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.

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].

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 a(p, p0) ,

where p0 = on-shell momentum, S' = full propagator, a = 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].

Water and Ice > s.a. crystals.
* Ice: Has 10 known crystal structures, and one more at high p (see the story of ice-nine); 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 forms above 160 K.
* 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].
@ 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)phy/05; Ball pw(06)apr; Esposito et al PhyA(07) [and phase transitions in water].
@ 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.
@ Related topics: Cho et al PRL(96) + pn(96)feb [warming and shrinking]; Jiang & Schrader PRL(98) + pn(98)nov [positronic water]; Bergeron & Quéré pw(01)may [bouncing droplets]; Pellicer et al AJP(02) [surface tension]; Waltham phy/02 [heavy water in Canada]; Mattsson & Desjarlais PRL(06) + pn(06)aug [conducting at T = 4000 K, p = 100 GPa].

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.
@ General references: Jánossy APH(52); Renninger ZfP(53) [translation De Baere phy/05]; Diner in(84); Bardou AJP(91); 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].
@ Special cases: Clifton PLA(00)qp/99 [spin-0, and Kochen-Specker arguments]; Hackermüller PRL(03) + pw(03)sep [large organic molecules].
@ Related topics: Kolar et al qp/05 [anomalies from entanglement]; Wesson GRG(06) [waves and particles in general relativity].

Wavelets > s.a. Cuntz Algebra; wave equations.
* Idea: Wavelet analysis is an alternative decomposition of waves, wrt Fourier analysis.
* Advantages: Localization.
@ General references: Strang AS(94); Han et al PLA(95) [photons]; Kaiser 94 [IIIa]; Holschneider 95; Walnut 02 [r BAMS(03)]; Addison pw(04)mar [applications]; Altaisky 05 [including applications].
@ 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 theory: Bagarello JPA(96) [pedagogic]; Steeb 98; Havukainen qp/00 [in QED]; > 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 Interactions > s.a. electroweak; 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.
* 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)RL; Cline ThSc(93); Greiner & Müller 96 [III]; Lee IJMPA(01) [history]; Anthony et al PRL(05) + pn(05)jul [measurement weak mixing angle over large distance range].

Weak Operator Topology on () > see topology.

Web > see 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 a0806 [conditions for geodesic flow].

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

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.

Weinberg-Salam Electroweak Theory

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.

Weingarten Matrix
* Idea: For a 2D surface patch in R3, it is given by qabKab.

Weitzenböck Connection > s.a. teleparallel gravity.
@ References: Bel a0805 [and Christoffel connection].

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

Weyl Algebra > s.a. knots; observables.
@ References: Thirring v3, Sec 3.1.

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 Quantization
@ References: Ozorio de Almeida PRP(98); Gelca & Uribe CMP(03)mp/02 [flat SU(2) connections].

Weyl Solutions > see solutions of general relativity with symmetries.

Weyl Manifold / Space
* 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].

Weyl Spinors > see 2-spinors.

Weyl Tensor

Weyl Transfom > see path integrals.

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.

White Dwarf > s.a. Chandrasekhar Limit.
* Idea: A star held in equilibrium by electron degeneracy pressure.

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).

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 Theory of Gravity
@ References: Coleman phy/05, a0704; Gibbons & Will SHPMP(08)gq/06 [truly dead].

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

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].

Wick Rotation > s.a. approaches to quantum field theory.
* 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 fock space [time-ordered product].

Widom Conjecture > see entropy in quantum theory.

Wieferich Primes > see number theory.

Wien's Law > see thermal radiation.

Wiener Measure > see measure.

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| (x)(x') |0 and G–(x,x'):= 0| (x')(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 Rotations > s.a. Rapidity.
@ References: Soo & Lin IJQI(04)qp/03.

Wigner Theorem
* Idea: A bijective transformation of the set of all one-dimensional linear subspaces of a complex 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 a0712 [simple proof, realization of symmetries in quantum mechanics and projective geometry].

Wigner Velocity > see quantum effects [tunneling].

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

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: 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 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 a0804 [worldlines as Wilson lines]; > s.a. loops, loop quantum gravity, loop variables.
@ Other theories: Tseytlin & Zarembo PRD(02), Drukker et al a0704 [N = 4 super-Yang-Mills]; > s.a. BF theory; chern-simons; lattice gauge theory; susy 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 ' A = 0, where DAA ' = AA '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; quantum field theory in curved spacetime; quantum geometrodynamics; semiclassical quantum mechanics.
* Idea: A method for finding approximate solutions of a second-order linear ordinary differential equation of the form ''(z) + k2 f(z) (z) = 0, when f vanishes at a point.
@ References: Gough AN(07)ap.

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-Energy Theorem > see classical mechanics; energy.

World Function

Worldline > s.a. Timelike Curve.
* Idea: A piecewise C2 curve in spacetime with timelike tangent vector, representing a particle/observer.

Wormholes

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

WZW Model > see under Wess-Zumino-Witten.

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 11 jul 2008