Topics, W

W Particle > see electroweak theory.

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

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

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

Wahlquist Metric
* Idea: A perfect fluid solution of which the Kerr metric is a vacuum subcase.
* Result: 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].

Walk > see random walk.

Walkers
* 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 a1201-conf].

Ward-Takahashi Identities > s.a. 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 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 + (p – p0)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].

Warp Drive > see causality violations.

Wasserstein Metric > see types of distances.

Water and Ice > s.a. crystals.
* Ice: Has 16 known crystal structures (as of 2009; s.a. the story of ice-IX); Close to 0 K, water molecules can't move very well and don't behave the way they do at warmer temperatures; If sprayed onto a platinum surface they tend to stay where they land, and additional molecules stick together wherever they can, forming amorphous ice, in which molecules don't have enough energy to line up to form a crystalline array; Just above 120 K, molecules have a chance to creep around enough to start assembling a proper crystal, with a cubic crystal structure; Common ice with its hexagonal structure is ice Ih (one of the two forms of ice I), and forms above 160 K; 2012, New phase in the 1–5 TPa pressure range.
* Unusual properties: At 10 GPa it remains frozen up to 320ºC! [@ Schwegler et al PRL(00) + pn(00)mar].
* Mpemba effect: The observation that initially hot water freezes faster than initially cold water.
@ General references: Eisenberg & Kauzmann 69; Caro 93; Denny 93 [and air]; Ball 99 [r pw(00)feb]; new ns(11)oct [quantum origin of water's properties].
@ Cold, non-crystalline states: Smith et al PRL(97) + pn(97)jul [amorphous solid water]; Debenedetti & Stanley PT(03)jun.
@ Mpemba effect: Jeng AJP(06)jun-phy/05; Ball pw(06)apr; Esposito et al PhyA(07) [and phase transitions in water]; Katz AJP(09)jan [suggested explanation in terms of solutes]; news ns(10)mar [explanation in terms of random impurities]; Brownridge AJP(11)jan [how to observe].
@ Ice: Choi et al PRL(05) + pn(05)aug [ice at room T with E fields]; Rosenberg PT(05)dec [slipperiness]; news pn(08)jun; news usn(09)sep [ice XV seen in the lab]; Hermann et al PNAS(12) + news cornell(12)jan [new high-pressure phase].
@ Warming and shrinking: Cho et al PRL(96) + pn(96)feb; news po(09)jul.
@ Related topics: Jiang & Schrader PRL(98) + pn(98)nov [positronic water]; Bergeron & Quéré pw(01)may [bouncing droplets]; Pellicer et al AJP(02)jul [surface tension]; Waltham phy/02 [heavy water in Canada]; Mattsson & Desjarlais PRL(06) + pn(06)aug [conducting at T = 4000 K, p = 100 GPa]; Hock et al PRL(09) + Fernández-Serra Phy(09) [small clusters and size-dependent phase diagrams]; Feibelman PT(10)feb [wetting of solids]; Knudson et al PRL(12) + Nellis Phy(12) + news sn(12)mar [high-pressure equation of state].

Wave Equation > 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].
@ 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].
@ Special cases: Clifton PLA(00)qp/99 [spin-0, and Kochen-Specker arguments]; Hackermüller PRL(03) + pw(03)sep [large organic molecules]; > s.a. types of particles [electrons]; Schilling & von Zanthier a1006 [in two-way interferometer with which-way detector].
@ Related topics: Kolář et al qp/05 [anomalies from entanglement]; Wesson GRG(06) [waves and particles in general relativity]; Davydov a1201-conf [in classical mechanics].

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.
@ 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]; > s.a. matter phenomenology in quantum gravity; quantum field theory techniques; stochastic quantization.
@ Other topics: Antoine & Vandergheynst JMP(98) [on Sn].

Weak Derivative > see tensor field.

Weak Interaction > s.a. electroweak theory; standard model.
* Idea: A nuclear interaction, now incorporated into the electroweak interaction, but initially described by the Fermi theory, an empirically successful but non-renormalizable theory; It is responsible for parity violation (> see Parity).
* Types: Charged currents, which change flavors within families and are mediated by W ± bosons; Neutral currents, mediated by Z bosons, which are responsible for example for neutrino-electron scattering.
@ 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]; Lee IJMPA(01) [history]; Anthony et al PRL(05) + pn(05)jul [measurement weak mixing angle over large distance range]; Lesov a0911 [history].

Weak Operator Topology on () > see topology.

Web > see Cosmic Web; foliation.

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-a0806 [conditions for geodesic flow].

Wehrl Entropy > see entropy in quantum theory.

Weierstraß Functions
* Idea: Functions that are everywhere continuous, but nowhere differentiable, such as W(x) = ∑k = 0∞ ak cos(bkx).
@ References: in Stromberg 81.

Weierstraß Theorem
* Idea: A result on uniform approximation of continuous functions by polynomials.

Weil Conjecture > see conjectures.

Weil Homomorphism
$ Def: A map w: I(G) → H*(M; 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-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 R3, it is given by qabKab.

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.
@ References: Bel a0805 [and Christoffel connection].

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.

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

Wetting > see Water.

Weyl Algebra > s.a. knots; observables.
@ References: Thirring 81 (v3, Sec 3.1); Arai LMP(08) [uniqueness of Weyl representation of commutation relations]; Grundling & Neeb RVMP(09) [C*-algebra for full set of regular representations].

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); Stoica a1203.

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

Weyl Invariance > same as conformal invariance [usually refers to metric as well as 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 a1106 [and timelike geodesics], a1109 [and fluid conservation laws] Poulis & Salim IJMPCS(11)-a1106 [and spacetime structure]; Romero et al a1106-conf [and general relativity]; Scholz a1111-proc [in late 20th-century physics];.

Weyl Quantization
@ References: Ozorio de Almeida PRP(98); Gelca & Uribe CMP(03)mp/02 [flat SU(2) connections]; Lein a1009-ln [and semiclassics].

Weyl Solutions > see solutions of general relativity with symmetries.

Weyl Spinors > see 2-spinors.

Weyl Tensor

Weyl Transfom > see path integrals; wigner function.

Weyl Tube Formula > see Lovelock Gravity.

Weyl Vector > see affine connection.

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.

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

Weyl-Schouten Tensor > see weyl tensor.

Wheeler-De Witt Equation > see geometrodynamics.

Which-Way Experiment > see interference.

White Dwarf > s.a. Chandrasekhar Limit; dark-matter types.
* Idea: A low-mass star in a late evolutionary stage, held in equilibrium by electron degeneracy pressure.
@ White dwarves: 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 a1202/ApJL [binary merger rate in the galactic disk]; Kundu & Mukhopadhyay MPLA-a1204 [highly magnetized white dwarfs exceeding the Chandrasekhar limit].

White Hole
* Idea: The time reversal of a spacetime in which gravitational collapse has occurred to form a black hole.
@ References: Wald & Ramaswami PRD(80) [particle production]; Barrabès et al PRD(93); Hsu a1007 [isolated white holes]; Retter & Heller NA-a1105 [white holes as small bangs].

Whitehead Continua
* Idea: There is an open, contractible 3D topological manifold W, which is not homeomorphic to R3, but such that R1 × W is R4.

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, E ⊕ F 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*(E ⊕ F) of the Cartesian product E × F, under the diagonal embedding d: B → B × 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 [and related stochastic processes].

Wick Rotation > s.a. approaches to quantum field theory; quantum dirac fields.
* Idea: The rotation of the time axis from real to imaginary times performed in field theory to calculate some integrals.
* In quantum gravity: Has been introduced as a mapping between real euclidean metrics and real lorentzian metrics.
@ References: Liu ht/97 [geometric aspects]; Dasgupta & Loll NPB(01)ht, Dasgupta JHEP(02)ht [in quantum gravity].

Wick's Theorem > see Time-Ordered Product.
@ References: Wick PR(50); Plimak & Stenholm PRD-a1104 [and causal signal transmission].
> 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's Theorem > see fourier analysis.

Wightman Axioms (For relativistic quantum field theory) > s.a. approaches to quantum field theory.
@ References: Streater & Wightman 64; Rehren CMP(96) [solutions].

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

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

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 Functions

Wigner Inequality
@ References: Nikitin & Toms PRA(10)-a0907 [in quantum field theory].

Wigner Rotations > s.a. Rapidity; special-relativistic kinematics.
@ References: Soo & Lin IJQI(04)qp/03; Saldanha & Vedral a1111 [physical interpretation].

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.
@ 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].

Wigner Velocity > see quantum effects [tunneling].

Wigner-Araki-Yanase Theorem
* Idea: A result in quantum mechanics which describes restrictions that conservation laws impose upon the physical measuring process.
@ References: Wigner ZP(52); Araki & Yanase PR(60); Kakazu & Pascazio PRA(95) [alternative formulation]; Miyadera & Imai PRA(06)qp; Meister qp/07-in [extension to multiplicative conserved quantities]; Busch & Loveridge PRL(11) + news physorg(11)mar [and position measurements].

Wigner-Eckart Theorem
@ References: Eckart RMP(30); Wigner 31; Narcowich & Fulling ed.

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

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 -valued connection A, the Wilson loop is the gauge-invariant functional

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

@ 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]; > 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 a1004 [lightlike, 3D Chern-Simons and ABJM 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]; Beckman et al PRD(02)ht/01 [measurability]; Freidel et al PRD(06)gq [as particles]; > s.a. Stokes Theorem.

WIMP > 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].

WKB Approximation > s.a. schrödinger equation; semiclassical quantum mechanics; pilot-wave quantum theory.
* Idea: 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.
@ References: Gough AN(07)ap.
> In quantum field theory: see quantum field theory in curved spacetime; quantum geometrodynamics.
> Online resources: see Wikipedia page.

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

Word
* 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; Virtual Work.
* 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.
@ References: Mallinckrodt & Leff AJP(92) [different definitions of work].
> Online resources: see Wikipedia page.

Work-Energy Theorem > s.a. energy.
* Work-energy theorem: 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]; Timelike Curve.
* Idea: A piecewise C2 curve in spacetime with timelike tangent vector, representing a particle/observer.

Wormholes > s.a. wormhole solutions.

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

WZW Model > see under Wess-Zumino-Witten.

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