Topics, W

W Particle > see electroweak theory.

@ And particles: Ramos & Roca NPB(95)ht.

* Idea: One in which there are no timelike lines of unbounded length, and there is a compact Cauchy hypersurface.
* Properties: It recollapses.

* Remark: "W" stands for weak-operator closed.
@ References: Sakai 71.

Wagner Conjecture > see graph theory.

Wahlquist Metric
* Idea: A stationary, axially symmetric perfect-fluid solution of which the Kerr metric is a vacuum subcase.
* Result: It cannot be smoothly joined to an exterior asymptotically flat vacuum region.
@ References: Bradley et al CQG(00)gq/99 [asymptotically flat matching no-go]; Mars PRD(01)gq [extension to "Wahlquist-Newman"]; Sarnobat & Hoenselaers CQG(06) [non-asymptotic flatness]; Hinoue PRD(14) [in all dimensions].

Walk > see random walk.

* Idea: Droplets that bounce on a vertically vibrating bath of the same fluid and can form wave-particle symbiotic structures with the surface waves they generate. Macroscopic walkers were shown experimentally to exhibit particle and wave properties simultaneously [@ in Davydov JPCS(12)-a1201].

Wallis Formula > see π.

Ward / Ward-Takahashi Identities > s.a. gauge [in quantum field theory]; noether theorem; quantization of constrained systems.
* Idea: Identities satisfied by the complete Green functions of quantum fields when the original classical Lagrangian system is degenerate, that represent the invariance of the theory under some gauge transformations, and come from compensating terms in the measure and integrand in the path integral.
* For QED: One form is

[S'(p)]−1 = [S'(p0)]−1 + (pp0)a Γa(p, p0) ,

where p0 = on-shell momentum, S' = full propagator, Γa = full QED vertex.
@ References: Ward PR(50); Takahashi NC(57); Danos FP(97)ht [mathematically rigorous]; Jackiw ht/97 [history, significance]; Dütsch & Boas RVMP(02)ht/01 [master Ward identity]; Prinz a2001 [generalized, as Hopf ideals]; Baloïtcha et al a2001 [in tensorial group field theories and matrix models, rev].

Warp Drive > see causality violations.

Wasserstein Metric > see types of distances.

water (and ice)

Waves > s.a. wave phenomena.

Wave Function > see foundations of quantum mechanics [reality]; wave-function collapse.

Wave-Particle Duality > s.a. Complementarity.
* Idea: Heisenberg's view that one can interpret the quantum-mechanical equation of motion in terms of either a wave ontology or a particle ontology; Can be resolved by the realization that both concepts are idealizations; Related to complementarity.
* History: Earlier thought to be a consequence of uncertainty, it is now recognized as independent of the latter.
* Observation: The classic signature is the interference pattern produced when partices pass through a double slit; It has been seen in electrons, atoms and small molecules, but never in the macroscopic world; 1999, Observed by Anton Zeilinger's group in Vienna with C60 − buckminsterfullerene − and C70 molecules, about 1 nm in diameter; 2003, Reported in 2-nm organic molecules; > s.a. interference.
@ General references: Jánossy APH(52); Renninger ZfP(53) [translation De Baere phy/05]; Diner in(84); Bardou AJP(91)may; Selleri 92; Comborieu & Rauch FP(92) [rev]; Busch & Lahti RNC(95); Buks et al Nat(98)feb; Freyberger PLA(98) [measurement]; Camilleri SHPMP(06) [and complementarity]; Blackman PE(13)-a1207; Rashkovskiy SPIE(13)-a1302 [proposed interpretation]; Qureshi AJP(16)jul-a1501 [quantitative]; Orefice et al JAMP(18)-a1701 [dynamics].
@ For light: Cormier-Delanoue FP(95); Duncan & Janssen a0709 [P Jordan's contribution]; Dimitrova & Weis AJP(08)feb [demonstration experiment]; Fick and Kant SHPMP(09) [Walther Bothe's contributions]; Huang et al PRA(13) [higher-order inequalities]; Sperling et al a1907, Ryff a2103 [neither wave nor particle].
@ Special cases: Clifton PLA(00)qp/99 [spin-0, and Kochen-Specker arguments]; Hackermüller PRL(03) + pw(03)sep [large organic molecules]; Schilling & von Zanthier a1006 [in two-way interferometer with which-way detector]; Dittel et al a1901 [many-body quantum states]; > s.a. types of particles [electrons].
@ Related topics: Kolář et al qp/05 [anomalies from entanglement]; Wesson GRG(06) [waves and particles in general relativity]; Davydov JPCS(12)-a1201 [in classical mechanics]; Siddiqui & Qureshi QSMF(16)-a1406 [non-local duality relation]; Siddiqui & Qureshi a2010 [wave-particle superposition and complementarity]; Popławski a2101, comment Diether & Christian a2103 [for spinors].

Wavelets > s.a. Cuntz Algebra; wave equations.
* Idea: Wavelet analysis is an alternative decomposition of waves, with respect to Fourier analysis.
* Advantages: Localization.
@ General references: Strang AS(94)apr; Han et al PLA(95) [photons]; Kaiser 94 [IIIa]; Holschneider 95; Walnut 01 [r BAMS(03)]; Addison pw(04)mar [applications]; Altaisky 05 [including applications]; Walker 08; Nickolas 17 [student guide].
@ Physical: Fujiwara & Soda PTP(96)ap/95 [cosmological perturbations]; Kaiser PLA(92)mp/01, ACHA(94)mp/01 [in electromagnetism]; Visser PLA(03); Kaiser JPA(03)mp [acoustic + electromagnetic, review].
@ In quantum (field) theory: Federbush PTP(95); Bagarello JPA(96) [pedagogic]; Steeb 98; Havukainen qp/00 [in QED]; Albeverio & Altaisky a0906 [gauge invariance]; Polyzou & Bulut in(13)-a1312; Brennen et al PRA(15)-a1412 [multi-scale quantum simulation]; Altaisky a1712-conf [and renormalization group]; Chawhan & Ratabole a2010; > s.a. matter phenomenology in quantum gravity; quantum field theory techniques; renormalization group; stochastic quantization.
@ Related topics: Antoine & Vandergheynst JMP(98) [on S\(^n\)].

Weak Coupling Limit of Gravity > see modified approaches to quantum gravity.

Weak Derivative > see tensor field.

Weak Gravity Conjecture > s.a. cosmic censorship.
* Idea: The suggestion that, in a self-consistent theory of quantum gravity, the strength of gravity is bounded from above by the strengths of the various gauge forces in the theory; A proposed constraint on gauge theories coupled to gravity, requiring the existence of light charged particles and/or imposing an upper bound on the field theory cutoff Λ; More specifically, an Abelian gauge theory coupled to gravity is inconsistent unless it contains a particle of charge q and mass m such that qm/mPl; That is, there exist superextremal charged particles, with mass smaller than or equal to their charge in Planck units.
* Motivation: With it, an extremal charged black hole can completely evaporate without leaving a dangerous stable extremal remnant while simultaneously not revealing a naked singularity along the way; In other words, it ensures that the charge is emitted faster than the mass of a black hole.
@ References: Arkani-Hamed et al JHEP(07)ht/06; Cheung & Remmen PRL(14)-a1402 [and naturalness], JHEP(14)-a1408 [and low-energy effective field theory]; Kooner et al PLB(16)-a1509; Hod IJMPD(17)-a1705-GRF [from Bekenstein's generalized second law of thermodynamics]; Cheung et al JHEP(18)-a1801 [proof from black-hole entropy]; Urbano a1810 [towards a proof]; de Rham et al a1812 [and spin-2 fields]; Montero JHEP(19)-a1812 [holographic derivation]; Shirai & Yamazaki a1904 [scalar weak gravity conjecture in de Sitter space].
@ Different versions: Heidenreich et al JHEP(17)-a1606 [evidence for a stronger statement]; Montero et al JHEP(16)-a1606 [3D]; Heidenreich a1906 [two precise formulations]; McInnes a2007 [holographic dual]; Solomon & Stojkovic a2008 [generalized, non-extremal black holes].

Weak Interaction > s.a. electroweak theory; history of particle physics; Fermi Theory; standard model.
* Idea: One of the four "fundamental" interactions, and one of the two nuclear forces; It affects all known fermions and is now incorporated into the electroweak interaction, but was initially described by the Fermi theory, an empirically successful but non-renormalizable theory; Phenomenologically, it is responsible for the decay of subatomic particles and for parity violation (> see parity), and initiates hydrogen fusion in stars.
* Types: Charged currents, which change flavors within families and are mediated by \(W^\pm\) bosons; Neutral currents, mediated by Z bosons, which are responsible for example for neutrino-electron scattering.
@ General references: Feynman & Gell-Mann PR(58) [4-fermion interaction]; Radicati ed-60; Bell yr(72); Commins 73; Holstein AJP(77)nov [RL]; Cline ThSc(93)nov; Greiner & Müller 96 [III]; Anthony et al PRL(05) + pn(05)jul [measurement of the weak mixing angle over large distance range].
@ And gravity: Alexander et al PRD(14)-a1212 [as the right-handed chiral half of the spacetime connection]; Onofrio MPLA(13)-a1412 [weak interactions as short-distance manifestations of gravity]; > s.a. unified theories.
> Online resources: see Wikipedia page.

Weak Measurements / Values of Observables > s.a. quantum measurement [for fields]; types of quantum measurements [sequential].
* Idea: A tool whereby the presence of a detector has an effect that is smaller than the level of uncertainty around what is being measured, so that there is an imperceptible impact on the experiment; Used to resolve Hardy's Paradox.
@ Reviews, intros: Diósi qp/05-en; Parrott a0908, a0909 [intros]; Shikano in(12)-a1110; Dressel et al RMP(14) [and applications]; Aharonov et al PRA(14) [foundations and applications]; Vaidman a1703 [controversy]; Sokolovski & Akhmatkaya a1705 [simple].
@ General references: Aharonov et al PRL(88), comment Leggett PRL(89) + Duck et al PRD(89) [spin]; Ruseckas & Kaulakys LJP(04)qp [and time]; Johansen & Luís PRA(04)qp [non-classicality]; Oreshkov & Brun PRL(05)qp; Tollaksen & Aharonov SPIE(07)qp/06 [non-statistical]; Davies PRA(09)-a0807 [time-dependent]; Katz et al PRL(08) + Bruder & Loss Phy(08) [reversibility, state recovery]; news nfr(09)jan [and Hardy's Paradox]; Lobo & Ribeiro PRA(09)-a0903 [and quantum phase space]; Dixon et al PRL(09) + Popescu Phy(09) [and ultrasensitive position and angle measurements]; Ashhab & Nori a0907; Dressel & Jordan PRL(12)-a1206 [universal nature of quantum weak values]; Aharonov et al a1207, a1207, PRA(14); Kofman et al PRP(12); Zhu et al PLA(13)-a1212 [and negative probabilities]; Hofmann AIP(14)-a1303 [complex conditional probabilities and fundamental laws]; Salvail a1310 [weak values beyond weak measurement]; Geelan a1306; Feyereisen FP(15)-a1503 [weak variance]; Hari Dass CS-a1509; Roik et al a1903 [and strong measurements]; > s.a. Quantum Trajectories.
@ Conceptual: Svensson FP(13)-a1301 [interpretation of weak values]; Sokolovski Quanta(13)-a1305 [critical view]; Ferrie & Combes PRL(14)-a1403, comment Vaidman a1409, Hofmann et al a1410 [classical analog]; Dressel PRA(15)-a1410; Mochizuki a1604, Cohen FP(17)-a1704 [physical meaning]; Hiley a1809 [weak values and the nature of quantum processes].
@ Experiments: Pan et al a1910 [weak-to-strong transition].
@ And photons: Kocsis et al Sci(11)jun + news nat(11)jun [photon two-slit experiment and uncertainty principle]; Hofmann a1311-proc [direct observations of photon wavefunctions in weak measurements, and complex probabilities]; Flack & Hiley a1611 [electromagnetic field momentum]; Hallaji et al a1612 [truly quantum example, with photon count].
@ Applications: Davies in(14)-a1309 [cosmology]; Jordan et al QS:MF(18)-a1811 [gravitational sensing]; Higashino et al a2011 [CP violation in B meson decays]; > s.a. gravitational-wave interferometers.
> Related topics: see uncertainty relations.

Weak Operator Topology on \(\cal B\)(\(\cal H\)) > topology.

Webs > see Cosmic Web [cosmology]; foliation [mathematics].
* In quantum field theory: Sets of Feynman diagrams that contribute to the exponents of scattering amplitudes, in the kinematic limit in which the emitted radiation is soft.
@ References: White JPG(16)-a1507-ln [pedagogical introduction]; > s.a. spin networks; Wilson Loops [gauge theories].

Weber Functions > see bessel functions.

Wegner's Flow Equations
* Idea: A powerful tool for diagonalizing a given Hamiltonian, widely used in various branches of quantum physics.
@ References: Itto & Abe FP(12)-a0806 [conditions for geodesic flow].

Wehrl Entropy > see entropy in quantum theory.

Weierstraß Elliptic Functions
@ References: Pastras a1706-ln [in classical and quantum mechanics].
> Online resources: see Wikipedia page.

Weierstraß Functions > s.a. Takagi Function.
* Idea: A 2-parameter family of functions that are everywhere continuous, but nowhere differentiable, given by W(x) = ∑k = 0 ak cos(bkx) with 0 < a < 1 and b a positive integer (or by some other similar family of functions); The graph of the function has detail at every level as one zooms in, and the function could perhaps be described as one of the very first fractals studied.
@ References: in Stromberg 81.
> Online resources: see Wikipedia page.

Weierstraß Theorem
$ Def: Every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function.
* Generalizations: The Stone-Weierstraß theorem extends the result in two ways, (1) The interval can be replaced by an arbitrary compact Hausdorff space X, and (2) The algebra of polynomial functions can be replaced by elements from more general subalgebras of C(X).
> Online resources: see Wikipedia page.

Weil Conjecture > see conjectures.

Weil Homomorphism
$ Def: A map w: I(G) → H*(M; \(\mathbb R\)) from the set of invariant Lie algebra polynomials to the set of all cohomology classes, which is a ring homomorphism.

Weil Representation > see representations in quantum theory.

Weinberg Paradox
* Idea: The amplitude for a single photon exchange between an electric current and a magnetic current violates Lorentz invariance.
@ References: Terning & Verhaaren a1809 [resolution].

Weinberg Theorem > see cosmological perturbations.

Weinberg-Witten Theorem
* Idea: The statement that no massless (composite or elementary) particles with spin j > 1 are consistent with any renormalizable Lorentz-invariant quantum field theory other than (non-renormalizable) theories of gravity and supergravity.
> Online resources: see Wikipedia page.

Weinberg-Salam Electroweak Theory

Weingarten Matrix
* Idea: For a 2D surface patch in \(\mathbb R\)3, it is given by qab Kab.

Weinhold Metric > s.a. thermodynamics; black-hole thermodynamics.
* Idea: A metric on the state space of a thermodynamical system, conformally related to the Ruppeiner metric.
@ References: Weinhold JChemP(75), JChemP(75).

Weiss Variational Principle > see variational principles in physics.

Weitzenböck Connection / Spacetime > s.a. Metric-Affine Gravity; teleparallel gravity; tensor fields.
* Idea: A flat connection Γabc := eaib eci with non-vanishing torsion T abc := Γabc − Γacb defined by a tetrad field eai on a parallelizable manifold, used in some gravity theories; The Weitzenböck connection is also defined on any Lie group.
@ References: in Bishop & Goldberg 68; in Goldberg 62; Hayashi & Shirafuji PRD(79) [and gravity theory]; Bel a0805 [and Christoffel connection].
> Online resources: see Wikipedia page on Roland Weitzenböck.

Welcher-Weg Experiment > see interference [German for "which-way experiment"].

Well-Ordered Set
$ Def: A totally ordered set in which every non-empty subset has a least member.
* Well-ordering principle: For any set X, there is an ordering that makes it well-ordered.

Well-Posed Problem
* Idea: In Jacques Hadamard's definition, a set of differential equations (representing a mathematical model for a physical phenomenon) together with a specification of data required to find a solution is well-posed if (a) A solution exists, (b) The solution is unique, and (c) The solution changes continuously with the initial conditions.
> Online resources: see Wikipedia page.

Wess-Zumino, Wess-Zumino-Witten Model > see types of supersymmetric theories.

Wetting > see water.

Weyl Algebra > s.a. knots; observables.
@ References: Thirring 81 (v3, §3.1); Arai LMP(08) [uniqueness of Weyl representation of commutation relations]; Grundling & Neeb RVMP(09) [C*-algebra for full set of regular representations]; Mnatsakanova et al IJMPA(16)-a1606 [representation in a Krein space]; Feintzeig et al a1805, Feintzeig & Weatherall a1805 [re regularity of representations].

Weyl Anomaly > see anomalies.

Weyl Curvature Hypothesis
* Idea: The hypothesis that the Weyl tensor vanished at the Big Bang singularity, introduced by R Penrose in an attempt to explain the high homogeneity and isotropy, and the very low entropy of the early universe, in conjunction with his proposal to use the Weyl tensor to define gravitational entropy.
@ References: Penrose in(79), in Penrose 04 [§§ 28.5, 28.8]; Stoica AP(13)-a1203; Okon & Sudarsky CQG(16)-a1602-GRF [dynamical justification]; Ashtekar & Gupt CQG(17) [quantum generalization].

Weyl Fermions > see 2D spinors.

Weyl Geometry > s.a. unified theories.
@ References: Romero et al CQG(12)-a1201 [and general relativity]; Barreto et al AIP(15)-a1503 [ADM formalism]; Scholz a1703-conf [resurgence]; Wheeler GRG(18)-a1801; Scholz a1911-conf [history].

Weyl Gravity > see unified theories of gravity and electromagnetism; tests of general relativity [light deflection].

Weyl Invariance / Symmetry > similar to conformal invariance [but it usually refers both the metric and matter]; s.a. mass.

Weyl Manifold / Space > s.a. unified theories.
* Idea: A differentiable manifold with compatible conformal and projective structures.
@ References: Ehlers, Pirani & Schild in(72); Hall JMP(92); Bokan et al PRS(97) [differential operators and invariants]; Fatibene & Francaviglia IJGMP(12)-a1106 [and timelike geodesics], a1109 [and fluid conservation laws]; Poulis & Salim IJMPCS(11)-a1106 [and spacetime structure]; Romero et al IJMPCS(11)-a1106 [and general relativity]; Scholz a1111-proc [in late 20th-century physics].

Weyl Quantization
* Idea: A prescription for promoting polynomial observables to operators in quantum theory, which consists in using the most symmetrical operator ordering; > different from Born-Jordan Quantization.
@ General references: Ozorio de Almeida PRP(98); Lein a1009-ln [and semiclassics]; He et al MPLA(14) [comparison using a ray function in classical phase space].
@ Special cases: Gelca & Uribe CMP(03)mp/02 [flat SU(2) connections]; Przanowski & Brzykcy AP(13) [on the cylinder]; Ligabò a1409 [on the torus phase space].
> Online resources: see Encyclopedia of Mathematics page; Terence Tao's page.

Weyl Solutions > see solutions of general relativity with symmetries.

Weyl Spinors / States > see 2-spinors.

Weyl Tensor

Weyl Transform > s.a. path integrals; wigner function; Wigner Transform.
* Idea: A mapping from phase-space functions to Hilbert-space operators in quantum mechanics.
> Online resources: see Wikipedia page.

Weyl Transverse Gravity > a theory without full diffeomorphism invariance.

Weyl Tube Formula > see lovelock gravity.

Weyl Vector > see affine connection.

Weyl's Raumproblem > see Raumproblem.

Weyl-Cartan Spacetime > s.a. Metric-Affine Gravity.
* Idea: A non-Riemannian manifold with non-metricity and torsion, as in metric-affine theories of gravity.
@ References: Haghani et al JCAP(12) [Weyl-Cartan-Weitzenböck gravity].

Weyl-Lanczos Equations
@ References: O'Donnell GRG(04) [in Schwarzschild spacetime].

Weyl-Schouten Tensor > see weyl tensor.

Weyl-Wigner-Moyal Formalism > s.a. contextuality.
* Idea: A prescription for associating with each operator describing a state, observable or transition in quantum theory, a function on phase space; This function is known as the Weyl symbol, or the Weyl transform of the corresponding operator; As first pointed out by Moyal, the Weyl transform generates a deformation of the classical Poisson brackets and of the usual commutative product on phase space; The deformed product is denoted by ∗ and is called the twisted product; The deformation of the Poisson bracket is known as the Moyal bracket.
@ References: Antonsen IJTP(98)qp/96 [on algebraic structures]; Li et al EPL(13)-a1210 [for spin].

Wheeler-De Witt Equation > see geometrodynamics.

Which-Way Experiment > see interference.

White Dwarf > s.a. Chandrasekhar Limit; dark-matter types; star types [including binaries].
* Idea: A low-mass star in a late evolutionary stage, held in equilibrium by electron degeneracy pressure.
* Super-Chandrasekhar white dwarfs: Highly magnetized white dwarfs whose mass can exceed the Chandrasekhar limit.
@ General references: Van Horn PT(79)jan; Isern et al JPCM(98); Kawaler & Dahlstrom AS(00); Hansen PRP(04); Napiwotzki JPCS(09)-a0903 [galactic population]; Blackman a1103/PT [history of white-dwarf mass limit]; Badanes & Maoz ApJL(12)-a1202 [binary merger rate in the galactic disk]; Kissin & Thompson ApJ(15)-a1501 [spin and magnetism]; Das & Mukhopadhyay IJMPD(15)-a1506-GRF [and modified gravity]; Kepler et al IJMPcs(17)-a1702 [results]; Smart PT(18)feb [chemistry]; Wilson PT(19)mar [crystallization phase transition].
@ And relativistic gravity: Mathew & Nandy RAA(17)-a1401, Boshkayev et al JKPS(14)-a1412 [in general relativity]; Das & Mukhopadhyay IJMPD(15)-a1506-GRF [in modified gravity]; Banerjee et al JCAP(17)-a1705 [constraints on modified gravity]; Carvalho et al GRG(18)-a1709.
@ Super-Chandrasekhar white dwarfs: Kundu & Mukhopadhyay MPLA(12)-a1204; Das & Mukhopadhyay MPLA(14)-a1304 [physics issues], IJMPD(13)-a1305 [new mass limit]; Chamel et al PRD(13) [stability]; Das & Mukhopadhyay a1406.
@ Related topics: Caiazzo & Heyl MNRAS(17)-a1702, Stephan & Zuckerman ApJL(17)-a1704 [atmosphere pollution by comets, asteroids and planetary bodies].

White Hole
* Idea: The time reversal of a spacetime in which gravitational collapse has occurred to form a black hole.
@ General references: Wald & Ramaswami PRD(80) [particle production]; Barrabès et al PRD(93); Hsu CQG(11)-a1007 [isolated white holes]; Bardeen a2007 [quasi-classical evolution]; Barrau et al a2101 [as black hole remnants].
@ Bouncing geometries with white and black holes: Barceló et al IJMPD(14)-a1407-GRF; Olmedo et al CQG(17) + CQG+ [in quantum gravity].
@ Phenomenology: Retter & Heller NA(12)-a1105 [white holes as small bangs]; Akhoury et al a1608 [as a result of gravitational collapse in Einstein-æther theory].

Whitehead Continua
* Idea: There is an open, contractible 3D topological manifold W, which is not homeomorphic to \(\mathbb R\)3, but such that \(\mathbb R\)1 × W is \(\mathbb R\)4.

Whitehead Theorem / Triangulation > s.a. types of manifolds [PL-manifolds].
* Idea: For each smooth manifold M, there exists a PL-manifold MPL, called its Whitehead triangulation, such that M is diffeomorphic to a smoothing of MPL; MPL is unique up to a PL-isomorphism.
@ References: Minian & Ottina JHRS-math/06 [generalization using CW(A)-complexes].

Whitehead Theory of Gravity
* Idea: A theoryof gravity with a flat background, non-dynamical metric that governs the propagation of gravitational waves and a curved, dynamical metric that governs the propagation of matter fields, such as electromagnetic waves.
@ References: Coleman phy/05, a0704; Gibbons & Will SHPMP(08)gq/06 [truly dead]; Desmet ln(10).
> Online resources: see Wikipedia page.

Whitney Duality Theorem
$ Def: If M is a manifold embedded in Euclidean space and N(M) its normal bundle, then w(N(M)) = w(T(M))−1.

Whitney Embedding Theorem > see embeddings.

Whitney Numbers
@ References: Baclawski AiM(75) [of geometric lattices].

Whitney Product Theorem > see stiefel-whitney classes.

Whitney Sum of Vector Bundles
* Idea: Given two vector bundles E and F over the same B, EF has as fiber the direct sum of the fibers, and similarly for the transition functions: g = diag(gE, gF).
$ Def: It can be defined as the pullback d*(EF) of the Cartesian product E × F, under the diagonal embedding d: BB × B, d(b):= (b,b).

Whitney Topology
@ References: in Mather AM(69).

Whittaker Functions
@ References: Lucietti JMAA(04)mp [and Bessel functions]; O'Connell a1201-proc [and related stochastic processes].

Wick Rotation > s.a. approaches to quantum field theory; quantum dirac fields.
* Idea: The rotation of the time axis in Minkowski space from real to imaginary times performed in field theory to calculate some integrals.
* In quantum gravity: It has been introduced as a mapping between real euclidean metrics and real lorentzian metrics, but no generalization of the flat spacetime Wick rotation is known that would map general curved, smooth lorentzian metrics to euclidean ones, or diffeomorphism equivalence classes of metrics of either signature, into each other; In fact, it is known that in 2D lorentzian and euclidean non-perturbative gravitational path integrals in general yield inequivalent results.
@ References: Visser GRF(91)-a1702 [as a complex deformation of the spacetime metric]; Liu ht/97 [geometric aspects]; Helleland & Hervik a1504 [Wick-rotatable metrics are purely electric].
@ In quantum gravity / curved spacetime: Dasgupta & Loll NPB(01)ht; Dasgupta JHEP(02)ht; Baldazzi et al CQG(19)-a1811 [difficulties]; Kontsevich & Segal a2105 [extension to complex-valued metrics].

Wick's Theorem > see Time-Ordered Product.
@ References: Wick PR(50); Plimak & Stenholm PRD(11)-a1104 [and causal signal transmission]; Beloussov SpMa(15)-a1501 [convenient algebraic formulation]; Schönhammer PRA(17)-a1707 [finite-temperature version, in the canonical ensemble]; Diósi JPA(18)-a1712 [for all orderings of canonical operators].
> Online resources: see Wikipedia page.

Widom Conjecture > see entropy in quantum theory.

Wieferich Primes > see number theory.

Wien's Law > see thermal radiation.

Wiener Measure > see measure theory.

Wiener Process > see stochastic processes.

Wiener's Theorem > see fourier analysis.

Wightman Axioms (For relativistic quantum field theory) > s.a algebraic quantum field theory; approaches to quantum field theory.
@ General references: Streater & Wightman 64; Rehren CMP(96) [solutions].
@ Generalized: Johnson a1205 [revised Wightman axioms and massless particles]; Morgan a1211 [non-linear maps]; Jäkel et al a2002 [interacting theories].

Wightman Functions > s.a. [green functions]; locality in quantum field theory; non-commutative field theory.
* For a scalar field: Defined by G+(x, x'):= \(\langle\)0| φ(x)φ(x') |0\(\rangle\) and G(x, x'):= \(\langle\)0| φ(x')φ(x) |0\(\rangle\).
* Properties: It satisfies the homogeneous field equation.
@ References: Ritter mp/04 [for gauge fields]; de Ramón et al a1806 [direct measurement].

Wigner 6j and 9j Symbols > see SU(2).

Wigner Crystal
* Idea: A solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934, formed by a gas of electrons moving in 2D or 3D in a uniform, inert, neutralizing background if the electron density is less than a critical value; The crystal lattice is body-centered cubic in 3D, and triangular in 2D.
@ References: Wigner PR(34); Grimes & Adams PRL(79), Fisher et al PRL(79) [first observation]; Dykman Phy(16); Lotfizadeh et al PRL(19).
> Online resources: see Wikipedia page.

Wigner Delay
* Idea: The time light spends outside a piece of transparent material when it undergoes total internal reflection.
* History: First suggested by Newton; Wigner made a prediction for the value in 1955; Measurements first reported in 2005, showing two different values depending on the polarization.
@ References: Chauvat et al PLA(05) + pw(05)mar [first measurement, and doubling].

Wigner Derivative > see SU(2) [asymptotics of 6j symbols].

Wigner Functions > s.a. specific systems and generalizations.

Wigner Inequality
@ References: Nikitin & Toms PRA(10)-a0907 [in quantum field theory]; Plick et al PRA(15)-a1304 [extension].

Wigner Rotations > s.a. Rapidity; special-relativistic kinematics.
@ References: Soo & Lin IJQI(04)qp/03; Rhodes & Semon AJP(04)jul [geometric approach]; Saldanha & Vedral NJP(12)-a1111 [physical interpretation], PRA(13) [and apparent paradox in relativistic quantum information].

Wigner Theorem > s.a. symmetries in quantum theory.
* Idea: A bijective transformation of the quantum state space (projective Hilbert space) which preserves orthogonality is induced by either a unitary or an anti-unitary operator.
* Non-bijective version: A map defined on the set of self-adjoint, rank-one projections (or pure states) of a complex Hilbert space which preserves the transition probability between any two elements, is induced by a linear or antilinear isometry.
@ General references: Györy RPMP(04) [elementary proof]; Chevalier IJTP(05) [lattice approach], IJTP(08) [Wigner-type theorem for projections]; Keller et al MSB(08)-a0712 [simple proof, realization of symmetries in quantum mechanics and projective geometry]; Buth a0802, Freed a1112 [proof using geometrical methods]; Simon et al PLA(08)-a0808 [simple proofs]; Mouchet PLA(13)-a1304 [elementary proof]; Brody JPA(13)-a1305 [complex extension]; Harding a1604 [for an infinite set]; Pankov & Vetterlein a2012 [geometric approach].
@ Non-bijective version: Gehér PLA(14)-a1407 [elementary proof].
@ Other generalizations: Chiribella a2101 [for quantum evolutions].

Wigner Transform > s.a. Weyl Transform.
* Idea: A mapping from Hilbert-space operators to phase-space functions in quantum mechanics.
@ References: Costa Dias et al RVMP(13) [metaplectic formulation]; Mann et al a1507 [family of Wigner transforms for continuous and finite-dimensional Hilbert spaces]; Sarbicki et al JPA(16)-a1602 [generalization]; de Gosson 17; Cai et al a1802 [discrete analog]; Sacchetti a2104 [rev, and quantum forced oscillator].
> Online resources: see Wikipedia page.

Wigner Velocity > see quantum effects [tunneling].

Wigner's Friend
* Idea: A paradox in quantum mechanics; Wigner's friend makes a measurement in a closed laboratory, notes the outcome, and assigns a state corresponding to that outcome; Wigner, outside the door, doesn't know the outcome and assigns the friend, the apparatus, and the system an entangled state that superposes all possible outcomes; Who is right? Quantum Bayesianism (QBism) says both are right.
@ References: Vedral a1603 [Schrödinger's cat meets Wigner's friend thought experiment]; Łukaszyk a1801; Yang FP(19)-a1812 [consistent descriptions]; Suarez a1906; Stacey a1907 [QBism and misapprehensions]; Bong et al nPhys(20)aug-a1907; Hansen & Wolf a1912; Castellani a2011 [quantum collapse is collapse of histories]; Lostaglio & Bowles a2101; Sokolovski & Matzkin a2102 [and the internal consistency of standard quantum mechanics]; Kastner a2105 [critical review of experiment].
> Online resources: see Wikipedia page.

Wigner-Araki-Yanase Theorem
* Idea: A result in quantum mechanics which describes restrictions that conservation laws impose upon the physical measuring process.
@ General references: Wigner ZP(52); Araki & Yanase PR(60); Kakazu & Pascazio PRA(95) [alternative formulation]; Miyadera & Imai PRA(06)qp; Meister qp/07-proc [extension to multiplicative conserved quantities]; Busch & Loveridge PRL(11) + news physorg(11)mar [and position measurements]; Loveridge a2006 [relational perspective].
@ Generalizations: Tukiainen PRA(17)-a1611 [without the assumption of additivity, in terms of quantum incompatibility].

Wigner-Eckart Theorem
* Idea: A result derived by Eugene Wigner and Carl Eckart as part of a formalism linking spatial symmetries and conservation laws; A theorem of representation theory stating that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other is a Clebsch-Gordan coefficient; Physically, it says that operating with a spherical tensor operator of rank k on an angular momentum eigenstate is like adding a state with angular momentum k to the state.
@ References: Eckart RMP(30); Wigner 31; Narcowich & Fulling ed; Dai et al PRD(13)-a1211 [in cosmology]; Sellaroli a1609-PhD [for non-compact groups]; Heissenberg & Strocchi Symm(20)-a2007 [corrections for systems with spontaneously broken symmetries].
> Online resources: see MathWorld page; Wikipedia page.

Wigner-Weyl-Moyal Formalism > see under Weyl-Wigner-Moyal.

Wigner-Yanase Information > see quantum states [space of states].

Wild Embeddings
@ References: Asselmeyer-Maluga & Król IJGMP(13) [as quantum states, example in 4D spacetime and consequences for cosmology].

Willmore Surfaces
* Idea: Surfaces of minimal total extrinsic curvature.
@ References: Willmore 93 [III]; Kuwert & Schatzle AM(04) [removability of point singularities].

Wilson Lines / Loops > s.a. path integrals for gauge theories.
$ Def: Given a loop c in M, and a \(\cal G\)-valued connection A, the Wilson loop is the gauge-invariant functional given by the trace of the holonomy,

Wc(A):= tr P exp{ A}    (depends on a choice of representation of \(\cal G\)) .

@ General references: Corrigan & Hasslacher PLB(79) [variation]; in Ramond 89 [P exp]; Lee & Zhu PRD(91) [holonomies and group representations]; Drukker JHEP(99)ht [lightlike]; Lévy JGP(04)mp/03 [spin networks, observables with various groups].
@ In Yang-Mills theory: Caselle et al NPB(94) [lattice, high-T phase]; Rajeev & Turgut IJMPA(95)ht/94; Ashtekar et al JMP(97)ht/96 [2D SU(N)]; Aroca & Kubyshin AP(00) [2D]; Brzoska et al PRD(05)ht/04 [distribution as diffusion on SU(2)]; Olesen PLB(08)-a0712 [linear equation, and confinement]; > s.a. gauge theories.
@ Non-commutative theories: Ishibashi et al NPB(00) [non-commutative gauge theory]; > s.a. non-commutative field theories.
@ Gravity: Modanese PRD(94) [general relativity]; Hamber & Williams PRD(07)-a0706 [correlation length in semiclassical form and effective curvature]; Green PRD(08)-a0804 [worldlines as Wilson lines]; Hamber & Williams PRD(09) [large-scale curvature], PRD(10)-a0907 [discrete gravity, strong coupling]; Melville et al PRD(14) [high-energy limit of gravitational scattering]; Ambjørn et al PRD(15)-a1504 [in causal dynamical triangulations]; > s.a. loops; loop quantum gravity; loop variables.
@ Other theories: Tseytlin & Zarembo PRD(02), Drukker et al PRD(07)-a0704 [N = 4 super-Yang-Mills]; Henn et al JHEP(10)-a1004 [lightlike, 3D Chern-Simons and ABJM theory]; Groeger CM-a1312 [super Wilson loops and holonomy on supermanifolds]; Chatterjee CMP(20)-a1811 [Ising lattice gauge theory]; > s.a. BF theory; chern-simons theory; lattice gauge theory; supersymmetric theories.
@ Related topics: Giles PRD(81), Brambilla & Vairo PRD(97)ht/96 [and potentials]; Chen et al MPLA(00)ht [and non-abelian Stokes theorem]; Beckman et al PRD(02)ht/01 [measurability]; Freidel et al PRD(06)gq [as particles]; Dukes et al JHEP(14) [generalised to correlators of multiple Wilson line operators; webs and posets]; Belitsky et al NPB(14) [null, polygonal]; Zucchini a1903-proc [2-connections, Wilson surfaces]; > s.a. holonomy; Stokes Theorem.
> Online resources: see Wikipedia page.

WIMPS > see types of dark matter.

Winding Number > a topological invariant used to classify kinks and topological defects.

Witt Algebra > see diffeomorphisms.

Witten Equation
* Idea: The equation DAA ' λA = 0, where DAA ' = σAA 'a Da is a spatial covariant derivative acting on spinors; It has a unique solution, and provides a way of parallel transporting a spinor from spacelike infinity inward.
* Use: It was introduced by Witten in his proof of the positive gravitational energy theorem.
@ References: Witten CMP(81) [proof of the positive-energy theorem]; Reula JMP(82) [existence of solutions].

WKB (Wentzel, Kramers & Brillouin) Approximation
* Idea: (Math) A method for finding approximate solutions of a second-order linear ordinary differential equation of the form Ψ''(z) + k2 f(z) Ψ(z) = 0, when f vanishes at a point; (Phys) A method for finding semiclassical solutions to the Schrödinger equation in quantum theory (a good approximation far from turning points).
* In quantum mechanics: It consists in writing the solution of the Schrödinger equation in the form ψ = A exp(iS), with S real, rapidly varying with respect to A; Then S satisfies the classical Hamilton-Jacobi equation, with any of whose solutions we associate a family of classical trajectories in configuration space; Sometimes equivalent to the stationary-phase or one-loop approximation; Fails in a neighborhood of the boundary between the classically allowed and forbidden regions.
@ References: Gough AN(07)ap.
@ In quantum mechanics: Lindblom & Robiscoe JMP(91); Bronzan PRA(96) [modified]; Romanovski & Robnik JPA(00) [convergence, examples]; Hyouguchi et al PRL(02), AP(04) [divergence-free modification]; Sergeenko qp/02 [0th-order]; Friedrich & Trost PRP(04) [far from semiclassical limit]; Voros mp/04-proc [1D, overview]; Fityo et al JPA(06) [with minimal length]; Bracken PJM-mp/06 [time-dependent version, for tunneling]; Carles CMP(07) [for non-linear quantum mechanics]; Tripathy a2006 [summation to all orders].
> In quantum mechanics: see schrödinger equation [methods]; classical limit; pilot-wave quantum theory; deformation quantization.
> In quantum field theory: see quantum field theory in curved spacetime; quantum geometrodynamics.
> In gravitational physics: see black-hole quasinormal modes; quantum geometrodynamics; semiclassical cosmology.
> Online resources: see Wikipedia page.

WMAP (Wilkinson Microwave Anisotropy Probe) > see cosmic microwave background.

Wodzicki Residue > s.a. non-commutative field theory.
@ References: Wang JGP(06) [for manifolds with boundary].

Wojcik Model
* Idea: A position-dependent 1D quantum walk with one defect.
@ References: Endo & Konno YMJ-a1412 [weak convergence].

Wolfram Model > see cellular automata; causal set dynamics; discrete models.

* Idea: A sequence of generators of a group, of the form w = ai ±1aj ±1 ... ak ±1.
* Equivalence: Given a group presentation G = (a1, a2, ...; r1, r2, ...), two words are equivalent if one can be converted to the other in a finite number of steps.
* Word problem: The problem of deciding, given a word w, whether w = 1 in some given presentation; It is equivalent to asking whether two words u and v are equivalent, since u = v iff u v−1 = 1.
* Status: It has been proved unsolvable in general, but it has been solved for all one-relator and knot groups.
@ References: Stillwell BAMS(82); Batty et al a0801 [Deutsch-Josza algorithm and word problem].

Work > s.a. laws of thermodynamics.
* Idea: The work done by a force F on an object when the point at which it is applied moves by a displacement ds is dW = F·ds.
* In quantum theory: Using coherence and entanglement, one can store more energy in quantum systems than in purely classical ones; However, this advantage decreases as the number of particles increases and macroscopic thermodynamics is probably insensitive to the underlying microscopic mechanics.
@ General references: Mallinckrodt & Leff AJP(92) [different definitions of work]; Fonteneau & Viard Physis-a1301 [history, in Daniel Bernoulli's works]; Gallego et al NJP(16)-a1504 [operational definition].
@ In quantum theory: Lostaglio et al PRL(15)-a1409 [from absence of correlations]; Perarnau-Llobet et al PRX(15) [extractable from correlations]; Jarzynski et al PRX(15)-a1507 [definition, and quantum-classical correspondence]; Talkner & Hänggi PRE(16)-a1512; Barbosa a1904-MS [definition].
> Related topics: see generalized thermodynamics [work from quantum coherence]; Virtual Work.
> Online resources: see Wikipedia page.

Work-Energy Theorem > s.a. energy.
* Idea: The net work done (by all forces) on an object under a displacement equals the object's change in kinetic energy.

World Function

Worldline > s.a. poincaré symmetry [deformed]; Simon Tensor; Timelike Curve.
* Idea: A piecewise C2 curve in spacetime with timelike tangent vector, representing a particle/observer.
@ References: Ballesteros et al PLB(19)-a1902 [non-commutative spaces of worldlines].
> Worldline approach for field teory: see approaches to quantum gravity; GUTs; quantum field theory techniques

Wormholes > s.a. wormhole solutions.

@ References: Berger & Prior JPA(06) [for open and closed curves].

Weyssenhoff Fluid > s.a. gravitational collapse.
* Idea: A perfect fluid in which the particles have intrinsic spin.
@ References: de Berredo-Peixoto & de Souza a1506 [coupled to gravity, with the Host action and torsion].

WZW Model > see under Wess-Zumino-Witten.

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