**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.

**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.

**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 JPCS(12)-a1201].

**Wallis Formula** > see π.

**Ward / Ward-Takahashi Identities**
> s.a. 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 transformations, and come from
compensating terms in the measure and integrand in the path integral.

* __For QED__: One form is

[*S'*(*p*)]^{−1}
= [*S'*(*p*_{0})]^{−1}
+ (*p* − *p*_{0})^{a
}Γ_{a}(*p*, *p*_{0}) ,

where *p*_{0} = on-shell momentum,
*S'* = full propagator, Γ_{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].

**Warp Drive** > see causality violations.

**Wasserstein Metric** > see types of distances.

**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 C_{60} −
buckminsterfullerene − and C_{70}
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 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].

@ __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 JPCS(12)-a1201 [in classical mechanics];
Siddiqui & Qureshi QSMF(16)-a1406
[non-local duality relation].

**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 16 [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]; > 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 Derivative** > see tensor field.

**Weak Gravity Conjecture**

* __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 *q* ≥
*m*/*m*_{Pl}; That is, there exist
superextremal charged particles, with mass smaller than or equal to their charge in Planck units.

@ __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;
Heidenreich et al a1606
[evidence for a stronger statement];
Montero et al JHEP(16)-a1606 [3D version];
Hod IJMPD(17)-a1705-GRF [from Bekenstein's generalized second law of thermodynamics].

**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*^{±} 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.

@ __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].

@ __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]; > s.a. gravitational-wave interferometers.

**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}^{∞} *a*^{k}
cos(*b*^{k}*x*) 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 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 *q*^{ab}*K*_{ab}.

**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 Γ^{a}_{bc}
:= *e*^{a}_{i}
∂_{b} *e*_{c}^{i}
with non-vanishing torsion *T*^{ a}_{bc}
:= Γ^{a}_{bc}
− Γ^{a}_{cb}
defined by a tetrad field *e*^{a}_{i}
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 a1801.

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

**Weyl Invariance** > similar
to conformal invariance [but it usually
refers to the 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 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 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.

* __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__: 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].

@ __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 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].

@ __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 *M*_{PL},
called its Whitehead triangulation, such that *M* is diffeomorphic to a
smoothing of *M*_{PL}; *M*_{PL}
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(g_{E},
*g*_{F}).

$ __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-proc [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__: Visser GRF(91)-a1702 [as a complex deformation of the spacetime metric];
Liu ht/97 [geometric aspects];
Dasgupta & Loll NPB(01)ht,
Dasgupta JHEP(02)ht
[in quantum gravity]; Helleland & Hervik a1504 [Wick-rotatable metrics are purely electric].

**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 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].

**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].

**Wigner 6 j 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).

> __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 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].

@ __ Non-bijective version__: Gehér PLA(14)-a1407 [elementary proof].

**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];
17; Cai et al a1802 [discrete analog].

> __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].

> __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].

@ __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 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].

> __Online resources__:
see MathWorld page;
Wikipedia page.

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

**Wigner-Yanase Information** > see quantum
[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,

*W*_{c}(*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 a1312
[super Wilson loops and holonomy on supermanifolds];
> 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]; > s.a. holonomy;
Stokes Theorem.

> __Online resources__:
see Wikipedia page.

**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 *D*_{AA '} *λ*^{A}
= 0, where *D*_{AA '} = *σ*_{AA
'}^{a} *D*_{a}
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*) + *k*^{2}
*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(i*S*), 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].

> __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].

**Word**

* __Idea__: A sequence
of generators of a group, of the form *w* = *a*_{i}^{
±1}*a*_{j}^{
±1} ... *a*_{k}^{
±1}.

* __Equivalence__:
Given a group presentation *G* = (*a*_{1},
*a*_{2}, ...; *r*_{1},
*r*_{2}, ...), 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 d**s** is d*W* =
**F**·d**s**.

* __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-a1512.

> __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.

**Worldline** > s.a. poincaré
symmetry [deformed]; Timelike Curve.

* __Idea__: A piecewise
C^{2} curve in spacetime with timelike
tangent vector, representing a particle/observer.

> __Worldline approach for field teory__:
see GUTs; quantum field theory techniques

**Wormholes** > s.a. wormhole solutions.

**Writhe**

@ __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.

main page
– abbreviations
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– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 16 may 2018