$ Def: For any open set
Rn
and N-function A, with complementary N-function A~,Topics, O
Objectivity > see decoherence; foundations of quantum mechanics; symmetries.
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]; Manogue & Dray a0911-in [particle physics]; > s.a. gravitational
instantons; formulations
of
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)nov
[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?", as opposed to "What do we know?" (epistemology).
@ 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.
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].
Oppenheimer-Snyder Solutions > see collapse; lattice gravity.
Optical Activity > see optics.
Optical Geometry > see electromagnetism in curved spacetime; optics [including specific spacetimes]; self-force.
Optical Theorem > see scattering.
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 mechanics]; orbits of test bodies, of newtonian and relativistic gravitating bodies; Trajectories.
Order > s.a. Disorder.
* 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]; Nussinov & Ortiz AP(09) [topological quantum order].
> Phenomenology: see Defects; diffraction; phase
transitions; thermodynamics [second law].
> Systems: see Extended
Objects; poset; 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.
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_{LA()} v_{LA~()}
.
* Relationships: This is a generalization of the Hölder inequality.
Orlicz Space
$ Def: A Banach space
LA():=
{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:=
infk>0 Omega A(u(x)/k)
dx 1.
@ References: in Adams 75; Streater mp/04-in
[quantum,
in information geometry]; Labuschagne & Majewski a0902 [non-commutative].
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.
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified
13 nov 2009