≥ 1 and u⁻¹ − 1 if u < 1, which appears in mixmaster dynamics.

FCI (Force Concept Inventory) > see physics teaching.

Fedosov Algebra / 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: The first ones are 3² + 4² = 5²; 5² + 12² = 13²; ... (there are infinitely many).
@ General references: Ribenboim 79; Goldfeld ThSc(96)mar [history]; Singh & Ribet SA(97)nov; Ribenboim 99 [advanced]; Mozzochi 00; Aczel 07.
@ Wiles' approach: Rubin & Silverberg BAMS(94); Wiles AM(95); Cornell et al ed-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 and physics.
* 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 γ_S (worldline of light source), in an arbitrary spacetime.
@ General references: Helfgott & Helfgott AJP(02)dec [and the law of refraction]; Ogawa qp/02 [and the wave equation]; Elsayed PLA(14)-a1302 [generalized to classical and quantum many-dimensional systems].
@ In general relativity: Kovner ApJ(90); Perlick CQG(90), CQG(90); Perlick JMP(95), Giannoni & Masiello GRG(96) [Morse theory for light rays]; Frolov PRD(13)-a1307 [in a curved spacetime, and effective Hamiltonian].
@ In general relativity, application to lensing: Nandor & Helliwell AJP(96)jan; Giannoni et al JMP(02).
> Online resources: see Wikipedia page.

Fermi Acceleration
* Idea: An unbounded growth of the energy of a system due to its interaction with the environment; It plays an important role in astrophysical models, such as shocks in solar flares and supernova remnants.
> Online resources: see Wikipedia page.

Fermi Energy > see Chemical Potential.

Fermi Function > s.a. phase space.
* Idea: A function g_F(q, p) on phase space that can be used to represent a quantum state, and is conceptually different from the Wigner function W(x, p).
@ For generalized coherent states: Benenti & Strini AJP(09)jun [and Gaussian wave packets], EPJD(10)-a0906 [and Wigner function, perturbed Gaussian wave packets]; de Gosson a1107 [for squeezed coherent states]; de Gosson & de Gosson a1301 [and symplectic capacities].

Fermi Gas > see gas.

Fermi Normal Coordinates > see coordinates.

Fermi Paradox > see civilizations.

Fermi Sea > see composite quantum systems; Dirac Sea.

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).
@ References: Derunova et al a2007 [mathematical].
> Online resources: see Wikipedia page.

Fermi Theory > s.a. Weak Interaction.
* Idea: A four-fermion interaction proposed by E Fermi in 1933 to explain neutron beta decay and model wea interactions.
> Online resources: see Wikipedia page.

Fermi Transport > see coordinates.

Fermi Two-Atom Problem
* Idea: A problem illustrating an apparent causality violation in quantum field theory, which has to do with the nature of the built-in correlations in the vacuum.
@ References: Zohar & Reznik NJP(11)-a1103 [in discrete systems, such as trapped atoms in optical lattices or trapped ions].

Fermi's Golden Rule > see Golden Rule.

Fermi-Dirac Statistics > see particle statistics.

Fermi-Einstein Condensation > s.a. quantum phase transitions.
* Idea: A phenomenon by which dressed fundamental fermions acquire Bose-like statistics and undergo condensation.
@ References: Langfeld et al AP(12) [in dense QCD-like theories]; Ghosh et al a1301 [coherent-state representation].

Fermi-Hubbard Model > see Hubbard Model.

Fermi-Pasta-Ulam-Tsingou Model / Paradox > s.a. 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]; Benettin et al JSP(09) [relevance of initial conditions]; Porter et al AS(09)may [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 CQG(08)-a0712 [coordinate transformations]; Maluf & Faria AdP(08)-a0804 [Fermi-Walker transported frames].

Fermion Doubling > see lattice field theory [non-commutative]; matter in lattice field theory; loop quantum gravity; non-commutative field theory.

Fermion Fields / Particles

Fermionic Projector
* Idea: A formulation of quantum field theory.
@ References: Finster JPCS(07)ht/06, LMP(11)-a0911, JPCS(11)-a1011.

Fermionization > s.a. particle statistics.
* Idea: A phenomenon in which strongly correlated bosonic, or fermionic but distinguishable particles exhibit the behavior of identical fermions, for example mimicking the exclusion principle.
@ References: Paredes et al Nat(04)may [in Tonks-Girardeau gas]; Zürn et al PRL(12) [fermionic Li-6 atoms with opposite spins]; Mankoč Borštnik & Nielsen a1602-proc [in arbitrary dimensions]; Botelho & Rodrigues a1611; Borstnik & Nielsen a1805-conf [number of families].

Ferroelectric Materials > see electricity.

Ferromagnetism > see magnetism.

Feshbach Resonance > see resonances.

Feshbach-Villars Equation > see spin-1/2 fields.

Feynman Checkers
* Idea: An elementary model of electron motion introduced by Richard Feynman.
@ References: Novikov a2010.

Feynman Clock > see clocks; quantum states.

Feynman Diagram / Graph > s.a. quantum field theory techniques [perturbative]; history of particle physics.
@ General references: 't Hooft & Veltman yr(73); Bilenky 74; Veltman 94; Frankel AS(03); Kaiser AS(05); Scadron 06; Gross SHPMP(12) [in Feynman's early lectures]; Weinzierl a1301-ln; Fried 14; Dorato & Rossanese a1711 [conceptual, nature of representation]; Passon EJPS(19)-a1901 [interpretation, and the Higgs discovery].
@ Theoretical techniques: Suzuki & Schmidt JHEP(97) [negative-dimensional technique]; Kreimer 00 [and knot theory]; Kreimer LMP(00)ht/99 [shuffling identities]; Larin PLB(99) [expansion method]; Connes & Kreimer AIHP(02)ht [as Lie algebra]; Argeri & Mastrolia IJMPA(07) [differential equation method]; Bogner & Weinzierl a0912-conf [mathematical structures]; Smirnov & Petukhov LMP(11)-a1004 [finiteness of the number of master integrals]; González NPPS(10)-a1008 [method of brackets and the master theorem of Ramanujan]; Yost et al PoS-a1101 [differential reduction techniques]; Caron-Huot JHEP(11) [relations between loop and tree amplitudes]; Fried & Gabellini AP(12) [summation of all Feynman graphs relevant to a particular process]; Rivasseau & Wang AHP(14)-a1304 [resumming]; Panzer a1407-proc, Todorov a1611-proc [hyperlogarithms].
@ Numerical techniques: Ohl CPC(95)hp [LaTeX drawings]; Easther et al PRD(00)hp/99; Preti a1811 [method of uniqueness, Mathematica package].
@ Special systems: Djah et al mp/05 [for non-Gaussian measures]; Baratin & Freidel CQG(07)gq/06, CQG(07) [and 3D-4D topological BF spin-foam model]; > s.a. spin-foam models.
@ In other areas: Mattuck 76 [many-body problem]; Penco & Mauro EJP(06)ht [in classical mechanics]; Cattaruzza et al AP(11) [classical field theory, diagrammar]; Jishi 13 [condensed matter physics]; > s.a. dynamics of gravitating bodies.
@ Generalizations: Marcolli & Rej JPA(08)-a0807 [and supermanifolds]; Brouder & Frédéric a1103 [non-local, non-commutative]; Martins & Biezuner a1903 [hypergraphs].
@ Related topics: De Pietri & Petronio JMP(00)gq [and manifolds]; Ogreid & Osland JCAM(02)mp/00 [and infinite series]; Davydychev NIMA(06)ht/05-in [N-point, geometrical method]; Baikov PLB(06) [irreducibility criterion]; Bender et al Twist-qp/06 [theories and combinatorics]; Casadio IJMPA(12)-a0806 [gravitationally modified]; Berghoff a1709 [on moduli spaces of graphs].
> Related topics: see regularization; renormalization.

Feynman Gauge > see gauge.

Feynman Integrals
* Idea: Integrals that appear in many quantum field theory calculations, and consist in momentum-space integrals of products of propagators.
@ References: Kastening & Kleinert PLA(00)qp/99; Henn JPA(15)-a1412 [computation using differential equations]; Mastrolia a1507-conf [algebraic patterns]; Re Fiorentin IJMPC(16)-a1507 [Mathematica package for tensor reduction]; Liu & Ma PRD(19)-a1801 [arbitrary integrals from vacuum ones]; Mastrolia & Mizera a1810 [and intersection theory]; Preti a1905-proc [Mathematica package]; Jing et al a2005 [graphic method]; > s.a. Motives.

Feynman's Operator Calculus > see modified quantum mechanics.

Feynman Propagator

Feynman's Rest of the Universe
* Idea: "When we solve a quantum-mechanical problem, what we really do is divide the universe into two parts–the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the entire universe. To motivate the use of density matrices, let us see what happens when we include the part of the universe outside the system."
@ References: Feynman 72, §2.1; Kim & Noz Sym-a1210 [and spacetime symmetry, Dirac matrices].

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.
@ General references: DeWitt-Morette & Zhang PRD(83)[in phase space, and coherent state transitions].
@ Variations, related topics: Turgeman et al PRL(09) [anomalous diffusion, fractional Feynman-Kac equations].
@ Examples: Dimock a2005 [charged quantum particle in the presence of a magnetic monopole].
> Online resources: see Wikipedia page; Answers.com page.

Fiber > s.a. fiber bundle; principal fiber bundle.
$ Def: The image of a curve γ: [0,1] → \(\mathbb R\)² which is C¹, has non-zero tangent vector, and does not self-intersect.
$ Fiber system: A closed subset of \(\mathbb R\)² which can be represented as a countable (at most) union of fibers \(\gamma^i\) which intersect only at endpoints, and such that any compact K ⊂ \(\mathbb R\)² intersects only finitely many γⁱs.
$ Fiber process: A random variable with values in the set of fiber systems as a σ-algebra.
@ References: Parkhouse & Kelly PRS(95) [random 3D packing of straight fibers].

Fiber Derivative > same as Legendre Transformation.

Fibonacci Numbers > see sequences; integration; / For the Fibonacci Operator, see quantum effects.

Fibration > s.a. bundle.
$ Def: A map π : X → B of a manifold X to a manifold B, such that (X, B, π) is a bundle.
@ References: Daverman T&A(05) [approximate fibrations, fibrator properties].

Fick's Law > see diffusion.

Fidelity > s.a. quantum phase transitions [fidelity-metric approach]; quantum states [quantum fidelity].
@ References: Bahder a1102 [and quality of physical measurements].

Fidelity Susceptibility > s.a. quantum phase transitions.
* Idea: A general purpose probe of phase transitions; Based on quantum information and the differential geometry perspective of quantum states, it can indicate the presence of a phase transition without prior knowledge of the local order parameter, as well as reveal the universal properties of a critical point.

Field > s.a. Galois Field [finite field]; ring; vector space.
$ Def: A commutative ring with inverses.
* Examples: The real and complex numbers.
@ General references: Kaplansky 72; Roman 06; Reis 11 [II].
> In physics: see modified quantum mechanics [over a finite field].

Field Lines > s.a. electromagnetism.
@ References: Belcher & Olbert AJP(03)mar [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)dec [simple derivation].

Fierz-Pauli Lagrangian > see under Pauli-Fierz.

Fifth Force

Fifth-Order Algebraic Equation > see elementary algebra [quintic].

Figurate Numbers > see number theory.

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; It 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 \(\mathbb N\).
* 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 61, sec9.1 [duality with covers]; in Dolecki & Mynard 16 [and approach to topology].

Filter Base for a Set X
$ Def: A family \(\cal F\) of subsets of X, such that (1) For all A and B in \(\cal F\), there exists C in \(\cal F\) with C ⊂ A ∩ B; (2) Ø is not in \(\cal F\).
* Example: Given any net \(\cal N\) = {x_k}_{k ∈ K} on a topological space X, the family \(\cal F\)= {F_k} of subsets of X, with F_k = {x_l | 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.

Final Cause > s.a. Teleology.
* Idea: The end for the sake of which a thing is what it is, or a change occurs.
> Online resources: see Wikipedia page.

Finality > see Teleology.

Fine-Structure Constant > s.a. bounds on fine-structure constant variations.

Fine Tuning > s.a. anthropic principle; Naturalness; physical theories [criteria].
* Idea: The fact that the values of the parameters in our basic physical theories must be adjusted very precisely in order for the predictions of those theories to agree even qualitatively with observations; Examples of how this manifests itself in various theories are the flatness problem in cosmology (addressed by inflationary scenarios), the cosmological constant problem, the hierarchy problem, and the strong CP problem.
* Explanations: The two main types of approaches are the ones in which the values of the parameters are actually constrained to be at least close to their observed values (e.g., arguments from design), and those in which they acan have any values but there are reasons why we observe the values we do (e.g., some versions of the multiverse and the anthropic principle).
@ References: Koperski BJPS(05) [and probabilistic arguments]; Grinbaum FP(12)-a0903 [naturalness and the Standard Model]; Landsman a1505 [critique of common approaches]; Alexander et al PLB(16)-a1507 [cyclic universe approach]; Barnes EJPS-a1707 [in the context of Bayesian theory testing]; Hossenfelder a1801 [critical analysis]; Wells FP(19)-a1809 [fine-tuned theories and improbable theories]; Azhar & Loeb PRD(18)-a1809 [mathematical framework]; Päs FP(19)-a1809-proc [and a fundamental theory]; Azhar & Loeb a1910 [and lack of explanatory depth]; Adams PRP(19); De Vuyst a2012 [introduction, and naturalness].
> Cosmology, in general: see cosmology; multiverse; quantum cosmology.
> Cosmology, specific issues: see acceleration and inhomogeneities; inflation (including phenomenology, planck-scale physics and scenarios).
> Other areas: see complexity; Hierarchy Problem [particle physics].
> Online resources: see Wikipedia page.

Fine's Theorem > s.a. CHSH Inequalities.
* Idea: A result concerning the conditions under which a certain set of probabilities for pairs of four bivalent quantities may be taken to be the marginals of an underlying probability distribution; It states that the eight CHSH inequalities are necessary and sufficient conditions.
@ References: Halliwell PLA(14) [two proofs].

Finitary Argument, Function, Operation > see Wikipedia page; s.a. discrete structures [finitary sets].

Finite-Element Method > s.a. lattice gauge theory.
* Idea: A numerical technique for finding approximate solutions to boundary-value problems for partial differential equations.
> Online resources: see Wikipedia page.

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: Ikeda AdP(90) [and general relativity]; Batten 97.

Finite Groups

Finite-Part Distribution > see distribution.

Finite-Size Effects > see Compressibility; Density of States; gas [photon gas]; lattice field theories.

Finite-Temperature (Thermal) Field Theory > see generalized quantum field theories and specific types [effects]; thermodynamic systems [specific theories].

Finkelstein Extension > see coordinates for schwarzschild spacetime.

Finsler Geometry > s.a. finsler geometry and physics.

Firewalls > s.a. black-hole geometry [interior]; event horizons.
@ References: Kaplan & Rajendran PRD(19)-a1812 [spherical solutions].

First Countable Topological Space > see 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 Distance / Metric > see distances; metrics; riemannian geometry.

Fisher Information > see information; quantum information.

Fisher-Rao Metric / Information Measure > see distances; mixed quantum states.

Fitting > see statistics in physics [curve fitting].

Fitting Problem in Cosmology > see cosmological models; general relativistic cosmology.

Five Lemma > see exact sequence.

Fixed-Point Theorems

Fizeau's Experiment
* Idea: An experiment carried out in 1851 by Hippolyte Fizeau, designed to measure the effect of movement of a medium upon the speed of light.
@ References: Lahaye et al AJP(12)jun-a1201 [in the undergraduate laboratory].
> Online resources: see Wikipedia page.

Flag
$ In a vector space V: A sequence W₁ ⊂ W₂ ⊂ ··· ⊂ W_k of subspaces of V.
$ In a topological space X: A sequence A₁ ⊂ A₂ ⊂ ··· ⊂ A_k of subsets of X with dim(A_i) < dim(A_i+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; It is used in Matroid Theory.

Flat Manifolds > see types of spacetimes.

Flatness Problem > s.a. inflation.
* The issue that the current energy density of the universe is observed to be very close to the critical value for which the universe is spatially flat.
@ References: Helbig MNRAS(12)-a1112 [the flatness problem does not exist].
> Online resources: see Wikipedia page.

Flavor Physics > see QCD.

Floating > s.a. Cheerios Effect.
@ References: Hueschen AJP(10)feb [and rising bubbles]; Straulino et al AJP(11)jan [and surface tension, Galileo's experiments and writings].

Floer Homology > see homology types.

Floquet Spectrum / Floquet's Theorem > s.a. types of quantum systems [periodic in time].
@ References: e.g., in Graffi & Yajima CMP(00)mp [forced harmonic oscillator].

Flow of a Vector Field > see vector fields; s.a. Flux [flow rate].

Fluctuations

Fluctuation-Dissipation Theorem > s.a. fluctuations; photons.
* Fluctuation-compressibility theorem: An example, stating that particle number fluctuations are proportional to the isothermal compressibility.

Fluid > s.a. perfect fluid.

Fluid-Gravity Correspondence > see solution-generating methods for einstein's equation.

Flux > s.a. connections [space of generalized fluxes]; phenomenology of magnetism; QED [flux quantization].
* Idea: The integral of a vector field, interpreted as flux density of some physical quantity, over a surface.
* Examples: The flux of a fluid's velocity is the volume flow rate through the surface; The flux of the electric current density is the total current through the surface; The electric field flux and magnetic field flux are used in the integral version of Maxwell's equations.
@ General references: Blumenhagen et al FdP(12)-a1205 [Bianchi identities, from quasi-Poisson structures to Courant algebroids].
@ In fluids: Mathwig et al PRL(12) [measuring picoliter-per-minute fluid flows in nanochannels]; > s.a. heat [heat flux].
@ In gauge theories: in Thiemann CQG(01)ht/00 [gauge-covariant fluxes]; Klitgaard et al CQG(20)-a2004 [gravitational Wilson loop]; > s.a. gauge theories [flux tubes].
> Online resources: see Wikipedia page.

Flux Tube
* Idea: A flexible strand of energy that binds elementary particles together; e.g., linking quarks and antiquarks with the help of gluons.
> Related topics: see gauge theories; QCD effects and phenomenology; QED.
> In cosmology: see cosmic strings; early universe, spacetime dimension [flux tube networks].

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 MNRAS(08)-a0806 [and modification of inertia]; Turyshev & Toth SSR(09)-a0907 [rev]; Adler IJMPA(10)-a0908, a0910, IJMPA(13) [and dark-matter scattering]; Martin Nieto & Anderson PT(09)oct-a0910; Bertolami et al IJMPD(12)-a1201 [space mission proposal]; Acedo ASR(14)-a1505 [not gravitomagnetism].

Foam > s.a. meta-materials; spacetime foam.

Fock Space

Fock Symmetry
* Idea: The SO(4) symmetry of a system with 1/r potential, which leads to additional degeneracies with respect to the expected ones in a spherically symmetric system; The main examples are the hydrogen atom (in which the quantum mechanical energy levels depend only on n and are n²-fold degenerate, rather than depending on l and being just (2l+1)-fold degenerate) and the Kepler problem (in which orbits are closed, and their energies depend only on their semimajor axis and not on their eccentricity).
@ References: Fock ZP(35); Meremianin & Rost JPA(06)mp.
> Online resources: see Harold McIntosh page.

Fock-Lorentz Symmetry
* Idea: The invariance corresponding to the Projective Lorentz transformations of a theory satisfying the principle of relativity, but not the principle of invariance of the speed of light.
> Online resources: see Wikipedia page.

Focusing of Geodesics > see geodesic.

Fokker-Planck Equation

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 > s.a. quantum dirac fields and klein-gordon fields.
@ References: Silenko PRA(15)-a1501 [general approach applicable to relativistic particles with any spin in arbitrarily strong external fields].

Foliation > s.a. embeddings [including webs]; extrinsic curvature [extremal surface].

Force

Force-Free Electrodynamics > see phenomenology of magnetism; black-hole phenomenology.

Forcing
* Idea: A technically difficult mathematical subject, used by Paul Cohen to prove the independence of the continuum hypothesis from the standard axioms of set theory.
@ References: Weaver 14.

Forecasting > see Prediction.

Forgetful Functor > see functors.

Form > see differential form; Integral Form.

Formal Cause
* Idea: "A change or movement's formal cause is a change or movement caused by the arrangement, shape or appearance of the thing changing or moving."
> Online resources: see Wikipedia page.

Formal Groups > see lie groups.

Forman Curvature > see curvature.

Fortran > see programming languages.

Foucault's Pendulum > see Pendulum.

Foundations > see foundations of quantum theory; physical theories.

Four-Color Theorem > s.a. Coloring.
* Idea: In coloring a 2D map on a surface [homeomorphic to \(\mathbb R^3\)], 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 Conductivity [thermal]; heat [flow equation].

Fourth-Order Algebraic Equation > see elementary algebra [quartic].

Fourth Quantization
@ References: Faizal PLB(13) [and creation of the multiverse].

FPU Paradox > see under Fermi-Pasta-Ulam Model.

Fractal > s.a. fractals in physics.

Fractal Dimension > see dimension of a space.

Fractional Charges > see charge.

Fractional Derivatives / Differential Equations > see fractional calculus; differential equations.

Fractional Statistics > see modified particle statistics.

Fractons
* Idea: Excitations of quantum matter that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions; The immobile excitations have been dubbed fractons because combining fractons tends to make them less constrained than a single fracton, as if a fracton by itself is only a fraction of a fully mobile particle.
* Examples: Higher-rank generalizations of electrodynamics can host fracton excitations, with connections to both fracton topological order and gravity.
@ References: Nandkishore & Hermele ARCMP(19)-a1803 [rev]; Pretko & Radzihovsky PRL(18) + viewpoint Slagle Phy(18) [duality with lattice defects in a 2D crystal]; Pretko et al IJMPA(20)-a2001 [rev]; Yan et al PRL(20) [proposal, in a rank-2 U(1) spin liquid].

Fragility in Cosmology > see cosmology; cosmic topology.

Fragmentation
> In astrophysics: see minor solar-system objects; types of stars.
> In particle physics: see cosmic-string phenomenology; Feynman-Fields Model and QCD phenomenology [quark fragmentation].

Frame > s.a. coordinates; reference frames [more physical point of view]; tetrads.
* Idea: The assignment of a basis for the tangent space at each point of a differentiable manifold.
* Types: Holonomic (one arising from a coordinate system), or non-holonomic.
@ Causal types: Morales AIP(06)gq; > s.a. special relativity.

Frame Bundle > s.a. principal fiber bundles.
* Idea: A principal fiber bundle whose base manifold is an n-manifold M, whose fibers are the sets of frames (sets of n linearly independent vectors) at p ∈ M, and whose structure group G = GL(n, \(\mathbb R\)).
@ References: Cordero & Dodson 88; Ståhl gq/00/JMP [over spacetime, geometry].
> Related topics: see approaches to quantum gravity [quantum frame bundles].
> Online resources: see MathWorld page; Wikipedia page.

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 in(81); Ehlers CQG(97).

Franck-Hertz experiment > see experiments in physics.

Frauchiger-Renner Paradox > see Unitarity in Quantum Theory.

Fréchet Algebra
* Idea: A complete, metrizable, topological algebra whose topology is defined by an increasing family {q_n} of multiplicative seminorms.

Fréchet Derivative > see Banach Space.

Fréchet Geometry / Manifolds
@ References: Müller JGP(08) [metric approach]; Dodson a1109 [rev]; Dodson IJGMP(14) [via projective limits].

Fréchet Space
$ Def: A complete metric vector space / A topological vector space that is locally convex, Hausdorff, metrizable and complete.
* Examples: Every Banach space is a Fréchet space; More generally, any countable cartesian product of Banach spaces is a Fréchet space.
* Motivation: Some different notions of differentiability coincide.
@ References: Nyikos T&A(10) [products of Fréchet spaces].

Fredholm Alternative > s.a. integral equations.

Free Action of a Group on a Manifold > see group action.

Free-Body Diagram > see physics teaching.

Free Energy > s.a. thermodynamics.
* Idea: A free energy captures the interplay between information and energy, which determines the amount of work that can be extracted from a system.
* Rem: There is a small set of usual free energies that do the job in the thermodynamic limit of a statistical system, but in the regime of small or strongly correlated systems there is an infinite family of free energies, related to the Renyi entropies, which constitutes a family of "second laws" constraining transitions in cyclic processes.
$ Helmholtz free energy: The thermodynamic quantity F:= E − TS. / It is the amount of energy that can be converted into work in a T = constant reversible (S = const) transformation; For a statistical mechanical system in a canonical ensemble with partition function Z, it can be calculated as F:= −kT log Z; > s.a. entropy [and Rényi entropy].
$ Gibbs free energy: The thermodynamic quantity G:= E − TS + pV. / A process is spontaneous if ΔG < 0.
@ General references: Coffey HSPBS(06) [historical, and the third law of thermodynamics]; Vemulapalli PhSc(10)dec [how G can predict properties of matter]; Sanami JMP(13) [characterization of the Helmholtz free energy]; Prentis & Obsniuk TPT(16) [simple model to illustrate the concept]; Geller & Daane AJP(19)jul [as useful energy].
@ Geometric: Pollicott & Weiss CMP(05) [for surfaces with variable negative curvature].
> Online resources: see Wikipedia page.

Free Fall > s.a. equivalence principle; Projectile Motion.
@ References: Spallicci a1005-ln [and self-force, historical perspective].

Free Group > see types of groups.

Free Mobility Postulate (Helmholtz) > see spacetime models.

Free Product of Goups > see group.

Free Will / Freedom of Choice > s.a. determinism; philosophy of physics; Retrocausality.
* Compatibilist position: Free will is compatible with complete determinism, the assumption that our actions are determined by the laws of nature.
* Seth Lloyd's position: What gives rise to our impression that we possess free will is the intrinsic computational unpredictability of our decision-making process.
* "Free will theorem": A result stating that if experimenters have free will in the sense that their choices are not a function of the past, so must some elementary particles.
@ General references: Hossenfelder a1202 [meaning, without employing metaphysics]; Svetlichny a1202 [physical and mathematical model]; Esposito a1202 [personal point of view]; Mandayam Nayakar et al a1202 [definition and model]; Mandayam Nayakar & Srikanth a1210 [and uncomputability]; Lloyd PTRS(12)-a1310 [Turing test, roles of quantum mechanics and computation]; Mandayam Nayakar & Srikanth a1401; Svozil a1405 [dualistic scenario]; Hashim & Srikanth a1503; Lopez-Corredoira a1612-in; Krumm & Müller a2101 [computational irreducibility and compatibilism].
@ And quantum theory: Nikolić ASL-a1006; Barrett & Gisin PRL(11)-a1008 [and locality]; Hall PRL(10) [partially giving up free will to maintain locality and realism]; Di Lorenzo a1105; Brassard & Raymond-Robichaud a1204 [and the theory of "parallel lives"]; Bisognano a1211 [and quantum measurement]; Ghirardi & Romano FP(13)-a1301 [and extensions of quantum theory]; Colbeck & Renner a1302; Aaronson a1306-in [Turing's ideas and related issues]; Khan a1604 [and quantum information]; Bednorz PRD(16)-a1605 [in relativistic quantum field theory]; Kochen a1710 [and Born's rule, EPR]; Boughn a2012.
@ "Free will theorem": Conway & Kochen a0807 [strengthening]; Hájíček GRG(09); Goldstein et al NAMS(10)-a0905 [criticism]; Suarez a1002, a1006; Reznikoff a1008 [logical proof]; Liu et al PRL(16)-a1603 [experimental test]; Dzhafarov & Kujala a2007; > s.a. relativistic quantum mechanics.
> Online resources: see Internet Encyclopedia of Philosophy page; Stanford Encyclopedia of Philosophy page; Wikipedia page.

Freezing > see condensed matter.

Frenet-Serret Curvature / Formulas > see lines and curves / classical relativistic particles; minkowski space.

Frequency > s.a. wave equations.
* Idea: The number of cycles per unit time in a time-dependent phenomenon of oscillatory type; For a physical quantity A(t) with sinusoidal time dependence the frequency f = 1/T is the inverse of the period; For a more general time dependence it is the variable conjugate to t in a Fourier series/transform.
* Rem: Frequency is the physical quantity that can be measured with the greatest accuracy, by far.
@ Frequency operator: Caves & Schack AP(05) [and quantum probability postulate].

Fresnel Integrals
* Idea: Integrals of the type ∫_−∞^∞ dx exp{±ix²} = π^1/2; Results like this can be proved taking the a → 0 limit of generalized Gaussian integrals.

Freud Pseudotensor / Superpotential > see stress-energy pseudotensors.

Friction

Friedmann Equation and Solutions

Friedmann-(Lemaître)-Robertson-Walker Spacetime > s.a. fields, geometry, 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]; Dereziński & De Roeck JMP(07)qp/06 [stochastic limit]; Courbage et al PLA(07) [and kaon phenomenology]; Akchurin TMP(10) [generalized, spectrum]; Gadella & Pronko FdP(11)-a1106 [and resonance phenomena, rev].

Frobenius Algebra
@ References: Kaufmann CMP(04) [second quantization]; Stigner PhD-a1210 [in conformal field theory].

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 1990s; 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 > see Division Algebra.
> Online resources: see MathWorld page [Frobenius theorem for matrix eigenvalues].

Frobenius-Perron Operator
$ Def: The operator U generating discrete time evolution for a classical distribution function ρ, i.e.,
ρ_n+1(x) = U ρ_n(x).

Froth > see Foam; tiling.

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.

Frozen Star > see black-hole phenomenology.

Frustration > s.a. spin models.
* Geometrical frustration: A situation in which a system of interacting particles is unable to find its lowest energy state because of how the particles are arranged; It can play an important role at microscopic scales in solids, in particular magnets, where the absence of a single, lowest-energy state can affect a material's conductivity; An example is a quantum spin liquid, in which a destruction of magnetism is exhibited by spins arranged in a triangular lattice because the QSL never enters into a long-range ordered phase; Instead, the electrons' spins remain fluid-like, even at 0 K.
@ Examples: Kang et al PRL(14) [macroscopic geometrical frustration in a lattice of triangular tiles]; > Spin Liquid.

Fubini-Study Metric > see types of metrics.

Fuchsian Analysis / Equations > see gowdy spacetime.

Fugacity
$ Def: The quantity z:= exp{βμ}, in terms of which the grand canonical partition function for a classical monatomic ideal can be written as Z = exp{zV/λ³}.
> Online resources: see Wikipedia page.

Full Chronological Space > see Chronological Space.

Functional Analysis, Derivative, Equations

Functionalism > see spacetime.

Functions > s.a. functions and maps on differentiable manifolds.

Functor

Fundamental Group > s.a. homotopy theory.

Fundamental Homology Class
$ Def: For a compact oriented n-manifold M, the unique μ in Hⁿ(M; \(\mathbb Z\)) such that ρ_x(μ) = μ_x, where ...

Fundamental Identity of Thermodynamics > see thermodynamics.

Fundamental Length > see under Minimal Length.

Fundamental Physics / Theory > see physical theories / paradigms in physics.

Fundamental Theorem of Algebra > see elementary algebra.

Fundamental Theorem of Calculus
$ Def: The statement that \(\int_a^b{\rm d}x = b-a\).

Fusion > see nuclear physics and technology.

Fusion Bases / Coefficients / 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-conf, JMP(00), JMP(00) [for affine Lie algebras], JMP(02)ht/01 [as facets of polytopes].
@ Fusion coefficients: Alesci et al CQG(10) [from SO(3) to SO(4)].
@ Fusion rules: Zimborás in(06)m.GR/05 [compact groups, information contained].

Future > see cosmology; spacetime subsets; time.

Future Included / Not Excluded Theory > see formulations of quantum theory.

Fuzzballs > s.a. event horizons; string phenomenology.
* Idea: The fuzzball construction is a way of resolving the black hole information paradox by making spacetime end just before the horizon is reached.
@ References: Mathur AP(12) [and black-hole information]; Mathur a1401 [and black-hole thermodynamics]; Hertog & Hartle a1704 [quantum dynamics of gravitational collapse, observational signatures]; Bianchi et al PRL-a2007 [multipolar structure].
@ And observations: Guo et al JHEP(18)-a1711; Mayerson GRG(20) [rev]; Ikeda et al a2103 [discrimination using gravitational-wave spectroscopy].

Fuzzy Geometry / Manifolds > see differential geometry; higher-dimensional spacetime; non-commutative geometry.

Fuzzy Set Theory > see set theory; logic.

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