**Topics, S**

**S Theorem**
> see lie algebra.

**S-Matrix (a.k.a. Scattering Matrix)**
> s.a. quantum field theory techniques; scattering
/ Coleman-Mandula Theorem; LSZ Formalism.

* __History__:
Introduced by J Wheeler in the context of nuclear physics.

$ __Def__: In quantum field theory,
the operator *S*:= lim *U*(*t*, *t*_{0})
for *t* → ∞, *t*_{0}
→ −∞, where *U* is the time evolution operator.

* __Idea__: The
scattering matrix relates the initial state and the final state of a
physical system undergoing a scattering process.

* __Assumptions__: Causality, unitarity, analiticity.

* __Properties__: Its
unitarity, to first order, is (formally) equivalent to conservation of
probability; To second order it is guaranteed by completeness of the
Hilbert space and self-adjointness of the (interaction) Hamiltonian.

* __Transition matrix__:
The matrix *T* related to the S-matrix by *S*_{fi}
= δ_{fi} − 2πi δ(*ω*_{f}
− *ω*_{i}) *T*_{fi}.

@ __General references__: Stern PT(64)apr [criticism];
White hp/00-ch [rev];
Kummer EPJC(01)ht [gauge invariance];
Colosi & Oeckl PLB(08)-a0710 [new approach];
Cachazo et al PRL(14)
+ Dixon Phy(14)
[new compact formula].

@ __Gravitational__: Giddings & Porto PRD(10)-a0908;
Giddings a1105-ln;
Wiesendanger CQG(13)-a1203.

> __Online resources__:
see Wikipedia S-Matrix page
and S-Matrix Theory page.

**Sachs-Wolfe Effect**
> s.a. CMB [non-Gaussianities];
gravitational-wave propagation.

* __Idea__: A
contribution to the cosmic microwave background anisotropy from changes in
the *γ*'s 4-momentum due to density perturbations; It allows us
to correlate features in the cmb to large-scale structure.

@ __General references__:
Sachs & Wolfe ApJ(67),
reprinted GRG(07);
Magueijo PRD(93);
Ferrando et al PRD(93);
Russ et al PRD(93);
Pyne & Birkinshaw ApJ(93)ap [null geodesics in perturbed spacetime];
White & Hu A&A(97)ap/96 [pedagogical];
Hwang & Noh PRD(99)ap/98;
Cooray PRD(02)ap/01 [integrated];
Aguiar & Crawford ap/01 [anisotropic spacetimes];
Multamäki & Elgarøy A&A(04)ap/03 [non-standard cosmology];
Mendonça et al CQG(08)ap/05 [new approach];
Granett et al ApJL(08)-a0805;
Roldan JCAP(17)-a1706 [non-linear generalization].

@ __Integrated, models__: Giovannini CQG(10)-a0907 [effect of magnetic fields and dark energy].

@ __Integrated, observation__:
Boughn et al NA(98),
Boughn & Crittenden NAR(05)ap/04-conf [consistent with Λ-CDM];
Cooray PRD(02) [large-scale structure];
Padmanabhan et al PRD(05)ap/04;
Pietrobon et al PRD(06)ap;
Pogosian NAR(06)ap;
Dupé et al A&A(11)-a1010;
> s.a. Copernican Principle [test].

> __Online resources__:
see Wikipedia page.

**Sackur-Tetrode Equation**
> s.a. gas.

* __Idea__: An explicit
expression for entropy of a monatomic ideal gas in terms of fundamental
constants, derived in 1912 by Otto Sackur and Hugo Tetrode.

@ __References__: Grimus a1112 [history].

> __Online resources__:
see Wikipedia page.

**Saddle-Point Approximation / Method**
> another name for the Stationary-Phase Approximation.

**Sagnac Effect**
> s.a. atomic physics; kinematics of
special relativity; tests of newtonian gravity.

* __Idea__: The fact that,
if we send two light rays in opposite directions around a rotating ring
(say, on the surface of the Earth), they return with a time difference
proportional to *ω* and the enclosed area given by the Sagnac formula,
Δ*t* = 4 **A** · **ω** /
*c*^{2}, and the interference depends
on *ω*.

* __Applications__: Laser
gyroscope, used for inertial guidance, based on beats between the two rays.

@ __Early work__: Michelson PM(04);
Sagnac CRAS(13);
Michelson et al ApJ(25) [experiment].

@ __General references__: Logunov & Chugreev SPU(88);
Anderson et al AJP(94)nov;
Rizzi & Tartaglia gq/98;
Klauber FPL(03)gq/02 [general case];
Bertocchi et al JPB(06)-a1312 [single-photon interferometer];
> s.a. Reference Frames [rotating].

@ __With matter waves__: Gustavson et al PRL(97) [atom interferometer and Earth's rotation];
Lenef et al PRL(97)
+ pn(97)feb;
Rizzi & Ruggiero GRG(03)gq,
in(04)gq/03 [and Aharonov-Bohm effect],
GRG(03)gq.

@ __In general relativity, curved spacetime__: Ashtekar & Magnon JMP(75);
Tartaglia PRD(98)gq;
Gogberashvili FPL(02)gq/01;
Sivasubramanian et al gq/03 [and gravitational waves];
Camacho GRG(04)gq/03 [non-Newtonian];
Ruggiero GRG(05) [and Aharonov-Bohm effect];
Maraner & Zendri GRG(12)-a1110;
Frauendiener GRG(18)-a1808 [general formula and simple cases],
a1808 [and gravitational waves];
Benedetto et al EPJC-a1902 [framework].

@ __Related topics__: Wucknitz gq/04|FP [and closed Minkowski spacetime];
> s.a. Galilean Group [boosts and Sagnac phase].

> __Online resources__:
see MathPages page;
Wikipedia page.

**Saha Equation**

* __Idea__: An
expression for the relative number densities of different ionization
levels in an ionized gas in thermodynamic equilibrium, in terms of the
temperature; It allows us to infer densities of various ions from spectral
line intensities.

@ __References__: Fowler a1209 [normalized by the total number density];
De & Chakrabarty Pra-a1412 [in a uniformly accelerated frame].

**Salpeter Equation** >
see modified quantum mechanics.

**Sampling** > s.a. information.

@ __Shannon sampling__: Kempf PRL(00)ht/99 [generalization, unsharp coordinates];
Smale & Zhou BAMS(04).

**Sand Pile** > see critical phenomena.

**Sandwich Conjecture**

* __Idea__: The
conjecture that, given two spatial metrics *q* and *q*' on
two hypersurfaces in spacetime, there is a unique solution of Einstein's
equation that will interpolate between them, up to gauge.

* __Thin sandwich__:
The hypersurfaces are infinitesimally close; One specifies the spatial
field configurations and their time derivatives.

* __Thick sandwich__:
The hypersurfaces are a finite distance apart.

@ __General references__:
in Beierlein et al PR(62);
Bergmann in(70);
Christodoulou & Francaviglia in(79),
RPMP(77);
Teitelboim in(82).

@ __Thin sandwich__: Bartnik & Fodor PRD(93)gq;
Giulini JMP(99)gq/98 [Einstein + gauge theory + scalar];
York PRL(99)gq/98 [and initial-value problem];
Komar; Bartnik & Isenberg gq/04-proc;
Pfeiffer & York PRL(05)gq [conformal, uniqueness];
Avalos et al JMP(17)-a1703 [in higher-dimensional theories].

**Satellites** > see solar planets.

**Scalar Fields** > s.a. klein-gordon fields.

**Scalar Product** > see vectors.

**Scalar Theory of Gravitation**
> s.a. matter phenomenology;
scalar-tensor theory.

* __History__: It
started with Nordström's attempt at developing a special relativistic
theory of gravity; Nordström's theory is massless, but there are also many
possible massive theories, including the Freund-Nambu theory; In this
theory, the equivalence principle is valid and one predicts a redshift of
the spectral lines from the Sun, but the perihelion precession of Mercury
is like the one predicted by Newton's theory, and it cannot explain the
deflection of light near the Sun.

@ __General references__: Giulini SHPMP(08)gq/06
[history and assessment];
Pitts GRG(11)-a1010 [massive, rev and history].

@ __Nordström's theory__: Nordström AdP(13);
Einstein & Fokker AdP(14);
in Pauli 58;
Wellner & Sandri AJP(64)jan;
Harvey AJP(65)feb;
Norton AHES(92) [history];
Bauer mp/04 [self-gravitating particles];
Boozer PRD(11) [2D, coupled to matter];
Deruelle GRG(11) [and the equivalence principle];
Weinstein a1205 [history, and the Einstein-Nordström theory];
> s.a. equivalence principle.

@ __Other theories__: Novello et al JCAP(13)-a1212 [geometric theory];
Giulini a1306-conf [history, Einstein's 1912 "Prague-Theory"];
Franklin AJP(15)-a1408 [self-consistent, self-coupled theory];
Pitts SHPMP(16)-a1509 [massive scalar gravity];
Bittencourt et al PRD(16)-a1605 [Schwarschild geometry, and Post-Newtonian approximation];
> s.a. history of relativistic gravity.

@ __Equations of motion__: Arminjon RJP(00)ap [with preferred frame];
Kaniel & Itin gq/99;
Beig et al PRL(07)gq/06 [helically symmetric *N*-particle solutions].

@ __PN approximation__:
Arminjon in(02)gq/01,
in(04)gq/03.

@ __As model__: Watt & Misner gq/99 [for numerical gravity];
Sundrum ht/03;
> s.a. modified general relativity
[analog]; spacetime singularities.

@ __Related topics__: Bezerra et al MPLA(02) [2+1, including black hole].

**Scalar-Tensor Theories of Gravity**

**Scalar-Vector Theories of Gravity**
> see theories of gravity.

**Scalar-Vector-Tensor Theories of Gravity**
> see MOG (STVG); MOND (TeVeS);
theories of gravity.

**Scale Relativity**

* __Idea__; A theory
based on the idea that physics must apply to coordinate systems in all
"states of scale"; Spacetime is described as a non-differentiable
continuum, a fractal which depends explicitly on internal scale variables.

@ __General references__: Nottale IJMPA(92);
Nottale 93;
Nottale CSF(94) [fractal spacetime];
Célérier & Nottale JPA(04)qp/06 [quantum mechanics and fields];
Nottale 11.

@ __Applications to various theories__: Castro ht/96 [strings];
Nottale et al JMP(06)ht [gauge theory];
Célérier & Nottale JPA(06)qp [Pauli equation];
Hammad JPA(08) [derivation of Pauli and Dirac equations];
Célérier JMP(09) [chaotic fluid motion];
Célérier & Nottale IJMPA(10)-a1009 [Maxwell, Klein-Gordon and Dirac equations];
Barbour et al GRG(13) [point-particle analog and time in quantum gravity];
Nottale & Célérier JMP(13) [complex and spinor wave functions].

**Scale Invariance / Symmetry**
> s.a. conformal symmetry.

* __Idea__: The
property of certain theories or their solutions of being invariant under a
transformation in which scales of length, time, energy, or other
variables, are multiplied by a common factor.

* __Examples of
scale-invariant systems__: Classical and quantum Bose and Fermi ideal gases.

* __Vs conformal symmetry__:
here is a conjecture that in unitary field theories scale invariance
implies conformality, and a proof by Zamolodchikov and Polchinski for 2D
theories, that is not valid in higher dimensions.

@ __Vs conformal symmetry__: Awad & Johnson PRD(00)ht,
IJMPA(01)ht/00-in [from AdS-cft correspondence];
Riva and Cardy PLB(05)ht [in 2D elasticity];
Dorigoni & Rychkov a0910 [conjecture that scale invariance and unitarity imply conformal invariance];
Nakayama IJMPA(10) [holographic approach];
Fortin et al PLB(11)-a1106,
JHEP(12)-a1107,
JHEP(12)-a1202 [unitary, scale-invariant but non-conformally-invariant model];
Nakayama IJMPA(12)-a1109 [supersymmetric theories];
Fortin et al JHEP(13)-a1208;
Nakayama a1302-ln;
Dymarsky et al a1309,
Farnsworth et al a1309-wd,
Dymarsky et al a1402
[a scale invariant, unitary 4D quantum field theory is conformally invariant];
Sachs & Ponomarev a1402-wd;
Sibiryakov PRL(14)
[1+1 scale invariance and standard assumptions leading to conformal algebra and Lorentz symmetry];
Delamotte et al PRE(16)-a1501 [in the 3D Ising model];
Dymarsky & Zhiboedov JPA(15)-a1505 [scale-invariant breaking of conformal symmetry];
Fareghbal et al PLB(17)-a1511 [in ultra-relativistic field theory];
Oz a1801 [in turbulence statistics];
Li et al a1812 [in Horndeski gravity].

@ __Spatial scale invariance__:
Westman a0910,
Westman & Zlosnik a1201
[as a local gauge symmetry].

@ __Breaking__: Camblong et al PRL(01) [in molecule + electron];
Marchais et al PRD(17)-a1702 [spontaneous breaking, using functional renormalisation].

@ __Related topics__: Belitz et al RMP(05) [and phase transitions];
Hill ht/05-talk [and
dimension, cosmological constant, physical scales];
Sochichiu JHEP(09)
[3D dilatation operator, perturbative];
Shaukat PhD(10)-a1003 ["unit invariance"];
Lesne & Lagües 12 [from phase transitions to turbulence];
Wetterich a1901 [in quantum field theory].

@ __In gravity__: Garfinkle PRD(97)gq/96 [and Choptuik scaling];
Jain et al a1010 [and the cosmological constant];
Blas et al PRD(11)-a1104 [massless particle spectrum];
Quirós a1405 [fake scale invariance];
Lasenby & Hobson JMP(16)-a1510;
Einhorn & Jones JHEP(16)-a1511 [dimensional transmutation and effective Einstein-Hilbert action];
Maeder ApJ(17) [as alternative to dark matter];
Casas et al a1811 [inflation and dark energy];
> s.a. gravity theories.

@ __Generalizations__: Gozzi & Mauro JPA(06)qp/05 [mechanical similarity as generalization].

> __Online resources__:
see Wikipedia page.

**Scaling**
> s.a. Critical Phenomena [scale-free networks];
Multiscale Physics; phase transition;
renormalization group.

* __Idea__: The *p*-point
correlation functions can be written in terms of the 2-point correlation
function or variance.

* __Scale-free distribution__:
One given by a power law, as opposed to an exponential with a scale in
the exponent; Power laws seem to be prevalent in nature, and may signal
an underlying universality.

* __In galaxy distribution__:
Expected if an initially Gaussian distribution of density fluctuations
evolves under the action of gravitational instability.

@ __General references__: Wiesenfeld AJP(01)sep [RL];
Henkel NPB(02) [in statistical mechanics];
West CSF(04) [renormalization group, complexity];
Gupta et al PhyA(08) [power law scaling and limitations in Tsallis statistics].

@ __In biological systems__:
Brown & West 00 [in biology];
West & Brown PT(04)sep.

@ __In other areas__: Peterson AJP(02)jun-phy/01 [Galileo and the geography of Dante's Inferno];
> s.a. galaxy distribution;
many-particle quantum systems; turbulence.

> __Related topics__:
see entropy; fractal; noether theorem;
poisson structure [change of description]; Zipf's Law.

**Scarring** > see quantum chaos.

**Scharnhorst Effect** > see casimir.

**Schemes** > s.a. Algebraic Geometry.

* __Applications__:
Used in algebraic topology, number theory, ...

@ __General references__: Eisenbud & Harris 92,
00.

@ __In physics__: Choi & Shrock PRD(16)-a1607
[scheme transformations in a quantum field theory].

**Schläfli Formula / Identity**

* __Idea__: A formula
relating the variations of the dihedral angles of a smooth family of
polyhedra in a space form to the variation of the enclosed volume;
It is important in Regge calculus and loop quantum gravity.

@ __References__: Souam DG&A(04) [for immersed piecewise smooth hypersurfaces in Einstein manifolds];
Haggard et al JPA(15)-a1409
[symplectic and semiclassical aspects].

**Schlegel Diagram**
> s.a. types of graphs.

* __Idea__: A
polytope in \(\mathbb R\)^{n}
obtained as a projection of a polytope in \(\mathbb R\)^{n+1}
using a point beyond one of its facets.

> __Online resources__:
see Menachem Lazar page;
Wikipedia page.

**Schmidt Decomposition**

* __Idea__: A result in linear algebra,
and a way of expressing a vector in the tensor product of two inner product spaces;
It has numerous applications in quantum information theory, for example in entanglement
characterization and in state purification.

@ __References__: Sciara et al sRep(17)-a1609 [and particle identity in multiparticle systems, universality].

> __Online resources__:
see Wikipedia page.

**Schnyder's Theorem**
> see types of posets.

**Schott Energy**
> see radiation [acceleration radiation].

**Schouten Bracket / Concomitant**

* __Idea__: A function
of two symmetric contravariant tensor fields, closely related to
the Poisson bracket.

@ __References__: Bloore & Assimakopoulos IJTP(79);
Kiselev & Ringers a1208-proc [definitions on jet spaces].

**Schouten Gravity**

* __Idea__: A (pure)
quadratic curvature three-dimensional model.

@ __References__: Deser et al JPA-a1208v2 [conformal vs coordinate invariance].

**Schouten Theory of Spin Densities**
> see 2-spinors.

**Schouten-Nijenhuis Bracket**
> see killing tensors [Killing-Yano].

**Schreier's Conjecture**

$ __Def__: The outer
automorphism group of any finite simple group is solvable; Has been proved.

**Schrödinger Representation of Quantum Theory**
> see representations of quantum theory.

**Schrödinger-Newton Equation**
> see quantum mechanics in curved spacetimes [tests];
quantum gravity [alternatives].

**Schrödinger's Cat**
> see experiments in
quantum mechanics; quantum states.

**Schrödinger's Hat**

* __Idea__: 2012, A
proposed device that could detect the presence of waves or particles while
barely disturbing them, based on the idea of interaction-free measurement.

@ __References__:
news pw(12)jun.

**Schubert Calculus**

* __Idea__:
Originally invented as the description of the cohomology of homogeneous
spaces, it has been generalized to the case of deformed cohomology
theories such as the equivariant, the quantum cohomology, K-theory, and cobordism.

@ __References__: Gorbounov & Petrov JGP(12) [and singularity theory].

**Schubert Cell**
> see grassmann.

**Schubert Symbol**

$ __Def__: Any
non-decreasing finite sequence of integers {*p*_{i}},
*i* = 1,..., *n*, i.e., *p*_{i}
in \(\mathbb N\), with 1 ≤ *p*_{1}
≤ ... ≤ *p*_{n} ≤ *m*.

**Schur's Lemma**

$ __Def__: In a
finite-dimensional irreducible representation of a group *G*, the
only elements which commute with all others are multiples of the identity.

**Schwarz Inequality**
> see inequalities.

**Schwarz Space**
> see distribution.

**Schwarz Transformation**
> see analytic functions.

**Schwarzschild Spacetime**
> s.a. coordinates and geometry;
fields and perturbations; particles
in schwarzschild spacetime.

**Schwarzschild-de Sitter Spacetime**

**Schwinger Effect / Mechanism / Pair Production** > see particle effects.

**Schwinger Function**
> see green functions in quantum field theory.

**Schwinger-Dyson Equation**
> s.a. gauge theory quantization; quantum gravity and renormalization.

* __Idea__: General relations
between Green functions in quantum field theories, corresponding to the
equations of motion for the Green's function.

@ __References__: Lyakhovich & Sharapov JHEP(06) [for non-Lagrangian field theory];
Tanasă & Kreimer JNCG(13)-a0907 [for non-commutative field theory].

> __Online resources__:
see Wikipedia page.

**Schwinger Model**
> see dirac fields; modified QED.

**Schwinger's Trick**
> see perturbative quantum field theory.

**Schwinger's Variational Principle / Quantum Action Principle**

* __Idea__:
The generalization of Hamilton's principle of stationary action
to quantum theory.

@ __References__: Popławski PRD(14)-a1310 [in Einstein-Cartan gravity];
Gu a1311 [and the generalized uncertainty principle];
Milton a1402/EPJH,
book(15)-a1503 [development];
Ciaglia et al MPLA(18)-a1807 [grupoid picture],
a1907 [algebras and observables].

**Screened Modified Gravity / Screening Mechanisms**

* __Idea__:
Mechanisms invoked to argue for the viability of certain modified gravity
theories, because they hide any effects of the modifications in our local
(high-density) environments, where high-precision gravity experiments have
been performed, while at the same time allowing for potentially large
deviations in regions of spacetime where the average density is much
lower, on cosmological scales.

* __Types__:
Screening mechanisms include chameleons, symmetrons, dilatons, MOND-like
dynamics, and the Vainshtein mechanism, and can be divided into three
types, relying on (i) the coupling to matter, e.g., the dilaton, (ii) a
mass term, e.g., the chameleon, and (iii) a kinetic term, e.g., the
Vainshtein mechanism; Some approaches, such as the symmetron, use
combinations of those types.

@ __References__: Brax a1211-ln [rev];
in Berti et al CQG(15)-a1501;
Perkins & Yunes CQG(19)-a1811 [tests with gravitational waves].

**Screw Theory**

@ __References__: Minguzzi EJP(13)-a1201
[application to classical mechanics, and the Lie algebra of the group of rigid maps].

**Scri** ("Penrose script I")
> see asymptotic flatness and null infinity.

**SDSS (Sloan Digital Sky Survey)**
> see galaxy distribution.

**Second-Countable Topological Space**
> see types of topologies.

**Second Fundamental Form**

> see extrinsic curvature.

**Second Law of Thermodynamics**
> see thermodynamics.

**Second-Order Equations**
> see elementary algebra.

**Second Quantization**
> s.a. quantum field theory.

* __Idea__: It is
a field quantization, not really a second quantization.

* __Motivation__: Seems
necessary in order to obtain a consistent Lorentz-covariant quantum
theory of particles.

* __Commutation relations__:
The commutation relations between creation and annihilation operators
corresponding to a given set of modes in a classical field theory are
related to properties of the classical modes by [*a*(*φ*),
*a*^{†}(*φ'*)] = \(\langle\)*φ*
| *φ'*\(\rangle\).

@ __References__:
Tasaki a1812 [intro].

**Sectional Curvature**
> see riemann tensor.

**Secular Equation**
> another name for the characteristic equation of a matrix.

**Seebeck Effect**
> s.a. electricity [thermoelectricity].

@ __References__: Uchida et al Nat(08)oct [spin Seebeck effect].

**Seesaw Mechanism**
> s.a. neutrinos; cosmological constant.

* __Idea__: A mechanism
by which a phenomenon with very high characteristic energy scales
can be seen at much lower energies.

**Segal-Bargmann Transform**
> s.a. coherent states; Holomorphic Functions.

@ __General references__: Hall JFA(94),
JFAA(01)mp;
Hall & Mitchell TJM-a0710;
Olafsson 14 [and Hilbert spaces of holomorphic functions].

@ __Specific types of systems__: Díaz-Ortiz & Villegas-Blas JMP(12) [on the *n*-sphere, and coherent states].

**Segre Classification of Traceless Ricci Tensors**
> see Ricci Tensor.

**Seiberg-Witten Map, Theory**
> s.a. non-commutative gravity.

@ __References__: Marcolli dg/95-ln;
Flume et al NPB(97)
[*L*_{eff} uniqueness],
ht/96 [rev];
Morgan 96;
Adam et al JMP(00) [solutions];
Ghosh JPA(03) [map, interpretation].

**Seifert Forms**

**Seifert Manifolds**

* __Idea__: Quotient manifolds,
for example of the form S^{3}/*G*,
where *G* is a finite subgroup of SU(2).

@ __References__: Hikami CMP(06) [quantum invariants].

> __Online resources__:
see Wikipedia page.

**Seifert-Van Kampen Theorem**
> see fundamental group.

**Selberg's Trace Formula** > see Trace Formulas.

**Self-Adjoint Operator** > see operators.

**Self-Dual Gauge Fields**
> s.a. differential forms [self-dual].

**Self-Dual Gravity** > see connection
formulation of general relativity; gauge gravity;
self-dual solutions; supergravity.

**Self-Energy** > see classical
field theory; energy.

**Self-Force**
> s.a. gravitational self-force / semiclassical
general relativity (back-reaction); energy-momentum tensor [post-Newtonian].

**Self-Organization**
> s.a. critical phenomena.

@ __General references__:
Nicolis & Prigogine 77 [non-equilibrium systems];
Olemskoi et al PhyA(04) [with order-parameter field].

@ __Specific areas__:
Bouchet & Venaille PRP(12) [2D and geophysical turbulent flows];
Aschwanden SSR(18)-a1708 [in astrophysics].

**Self-Similarity**

* __For solutions of Einstein's
equation__: In the spherically symmetric case, a spacetime in which all
dimensionless variables depend only on *z*:= *r*/*t*.

@ __General references__: Embrechts & Maejima 02 [self-similar processes].

@ __For spacetime metrics, kinematical__: Coley CQG(97)gq/96;
Carr & Coley CQG(99) [rev];
> s.a. spherical symmetry.

@ __For spacetime metrics, in general relativity__: Carot & Sintes
in(97)gq/00 [fluid];
Harada CQG(01) [pfluid, stability criterion];
Martín-García & Gundlach PRD(03)gq [scalar];
Harada & Maeda CQG(04) [scalar, stiff fluid, stability];
Maeda & Harada gq/04-ch;
Carr & Coley GRG(05)gq [similarity hypothesis];
> s.a. bianchi IX; bianchi models;
critical collapse; spherical symmetry.

**Selleri's Paradox**
> see reference frames [rotating].

**Semialgebraic Geometry**
> see rings [partially ordered].

**Semiclassical Physics**

* __Idea__: A
semiclassical theory is one in which one part of a system is described
quantum-mechanically while the other is treated classically.

> __For quantum mechanics__:
see classical-quantum relationship;
quantum-to-classical transition.

> __For field theories__: see
QED; semiclassical general
relativity; states in quantum field theory.

> __Online resources__:
see Wikipedia page.

**Semiconductors** > see electricity.

**Semicontinuity, Upper / Lower**

$ __Def__: A function
is said to upper/lower semicontinuous at a point *x*_{0}
if for every *ε* > 0 there exists a neighborhood *U* of
*x*_{0} such that *f*(*x*)
≤ *f*(*x*_{0}) + *ε*
(resp., *f*(*x*) ≥ *f*(*x*_{0})
− *ε*) for all *x* in *U*.

> __Online resources__:
see Wikipedia page.

**Semidirect Product of Groups**

$ __Def__: Given a group *G*
and an Abelian group *V*, with a *G*-action on *V*, their
semidirect product *G* ⊗_{s}
*V* is the set *G* × *V* with the composition law
(*g*_{1}, *v*_{1})
(*g*_{2}, *v*_{2}):=
(*g*_{1} *g*_{2},
*v*_{1} + g_{1}
*v*_{2}).

* __Remark__: We can thus
get a new group from every representation of *G*, with *V*
⊗_{s} *G* / *V* = *G*.

@ __References__:
Geroch & Newman JMP(71).

> __Examples__: see the
poincaré group and the BMS Group.

**Semigroup**
> s.a. markov processes; poincaré group.

$ __Def__: A set with
an associative composition law (an associative groupoid).

* __Relationships__:
If it has an identity it is a monoid; If it has an identity and
an inverse for each element, a group.

* __Special types__:
Additive or Abelian if commutative; Cancellative if *a*
+ *c* = *b* + *c* implies *a*
= *b*; > s.a. Monoid;
Semiring.

* __Topological semigroup__:
Theory created by A D Wallace.

* __Applications__:
Irreversible dynamics such as random walks, both in classical mechanics
(> see Transport) and in quantum
mechanics (& Prigogine, Bohm, > see dissipation,
modified quantum mechanics); Non-deterministic
dynamics (Blanchard & Jadczyk, > see stochastic
processes); > s.a. arrow of time.

@ __General references__:
Wallace BAMS(55);
Carruth, Hildebrant & Koch 83;
Belleni-Morante 94 [and evolution equations];
Lawson 98 [inverse semigroups, and partial symmetries];
Steinberg JCTA(06) [representations, and Möbius functions];
Högnäs & Mukherjea 11 [probability measures, and applications].

@ __Quantum dynamical semigroups__: Davies JFA(79) [generators];
Alicki qp/02-ln;
Antoniou et al OSID(02) [implementability];
Courbage IJTP(07) [unstable states];
Harshman IJTP(07) [from underlying Poincaré symmetry];
Bohm et al IJTP(07) [from causal symmetries];
Baumgartner & Narnhofer RVMP(12)-a1101 [structures of state space];
> s.a. neutrons [interferometry].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Semimetals** > see Metals.

**Seminorm** > see norm.

**Semiorder** > see posets [generalizations].

**Semi-Riemannian Manifold** > same as lorentzian manifold.

**Semiring** > s.a. Burnside Ring.

$ __Def__: A semigroup with
distributive multiplication.

* __Of subsets of a set__:
A collection *R* of subsets of a set *X* such that Ø,
*X* in *R*, and *R* is closed under intersection.

> __Online resources__:
see Wikipedia page.

**Semisimple Group**
> s.a. lie groups and representations.

$ __Def__: One
with no (proper, non-trivial) invariant Abelian subgroup.

* __And other structure__:
A n.s.c. for them to have a non-degenerate metric is that *k*_{ab}:=
*C*_{am}^{n}
*C*_{bn}^{m}
be non-singular.

* __Semisimple Lie
groups__: They are locally isomorphic to products of simple groups;
These groups have a very rich structure and have been completely classified early.

**Separability** > s.a. Banach
Space; C*-Algebra; set of posets.

* __For a partial
differential equation__: The ability to write it as an equivalent
set of uncoupled ordinary differential equations.

* __In quantum theory__:
All events associated to the union of some set of disjoint regions are
combinations of events associated to each region taken separately.

@ __In quantum theory__:
Wootters & Zurek PRD(79);
d'Espagnat PRP(84);
Schumacher PRA(91);
Costa de Beauregard qp/98;
Henson FP(13)-a1302 [and Bell's theorem].

@ __For field equations__: Unruh PRL(73).

> __In quantum theory__:
see causality; Cluster Separability;
Contextuality; correlations;
entanglement; entropy;
Gleason's Theorem; formulations;
mixed states; quantum chaos;
Superseparability; types of quantum states.

> __Other areas of physics__: see formulations
of electromagnetism; information theory; loop
quantum gravity; types of dark matter.

**Separable Hilbert Space** > see hilbert space.

**Separable Quantum State** > see types of quantum states.

**Separable Topological Space** > see types of topologies.

**Separate Universe Problem** > see Baby Universes.

**Separation Axioms (T _{0}, T_{1},
T_{2}, T_{3},
T_{4} Spaces)** > see types
of topological spaces.

**Separation of Variables**
> see hamilton-jacobi; schrödinger equation.

**Separatrix**

* __Separatrix mapping__:
The mapping that gives the energy and phase of a perturbed non-linear
pendulum near the separatrix after a velocity pulse (swing), in terms of
their values before; It shows that the reason for the emergence of local
instability is the sensitivity of the variation in phase on the orbit.

@ __References__: in Zaslavskii et al 91, p39;
Wiesenfeld JPA(04)
[Hamiltonians with symmetries, existence].

**Sequence Transformation**

@ __References__: Wimp 81.

**Sequential Dynamical Systems**
> s.a. causal sets [sequential growth dynamics].

* __Idea__: A class of
discrete dynamical systems which significantly generalize many aspects
of systems such as cellular automata, and provide a framework for studying
dynamical processes over graphs.

@ __References__:
Mortveit & Reidys 07 [intro].

**Sequential Space** > see types of topological spaces.

**Sequestering**
> see vacuum phenomenology [vacuum energy sequestering].

**Series** > s.a. summations.

**Serret-Frenet Equations** > see under Frenet-Serret.

**Sesquilinear Form** > see Quadratic Form.

**Sextic Equation** > see elementary algebra.

**Shadow of a Black Hole**
> see black-hole phenomenology.

**Shannon Coding, Information, Sampling**
> see information; Sampling.

**Shannon-Khinchin Axioms**
> s.a. entropy.

* __Idea__: A set of
axioms for statistical systems under which Shannon in 1948 and Khinchin in
1953 proved that the entropy must be of the Boltzmann-Gibbs form.

**Shape** [antisymmetric function used to describe many-fermion wave functions]
> see composite quantum systems.

**Shape Dynamics**

* __Idea__: A theory of
gravity dynamically equivalent to general relativity in 3+1 (ADM) form,
which does not possess foliation invariance as does the ADM formulation of
general relativity but replaces that symmetry by local spatial conformal
invariance; A theory of evolving conformal geometries; It is inspired by
adherence to Mach's Principle, and is important for the relational
formulations of classical mechanics and gravity.

@ __References__: Barbour & O'Murchadha gq/99;
Anderson et al CQG(05)gq/04 [evolving conformal geometry];
Gomes & Koslowski CQG(12)-a1101 [and general relativity];
Barbour a1105-proc [introduction];
Budd & Koslowski GRG(12)-a1107 [in 2+1 dimensions];
Gomes PhD-a1108;
Koslowski JPCS(12)-a1108 [constraints and Hamiltonian];
Gryb & Thébault FP(12)-a1110 [time and dynamical evolution];
Gomes a1201 [Hamiltonian];
Gryb PhD-1204;
Gomes & Koslowski FP(13)-a1211 [FAQs];
Koslowski a1301-conf;
Gomes & Koslowski a1303-MG13
[differences and similarities with general relativity];
Barbour et al CQG(14)-a1302 [solution to the problem of time];
Koslowski IJMPA(13) [rev];
Carlip & Gomes CQG(15)-a1404 [Lorentz invariance];
Smolin PRD(14)-a1407 [Ashtekar-variables-type formulation];
Mercati a1409 [tutorial];
Gryb ch(15)-a1501
+ blog(15) [no fundamental discreteness];
Koslowski a1501-proc;
Anderson a1503 [configuration spaces for various theories];
Anderson a1505 [foundations];
Anderson a1812 [background independence].

@ __Black holes__: Gomes & Herczeg CQG(14)-a1312;
Herczeg & Shyam CQG(15)-a1410 [entropy];
> s.a. Birkhoff's Theorem.

@ __Quantum__: Koslowski a1302-MG13,
Wong IJMPD(17)-a1701 [loop quantization];
Dündar & Tonguç a1511-proc,
a1511-wd [emergence of spacetime];
Dürr et al a1808
[quantum motion and emergence of absolute space and time].

@ __Related topics__: Gomes & Koslowski GRG(12)-a1110 [coupling to matter and spacetime interpretation];
Gomes et al EPJC(13)-a1105 [and gravity/CFT correspondence];
Gomes & Koslowski a1206 [doubly general relativity];
Gomes PRD(13)-a1212 [Poincaré invariance and asymptotic flatness],
JMP(13) [Weyl anomalies];
Guariento & Mercati PRD(16)-a1606 [cosmological fluid solutions];
Anderson a1810 [relationally-invariant derivatives];
> s.a. time in gravity.

@ __Related theories__:
Anderson a1803 [Topological Shape Theory];
Anderson a1811;
> s.a. conformal gravity; conformal
invariance; Scale Invariance.

> __Online resources__:
see Wikipedia page;
SETI Institute talk
by Henrique Gomes.

**Shapiro Time Delay**
> see gravitational tests with light.

**Shear of a Congruence of World-Lines**

$ __Def__:
If *u*^{a} is
the unit timelike tangent vector to the congruence, one defines
the traceless shear tensor and the shear scalar as

*σ*_{ab}:=
*θ*_{ab} − \(1\over3\)*θ*
*q*_{ab}
, *σ*:=
(\(1\over2\)*σ*_{ab}
*σ*^{ab})^{1/2} .

where *θ*_{ab} is the
expansion tensor and *θ* the expansion scalar of the congruence,
and *q*_{ab} the spatial
metric *q*_{ab}
= *g*_{ab}
+ *u*_{a}
*u*_{b}.

**Shear of a Vector Field**
> see vector calculus.

**Shear, in Cosmology**
> see cosmological expansion, averaging
and parameters; cosmological tests of gravity;
observational cosmology.

**Shell, Gravitating** > see gravitating
matter; metric matching; models in canonical gravity;
semiclassical general relativity; spherical symmetry.

**Shell Model** > see nuclear physics.

**Shell Theorem** > see Newton's Theorem.

**Shift Vector** > see ADM formulation;
initial-value formulation of general relativity;
metric decomposition.

**Shimura-Taniyama-Weil Conjecture** > see number theory.

**Shock Waves** > see Gastrophysics;
foliations; numerical
general relativity [gauge shocks]; gravitational
radiation; phenomenology
of higher-order gravity; velocity.

**Shor's Algorithm** > see quantum computing.

**Shore-Johnson Axioms** > see entropy.

**Short Exact Sequence** > see exact sequence.

**Shot Noise** > see Noise.

**Shtuka**

* __Idea__: A special
kind of module with a Frobenius-linear endomorphism attached to a curve
over a finite field.

@ __References__: Goss NAMS(03) [intro].

**Sierpiński Carpet / Sieve / Triangle **
> s.a. fractals; ising model.

* __Idea__: A fractal
of Hausdorff dimension log 3 / log 2 ≈ 1.585.

@ __References__: Sergeyev CSF(09)-a1203 [area as infinitesimal].

> __Online resources__:
see Wikipedia page.

**Sigma-Algebra ( σ-Algebra)**

$

*

*

**Sigma-Complex ( σ-Complex)**
> s.a. reconstruction of quantum theory.

*

**Sigma-Field** (*σ*-Field) > see ring.

**Sigma Ring** (*σ*-Ring) > see ring.

**Sigmoid Function** > see
MathWorld page;
Wikipedia page.

**Signal Retardation** > see gravitational redshift.

**Signaling**
> see causality in quantum theory; information.

**Signature of a Metric** > see metric;
gravity theories with extended signatures [including signature change];
spacetime models and dynamical spacetime models.

**Silent Universe**

@ __References__: Bruni et al ApJ(95)ap/94,
gq/96-proc
[Bianchi I with magnetic field, singularities],
Mars CQG(99)gq [3+1 description];
Van den Bergh & Wylleman CQG(04)gq [Petrov I with cosmological constant].

**Silver Mean
***

**Simon (Simon-Mars) Tensor**

* __Idea__: A tensor
on the manifold of trajectories in spacetime; It has the property of being
identically zero for a vacuum and asymptotically flat spacetime if and
only if the latter is locally isometric to the Kerr spacetime.

@ __References__: Bini et al CQG(01)gq [congruence approach];
Bini & Jantzen NCB(04)gq-proc [stationary spacetimes];
Somé et al a1412/PRD;
Bini & Geralico CQG(18)-a1808 [generalization].

**Simple Algebra**

$ __Def__: An algebra
that does not have any non-trivial ideals (i.e, other than 0 and the
algebra itself).

**Simple Group** > see group types.

**Simplicial Complex** > see cell complex.

**Simplicity Constraints**
> s.a. BF theory; spin-foam models.

* __Idea__:
Constraints imposed on the Lie-algebra valued 2-form *B* of a 4D *BF*
theory which enforce the condition that *B* be determined by a tetrad, as

*B* = *(*e* ∧ *e*) + (1/*γ*)
(*e* ∧ *e*) ;

With these constraints the theory becomes equivalent to general
relativity, and the *BF* action becomes the Holst action.

@ __References__: Dupuis et al JMP(12)-a1107 [holomorphic, commuting Lorentzian simplicity constraints].

**Simplicity of a System / Theory**
> s.a. physical theories.

* __Idea__: The
simplicity of a systems is a measure of its minimal structure or memory;
It can be used as a means for comparing alternative theories.

@ __References__: Aghamohammadi et al a1602 [classical and quantum simplicity;
simplicity is ambiguous, and not a total order on theories].

**Simply and Multiply Connected Spaces** > see connectedness.

**Simply Transitive Action** > see group action.

**Simulated Annealing**

* __Idea__: A method
to find a configuration of a system with many degrees of freedom that
minimizes a given function, based on the thermal Metropolis algorithm.

@ __References__: Contucci et al mp/04.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Simulations of Physical Systems**
> s.a. approaches to quantum gravity [analogs];
black-hole analogs; Models in Physics.

* __Idea__: Examples
of simulations are numerical ones by computer, simulations of curved
geometries with moving fluids or dielectrics, simulations of interacting
quantum field theories by cold-atom systems.

@ __General references__: Gershenfeld 11.

@ __For quantum systems__: Cirac et al PRL(10)-a1006 [cold-atom systems and interacting-fermion quantum field theories];
Hangleiter et al a1712 [analog quantum simulations, simulation vs emulation].

**Simultaneity** > s.a. kinematics of special
relativity; hidden variables; types of gauge
theories [fiber bundle formulation].

@ __References__: Jammer 06;
Kim & Noz AIP(06)qp [in relativity and quantum theory];
Mamone-Capria FP(12)-a1202 [various theories, as an invariant equivalence relation];
Rynasiewicz SHPSB(12) [distant simultaneity is conventional].

> __Online resources__:
see Wikipedia page.

**Sinai's Theorem**

* __Idea__: A box
of hard spheres is a chaotic system.

@ __References__: Sinai UMN(70).

**Sine-Gordon Equation** > s.a.
partial differential equations.

* __Idea__: An
equation for a (1+1)-dimensional field with solitonic solutions.

@ __General references__: Schief PRS(97) [2+1, integrable];
Dorey & Miramontes NPB(04) [homogeneous, mass scales and crossover];
Aktosun et al JMP(10)-a1003 [exact solutions];
Mikhailov JGP(11) [non-local Poisson bracket].

@ __Solitons__: Gegenberg & Kunstatter PLB(97)ht,
ht/97-proc [and dilaton gravity];
Christov & Christov PLA(08) [description as point particles, and quantization].

@ __Generalized__: Matsuno JPA(10),
JPA(10) [integrable, solution method].

**Single-World Interpretation of Quantum Theory**
> see many-worlds interpretation.

**Singletons**

* __Idea__: Unitary
non-decomposable representations of the (3+2)-dimensional de Sitter group;
They have strange gauge transformation properties and can be gauged away to
zero on any compact set, so they really live at infinity; Spin 0 or 1/2.

* __Uses__: Frønsdal
has proposed that leptons are made of a Fermi singleton ("Di") and a Bose one ("Rac").

@ __References__: Flato & Frønsdal CMP(87),
JGP(88);
Flato et al ht/99-in [rev];
Frønsdal LMP(00)ht/99 [and neutrinos].

**Singular Values**

* __For linear maps__:
A Generalization of the concept of eigenvalues.

@ __References__: Vandebril et al 08.

**Singularities for Differential Equations** > see partial differential equations;
solutions in electromagnetism; wave phenomena.

**Singularities for Mappings**
> s.a. Catastrophe; Cusp; Fold.

@ __General references__: Whitney AM(55);
Golubitsky & Guillemin 73;
Arnold 91;
Izumiya et al 15 [and the differential geometry of surfaces].

@ __Surface singularities__:
Kiyek & Vicente 04 [resolution, in characteristic zero].

**Singularities in Spacetime**
> see censorship; cosmological singularities
and other types of singularities; singularity theorems.

**Sinh-Gordon Equation**

@ __References__: Xie & Tang NCB(06) [solution method].

**6 j-Symbols**
> see SU(2).

**Sixth-Order Equations**
> see algebra [sextic].

**SKA (Square-Kilometer Array)**

* __2015__: The Square
Kilometre Array (SKA) project is an international effort to build the
world's largest radio telescope, with a square kilometre (one million
square metres) of collecting area.

@ __References__: Camera et al a1501-conf [cosmology];
Maartens et al a1501-conf [overview];
Bengaly et al a1810 [cosmic radio dipole];
news pt(19)jan [powering the SKA].

> __Online resources__:
see SKA website.

**Skein Relations**
> see knot theory and physics.

**Skein Space**
> see spin structures.

**Skeleton of a Simplicial Complex**

$ __Def__: Given a
simplicial complex *K* in \(\mathbb R\)^{n},
its *p*-skeleton *K*^{(p)} is the
union of the simplices *σ* in *K* of dimension ≤ *p*.

* __Example__: The elements of
*K*^{(0)} are the vertices of *K*.

**Sky** > see null geodesics.

**Skyrmion Model** > s.a. QCD
phenomenology / astronomical objects [skyrmion stars].

* __Idea__: A phenomenological
model for QCD that contains the *π* fields as basic fields,
and constructs the nucleons as solitonic solutions in the pion fields,
corresponding to bound states of pions; A "Skyrme term" has to be present
in the Lagrangian for stability, and the collective coordinate method is
used for quantization; > s.a. black-hole
solutions; black-hole hair.

@ __General references__:
Gisiger & Paranjape PRP(98);
Cho et al ht/99;
Abbas PLB(01) [and hadrons];
Wong hp/02,
hp/02,
hp/02;
Cho et al IJMPA(08)ht/04 [interpretation];
Rajeev AP(08)-a0801 [relativistic wave equation];
Ioannidou & Kevrekidis PLA(08)-a0807 [2+1 and 3+1 lattice versions];
Brown & Rho ed-10;
Boschi et al a1211-conf [relativistic].

@ __Quantization__: Jurciukonis et al JMP(05)nt [SU(3) model, canonical quantization];
Krusch ht/06 [overview].

@ __Skyrme black holes__: Zaslavskii PLA(92) [first law of thermodynamics];
Shiiki & Sawado CQG(05)gq [Λ < 0];
Brihaye & Delsate MPLA(06)ht/05 [in de Sitter];
Nielsen PRD(06)gq [isolated horizons];
> s.a. black-hole hair.

@ __And gravity__: Ioannidou et al PLB(06)gq [gravitating],
PLB(06)gq [spinning];
Dunajski PRS(13) [from gravitational instantons];
Klinkhamer PRD(14)-a1402,
Klinkhamer & Queiruga MPLA-a1805 [spacetime defects];
> s.a. bianchi
IX models; topology change.

@ __Applications, experiment__: Leslie et al PRL(09) [Skyrmions and half-Skyrmions in a spin-2 Bose-Einstein condensate, realization];
news pw(15)jun [skyrmions as magnetic bubbles in computers];
news sn(18)feb [use for data storage];
news pt(18)may
[how to create and destroy a skyrmion using electric currents];
news sn(18)nov [in nuclear physics];
Kim JPCM(19)-a1907 [Skyrmion Hall transport, rev].

**Slater Determinant**
> see Wikipedia page.

**Slice**

$ __Def__: A closed
achronal subset of spacetime without edge.

**Slicing**
> see foliation.

**Slingshot Effect**
> see orbits in newtonian gravity.

**Smale Conjecture**
> see diffeomorphisms.

**Smarr Formula**
> s.a. non-commutative gravity.

* __Idea__: A formula
that gives the mass of a stationary black hole in terms of quantities
defined on its horizon, such as area and surface gravity; For Kerr-Newman
black holes,

* __Remarks__: It looks like
the "integrated version" of the first law, but the latter holds for any
perturbation, not just stationary ones; For black holes with matter fields
a more suitable mass definition is the Tolman mass, which requires that
the spacetime be static or stationary.

@ __General references__: Smarr PRL(73) [Kerr black holes];
Breton GRG(05)gq/04-fs [in non-linear electromagnetism];
Barnich & Compère PRD(05)gq/04 [higher-dimensional Kerr-AdS];
Clément & Gal'tsov PLB(17)-a1707 [rotating dyonic black holes];
Lemos & Zaslavskii PRD(18)-a1712 [in the membrane paradigm].

@ __Generalized versions__: Kastor et al CQG(10)-a1005 [in Lovelock gravity];
Banerjee et al PRD(10)-a1007 [(*N*+1)-dimensional charged Myers-Perry spacetime];
Pradhan EPJC(14)-a1310;
Haas CQG(18)-a1405 [in 11-dimensional supergravity];
Villalba & Bargueño PRD(19)-a1902 [quasilocal, for static black holes].

**Smith Cloud** > see milky way galaxy.

**Smith Conjecture / Theorem** > see spheres.

**Smooth Particle Hydrodynamics** > see fluid.

**Smoothing** > see Coarse-Graining;
averaging in cosmology; dynamics
of gravitating bodies.

**Snark** > A type of graph.

$ __Def__: A
non-trivial 3-regular graph which cannot be 3-edge coloured.

@ __References__: Brinkmann et al JCTB(13) [generation and properties].

> __Online resources__:
see MathWorld page.

**Snell's Law** > s.a. refraction.

* __Idea__: The
equation reating the angle of incidence and the angle of refraction for
light crossing a smooth boundary between two transparent media.

@ __General references__: Heller AJP(48)sep [teaching];
Drosdoff & Widom AJP(05)oct,
comment Pérez AJP(06)sep [photon beam point of view].

@ __Related topics__: De Leo & Ducati JMP(13) [for quantum particles with quaternionic potentials].

> __Online resources__:
see Wikipedia page.

**Snyder Spacetime** > see non-commutative
geometry, spacetime and field
theory; minkowski spacetime [deformed];
types of quantum spacetime.

**SO( n) Group** > see examples of lie groups.

**Soap** > see meta-materials [foam].

**Sobolev Space** > s.a. *p*-Adic Numbers.

$ __Def__: The Sobolev space
W_{p}^{m}(*U*)
is the space of all functions which belong, together with their derivatives up to the
*m*-th order, to L^{p}(*U*):

W_{p}^{m}(*U*):=
{*f* | D^{j} *f* ∈
L^{p}(*U*) for all *j*
such that | *j* | ≤ *m*} .

* __Special case__:
For *p* = 2, we call H^{m}(*U*):=
W_{2}^{m}(*U*).

@ __References__: Adams 75;
Maz'ya 11;
Diening et al 11 [with variable exponents].

**Soccer Ball Problem** >
s.a. momentum-space geometry.

@ __General references__: Magueijo PRD(06)gq;
Olmo a1101,
JPCS(12)-a1111;
Hossenfelder Sigma(14)-a1403 [rev].

@ __And DSR__: Girelli & Livine gq/04,
BJP(05)gq/04;
Girelli & Livine JPCS(07)gq/06;
Hossenfelder PRD(07)ht.

@ __And relative locality__:
Amelino-Camelia et al PRD(11)-a1104,
comment Hossenfelder PRD(13)-a1202,
reply Amelino-Camelia PRD(13)-a1307.

**Soft Gravitons** > s.a. Gravitational Memory.

@ __Soft graviton theorem__: Laddha & Sen a1906 [classical proof, *D* > 4].

**Soft Matter**
> see condensed matter.

**Solar System**
> s.a. planets and minor objects.

**Soldering Form**
> s.a. spin structure.

* __Idea__: A
"disguised identity", also called Infeld-Van der Waerden Symbol, that
establishes an isomorphism between spin tensors and spacetime tensors.

* __SL(2, C)
spinors__: The objects that correspond to spacetime vectors are the
self-conjugate spinorial 2-tensors, and the soldering form takes

*V*^{a}
→ *V*^{AA'},
with *V*^{a}
= *σ*^{a}_{AA'}
*V*^{AA'}, or
*V*^{AA'}
= *σ*_{a}^{AA'}
*V*^{a} ;

With the right choice of basis, these *σ*s can be thought
of as the unit 2 × 2 matrix and the Pauli matrices.

* __SU(2) spinors__:
Objects corresponding to spacetime vectors are symmetric spinorial
2-tensors, and the soldering form takes

*V*^{a} →
*V*^{AB}, with
*V*^{a}
= *σ*^{a}_{AB}
*V*^{AB}, or
*V*^{AB}
= *σ*_{a}^{AB}
*V*^{a} ;

With the right choice of basis, these *σ*s can be thought of
as the three Pauli matrices.

* __4-spinors__: The soldering
form corresponds to the Dirac matrices.

* __Applications__: The
(complexified) SU(2) soldering form has been used as a variable for gravity.

**Solenoidal Vector Field** > see vector field.

**Solid Light**
> see Wikipedia page.

* __Idea__: A phenomenon by which
photons interact with and repel each other in a macroscopic, strongly-correlated way
[@ news sd(07)may].

**Solid Matter / Solid-State Physics**

**Solutions of Einstein's Equation**

**Solvability, Solvable Equation**
> s.a. classical systems; types
of waves [exactly solvable wave equations].

@ __References__: Pešić 03 [Abel and the quintic].

**Solvable Group**

$ __Def__: *G* is solvable
if it has a normal series whose factors are Abelian (solvable series); Or,
if the chain *G* = *Q*_{0} ⊃
*Q*_{1} ⊃ *Q*_{2}
⊃ ..., where *Q*_{i} is
the commutant of *Q*_{i−1},
has *Q*_{m} = {*e*}
for some *m* (the *height* of *G*).

* __Properties__:
A solvable group always has a commutative invariant subgroup (the
*Q*_{m−1} above).

* __Examples__:

- The 2D Euclidean group,
of height 2, *E*_{2}
= *T*_{1,1}
×_{s} SO(2) ⊃
*T*_{2} ⊃ {*e*}.

- The 2D Poincaré group:
*P*_{2}
= *T*_{1,1}
×_{s} SO(1,1) ⊃
*T*_{1,1} ⊃ {*e*}.

- The Heisenberg group.

**Sommerfeld Paradox**

* __Idea__:
Mathematically, the Couette linear flow is linearly stable for all
Reynolds numbers, but experimentally arbitrarily small perturbations can
induce the transition from the linear shear to turbulence when the
Reynolds number is large enough.

@ __References__: Li & Lin a0904 [proposed resolution];
Lan et al a0905.

**Sorkin-Johnston States** > s.a. quantum field theory
in curved spacetime; fields in de sitter spacetime.

@ __General references__: Johnston PRL(09)-a0909,
PhD(10)-a1010;
Sorkin JPCS(11)-a1107;
Afshordi et al JHEP(12)-a1205,
JHEP(12)-a1207.

@ __Related topics__: Avilán et al PRD(14)-a1408 [coupling to gravity];
Mathur & Surya a1906 [massive scalar field in a 2D causal diamond].

**Sp(2 n) Group**
> see under Symplectic Group.

**Space in Mathematics**

$ __Def__: (Souriau)
A set *E* is a space if there is a recueil *R* (of
"glissements") acting on *E*.

* __And other structure__:
A space has a natural topology, in which *F* ⊂ *E* is
open if id_{F} in *R*.

**Space in Physics** > s.a. Raumproblem
[problem of space]; spacetime models [absolute space];
tensor decomposition [for spacetime metric].

* __Idea__: Given a
spacetime manifold (*M*, *g*) and a time function *f*
on *M*, space is a level set for *f*.

@ __References__: Lachièze-Rey A&A(01) [for an arbitrary observer].

> __Space of possible
spatial structures__: see geometrodynamics [including generalizations].

**Space of Theories**
> see theory of physical theories.

**Spaceflight**
> see cosmic civilizations.

**Spacetime**
> s.a. decomposition; important subsets;
models in general and discrete models;
topology; types of spacetimes.

**Spacetime Algebra**
> see Geometric Algebra.

**Spacetime Crystal**
> see crystals.

**Spacetime Diagram**
> see Penrose Diagram; special-relativistic kinematics.

**Spacetime Reconstruction Problem**
> s.a. multipole moments.

@ __References__: Anderson & Mercati a1311 [classical Machian resolution].

**Sparking of the Vacuum**
> see vacuum [QED effect].

**Sparling Forms**
> s.a. stress-energy pseudotensors.

* __Real 2-forms__: The
set of four 2-forms given by

*σ*_{I} :=
\(-\frac12\)*ε*_{IJKL}
Γ^{JK}
∧ *e*^{L} ,

where *e*^{L} is a tetrad field,
and Γ^{J}_{Ka}
= *e*^{J}_{b}∇_{a}
e^{b}_{K} its Levi-Civita connection.

* __Complex 2-forms__:
The two sets of forms

*σ*^{(±)}_{I}
:= −*ε*_{IJKL}
Γ^{(±)
JK} ∧ *e*^{L}

where Γ^{(±) JK}:=
\(\frac12\)(Γ^{JK}
\(\mp\frac12\)i *ε*^{JKLM}
Γ_{LM}).

* __3-form__: A
tetrad-dependent 3-form *σ*_{I} or
*σ*^{(±)}_{I}
on the bundle of orthonormal frames over spacetime, which is a potential for a local
energy-momentum density *τ*_{I}
for the gravitational field; If *e**_{J}
is a basis of 3-forms, and *G*_{IJ}
the Einstein tensor,

d*σ*_{I} =
d*σ*^{(±)}_{I}
= *τ*_{I}
+ *G*_{I}^{J}
*e**_{J} .

@ __References__: Dubois-Violette & Madore CMP(87);
Goldberg PRD(88);
Frauendiener CQG(89),
GRG(90).

**Special Functions**
> s.a. Integral Transforms; representations of lie groups.

* __Idea__: Usually,
complete orthonormal sets of functions on some set *X* (often,
an interval *X* = [*a*,* b*]), with which we
approximate a function by a finite sum *f*(*x*) ≈
Σ_{n=1}^{N}
*a*_{n}*U*_{n}(*x*),
where the coefficients are calculated by *a*_{n}
∫_{X} d*x*
*U*_{n}*(*x*) *f*(*x*) and the
finite sum minimizes ∫_{X} d*x*
|*f*(*x*) − ∑_{n}
*a*_{n}
*U*_{n}|^{2}.

* __History__: The study
of orthogonal polynomials can be traced to the XVIII century, when
Legendre studied the motion of heavenly bodies.

* __Group theoretic
approach__: Most special functions are connected with the
representation of Lie groups; The action of elements *D* of the
associated Lie algebras as linear differential operators gives relations
among the functions in a class – for example, their differential
recurrence relations; & Gelfand, Naimark, N Ya Vilenkin.

* __Bochner's problem__:
The characterization of classical orthogonal polynomial systems as
solutions of second-order eigenvalue equations.

@ __Textbooks and reviews__: Rainville 63;
in Abramowitz & Stegun ed-65;
Wang & Guo 89;
Temme 96 [intro];
Lorente JCAM(03)mp/04 [applications];
Totik SAP(05)math [for non-experts];
Dunkl & Xu 14 [in several variables].

@ __General references__: Batterman BJPS(07) [what makes them special];
Celeghini & del Olmo AP(13)-a1205 [orthogonal polynomials and Lie algebras];
Schneider et al PT(18)feb [NIST's Digital Library of Mathematical Functions];
Celeghini et al a1907
[and Lie groups and rigged Hilbert spaces].

@ __And representation theory__:
Etingof & Kirillov Jr ht/93;
Wasson & Gilmore a1309-ug [rev].

@ __Related topics__: Lucquiaud JMP(90) [in curved space];
Peherstorfer mp/02 [zeros];
Gurappa et al mp/02 [new approach];
Eynard mp/05-proc [asymptotics];
Giraud JPA(05)mp [vanishing average];
Simon BAMS(05) [on S^{1}];
Alhaidari AML(07)mp/05 [integrals];
Coftas CEJP(04)mp/06 [from hypergeometric equations];
Bruschi et al JPA(07) [from Diophantine conjectures];
Gòmez-Ullate et al JAT(10)-a0805 [generalized Bochner problem];
Dunkl Sigma(08)-a0812 [in four variables];
Doria & Coelho RPMP(18)-a1703 [in *D* dimensions].

@ __Specific functions__: Raposo et al CEJP(07)-a0706 [Romanovski polynomials];
Vinet & Zhedanov JPA(11);
Alhaidari a1709 [two new classes];
> s.a. Airy; bessel;
Dirichlet Eta Function; Elliptic;
Gamma; Hypergeometric;
Jost; Mathieu;
Struve; Whittaker; Zeta
Function; spherical harmonics; other
functions.

> __Other polynomials__:
see Chebyshev, Gegenbauer,
Hermite, integral equations,
Jack, Laguerre
and legendre polynomials; graph
and knot invariants.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Special Relativity** > s.a. doubly
special relativity; special-relativistic kinematics.

**Species (Combinatorial Species)**

* __Idea__: A functor *F*
: \(\cal B\) → \(\cal B\), where \(\cal B\) is the category of finite sets
and bijections which gives, for every object *A* ∈ \(\cal B\), the
set *F*[*A*] of *F*-structures on *A*.

> __Online resources__:
see Wikipedia page.

**Species Problem**
> see origin of black-hole entropy.

**Spectral Action** > see non-commutative
physics; Non-Associative Geometry.

**Spectral Decomposition**
> see hilbert space.

**Spectral Dimension** \ s.a. dimension of a space.

$ __Def__: The expression \(d:= -2\,{\rm d}\log P(t)/{\rm d}\log t\), where *P*(*t*) is the return probability for a diffusion process in time *t*, which is also the trace of the heat kernel on the space; It is scale-dependent, but on an infinite flat space its value coincides with the topological dimension, for all scales.

@ __General references__: Sotiriou et al PRD(11)-a1105 [and dispersion relations];
Calcagni et al IJMPD(16)-a1408 [interpretation, in quantum field theory].

@ __And quantum gravity__: Rhodes & Vargas AHP(14)-a1305 [Liouville quantum gravity];
Alkofer et al PRD(15)-a1410 [from the spectral action, for almost-commutative geometry];
Muniz et al PRD(15)-a1412 [and relativistic diffusion].

> __Examples, special
cases__: see Triangulations; graph
invariants; minkowski spacetime [*κ*-deformed].

> __And quantum
gravity__: see causal sets; dimensionality
of quantum spacetime; geometry and
quantum gravity; dynamical triangulations;
2D quantum gravity.

**Spectral Distance / Geometry**
(a.k.a. Connes Distance)

> s.a. non-commutative
geometry / geometry of graphs; graph
theory in physics / coherent states.

**Spectral Function**

@ __References__: Kirsten ht/00-wd [review].

**Spectral Methods**
> see partial differential equations.

**Spectral Sequence**

@ __References__: in Spanier 66.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Spectral Theorem**

@ __References__: Gill & Williams a1211 [two representations];
Riechers & Crutchfield a1607 [extended to arbitrary functions of non-diagonalizable linear operators].

> __Online resources__:
see Wikipedia page.

**Spectral Theory / Analysis ** > see operator theory.

**Spectral Triple**
> s.a. holonomy; non-commutative geometry.

* __Idea__: A set of
data which encodes a non-commutative geometry; It includes a Hilbert
space, an algebra of operators on it and an unbounded self-adjoint
operator, endowed with supplemental structures.

@ __References__: Aastrup et al CMP(09)-a0807 [over a holonomy algebra];
Franco RVMP(14)-a1210 [temporal Lorentzian spectral triple];
Falk JGP(17)-a1602 [integration of spectral triples];
Bizi a1812-PhD [Lorentzian].

> __Online resources__:
see Wikipedia page.

**Spectrometer**
> see experiments in physics.

**Spectroscopy**
> see atomic physics; astronomy
at various wavelengths; optical technology.

**Spectrum of an Algebra**

$ __Def__: The set of its characters.

**Spectrum of an Algebra Element**

$ __Def__: The spectrum
of an element *a* of an algebra *A* over *K*
is the set of *λ* ∈ *K* such that
*a*−*λ*I is not invertible,

*σ*(*a*):= {*χ*(*a*) | *χ* a character of *a*} .

**Spectrum of an Operator** > see operator theory.

**Speed**
> see velocity; constants [speed of light];
tests of general relativity [speed of gravity].

**Speed of Quantum State Evolution**
> see quantum state evolution.

**Spence's Function** > see under Dilogarithm Function.

**Sphaleron** > see solutions of gauge theories.

**Sphere** (including Sphere Packings).

**Spherical Symmetry**
> s.a. spherical symmetry in general relativity;
gauge theory solutions.

**Spheroidal Harmonics**
> see spherical harmonics.

**Spi (Spatial Infinity)**
> see asymptotic flatness at spatial infinity.

**Spin / Spinors**
> s.a. 2-spinors; 4-spinors; spinors
in field theory; types of spinors [including ELKO and Kähler].

**Spin-Charge Separation**

@ __References__: Fiete Phy(11) [for photons].

**Spin-Echo Experiment**

@ __References__: Ainsworth FPL(05) [and approaches to statistical mechanics];
Anastopoulos & Savvidou PRE(11)-a1009 [and thermodynamics].

**Spin Glasses and Models** [including spin chains]
> s.a. quantum spin models.

**Spin Liquid**
> s.a. Frustration; quantum spin models.

* __Idea__: A material that
resists magnetic ordering down to absolute zero, i.e., a substance in
which the orientation of the magnetic dipole moments of the atoms
remains in a constant state of flux (although the positions of those
same atoms may be fixed if the substance is a solid); A frustrated
magnetic material.

* __History__: Lattices of
connected triangles, giving rise to frustration in antiferromagnetic
Heisenberg models, have been the subject of searches for spin liquids
ever since Anderson's suggestion in 1973; These materials are rare;
First observed in the greenish mineral called herbertsmithite and in
Ba_{3}CuSb_{2}O_{9}, as confirmed by neutron scattering data;
Also lithium iridate and a sodium iridate have the honeycomb structure
that Kitaev predicted in 2006 could make for ideal quantum spin liquids;
2017, copper iridium binary metal oxide is an even better spin liquid.

* __Applications__:
They could help develop a large-scale quantum computer.

@ __References__: news NIST(12)may [first observation];
news Phy(13) [vanadium compound as new candidate];
Imai & Lee PT(16)aug [do they exist?];
Clark Phy(17) [antiferromagnetic Heisenberg model for the kagome lattice];
news cosmos(17)oct.

**Spin Networks**
> s.a. connection representation of quantum gravity,
other spin models.

**Spin-Orbit Interaction**
> see atomic physics; Precession.

**Spin-Statistics Theorem**
> s.a. particle statistics.

**Spin-Weighted Spherical / Spheroidal Harmonics**
> see spherical harmonics.

**Spincube Models**
> see spin-foam models.

**Spinon** > see Luttinger Liquid.

**Spintessence** > see quintessence.

**Spiral, Logarithmic** {# s.a. Bernoulli.}

* __Examples in nature__: Galaxies, Nautilus.

@ __References__: in Thompson; in Maor ThSc(94)jul.

**Spline**

* __Idea__: A
continuous curve constructed so as to pass through a given set
of points and have a given number of continuous derivatives.

@ __References__: de Boor 78.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Splitting of Spacetime** > see decomposition.

**Splitting Sequence** > see exact sequence.

**Splitting Theorem**

@ __Lorentzian geometry__: Yau 82;
Galloway CMP(84),
JDG(89);
Ehrlich & Galloway CQG(90);
Newman JDG(90);
Galloway AHP(00)m.DG/99,
in(02)gq [null].

**Spontaneous Emission**

* __Idea__: The
process by which an atom or other quantum system in an excited state emits
a photon while undergoing a transition to a lower-energy state.

@ __General references__:
Crisp & Jaynes PR(69),
Leiter PRA(70) [in semiclassical radiation theory];
Cray et al AJP(82)nov [in terms of interference];
Milonni AJP(84)apr [and fluctuation dissipation];
Olsen et al OC(05)qp [2-level bosonic atom, phase space approach];
Kleppner PT(05)feb [and stimulated emission, Einstein's 1917 paper].

@ __Based on electron self-energy, without field quantization__:
Barut & Van Huele PRA(85),
& Dowling PRA(87),
& Salamin PRA(88).

@ __Related topics__: Jorgensen et al PRL(11)
+ Ning & Braun Phy(11)
[optical spontaneous emission control].

> __Online resources__:
see Wikipedia page.

**Spontaneous Process / Spontaneity**
> see Free Energy [*G*].

**Sporadic Groups**
> see finite groups.

**Sprinkling of Points in a Manifold**
> see statistical geometry.

**Square** (magic square, ...) > see number theory.

**Square Roots** > see elementary algebra.

**Squeezed States** > s.a. coherent
states; distance; QED;
symplectic structure [squeezing].

* __Idea__: A quantum
minimum-uncertainty (Δ*x* Δ*p* = \(\hbar\)/2)
state of an oscillator/field in which the complementary operators do not
have the same variance; The product of the variances of course satisfies
the uncertainty relation, but one of them is lower than the coherent
state value, the one predicted by semiclassical models.

* __Examples__:
Squeezed light may be applied in data transmission and high-precision
metrology; In gravitational-wave detectors, squeezed states of light are
used which have a lower uncertainty in their phase at the expense of a
higher uncertainty in their amplitude.

@ __General references__: Yuen PRA(76);
Yuen & Shapiro OL(79);
Caves PRD(81);
Henry & Glotzer AJP(88)apr;
Muñoz-Tapia AJP(93)nov [properties];
Nieto qp/97-proc [history];
Beckers et al PLA(98) [new sets];
Trifonov PS(98) [for *n* observables];
Saxena JPA(02) [eigenvalue equation];
Honegger & Rieckers PhyA(04) [non-classicality and coherence];
Sträng JPA(08)-a0708 [semiclassical evolution];
Fujii & Oike IJGMP(14) [rev].

@ __On S__^{1}:
Kowalski & Rembieliński JPA(02)qp,
JPA(03)qp;
Trifonov JPA(03)qp/02.

@ __For QED, light__: Loudon & Knight JMO(87) [light];
Slusher & Yurke SA(88)may [light];
Putz & Svozil NCB(04)ht/01 [vacuum, electron mass shift];
Popp et al PLA(02) [in biological systems];
Petersen et al PRA(05)qp;
Bachor et al CP(05);
Biswas & Agarwal PRA(07) [photon-subtracted, non-classicality];
Chua et al CQG(14) [in gravitational-wave detectors];
Lvovsky ch-a1401;
Andersen et al PS(16)-a1511 [rev];
> s.a. types of coherent states.

@ __Other systems__: Burgess PRD(97) [non-equilibrium quantum field theory];
Tavassoly JPA(06) [solvable];
Marchiolli et al PRA(07)qp [discrete];
Wollman et al Sci(15)aug
+ news pt(15)oct [micron-scale mechanical resonator];
> s.a. quantum oscillators.

@ __Squeezed number states__: Nieto PLA(97)qp/96;
Albano et al JOB(02)qp/01.

@ __Related topics__: Seroje et al EJP(15)-a1507 [effective thermodynamics].

@ __Generalized__: Marchiolli & Galetti PS(08)-a0709;
Shchukin et al a0712;
Thirulogasanthar & Muraleetharan a1706 [on a right quaternionic Hilbert space];
Zelaya et al PLA(18)-a1810 [and Wigner functions].

> __Related states and
generalizations__: see Fermi Function; fock space;
Kerr State; vacuum.

**SQUID (Superconducting Quantum Interference
Device)** > see superconductivity.

**Stability in Physics**

> __In general__:
see classical systems; higher-order
lagrangian systems; physical theories.

> __Gravitation__:
see black-hole perturbations; cosmological
perturbations; perturbations in general relativity.

> __Quantum gravity__: see linearized
quantum gravity; quantum cosmological perturbations.

> __Other theories__: see matter;
condensed matter [thin liquid films].

**Stability Theory in Mathematics**
> s.a. Bifurcation Theory; mappings between manifolds.

* __Stable property__:
A property of an object (or a subset) in a topological space is stable if
there is an open set containing it, all of whose members also have the property.

@ __References__: Yoshizawa 75;
Rouche et al 77.

**Stabilizer of a Group Element**
> see group action.

**Stacks**
> see category theory in physics.

**Standard Map**
> s.a. chaotic systems.

* __Idea__: A chaotic,
area-preserving discrete map of the unit square map onto itself used to
model a kicked rotator; Also called Taylor-Greene-Chirikov map; Defined by

*p _{n}*

@ __References__:
Shevchenko PhyA(07).

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Standard Model**
> see in cosmology; in particle
physics, and beyond the standard model.

**Standard Model Extension**
> see lorentz-violating theories.

**Standard Sirens**

@ __References__: Holz et al PT(18)dec [and cosmic distances].

**Star-Algebra**
> see abstract algebra.

**Star-Convex Subset of an Affine Space**
> see affine structures.

**Star Product (Non-Commutative Geometry / Phase Space)**
> s.a. non-commutative geometry.

* __Idea__: An antisymmetric tensor
*θ*^{mn} used to define
non-commutative geometrical structures, such that for two *f*and *g*,

(*f ***g*)(*x*):=
exp(\(1\over2\)i *θ*^{mn}
{∂/∂*y*^{m}}
{∂/∂*z*^{n}})
*f*(*y*) *g*(*z*)|_{y=z=x}
= *f*(*x*) *g*(*x*)
+ \(1\over2\)i *θ*^{mn}
∂_{m}* f*(*x*)
∂_{n} *g*(*x*) + h.o.t.

* __Remark__: This structure
is not Lorentz-invariant.

* __Types__: Two known star
products are the Husimi product of coarse-grained quantization, and a
damped star product for the harmonic oscillator.

@ __General references__: Zachos JMP(00)ht/99 [evaluation];
Gammella LMP(00) [tangential];
Man'ko et al PLA(05)ht/04 [dualities];
Pinzul & Stern NPB(08) [gauging];
Kupriyanov & Vassilevich EPJC(08)-a0806 [friendlier approach];
Aniello JPA(09)-a0902 [group-theoretical point of view];
Bratchikov IJGMP(13) [computation].

@ __Special types__: Aniello et al PLA(09) [on finite and compact groups];
Vassilevich CQG(09)-a0904 [diffeomorphism-covariant, and non-commutative gravity];
Chaichian et al IJMPA(10)-a1001 [covariant];
Långvik & Zahabi IJMPA(10)-a1002 [modified Weyl-Moyal for finite range of non-locality];
Vassilevich a1101-conf [covariant];
Soloviev PS(15)-a1802
[from *s*-ordering of the creation and annihilation operators, integral representation];
Robbins & Walton a1806 [and modified classical equations of motion].

@ __Special contexts__: Freidel & Krasnov JMP(02) [and spin networks];
Tagliaferro a0809 [differential forms on symplectic manifolds];
Filippov & Man'ko JPA(12)-a1108
[photon creation and annihilation operators].

@ __Related topics__: Waldmann a1012-proc [Morita-equivalent star products].

> __In physics__: see non-commutative
field theory; types of quantum field theories.

**Star Product (Poset Theory)**

> __Online resources__:
see Wikipedia page.

**Stark Effect**
> see atomic physics.

**Starobinski Model** > see types of inflationary models.

**Stars**
> s.a. star types.

**State of a System** > s.a. quantum state.

**State Sum Models** > s.a. spin-foam models.

@ __References__: Barrett et al JPA(13)-a1211 [for fermions on the circle].

**Static Spacetime** > see general relativity
solutions with symmetries; types of spacetimes.

**Stationary-Phase Approximation**
> s.a. Steepest-Descent Approximation.

* __Idea__: An
approximation used to calculate the leading-order behavior of integrals of
the type \(\int_{-\infty}^\infty {\rm d}x\, f(x) \exp\{{\rm i}\phi(x)/\hbar\}\)
in the limit of small \(\hbar\); It consists in taking into account only the
contribution from the critical points of *φ*(*x*), and
is related to the steepest-descent approximation.

* __In path integrals__:
The approximation of writing the field as the classical solution plus a
small perturbation; Sometimes known as WKB or one-loop approximation.

@ __General references__: Kamvissis CM(08)mp/07 [and steepest descent];
in Alastuey et al 16.

@ __In quantum-mechanical path integrals__: Sorkin a0911-in [saddle-point approximations and tunneling];
Smirnov JPA(10).

**Stationary Spacetime** > see general
relativity solutions with symmetries; types of spacetimes.

**Statistics**
> s.a. error analysis in physics;
particle statistics; probability.

**Statistical Mechanics**
> s.a. non-equilibrium, systems.

**Steady State** > see states of a system.

**Steady-State Cosmology**
> s.a. Continuous Matter
Creation; cosmological models
and general relativistic models.

* __History__: First
proposed in 1948 by H Bondi, then T Gold and F Hoyle (and Littleton?);
Despite its loss of mainstream favor, to some extent the idea has been
incorporated into some versions of inflation.

* __Idea__: It postulates
that the universe is always expanding, and matter is created at
precisely the rate required to maintain a constant spatial density, about
\(10^{-43}\) kg/m\(^3\)·s (equivalent to one hydrogen atom per cubic
meter in half a billion years); A steady-state universe has no beginning
or end, and its overall properties are constant in time.

* __And observation__:
These models don't have the singularity and flatness problems of the
standard model, but they are considered ruled out by observations on radio
sources by M Ryle et al at Cambridge in the 1950s and early 1960s, and by
the discovery of the microwave background (although 2.75-K radiation was
predicted by Gold from thermalising starlight, produced by assuming all
helium was made in stars).

* __Quasi-steady state
variant__: Matter is created only near very compact dense objects,
because it has to be created in units of the Planck mass; Interaction
between matter creation and the expansion or contraction of space produces
then universal oscillations with a period of 50 billion years,
superimposed on a general expansion; One inspiration for the theory was
observations by Ambartsumian of pockets of explosions that suggest
localised matter creation.

@ __General references__: Hoyle in(58);
Arp et al Nat(90)aug;
Andrews ap/01;
Altaie a0907
[from back-reaction effect of quantum fields];
Narlikar & Burbidge 10;
Kragh a1201 [historical review];
O'Raifeartaigh et al EPJH(14)-a1402,
O'Raifeartaigh & Mitton a1506-proc [Einstein's theory].

@ __Quasi-steady state__: Hoyle et al PRS(95) [comment Wright MNRAS(95)];
articles by Narlikar, Burbidge, and Arp in Sato 99;
Hoyle et al 00;
Burbidge et al PT(99)apr
[and reply by Albrecht PT(99)apr];
Burbidge ap/01-proc;
Narlikar et al PASP(02)ap [acceleration],
ApJ(03)ap/02 [and cmb];
Vishwakarma & Narlikar JAA(07)-a0705 [and repulsive gravity];
Narlikar et al JAA(07)-a0801 [and cyclic universe];
Narlikar et al MNRAS-a1505 [gravitational-wave background].

@ __Criticism of Big Bang__: Arp & Van Flandern PLA(92);
Arp ap/98-PASP;
López-Corredoira in(03)ap.

**Steady-State Equation**
> see partial differential equations.

**Stealth Fields** > s.a. scalar fields;
supersymmetric theories.

* __Idea__: Fields that
are not coupled to gravity, i.e., fields with vanishing energy-momentum tensor.

@ __References__: Ayón-Beato et al PRD(13)-a1307 [conformally invariant scalar field, and cosmology];
Smolić PRD(18)-a1711 [non-linear electromagnetic fields].

**Steepest-Descent Approximation**
> see integration.

**Steering (Quantum)**
> s.a. non-locality in quantum mechanics.

* __Idea__: The ability
of one party to "steer" the states of a remote party by performing
measurements on their half of an entangled set of particles; There are
situations in which steering can only occur in one direction.

@ __References__: Wollmann et al PRL(16) [observation of one-way Einstein-Podolsky-Rosen steering];
Liu et al AQT(18)-a1808
[influence of spacetime curvature on satellite-based steering];
Uola et al a1903 [rev].

**Stefan-Boltzmann Law** > see thermal radiation.

**Stein Structure** > see 4D manifolds.

**Stem** > see posets.

**STEP** > see tests of the equivalence principle.

**Stephani Universe / Model**

* __Idea__: A
spherically symmetric, inhomogeneous cosmological model, recently
used as a possible explanation of the cosmic acceleration.

@ __General references__: Stelmach & Jakacka CQG(06) [angular sizes];
Pedram JCAP(08)-a0806 [+ scalar, classical and quantum];
Balcerzak et al PRD(15)-a1409 [inhomogeneous pressure].

@ __And acceleration__: Stelmach & Jakacka CQG(01)-a0802;
Godlowski et al CQG(04)ap.

**STE-QUEST Space Mission** > see tests of the equivalence principle.

**Stern-Gerlach Experiment** >
s.a. experiments in quantum mechanics.

* __Idea__: An experiment
that demonstrates the quantization of electron spin, in which
silver atoms boiled off from a furnace are sent through a non-uniform
magnetic field and impinge on a photographic plate; Instead of a
continuous distribution of spots one sees two spots, corresponding
to spin up and spin down relative to the magnetic field axis.

@ __ General references__: Alstrøm et al AJP(82)aug;
Batelaan et al PRL(97) [electrons];
Hannout et al AJP(98)may;
Cruz-Barrios & Gómez-Camacho PRA(01) [semiclassical description];
Porter et al AJP(03)nov [transverse, demonstration];
Ashmead qp/03 [no collapse];
Frasca qp/04 [analysis];
Dugić et al a0812 [interpretation].

@ __Full quantum description__: Reinisch PLA(99) [entanglement];
Potel et al PRA(05)qp/04;
de Oliveira & Caldeira qp/06 [coherence and entanglement];
Hsu et al PRA(11);
Benítez et al EJP(17);
Mochizuki a1704
[disappearance of interference terms in the quantum measurement process].

@ __History__: Friedrich & Herschbach PT(03)dec; Bernstein a1007 [and analysis];
Schmidt-Böcking et al EPJH(16)-a1609;
Margalit et al a1801 [realization];
Pakvasa a1805.

@ __Variations__: França FP(09) [in classical electrodynamics];
Tekin EJP(16)-a1506 [with higher spins];
Björnson & Black-Schaffer a1509 [solid state Stern-Gerlach spin-splitter].

> __Online resources__:
see Wikipedia page.

**Stiefel Manifold of k-Frames**
>see differentiable manifolds.

**Stiefel-Whitney Classes / Numbers**

**Stieltjes Constants**

* __Idea__: The expansion coefficients
in the Laurent series for the Hurwitz zeta function about *s* = 1.

@ __References__: Coffey JMAA(06)mp/05 [evaluation],
PRS(06) [summation relations],
a0706
[*η*_{j} coefficients, Hurwitz zeta function],
a0706 [series representations],
a1008 [double-series expression];
Adell PRS(11)
[asymptotic estimates, probabilistic approach];
Coffey a1106 [hypergeometric summation representations];
> s.a. MathWorld page.

**Stieltjes Integral** > see integration.

**Stieltjes Moment Problem** > see types of coherent states.

**Stieltjes Transform**

@ __References__: Schwarz JMP(05)mp/04 [generalized];
> s.a. MathWorld page.

**Stiffness** > see Elasticity.

**Stimulated Emission**
> see quantum field theory in curved
backgrounds [black holes]; Spontaneous Emission.

**Stirling Formula** > s.a. Factorial Function.

* __Idea__: An approximate
expression for *n*!, or for ln *n*!; For *n* →
∞, *n*! ~ (*n*/e)^{n}
(2π*n*)^{1/2}, or ln *n*! ~
(*n*+\(\frac12\)) ln *n* − *n* + \(\frac12\)ln(2π).

**Stirling Numbers**

@ __References__: Branson DM(06) [representation in terms of recurrence relations].

**Stochastic Calculus**

* __Idea__: A branch of
mathematics that is used to treat stochastic processes, and can be
described as calculus on non-differentiable functions; Its main variants
are Itō Calculus and Malliavin calculus.

@ __References__: Klebaner 12 [and applications];
Gauthier a1407 [algebraic, categorical version].

> __Online resources__:
see Wikipedia page.

**Stochastic Electrodynamics**
> s.a. modified electromagnetism; modified
versions of QED [without second quantization].

* __Status__: A theory
in which a classical Lorentz-invariant radiation field has observable
consequences similar to those of the zero-point fluctuations in QED.

* __Motivation__: Avoid
having to quantize the electromagnetic field; It has also been used to
propose a classical origin for gravity and inertia.

* __History__: 2005,
Developed over the past few decades, with a view to establishing it as the
foundation for quantum mechanics; The theory had several successes, but
failed when applied to the study of particles subject to non-linear
forces; An analysis of the failure showed that this was due to the methods
used to construct the theory, particularly the use of a Fokker-Planck
approximation and perturbation theory; A new, non-perturbative approach
has now been developed, called linear stochastic electrodynamics.

@ __General references__: Boyer PRD(75),
PRD(75);
de la Peña & Cetto 96;
Rosen PT(13)may [letter, summary].

@ __Hydrogen atom__: Claverie et al PLA(80);
Claverie & Soto JMP(82);
Cole & Zou PLA(03)qp [ground state];
Nieuwenhuizen & Liska PS(15)-a1502 [ground state].

@ __Related topics__: Boyer PRD(80) [and acceleration radiation];
de la Peña-Auerbach & Cetto pr(84);
Ibison & Haisch PRA(96);
de la Peña & Cetto qp/05 [and quantum mechanics],
FP(06);
Cetto et al QSMF(17)-a1707
[possible physical explanation for electron spin and antisymmetry of the wave function];
> s.a. hidden variables [tests];
quantum oscillators.

> __Online resources__:
see Wikipedia page.

**Stochastic Gravity**
> s.a. Induced Gravity.

* __Idea__: A classical
theory of gravity, in which the metric is subject to stochastic
fluctuations motivated by features of the quantum theory.

* __Hu & Verdaguer
approach__: Based on the Einstein-Langevin equation, which has in
addition sources due to the noise kernel, the expectation value of the
stress-energy bi-tensor which describes the quantum matter fluctuations.

@ __General references__: Ross & Moreau GRG(95);
Moffat PRD(97)gq/96;
Zakir in(03)ht/98;
Hu IJTP(99)gq;
Cole et al PRA(01) [as residual van der Waals force];
Hu & Verdaguer gq/01-ln,
CQG(03)gq/02,
LRR(04)gq/03
+ LRR(08)-a0802,
et al SPIE(03)gq;
Dzhunushaliev IJGMP(11)-a1008 [with probability density related to Perelman's entropy functional];
Satin GRG(18)-a1509

@ __Applications__: Verdaguer JPCS(07)gq/06;
> s.a. cosmological perturbations.

**Stochastic Layer / Region in Phase Space** > see phase space.

**Stokes' Law**

* __Idea__: The
friction force on a small sphere of radius *r* moving with
terminal speed *v* in a homogeneous fluid of viscosity coefficient
*η* is *F* = 6π*rηv*.

**Stokes Parameters** > see polarization.

**Stokes' Theorem** > see integration on manifolds.

**Stone Space** > see types of topologies.

**Stone's Theorem**

* __Idea__: It says or implies
that exp(i*tH*/\(\hbar\)) is unitary if *H* is self-adjoint,
even if densely defined unbounded, on an infinite-dimensional space.

**Stone-von Neumann Theorem**
> s.a. representations of quantum mechanics.

* __Idea__: Every
irreducible regular representation of the canonical commutation relations
in Weyl form for conventional quantum theory with configuration space
\(\mathbb R\)^{n} is unitarily
equivalent to the Schrödinger representation on L^{2}(\(\mathbb
R\)^{n}).

$ __Def__: All representations
of the finite-dimensional Heisenberg algebra are unitarily equivalent.

@ __References__: von Neumann MA(31);
Grosse & Pittner pr(87) [for supersymmetric quantum mechanics];
Cavallaro et al LMP(99) [non-regular representations];
Huang a1704 [infinitesimal version].

**Stone-Weierstrass Theorem** > see Weierstrass Theorem.

**Stoney Units** > see units.

**Strain Tensor**
> s.a. spacetime [spacetime as a strained material].

* __Poisson's ratio__:
The negative ratio of transverse to axial strain of a material;
Penta-graphene has a negative Poisson ratio;
> s.a. Wikipedia page.

@ __References__: de Prunelé AJP(07)oct [in spherical coordinates].

**Strange Attractor** > see Attractors.

**Strange Quark Matter / Nugget**
/ **Strangelet** > see astronomical
objects; experimental particle physics;
QCD phenomenology.

**Strange Star** > see star types.

**Stratified Manifold** > see types of manifolds.

**Stratum** (Plural: Strata)

* __Idea__: The set of all
orbits of the same topological type for the action of a group on a manifold.

@ __References__: Sartori & Valente JPA(03) [compact linear *G*
on \(\mathbb R\)^{n}].

**Stress / Stress Tensor** > s.a. Elasticity;
stress-energy pseudotensors.

@ __General references__:
Azadi a1706 [history, Cauchy's tetrahedron argument].

@ __In mechanics and relativistic field theory__: Gronwald & Hehl
gq/97-conf;
Medina AJP(06)nov
[contribution to energy and momentum].

**Stress-Energy Tensor** > see energy-momentum tensor.

**String Bit Model**

@ __References__: Thorn a1507
[at finite temperature, Hagedorn phase].

**String Field Theory** > s.a. renormalization;
string phenomenology [superstring field theory].**
***

@

@

@

>

>

**String Theory**
> s.a. phenomenology; or under cosmic strings.

**String-Net Condensation** > see gauge theories
and particle models [collective excitations as emergent particles].

@ __References__:
PhysForum(07)apr [Wen's spin lattice and spin foams].

**Strong Coupling Limit of Gravity**
> see modified versions of general relativity,
modified approaches to quantum gravity.

**Strong CP Problem** > see CP violation.

**Strong Interaction**
> s.a. particle physics; QCD;
history of particle physics.

* __Idea__: One of the
four "fundamental" interactions, and one of the two nuclear forces; It is
currently modeled by QCD, according to which it acts between quarks and is
mediated by gluons.

@ __ References__: Chew 62.

**Strong Rigidity Theorem** > see Rigidity.

**Strongly Asymptotically Predictable Spacetime**
> see types of spacetimes.

**Structural Realism, Structuralism**
> s.a. heat [structural realism and
theories of heat]; realism.

* __Idea__: The view
that scientific theories at best reveal only structural features of the
unobservable world.

* __Ontic structural realism__:
The view that structures are all there is, there are no objects; Relations do
have relata, but interpreted in structural terms.

* __Moderate ontic structural realism__:
Objects only have relational but no intrinsic properties; An even more moderate position
is the claim that at the most fundamental level of reality there are only relational properties.

@ __References__: van Fraassen BJPS(06);
French SHPSB(12)
[and unitarily inequivalent representations in quantum field theory];
Lam & Wüthrich a1306
[no support for radical ontic structural realism from category theory].

> __Online resources__:
see Stanford Encyclopedia of Philosophy page.

**Structure Equations**
> see affine connection.

**Structure Formation in Cosmology**
> see early-universe cosmology.

**Structure of Matter** > see matter.

**Structure of Physical Theories** > see physical theories.

**Structure Sheaf** > see sheaf.

**Struve Function**

* __Idea__: The function
*H _{n}*(

**Stückelberg Extension** > see particle physics.

**Stückelberg Mechanics**
> s.a. classical particles; quantum particles.

* __Idea__: A
manifestly covariant formalism for relativistic particle dynamics.

@ __References__: Aharonovich & Horwitz JMP(10) [radiation from a uniformly accelerating point source].

**Stückelberg Mechanism / Model / Trick **
> s.a. classical particles [and Lorentz force]; massive gravity;
particle physics [standard model extension].

* __Idea__: A mechanism,
proposed in 1938 by Stückelberg, for making an abelian gauge
theory massive while preserving gauge invariance, by introducing an
additional scalar field; 2004, Numerous generalizations have been proposed
for the non-abelian case, but the Higgs mechanism in spontaneous symmetry
breaking remains the only known way to give masses to non-abelian vector
fields in a renormalizable and unitary theory.

* __Action__: It describes an
electromagnetic field *A* coupled with a scalar field *φ*,

\(\cal L\) = −|*g*|^{1/2}
*g*^{ac}
*g*^{bd}
∇_{[a}
*A*_{b]}∇_{[c}*
A*_{d]}
+ \(\frac12\)*g*^{ab}
(∇_{a}*φ*
+ *m* *A*_{a})
(∇_{b}*φ*
+ *m* *A*_{b}) .

@ __General references__: Dragon et al NPPS(97)ht [variation – BRS-invariant polynomial form];
Ruegg & Ruíz-Altaba IJMPA(04);
Cianfrani & Lecian IJMPA(08)-a0803-proc [historical].

@ __Quantization__: Horwitz ht/98;
Oron & Horwitz FP(03)gq;
Marshall & McKeon IJMPA(08)ht/06 [renormalization and gauge invariance];
Escalante & Zárate a1406
[5D theory with a compact dimension, Dirac and Faddeev-Jackiw quantization].

@ __And gravity__: Hinterbichler & Saravani PRD(16)-a1508
[applied to curvature-squared theories].

> __Online resources__:
see Wikipedia page.

**Student's t-Distribution / Test** > s.a. statistics.

>

**Sturm-Liouville Theory**
> s.a. ordinary differential equations
/ matrices [determinants].

* __History__: Started in the 1830s
with Sturm and Liouville's generalization of the Fourier sine series to expansions
in terms of eigenfunctions of some ordinary differential equations; The hardest
questions were those of convergence, resolved after 1900.

@ __References__: Azad & Mustafa a0906 [and orthogonal functions].

**SU( n) Group**
> see examples of lie groups; SU(2) group.

**Subbase for a Topology τ on a Set X**
> s.a. topology / Base.

*

$

>

**Subdifferential** > a generalized Derivative.

**Subentropy**

* __Idea__: An
entropy-like quantity that arises in quantum information theory.

@ __References__: Nichols & Wootters qp/02 [intermediate quantities];
Datta et al JMP(14)-a1310 [properties and operational interpretation].

**Subfactor Theory** > see topological field theories.

**Subgroup** > see group theory.

**Sublimation** > see phase transition.

**Submanifold** > s.a. curves and lines;
embedding; extrinsic
curvature [including extremal surface]; Hypersurface;
manifolds; spacetime subsets.

**Submarine Paradox** > see special relativity.

**Submersion**

$ __Def__: A smooth
mapping *f* : *M* → *B* which is onto, with
*f*_{*} onto for all *p* in *M*.

**Subnormal Matrix / Operator**

$ __Def__: (Halmos) A
non-square matrix *A* is subnormal if it can be completed to
a (square) normal matrix.

* __Topology__: The set of
such *A*'s is not closed (can give example of *A*(*t*)
subnormal for all *t* > 0 but not for *t* = 0).

* __Problem__: Is there
an intrinsic characterization of such matrices?

**Subobject of an Object A**

$

**Sub-Riemannian Geometry / Manifold**

* __Idea__: A
sub-Riemannian manifold is a generalization of a Riemannian manifold,
in which to measure distances you are allowed to go only along curves
tangent to so-called horizontal subspaces.

* __Properties__:
Sub-Riemannian manifolds carry a natural intrinsic metric called the
Carnot-Carathéodory metric; Their Hausdorff dimension is always
an integer and larger than their topological dimension (except in the
case of a Riemannian manifold).

* __Applications__:
Found in the study of constrained systems such as the motion of vehicles
on a surface and the orbital dynamics of satellites in classical
mechanics, and geometric quantities such as the Berry phase; The
Heisenberg group, carries a natural sub-Riemannian structure.

@ __References__: Calin & Chang JDG(08);
Calin & Chang 09;
Agrachev et al
19.

> __Online resources__:
see Wikipedia page.

**Subspace of a Vector Space**

* __Idea__: A subset which
is closed under the vector space operations; It can be characterized by a multivector.

**Substance**
> see Ontology.

**Substantialism**
> see spacetime.

**Subsystems in Physics ** > s.a. quantum
field theory formalism; composite quantum systems.

@ __References__: Healey & Uffink SHPMP(13) [part and whole];
Donnelly & Freidel JHEP(16)-a1601 [in gauge theory and gravity, and entanglement entropy];
Chiribella a1804-Ent [agents, subsystems and information].

**Sudden Singularity** > see types of spacetime singularities.

**Sufficient Reason, Principle of**
> s.a. Retrocausation.

* __Idea__: It asserts
that anything that happens does so for a reason; No definite state of
affairs can come into being unless there is a sufficient reason why that
particular thing should happen; The principle is usually attributed to
Leibniz, although the first recorded Western philosopher to use it was
Anaximander of Miletus; It seems to be contradicted by contemporary
quantum theory.

@ __References__: Romero FS-a1410
[analysis, and relevance for the scientific endeavour].

**Suicide, Quantum** > see many-worlds
interpretation; types of measurements.

**Sullivan-Baas Singularities** > see riemannian geometry.

**Sum Rules** > s.a. cosmic rays;
lattice gauge theories; [standard model of particle physics].

* __Idea__:
Relationships between structure functions for different particles, or
expressions for them derived or guessed on the basis of their constitution
(hadrons in terms of quarks); Examples are the Bjorken sum rules (no
evidence of any violation, but if found it could be serious) and Ellis-Jaffe
sum rules (seem to be violated, but it's no big deal); To verify them, use deep
inelastic scattering.

@ __References__: Adler a0905-en [Adler sum rule];
Visser a1808 [Pauli sum rules and BSM physics].

**Summations** > s.a.
series.

**Sunyaev-Zeldovich Effect** > see cosmic
microwave background.

**Superalgebras** > see poincaré algebra.

**Superbradyons** > see causality violations.

**Superconductivity**
> s.a. types of superconductors.

**Superdeterminism**
> see bell inequalities.

**Superenergy Tensor**
> see stress-energy pseudotensors.

**Superfields**
> see BRST; supersymmetric field theory.

**Superintegrable Systems**
> see integrable systems.

**Super-Kamiokande (Super-Kamioka Neutrino Detection Experiment,
Super-K) Experiment** > s.a. neutrino experiments.

* __Idea__: An enormous
underground neutrino detector under Mount Kamioka in Japan containing
50,000 tons of ultrapure water and outfitted with thousands of photomultiplier
tubes, designed to search for proton decay, study solar and atmospheric
neutrinos, and watch for supernovae in the Milky Way Galaxy.

> __Online resources__:
see official website;
Wikipedia page.

**Superluminal Communication / Propagation / Travel **
> see causality; causality violations;
light; photons; tachyons;
wave phenomena.

* __Idea__: Phenomena
involving motion of speeds faster than the speed of light; In Minkowski
space it can be used to roduce causality violations.

@ __Books__: Fayngold 02;
Tiwari 03;
Nahin 11 [I, writer's guide].

@ __And special relativity__:
Recami et al IJMPA(00);
Geroch a1005/JLG [viability of special relativity];
Székely RPMP(13) [consistency of superluminal particles];
Peacock LS-a1301 [and the principle of relativity];
Grössing et al JPCS(16)-a1603
[Lorentz-invariant superluminal information transfer without signaling].

@ __General references__: Svozil PLA(95) [paradoxes];
Recami FP(01)phy [review];
Zhou PLA(00)
[*v*_{g} > *c*, numerical];
Liberati et al AP(02)gq/01 [Scharnhorst effect];
Van Flandern & Vigier FP(02) [support];
Lobo & Crawford LNP(03)gq/02 [definitions];
Krasnikov PRD(03)gq/02 [quantum inequalities and shortcuts];
Buenker SJCP(04)phy [??];
Nimtz FP(04);
Bonvin et al a0706
[superluminal motion and causality];
Lüst & Petropoulos CQG(12)-a1110 [in general relativity];
Andréka et al CQG(14)-a1407 [superluminal motion does not imply time travel];
Zhao a1405 [and wave-particle duality].

@ __Specific theories__: Hashimoto & Itzhaki PRD(01) [solitons in non-commutative gauge theory];
Borghardt et al PLA(03)qp
[*v*_{g} > *c* in Klein-Gordon theory];
González-Mestres ap/04-proc [non-tachyonic "superbradyons"];
Cocciaro a1209
[entanglement and superluminal signals whose propagation is regulated by a non draggable ether];
Weatherall SHPMP-a1409
[in some cases, electromagnetic fields propagate superluminally in the Geroch-Earman sense];
Ghirardi a1411 [comments on a proposal];
> s.a. causality in quantum (field) theory;
clifford spaces; non-commutative geometry;
Pauli-Fierz Theory; Yukawa Theory.

**Supermanifold** > see manifolds.

**Supermassive Objects** > see black holes [alternatives].

**Supermetric** > see geometrodynamics.

**Superoscillations**
> s.a. schrödinger equation; types of waves.

* __Idea__: The
phenomenon by which differentiable functions can locally oscillate on
length scales that are smaller than the smallest wavelength contained
in their Fourier spectrum.

@ __References__: Kempf & Prain a1510 [and driven quantum systems];
Aharonov et al MAMS-a1511 [mathematical aspects];
Chojnacki & Kempf JPA(16)-a1608 [new method for constructing superoscillations];
Kempf a1803-talk.

**Superparticle**
> see quantum particles.

**Superposition Principle**
> related to Linearity; s.a. mixed
quantum states [coherent superposition vs statistical mixture].

* __In classical field theory__:
It holds when the field equations are linear, so that a linear combination of
solutions is a solution; If the theory is the classical limit of a quantum field
theory, it corresponds to the case in which the particles do not interact.

* __In quantum mechanics__:
The space of pure states of quantum theory is a vector space; Linear (coherent)
superpositions of states are also allowed states; States can also be combined
as incoherent statistical mixtures; The only known exception is associated
with superselection rules.

@ __Classical__:
Notte-Cuello & Rodrigues RPMP(08)mp/06 [and energy-momentum conservation].

@ __Quantum__: Pulmannová IJTP(79) [and quantum logic];
Károlyházy in(90) [breakdown];
Greenberger et al PT(93)aug [and interferometry];
Cisneros et al EJP(98),
comment Anand EJP(16)-a1507
[on limitations from superselection rules];
Cirelli et al JGP(99) [extension];
Bassi & Ghirardi PLA(00)qp [against],
d'Espagnat PLA(01)qp [reply];
Peacock qp/02 [suggested explanation];
Lan IJTP(08)qp/03 [superposition ≠ mixture];
Corichi GRG(06)qp/04 [and geometrical formulation];
Lynn & Caponigro qp/06 [epistemological];
Bassi et al a1212-FQXi
[and quantum theory as an approximation to a stochastic non-linear theory];
da Costa & de Ronde FP(13) [interpretations of superpositions];
Hari Dass a1311-proc [Bohr's and Dirac's attitudes];
de Ronde a1404 [paraconsistent approach];
de Ronde a1603 [and the representation of physical reality];
Theurer et al PRL(17)-a1703 [resource theory];
Bera a1809 [between quantum evolutions].

@ __Quantum, systems / states / experiments__:
Dowling et al PRA(06) [atom and molecule];
Day PT(09)sep
[chiral molecule, and quantum-to-classical transition];
Sinha et al SRep(15)-a1412 [in interference experiments];
Filan & Hope a1509 [how one could tell];
Vavilov tr-a1708 [for light in vacuo];
Elitzur et al SRep(18)-a1707
+ news sa(18)may [photonic quantum routers];
Zych et al a1809 [relativity of quantum superpositions];
Lake et al a1812 [between spacetime geometries].

@ __Quantum, macroscopic systems__:
Morimae & Shimizu PRA(06) [macroscopically distinct states];
Weiss & Castin PRL(09);
Fröwis & Dür PRL(11)-a1012 [stable macroscopic superpositions];
De Martini & Sciarrino RMP(12) [multiparticle superpositions];
Johnsson et al a1412 [and gravimetry];
Mari et al SRep(16)-a1509 [experiments];
Park & Jeong a1606 [macroscopic superpositions are destroyed by thermalization processes];
> s.a. Schrödinger's Cat.

@ __Quantum, violations__: Bahrami et al PRA(14) [and possible experiments];
Stoica a1604 [and the emergence of classicality];
Rengaraj et al a1610 [in interference experiments];
> s.a. superselection rules.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Superpotential** > see conservation laws.

**Super-Quantum Theory**
> s.a. quantum gravity.

* __Idea__: A theory whose
non-local correlations are stronger than those of canonical quantum
theory.

@ __References__: Ghirardi & Romano PRA(12)-a1203 [model with super-quantum correlations].

**Superradiance / Superradiant Scattering**
> s.a. black-hole analogs; black-hole radiation;
matter near black holes [instabilities].

* __Idea__: A radiation
enhancement process that involves dissipative systems.

* __Black-hole superradiance__:
The amplification of a wave scattering off a rotating black hole, a wave
analog of the Penrose process for energy extraction, which can be
interpreted as stimulated emission.

* __Conditions__: It occurs
only for bosonic fields.

@ __General references__: Arderucio a1404;
Brito et al LNP(15)-a1501;
Rajabi & Houde ApJ(16)-a1601 [in astrophysics];
Endlich & Penco JHEP(17)-a1609 [modern discussion];
Pleinert et al Opt(17)-a1702 [hyperradiance, from coherently driven atoms].

@ __Black-hole superradiance__: Zeldovich JETP(72);
Starobinskii JETP(73);
Bekenstein PRD(73);
Wald PRD(76);
& Misner; Bekenstein & Schiffer PRD(98)gq;
Finster et al CMP(09) [rigorous treatment];
Richartz et al PRD(09) [conditions for occurrence, generalized];
Richartz & Saa PRD(13)-a1306 [off rotating stars without event horizons];
Boonserm et al PRD(14)-a1407 [and flux conservation];
East et al PRD(14)-a1312 [including back-reaction effects];
Rosa PLB(15)-a1501 [tests with pulsar companions].

@ __Specific types of black holes__: Winstanley PRD(01)gq [scalar field in Kerr-Newman-AdS black holes];
Ortíz PRD(12)-a1110 [none in the rotating BTZ black hole];
Kiorpelidi et al a1803 [in an expanding universe].

**Superrotations**
> s.a. Supertranslations.

* __Idea__: A kind of
symmetry at infinity for black-hole horizons in which light rays are
moved relative to one another and interchanged; 2016, They are a much
newer concept than supertranslations, and not as well understood
[@ see Strominger interview sa(16)jan].

@ __References__: Carlip a1608 [in 2+1 dimensions];
Pasterski a1905 [implications for quantum field theory].

**Superscattering Matrix**

**Superseparability**
> s.a. superselection.

* __Idea__: The fact
that in quantum theory states of a single particle belonging to
inequivalent representations are always mutually orthogonal, and
do not interfere with each other.

@ __References__: Sen in(10),
a1201.

**Supersolids**
> see solid matter.

**Superspace**
> for space of geometries, see geometrodynamics; for a space with bosonic
+ fermionic coordinates, see manifolds [supermanifolds].

**Superspinars**
> see astrophysics [compact objects].

**Superstatistics**
> see statistics.

**Supersymmetry**

* __Supersymmetry group__:
An extension of the Poincaré group of flat spacetime isometries to
symmetry transformations between integer and half-integer spin fields; Its
generators *Q* change the spin by 1/2, and the number *N*
that classifies supersymmetric theories is like a "degree of kinship"
between bosons and fermions.

* __Supersymmetry
algebra__: A graded Lie algebra, with generators
{*Q*_{i}^{A},
*Q**_{j'B},
*P*_{a}},
with *i*, *j* ' = 1, 2 (spinor indices), *a*, *b*
= 1, ..., 4 (spacetime indices), and *A*, *B* = 1, ..., *N*,
with commutation relations

{*Q*_{i}^{A},
*Q**_{j'B}}
= 2 *σ*_{ij'}^{a}*
P*_{a}
δ^{A}_{B}
, {*Q*_{i}^{A},
*Q*_{j}^{B}}
= {*Q**_{i'A}, *Q**_{j'B}}
= 0 , [*P*_{a},
*Q*_{i}^{A}]
= [*P*_{a}, *Q**_{i'A}]
= 0 , [*P*_{a},
*P*_{b}] = 0 .

@ __General references__: Łopuszański 90 [lecture notes];
Cornwell 92;
Jolie SA(02)jul;
Ichinose ht/06,
ht/06 [graphical representation];
McKeon CJP(12)-a1203 [fermionic first-class constraints as generators];
Ivanov a1403-ln [elementary intro].

@ __Generalizations__: Dzhunushaliev AACA(15)-a1302,
a1509 [non-associative].

> __Mathematical
aspects__: see Adinkras; lie
algebras [superalgebras]; manifolds [supermanifolds].

> __In physical systems__:
see modified quantum mechanics;
supersymmetry in field theory [including supersymmetry
breaking and modified theories]; supersymmetry
phenomenology; supersymmetric theories.

**Supertask**

@ __References__: Manchak FP(08) [in general relativity]

**Supertranslations**
> s.a. asymptotic flatness / s.a. Superrotations.

* __Idea__:
Symmetries of a black hole geometry in which the individual light rays are moved
up and down; They can be thought of as the result of adding soft gravitons.

@ __References__: Compère & Long CQG(16) & CQG+ [and the final state of gravitational collapse].

**Supervenience**
> see Emergence.

**Surface**
> s.a. Area; dynamical triangulations [random];
Singularities.

* __Flexible__: A surface in
a smooth manifold *M* is called flexible if, for any diffeomorphism
*φ* on the surface, there is a diffeomorphism on *M* whose
restriction on the surface is *φ* and which is isotopic to the identity.

@ __Differential geometry in general__: Toponogov & Rovenski
05 [3D];
Izumiya et al 15 [and singularities].

@ __In 3D euclidean space__: Guzzardi & Virga PRS(07) [constant mean curvature].

@ __In 4D manifolds__: Hirose & Yasuhara Top(08) [flexible surfaces].

@ __Deformations__: Capovilla & Guven CQG(95).

**Surface Gravity**
> s.a. laws of black-hole dynamics.

* __In Newtonian gravity__:
The quantity *g* = *GM*/*r*^{2},
for a spherical body of mass *M* and radius *r*.

$ __For a black hole__: If *l*
is the stationary Killing vector field of a black hole, normalized at infinity, then
*κ* is defined by *l*^{ b}
∇_{b} *l*^{
a} = *κ* *l*^{
a}; It is constant over the horizon surface.

* __Schwarzschild black hole__:
Given by *κ* = *GM*/(2*GM*/*c*^{2})^{2}
= *c*^{4}/4*GM* .

* __Kerr black hole__:
Given by *κ* = (*r*_{+}
+ *r*_{−})/4*α*, where
*α*:= *A*/4π, *r*_{±}:=
*M* ± (*M*^{2}
− *Q*^{2} −
*a*^{2})^{1/2}
and **a**:= **L**/*M*; It vanishes only
in the extreme case *M*^{2}
= *Q*^{2}
+ *a*^{2}
(which does not mean *A* = 0).

> __Other situations__:
see horizons [isolated horizons]; killing horizons.

**Surface Physics**
> see condensed matter.

**Surface Tension**
> s.a. condensed matter; Floating;
thermodynamics; water.

* __Idea__: The energy
required to increase the surface area of a liquid by one unit; Its
effect is to resist surface deformations.

* __Rem__: One difference
with respect to gravity is that surface tension scales like the surface
area as opposed to volume, so it becomes the dominant force either for
very small amounts of liquid (drops) or in microgravity situations,
such as in orbiting spacecraft.

@ __General references__: Marchand et al AJP(11)oct;
Høye & Brevik PRA(17)-a1705 [and Casimir forces].

@ __Examples__: Behroozi & Behroozi AJP(11)nov [soap bubbles];
news sci(14)mar [and insects walking on water];
Riba & Esteban EJP(14) [simple measurement];
Meseguer et al EJP(14) [and microgravity].

@ __In gravitational theory__: Callaway PRE(96) [using the black-hole analogy];
> s.a. metric matching.

> __Online resources__:
see HyperPhysics page;
Wikipedia page.

**Surfactant**
> see condensed matter [soft matter].

**Surgery**
> see algebraic topology; tensors
[tensor surgery]; topology change [cobordism].

**Surreal Numbers**
> see types of numbers.

**Surveyor's Formula**

* __Idea__: A formula for
calculating the area inside a polygon in plane Euclidean geometry as a
sum of contributions from its sides; One chooses an origin, and writes the
area of the triangle formed by each side and the origin as a determinant;
The sum of those triangle areas (taking into account their signs) is the
area of the polygon [> in Wikipedia page on polygons].

**Susceptibility**

* __Idea__: The
susceptibility of a material is a parameter (linear response function)
characterizing its response to a small variation in an applied field; For
example, the magnetic susceptibility *χ* = ∂*M*/∂*B*.

@ __Magnetic__: Bosse et al PhyA(10) [for quantum gases of particles with charge and spin];
> s.a. 2D ising model.

@ __Topological__: Del Debbio et al PRL(05)ht/04 [SU(3) gauge theory],
JHEP(04)ht [SU(*N*) for large *N*, finite *T*].

> __Online resources__:
see Wikipedia page.

**Suspension of a Topological Space**
> see topology.

**Suspension of Particles in a Fluid**
> see fluids [complex fluids]; metamaterials.

**Sutherland Model**
> see integrable systems.

**Swampland Conjecture / Program**

@ __References__: Danielsson JHEP(19)-a1809 [quantum version];
Palti a1903-FdP [intro and review].

**Swimming in Curved Spacetime** > see Extended Objects;
test-particle orbits; s.a. Quasiparticle.

**Swiss-Cheese Cosmological Models**
(Einstein-Straus, Lemaître-Tolman-Bondi, Szekeres)

* __Idea__: A set of
models in which the universe is assumed to be homogeneous on the largest
scales, but filled with holes or voids with less matter than other regions
at some scale larger than galactic scales; These voids affect both the
evolution of the universe as a whole, and our observations through their
effect on propagating light and matter.

* __Einstein-Straus model__:
A model consisting of a Schwarzschild spherical vacuole in a FLRW dust
spacetime; It is widely used as a toy model for addressing the issue of
the local effects of the global cosmological expansion.

@ __References__:
Bolejko & Célérier PRD(10)-a1005 [Szekeres model and supernova observations];
Flanagan et al PRD(12)-a1109 [and fluctuations in luminosity distances];
Fleury et al PRL(13)-a1304,
Fleury a1511-PhD [and cosmological observations];
Lavinto et al JCAP(13)-a1308 [cosmological "Tardis" spacetime].

@ __Einstein-Straus model__:
Einstein & Straus RMP(45);
Mars PRD(98)gq/02 [axisymmetric],
CQG(01);
Mena et al PRD(02) [anisotropic];
Grenon & Lake PRD(10)-a0910,
PRD(11) [generalized];
Mars et al GRG(13)-a1307 [exact and perturbative, realistic, non-spherical deformations].

> __Related models__:
see brane cosmology, general-relativistic models;
Lemaître-Tolman-Bondi; perturbations in general;
Szekeres Model.

> __Effects__: see cmb anisotropy;
theory of cosmological acceleration; cosmological expansion;
lensing; light [propagation in curved spacetime].

**Sylow Subgroup, Theorems** > see finite group.

**Sylvester Graph** > see group theory.

**Sylvester's Theorem** > see laplacian.

**Symbolic Dynamics**

* __Idea__: A coarse-grained
description of dynamics.

@ __References__: Hao & Zheng 98 [and chaos].

**Symbolic Logic** > see logic.

**Symmetric Criticality Principle**
> see lagrangian dynamics.

**Symmetric Group** > see finite groups.

**Symmetric Operator or Matrix**
> s.a. operator theory.

* __Remark__: An operator is usually an object of the type
*A*^{a}_{b},
so we need a metric in order to ask whether it is symmetric, or *A*_{ab} = *A*_{ba};
In expressions like \(\langle\) *f* | *Av*\(\rangle\) = \(\langle\)*Af*
| *v*\(\rangle\), we are implicitly using the metric given by the
Hilbert-space inner product.

**Symmetric Space**
> s.a. matrices [random].**
***

*

*

@

**Symmetrization Operator** > see tensors.

**Symmetron Field** > s.a. Screening.

* __Idea__: A scalar field
associated with the dark sector whose coupling to matter depends on the
ambient matter density; It is decoupled and screened in regions of high
density through a symmetry restoration, thereby satisfying local
constraints from tests of gravity, but couples with gravitational strength
in regions of low density, such as the cosmos, where the symmetry is
broken and the field mediates a "fifth force".

@ __References__: Hinterbichler et al PRD(11)-a1107 [cosmology];
Upadhye PRL(13)
+ news PhysOrg(13)jan [lab experiments].

**Symmetry Properties of a Tensor**
> see tensors.

**Symmetry Reduction**
> see Reduction of a Dynamical System.

**Symplectic Capacity**
> s.a. Fermi Functions.

* __Idea__: A topological notion
in symplectic geometry, closely related to Gromov's non-squeezing theorem.

@ __References__: de Gosson a1203/PRL [and quantum universal invariants].

**Symplectic Group**
> see examples of lie groups.

**Symplectic Integrators**
> s.a. Perturbation Methods.

* __Idea__: A method to
evolve dynamical systems according to modified Hamiltonians whose error
terms are also well-defined Hamiltonians

@ __General references__: Fleck et al ApplP(76);
Suzuki PLA(90),
PLA(92);
& B K Berger et al;
Donnelly & Rogers AJP(05)oct [intro];
Brown PRD(06) [and midpoint rule for Hamiltonian systems];
Chin PLA(06) [theorem];
Kobayashi PLA(07);
Blanes et al ANM(13)-a1208
[new symplectic splitting methods for near-integrable Hamiltonian systems];
Jiménez-Pérez et al a1508 [on numerical errors].

@ __Examples, applications__: Chin PRE(07)mp/06 [and perihelion advance in the Kepler problem];
Frauendiener JPA(08)-a0805,
Richter & Lubich CQG(08)-a0807 [in numerical relativity];
McLachlan et al PRE(14)-a1402 [spin systems];
> s.a. computational
physics; newtonian gravity.

**Symplectic Structure**
> s.a. symplectic geometry; in physics;
variations.

**Synchronization** > s.a. chaos;
clocks; special-relativistic kinematics.

@ __Non-chaotic dynamical systems__: Bagnoli & Cecconi PLA(01).

**Synchrotron Radiation**
> see acceleration radiation.

**Synge's Theorem**
> see orientation.

**System Theory**
> s.a. classical and quantum
systems; state of a system.

**Syzygies**

* __In astronomy__:
Nearly straight-line configurations of three celestial bodies (as the Sun,
Moon, and Earth during a solar or lunar eclipse) in a gravitational system.

* __In mathematics__: A
relation between the generators of a module.

@ __References__: Evans & Griffith 85;
Eisenbud 05;
Johnson 12 [and homotopy theory].

> __Online resources__:
see Wikipedia pages on syzygies in astronomy
and mathematics;
MathWorld page.

**Szekeres Model / Spacetime**
> s.a. cosmological acceleration;
types of singularities.

* __Idea__: The
quasispherical Szekeres model is an exact solution of the Einstein equation
which represents a time-dependent mass dipole superposed on a monopole,
and is suitable for modelling double structures such as voids and adjourning
galaxy superclusters.

@ __References__: Bolejko ap/06-proc [and cosmology];
Krasiński PRD(08)-a0805 [properties of the quasi-plane model];
Apostolopoulos CQG(17)-a1611 [covariant approach];
Paliathanasis et al CQG(18)-a1801 [quantization].

**Szilard's Demon / Engine**
> s.a. laws of thermodynamics; Maxwell's Demon.

* __Idea__: The Szilard
engine is a stylized version of Maxwell's demon, where a yes/no
measurement of a classical single-particle system allows one to extract
a tiny amount of energy *kT* ln2 from a thermal reservoir at
temperature *T*; It has furnished insight into the foundations
of statistical mechanics, become the canonical model for investigations
of feedback-controlled systems, and spurred the creation of the field
of thermodynamics of computation.

@ __References__: Berger IJTP(90);
in Leff & Rex 03;
Kim et al PRL(11)
+ Parrondo & Horowitz Phy(11)
[and quantum statistics of indistinguishable particles];
Beyer et al a1906 [truly quantum].

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 1 aug 2019