Topics, O

Objectivity > see decoherence; foundations of quantum mechanics.

Observables > s.a. observable algebras.

Observers > s.a. observables [non-inertial]; phenomenology of gravity; Reference Frame [including accelerated].
* Types: Lagrangian (non surface-forming observers) or Eulerian (surface-forming ones); inertial or non-inertial.
@ In curved spacetimes: Page CQG(98)gq/97 [stationary axisymmetric, maximal acceleration].

Occam's Razor > see physical theories.

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

Octonions > s.a. dirac field theory; lorentz group; spin.
* Idea: The largest of the four normed division algebras; An example of elementary algebra with non-associative composition; 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.
@ And spinors: Carrion et al JHEP(03)ht [classification]; Boya mp/04-in.
@ Other physics: Schray CQG(96)ht/94 [superparticle]; Okubo 95; Manogue & Dray MPLA(99); Boya ht/03-in [M-theory]; Toppan ht/03-in [and exceptional structures]; Gogberashvili AACA(05)ht/04; Fredsted a0707 [generalization of general relativity]; > s.a. gravitational instantons; modified electromagnetism, modified quantum mechanics.
> Online resources: Tevian Dray's page.

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; Pesic AJP(98) [average brightness]; Arpino & Scardigli EJP(03)ap/00 [Newtonian cosmology].

Omega Number
@ References: Chaitin 05.

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

One-Parameter Subgroup > see lie group.

Onsager Solution > see ising model.

Ontology > s.a. foundations of quantum mechanics; quantum field theory, radiation.
* Idea: The study of the question, "What exists?".
@ References: Roemer qp/06 [substance vs process ontology]; Maudlin 07; Lokajicek CoP(07)-a0710 [phenomenological and ontological models].

Oort Cloud > see solar system.

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

Open System > s.a. [system theory]; quantum systems and composite systems; modified quantum mechanics [non-Hamiltonian].
* Idea: Originally, one that exchanges energy with a bath; Now, one that exchanges information with an environment, which can lead to the decoherence of a quantum state.
* Quantum: Usually described by mixed states of the type (t) = tr_bath |_{whole whole}|, and linear quantum state diffusion (LQSD) stochastic Schrödinger equations.
@ General references: Li PRP(86).
@ Quantum: Isar et al IJMPE(94)qp/04; Klimontovich PS(00); Calzetta et al PhyA(03)qp/00 [stochastic description]; Gambetta & Wiseman PRA(01)qp; Stelmachovic & Buzek PRA(01)qp [entangled with environment]; Breuer & Petruccione 02; Mensky PLA(03)qp/02 [evolution as measurement]; Okolowicz et al PRP(03); Ollivier et al PRL(04)qp/03, PRA(05)qp/04 [environment and objective properties]; Nicolosi OSID(05)qp; Jordan et al PRA(06)qp/05 [Schrödinger picture]; Vol PRA(06)qp/05 [semiclassical quantization]; Mohseni & Lidar PRL(06)qp [dynamics]; Bodor & Diósi PRA(06) [conserved current]; Crooks a0706 [time reversal of a quantum operation]; Yu PLA(08) [environment and entropy production]; Nesterov & Ovchinnikov PRE-a0806 [geometric phases, quantum phase transitions]; > s.a. types of quantum field theories.
@ Quantum, non-Markovian: Strunz et al PRL(99) [stochastic Schrödinger equation]; Breuer a0707; Fischer & Breuer PRA(07)-a0708 [spin + spin-bath]; Rodríguez-Rosario & Sudarshan a0803; Piilo et al PRL(08).
@ States: Klimontovich PS(98) [information]; Isar RJP(98)qp/06 [pure states]; > s.a. generalized coherent states.
@ Decoherence: Dugic IJTP(06)qp/99; Monteoliva & Paz qp/01 [classically chaotic]; Alicki qp/02, et al JPA(04)qp/03.
> Related topics: see geometric phase; lorentz transformations; path integrals; wigner function.

Operad > s.a. Module.
@ And physics: Paal CzJP(01)mp; Paal gq/02 [curvature]; Zois RPMP(05) [and quantum gravity].

Operation on a Set > see sets [associative].

Operationalism
* Idea: Physical quantities are completely defined by the series of operations with which one measures them.
* Example: No absolute time in physics.
@ References: Delaney IJTP(99) [limitations].

Operator Theory

Oppenheimer-Snyder Solutions > see collapse.

Optical Activity > see optics.

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

Optical Theorem > see scattering.

Optics

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].

Orbits of a Group Action > see group action; lie group examples.

Orbits in Mechanics > see [classical mechanic]; orbits of test bodies, of newtonian and relativistic gravitating bodies; Trajectories.

Order > s.a. Defects; Disorder; Extended Objects; phase transition; poset; random systems; thermodynamics.
* Remark: In quantum statistics, it is described in terms of symmetry breakdown, which is not antithetical to the notion of disorder.
@ 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-in [in quantum statistical mechanics, survey].
@ 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].

Order Parameter > see Potts Model.

Orientability, Orientation

Orientifold > see types of manifolds.

Orlicz Inequality
$ Def: For any open set Rⁿ and N-function A, with complementary N-function A~,

| _Omega u(x) v(x) dx | 2 u_{L_A()} v_{L_A~()} .

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

Orlicz Space
$ Def: A Banach space L_A():= {set of all functions u such that _Omega A(u(x)) dx < , or u is a (real) multiple of such a function}, where is an open set in R and A an N-function, and the norm u:= inf_{k>0 Omega} A(u(x)/k) dx 1.
@ References: in Adams 75; Streater mp/04-in [quantum, in information geometry].

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].

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 Theorem > see higher-order gravity.

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

Overhang Problem > see Center of Mass.

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