Topics, O

O'Raifeartaigh Model > see supersymmetry in field theory [supersymmetry breaking].

Objectivity > see Reality / decoherence; foundations of quantum mechanics; symmetries.

Observables > s.a. Covariance; observables in classical gravity; observables in quantum theories; observable algebras.

Observation of Physical Phenomena
@ General references: Bondoni a1005-wd [measurements as meta-sentences on real events]; Jaroszkiewicz PRS(10) [dynamical theory of observation]; Sassoli de Bianchi FoS(13)-a1109; Nyman GT-a1309 [psychologically based general theory of observation].
@ Observability in particle physics: Fox PhSc(09) [definitions of observation, and quarks]; Cabbolet a1511-conf [claim about short-lived particles].

Observers > see observers and observables.

Occam's Razor (Ockham's Razor) > s.a. physical theories.
* Idea: The principle according to which scientific explanations should be neat, with as few arbitrary assumptions as possible; Avoid doing with more what can be done with less; A.k.a. principle of economy; > s.a. Explanations [truth and simplicity].
@ References: Sorkin IJTP(83)ap/05; Garrett PW(91)may; Jefferys & Berger AS(92); Soklakov mp/00; Standish FPL(04) [justification]; Page a0811-conf; Gu et al nComm(12)-a1102 [sharpening, using quantum mechanics]; Gürel & Gürel a1107 [and special relativity]; Sober 15; Krizek a1701 [history, and the interpretations of quantum mechanics].

Ocneanu Cells > see SU(2) [6j symbols].

Octonions > s.a. dirac field theory; lorentz group; Non-Associative Geometry; spin.
* Idea: The largest of the four normed division algebras; An example of elementary algebra with non-associative composition; It is related to many interesting fields of mathematics.
@ General references: Porteous 69; Daboul & Delbourgo JMP(99)ht [matrix representation]; Baez BAMS(02)m.RA/01 [review]; Conway & Smith 03; Hackett & Kauffman a1010 [topological model]; Baez & Huerta SA(11)may; Dray & Manogue 15.
@ And spinors: Carrion et al JHEP(03)ht [classification]; Boya mp/04-proc.
@ Octonionic quantum mechanics: De Leo & Abdel-Khalek PTP(96)ht, IJTP(98)ht/99; Dzhunushaliev FPL(06)ht/05, ht/06, JGLTA(08)qp/07, AHEP(07)-a0706 [non-associative]; De Leo & Ducati JPA(12)-a1208 [eigenvalue problem for octonionic operators].
@ Field theory: Fredsted a0707 [generalization of general relativity]; Pushpa et al IJTP(11)-a1006 [Yang-Mills theory, QCD]; Restuccia & Veiro JPCS(16)-a1412 [Yang-Mills connections]; > s.a. gravitational instantons; formulations of electromagnetism; types of yang-mills theories.
@ Other physics: Schray CQG(96)ht/94 [superparticle]; Okubo 95; Manogue & Dray MPLA(99); Boya ht/03-conf [M-theory]; Toppan ht/03-proc [and exceptional structures]; Gogberashvili AACA(05)ht/04; Manogue & Dray JPCS(10)-a0911 [particle physics]; Dorofeev a1003; Gogberashvili & Sakhelashvili AMP(15)-a1506, Gogberashvili IJGMP(16)-a1602 [spacetime geometry]; Todorov & Drenska AACA(18)-a1805 [particle physics]; Freedman et al a1811 [and quantum computation].
> Online resources: see Tevian Dray's page; Wikipedia page.

Off-Shell Dark Matter > see dark-matter models.

OGLE (Optical Gravitational Lensing Experiment) > see gravitational lensing.

Ohm's Law > see electricity.

Olbers' Paradox
* Idea: The sky looks dark at night, contrary to what an infinite uniform universe would imply.
* Solution: Existence of a particle horizon (finite age of the universe) and, to a lesser degree, expansion.
@ References: Wesson ApJ(91); Maddox Nat(91)jan; Pešić AJP(98)nov [average brightness]; Arpino & Scardigli EJP(03)ap/00 [Newtonian cosmology]; Couture EJP(12).

Old Quantum Theory > s.a. history of quantum physics.
* Idea: A set of heuristic prescriptions developed in the early 1900s that introduced the first quantum corrections to classical mechanics; Its main tool was Bohr-Sommerfeld quantization, a procedure for selecting the allowed orbits in the motion of a particle subject to an attractive potential through a quantization condition on the orbit's action.
@ References: Garon et al AJP(13)feb [comparison with known analytic results for simple quantum mechanical systems].
> Online resources: see Wikipedia page.

Ollivier Curvature > see curvature of graphs; graph theory in physics.

Omega Number
@ References: Chaitin 05.

One-Loop Approximation > see schrödinger equation [WKB approximation].

One-Parameter Subgroup > see lie group.

One-Shot Statistical Mechanics > see statistical mechanics.

Onsager Solution > see ising model.

Onsager-Machlup Theory > see non-equilibrium statistical mechanics.

Ontology > s.a. Epistemology; foundations of quantum mechanics; Structural Realism.
* Idea: The study of the question, "What exists?", as opposed to "What do we know?" (epistemology); In quantum theory in practice it means the question of the reality of the wave function, in gravity it often refers to the debate between spacetime substantialism and relationalism.
@ General references: Roemer M&M(06)qp [substance vs process ontology]; Maudlin 07; Lokajíček CoP(07)-a0710 [phenomenological and ontological models]; Merrill Topoi(11)-a1903 [relation between philosophical ontology and scientific ontology].
@ Matter, particles: Esfeld et al a1510 [atomism and structural realism]; > s.a. particles.
@ Spacetime: Mohrhoff qp/03-ch; Lorente a0712-conf [and causal spin foams]; Sutherland a1904 [realist]; > s.a. Event; general relativity; spacetime.
@ In quantum theory: Margenau PhSc(49)oct [quantum objects as 'onta']; Albert et al PRL(85) [properties and predictions]; Peruzzi FP(90) [objects]; Pusch SHPMP(02) [classical vs quantum]; Hardy SHPMP(04) [ontological embedding]; Singh qp/04 [general]; De Baere qp/05 [Renninger's thought experiment]; Caponigro & Lynn qp/06; Verelst & Coecke phy/06 ['system', 'entity', and being vs non-being]; Monton PhSc(07)dec; de la Torre FP(07) [position and momentum have no values]; Tung SHPMP(10)-a1005 [Hardy's quantum ontological excess baggage]; Kuhlmann 10; Mansfield a1306 [ontic and epistemic interpretations]; Griffiths SHPMP(13) [consistent, from histories interpretation]; Esfeld a1406 [assessment of some proposals]; Hari Dass Quanta(14)-a1406; Esfeld Synth(14)-a1408 [and non-locality]; Herbut a1409, a1409 ["ontic breakthrough"]; Auffèves & Grangier FP(16) [physical properties attributed jointly to system and context]; Gambini & Pullin IJQF-a1605 [even ontology and downward causation]; Tung a1704 [generalized]; Klevgard a1802; Allen a1901-PhD; Batard a1908; Lombardi et al ed-19; Oldofredi & Lopez FP-a2006 [on Harrigan and Spekkens' classification]; Gao FP-a2011 [particle ontology]; > s.a. correlations; pilot-wave interpretation; ψ-Ontic Theories; quantum field theory.
@ In other physics: Caulton & Butterfield BJPS-a1002 [diffeomorphism invariance, permutation symmetry and structuralist ontology]; Page a1412-proc [cosmology]; > s.a. Maxwell-Lorentz Equations; quantum field theory; radiation.

Oort Cloud > see solar system.

Open Mapping
$ Def: A mapping f : XY between two topological spaces is open if the image of any open set in X is open in Y.

Open Set > see topological space.

Open System > s.a. system theory / composite systems; dissipative systems; open quantum systems.
* Idea: Originally, an open system was one that exchanges energy and possibly matter with a bath; Now, the term has come to denote one that exchanges information with an environment.
@ References: Li PRP(86) [general physics of open systems]; Li et al PhyA(11) [Tsallis statistical distribution].
> States: see states in statistical mechanics and non-equilibrium statistical mechanics.

Operad > s.a. Module.
* Idea: A set of operations, each one with a fixed finite number of inputs (arguments) and one output, which can be composed one with others; A category-theoretic analog of universal algebra; Algebras are to operads as group representations are to groups.
@ General references: Vallette a1202 [introduction, and application to homotopical algebra].
@ And physics: Paal CzJP(01)mp; Paal gq/02 [curvature]; Zois RPMP(05)ht/03 [and quantum gravity]; Paal RPMP-a1402 [and Yang-Mills gauge theories]; Benini et al HBM-a1709 [for algebraic quantum field theory].
> Online resources: see Wikipedia page.

Operation on a Set > see sets [associative].

Operational Theories > s.a. probabilities in physics; quantum mechanics and axioms for quantum theory.
@ References: Hardy a2104 [time-symmetric framework].

* Idea: Physical quantities are completely defined by the series of operations with which one measures them; In other words, an operationalist theory is defined by entities external to the system, or requires them in addition to entities and concepts pertaining to the system (as opposed to realist theories).
* Example: No absolute time in physics.
@ References: Delaney IJTP(99) [limitations].

Operator Product Expansion > see quantum field theory techniques.

Operator Theory

Oppenheimer-Snyder Solutions > see collapse; lattice gravity.

Oppenheimer-Volkoff Equations / Solutions > see under Tolman-Oppenheimer-Volkoff Equations / Solutions.

Optical Activity > see optics.

Optical Geometry > see electromagnetism in curved spacetime; optics [including specific spacetimes]; self-force.

Optical Lattice > see optics.

Optical Theorem > see scattering.

Optics > s.a. optical technology.

Orbifold, Orbispace \ s.a. types of manifolds.
* Idea: The quotient space M/G of a manifold by a group action; If G has fixed points it is not a manifold.
@ References: Pflaum mp/02 [deformation quantization]; Martin ht/04 [fuzzy]; Borzellino & Brunsden T&IA(12) [differential topology].

Orbit in Mathematics > see group action; lie group examples; vector field.

Orbit in Mechanics > see classical-mechanical systems / orbits of test bodies, of newtonian and relativistic gravitating bodies; Trajectories.

* Idea: One of the three electron-like quasiparticles predicted to exist in a 1D solid; A collective excitation of electrons in a 1D solid with orbital angular momentum but with no spin or electric charge.
@ References: news pw(12)apr [experimental discovery].

Order > s.a. Disorder; posets [partial order relations].
* Remark: In quantum statistics, it is described in terms of symmetry breakdown, which is not antithetical to the notion of disorder.
* In matter: A gas has no short-range or long-range order; A liquid has short-range but no long-range order; A crystalline solid has both short-range and long-range order.
@ General references: Suzuki PLA(80) [microscopic theory]; Strogatz 03 [spontaneous, r PT(04)jun]; Gorbban PhyA(07) [ordered and disordered states in phase space]; Sewell a0711-en [in quantum statistical mechanics, survey].
@ Topological: Nussinov & Ortiz AP(09) [quantum]; Juzeliūnas & Spielman Phy(11) [order-disorder]; Jamadagni & Weimer a2005 [definition]; Wang a2104-GRF [black holes, spin network states in lqg].
@ Other types: Baake et al SdW(02)m.HO [aperiodic, introduction]; Wen AP(05) [quantum, and forms of matter]; Gallas & Herrmann PhyA(05) [emergence, in class-4 cellular automaton].
> Phenomenology: see Defects; diffraction; phase transitions [topological order]; thermodynamics [second law].
> Systems: see Extended Objects; lattice gauge theories [QCD]; random systems; spin models.

Order Parameter > s.a. Potts Model.
* Idea: A thermodynamic function of state associated with a phase transition, whose value in a state indicates what phase it corresponds to; For example, for a ferromagnetic system the order parameter is the net magnetization, and for solid-liquid or liquid-gas transitions the density.

Organized States > see non-equilibrium statistical mechanics; Order.

Orientability, Orientation

Orientifold > see types of manifolds.

Orlicz Inequality
$ Def: For any open set Ω ⊂ \(\mathbb R\)n and N-function A, with complementary N-function \(\tilde A\),

| Ω u(x) v(x) dx | ≤ 2 ||u||LA(Ω) ||v||LA~(Ω) .

* Relationships: This is a generalization of the Hölder inequality.

Orlicz Space
$ Def: A Banach space LA(Ω):= {set of all functions u such that Ω A(u(x)) dx < ∞, or u is a (real) multiple of such a function}, where Ω is an open set in \(\mathbb R\) and A an N-function, and the norm ||u||:= infk > 0 Ω A(u(x)/k) dx ≤ 1.
@ General references: in Adams 75.
@ And physics: Streater mp/04-conf [quantum, in information geometry]; Labuschagne & Majewski a0902 [non-commutative]; Majewski & Labuschagne a1502, a1605 [and statistical physics]; > s.a. statistical mechanics.

Ornstein-Uhlenbeck Process > see diffusion.

Orthode > see states in statistical mechanics.

Orthogonal Groups > see examples of lie groups.

Orthogonal Polynomials > see Special Functions.

Orthogonalization Methods
@ References: Lee PRL(82) [simpler than Gram-Schmidt]; Chaturvedi et al JPA(98)qp [overview and proposal]; Rebollo-Neira mp/02, mp/02 [biorthogonalization].

> In particle theory: see axions [photon-axion oscillation]; gravity and matter fields [flavor oscillations]; hadrons [meson oscillations]; neutrino oscillations; neutron; standard model.
> In astronomy and astrophysics: see gravitating bodies; milky-way galaxy [stellar motion]; solar system [solar oscillations]; stellar properties.
> In cosmology: see cosmological acceleration, dark energy and large-scale geometry [baryon oscillations]; semiclassical cosmology.
> Other systems: see Bloch Theory [Bloch oscillations]; oscillators; Superoscillations; test-body orbits.

Oscillators > s.a. quantum oscillators.

Osserman Conditions / Manifold > s.a. lorentzian geometry; riemannian geometry.
* Idea: Study the question of how much information the sectional curvatures in a semi-Riemannian manifold give us on the Riemann tensor.
@ References: Bonome et al CQG(01) [generalized, 4D]; García-Río et al 02; Nikolayevsky DG&A(03); Alekseevsky et al JGP(05) [symmetric spaces].

Osterwalder-Schrader Construction > see algebraic quantum field theory.

Ostrogradski Instability > see higher-order lagrangian systems.

Ostrogradski Theorem > see higher-order lagrangian systems; types of higher-order gravity theories.

Out-of-Time Order Operators > see quantum chaos.

Outer Measure of a Set in a Metric Space > see distance.

Overhang Problem > see Center of Mass.

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