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, t0) for t → ∞, t0 → −∞, 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 Sfi = δfi − 2πi δ(ωfωi) Tfi.
@ 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, Δt = 4 A · ω / c2, 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.
@ 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: There 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].
@ 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].
@ 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]; > 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; 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 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 .

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 Equation

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 {pi}, i = 1,..., n, i.e., pi in \(\mathbb N\), with 1 ≤ p1 ≤ ... ≤ pnm.

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 > 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].

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.

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'\(\rangle\).

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) [Leff 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 S3/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

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 a1708 [in astrophysics].

* 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 x0 if for every ε > 0 there exists a neighborhood U of x0 such that f(x) ≤ f(x0) + ε (resp., f(x) ≥ f(x0) − 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 Gs V is the set G × V with the composition law (g1, v1) (g2, v2):= (g1g2, v1+g1v2).
* Remark: We can thus get a new group from every representation of G, with Vs 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].

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 kab:= Camn Cbnm 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 (T0, T1, T2, T3, T4 Spaces) > see types of topological spaces.

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

* 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.

Set Theory

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.
@ 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].
@ 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].
@ 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]; > s.a. time in gravity.
> Related theories: see 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.

Sheaf, Sheaf Cohomology

Shear of a Congruence of World-Lines
$ Def: If ua is the unit timelike tangent vector to the congruence, one defines the traceless shear tensor and the shear scalar as

σab:= θab − \(1\over3\)θ qab ,      σ:= (\(1\over2\)σab σab)1/2 .

where θab is the expansion tensor and θ the expansion scalar of the congruence, and qab the spatial metric qab = gab + ua ub.

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.

* 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)
$ Def: A collection \(\cal A\) of subsets of a set X with three properties: (a) The empty set is in the collection; (b) The complement X\A of any set A in \(\cal A\) is also in \(\cal A\); (c) The union of countably many sets in \(\cal A\) is also in \(\cal A\).
* Relationships: A σ-algebra is a σ-ring with the added requirement of property (a).
* Generating a sigma algebra: Given any collection \(\cal C\) of subsets of X, there exists a unique σ-algebra generated by it, defined as the intersection of all σ-algebras that contain \(\cal C\) (this set is not empty, since the power set of X is in it, for example); It is easy to verify that this object is in fact a σ-algebra, and it is also clearly minimal.

Sigma-Complex (σ-Complex) > s.a. reconstruction of quantum theory.
* Idea: A union of σ-algebras.

Sigma-Field (σ-Field) > see ring.

Sigma Models

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
* Value: The number √2 + 1 = 2 + 1/(2 + 1/(2 + ...)).

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.

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 = *(ee) + (1/γ) (ee) ;

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].

* 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.

Sinh-Gordon Equation
@ References: Xie & Tang NCB(06) [solution method].

6j-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].
> 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 [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].

$ 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,

M = \((\kappa/4\pi)\) A + Ω · J + Φ Q .

* 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]; Lemos & Zaslavskii 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 a1405 [in 11-dimensional supergravity].

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 Wpm(U) is the space of all functions which belong, together with their derivatives up to the m-th order, to Lp(U):

Wpm(U):= {f | Dj f ∈ Lp(U) for all j such that | j | ≤ m} .

* Special case: For p = 2, we call Hm(U):= W2m(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 > see Gravitational Memory.

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

VaVAA',   with    Va = σaAA' VAA',    or    VAA' = σaAA' Va ;

With the right choice of basis, these σs 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

VaVAB,    with    Va = σaAB VAB,    or    VAB = σaAB Va ;

With the right choice of basis, these σs 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 = Q0Q1Q2 ⊃ ..., where Qi is the commutant of Qi−1, has Qm = {e} for some m (the height of G).
* Properties: A solvable group always has a commutative invariant subgroup (the Qm−1 above).
* Examples:
- The 2D Euclidean group, of height 2, E2 = T1,1 ×s SO(2) ⊃ T2 ⊃ {e}.
- The 2D Poincaré group: P2 = T1,1 ×s SO(1,1) ⊃ T1,1 ⊃ {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.
@ 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].

Sound > s.a. music.

Sp(2n) 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 FE is open if idF 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].

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

σI := \(-\frac12\)εIJKL ΓJKeL ,

where eL is a tetrad field, and ΓJKa = eJba ebK its Levi-Civita connection.
* Complex 2-forms: The two sets of forms

σ(±)I := −εIJKL Γ(±) JKeL ,

where Γ(±) JK:= \(\frac12\)(ΓJK \(\mp\frac12\)i εJKLM ΓLM).
* 3-form: A tetrad-dependent 3-form σI or σ(±)I on the bundle of orthonormal frames over spacetime, which is a potential for a local energy-momentum density τI for the gravitational field; If e*J is a basis of 3-forms, and GIJ the Einstein tensor,

dσI = dσ(±)I = τI + GIJ 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) ≈ Σn=1N anUn(x), where the coefficients are calculated by an = X dx Un*(x) f(x) and the finite sum minimizes X dx |f(x) − n anUn|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].
@ 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 S1]; 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 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.

Specific Heat

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

Spectral Decomposition > see hilbert space.

Spectral Dimension
@ 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 a1602 [integration of spectral triples].
> 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 aλI 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 Harmonics

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

Spheroidal Harmonics > see spherical harmonics.

Spi > see asymptotic flatness.

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-Coefficient Formalism

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

Spin-Foam Models

Spin Glasses and Models > 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 Ba3CuSb2O9, 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 Structure

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.

* 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 (Δx Δp = \(\hbar\)/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 S1: 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].
@ 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 a1705 [on a right quaternionic Hilbert space].
> 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.
> Other theories: see condensed matter.

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

pn+1 = pn + K sin(θn) ,   θn+1= θn + pn+1.

@ 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.

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 θmn used to define non-commutative geometrical structures, such that for two fand g,

(f *g)(x):= exp(\(1\over2\)i θmn {∂/∂ym} {∂/∂zn}) f(y) g(z)|y=z=x = f(x) g(x) + \(1\over2\)i θmnm f(x) ∂n g(x) + h.o.t.

* Remark: This structure is not Lorentz-invariant.
@ 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].
@ 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 −∞ dx f(x) exp{iφ(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/m3·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 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].

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 experiments 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].
@ 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 [ηj 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.

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 (2πn)1/2, or ln n! ~ (n+\(\frac12\)) ln nn + \(\frac12\)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 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 a1509 [classical Einstein-Langevin equation].
@ Applications: Verdaguer JPCS(07)gq/06; > s.a. cosmological perturbations.

Stochastic Processes

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

Stochastic Quantization

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 = 6πrηv.

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(itH/\(\hbar\)) 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 L2(\(\mathbb R\)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.
* Idea: A perturbative theory of (open, closed, or open-closed) strings based on the classical string action with the addition of interaction terms, which describes quantum string dynamics including scattering, joining and splitting of strings; A classical configuration or string field is given by an element of the free string Fock space.
@ Reviews: Kaku IJMPA(87); Berkovits ht/01-ln [open superstrings]; Siegel 88-ht/01; Thorn PRP(89); Rastelli ht/05-en; Taylor ht/06-ch; Aref'eva TMP(10).
@ General references: Green & Schwarz PLB(84); Hata et al PRD(86) [covariant]; Witten NPB(86) [and non-commutative geometry], NPB(96) [open], pr(87); Bowick & Rajeev PRL(87), NPB(87); Strominger PRL(87); Horowitz & Witt PLB(87); Bordes & Lizzi IJMPA(90); Hashimoto & Itzhaki JHEP(02) [observables]; Kling et al PLB(03)ht/02 [non-perturbative solutions]; Bars ht/02 [Moyal star formulation]; Drukker JHEP(03)ht [actions]; Okawa & Zwiebach JHEP(04) [heterotic]; Taylor ht/04-ln [perturbative computations]; Crane RVMP(13)-a1201 [propagating on a discretized quantum spacetime]; Bars & Rychkov a1407 [background-independent].
@ Action: Horowitz et al PRL(86) [cubic]; Sen JHEP(16)-a1606 [reality conditions].
> Related topics: see causal dynamical triangulations.
> Online resources: see Wikipedia page.

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 > see modified versions of general relativity.

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 Hn(z) which satisfies the inhomogeneous Bessel equation z2 Hn''(z) + z Hn'(z) + (zn) Hn = (2/π) z/(2n−1)!!

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 gac gbd[a Ab][c Ad] + \(\frac12\)gab (∇aφ + m Aa) (∇bφ + m Ab) .

@ 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.
> Online resources: see Wikipedia page.

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.
* Idea: A collection of subsets of X which generates τ, in the sense that τ is the smallest topology containing it.
Def: A collection of subsets of X such that all open sets in τ are ∅, X, plus arbitrary unions of finite intersections of those subsets.
> Online resources: see Wikipedia page.

Subdifferential > a generalized Derivative.

* 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.

$ Def: A smooth mapping f : MB 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
$ Def: An object A' in the same category, with a monomorphism f : A' → 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.
> 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].

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, could be serious) and Ellis-Jaffe sum rules (seem to be violated; no big deal); To verify them, use deep inelastic scattering.
@ References: Adler a0905-en [Adler sum rule].

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) [vg > 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 [vg > 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].

Superparticle > see quantum particles.

Superposition Principle > related to Linearity; s.a. mixed quantum states [coherent superposition vs statistical mixture].
* In classical field theory: 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].
@ 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].
@ 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 a1601/ApJ [in astrophysics]; Endlich & Penco a1609 [modern discussion]; Pleinert et al 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].

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].

Superscattering Matrix

Superselection Rules

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 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 {QiA, Q*j'B, Pa}, with i, j ' = 1, 2 (spinor indices), a, b = 1, ..., 4 (spacetime indices), and A, B = 1, ..., N, with commutation relations

{QiA, Q*j'B} = 2 σij'a Pa δAB ,   {QiA, QjB} = {Q*i'A, Q*j'B} = 0 ,   [Pa, QiA] = [Pa, Q*i'A] = 0 ,   [Pa, Pb] = 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.

@ 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/r2, 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 bb l a = κ l a; It is constant over the horizon surface.
* Schwarzschild black hole: Given by κ = GM/(2GM/c2)2 = c4/4GM .
* Kerr black hole: Given by κ = (r+ + r)/4α, where α:= A/4π, r±:= M ± (M2Q2a2)1/2 and a:= L/M; It vanishes only in the extreme case M2 = Q2 + a2 (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].

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].

* 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.

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 Aab, so we need a metric in order to ask whether it is symmetric, or Aab = Aba; 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].
* Idea: A Riemannian manifold whose curvature is invariant under all parallel translations.
* History: The theory was developed by Cartan in the 1920s.
* As coset space: A symmetric space is diffeomorphic to G/H, where H is associated with a given involutive inner automorphism of G, as the subgroup generated by all the Lie-algebra elements which are eigenvectors with eigenvalue +1 of the differential of the inner automorphism, considered as an operator on the Lie algebra.
@ References: Helgason 78; Anker & Orsted ed-05 [reductive, Plancherel theorems]; Borel & Li JDG(07) [compactifications].

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 Breaking

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.

* 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 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].

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