**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. 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'*(*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];
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.

**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 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 [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].

**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];
> 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 *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 JHEP(17)-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];
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];
Heidenreich a1906 [two precise formulations].

**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];
> 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}^{∞}
*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 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 *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 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 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].

@ __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 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].

**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 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);
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 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];
de Gosson 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];
Yang FP(19)-a1812 [consistent descriptions];
Suarez a1906;
Stacey a1907 [QBism and misapprehensions];
Bong et al a1907;
Hansen & Wolf a1912.

> __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 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,

*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];
Chatterjee CMP-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 *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;
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.

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

* __Idea__: A piecewise
C^{2} 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 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
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 25 apr 2020