Topics, A

Abel's Theorem > see elementary algebra.

Abelian Group > see types of groups.

Aberration of Light
* Stellar aberration: An apparent displacement in the position of celestial objects, due to the finiteness of the speed of light and Earth's motion (analogous to the fact that vertically falling raindrops appear to be coming at a different angle to a running observer); Discovered by Astronomer Royal James Bradley in 1725, in an attempt to see stellar parallax; Values vary with a period of 1 year, with a maximum of about 20".
@ Relativistic: Gjurchinovski EJP(06)phy/04; Beig & Heinzle AJP(08)-a0708 [accelerating observer].
@ Gravitational: & Van Flandern; Carlip PLA(00)gq/99 [and speed of gravity]; Turyshev gq/02, gq/02, gq/02 [for SIM].
> Online resources: see Wikipedia page.

Aboav-Weaire Law > see random tiling.

Abraham Tensor > see energy-momentum.

Abraham-Lorentz Formula > see self-force.

Absolute Object > same as Ideal Element.

Absolute Space > see models of spacetime.

Acceleration > s.a. physics teaching; Reference Frame; rindler space [uniform acceleration].
* Covariant definition: Acceleration is a well-defined concept; It can be measured with a box and a mass with springs, and does not need a specification of "wrt what", contrary to the situation with velocity.
$ Definition: The 4-vector A^a:= u^b _b u^a, perpendicular to the world-line, A^a u_a = 0.
* In general relativity: A world-line accelerates only if subject to nongravitational forces; Objects in free fall follow geodesics A^a = 0.
@ Relativistic: Rindler & Mishra PLA(93) [relative acceleration in special relativity]; Bini et al CQG(95) [transformation law in general relativity].
@ Maximal acceleration: Brandt FPL(89) [and 4-velocity fiber bundle over spacetime]; Papini NCB(02); Feoli IJMPD(03) [different values]; Papini qp/04 [and superconductors]; Gallego gq/05 [Finsler models]; > s.a. kerr spacetime, modified lorentz symmetry.
> Related phenomena: see anomalous acceleration [Pioneer effect]; cosmic acceleration; radiation [accelerated charges]; quantum field theory effects in curved spacetime.

Accretion Disk, Process > see astrophysics; black hole perturbations and phenomenology.

Accumulation Point
$ Def: Given a topological space (X, ) and a sequence {x_n}, a point x X is an accumulation point for the sequence if every neighborhood of x contains infinitely many points of the sequence; Remark: It may not be possible to find a subsequence converging to an accumulation point, unless the space X is first countable.

Achronal Set > see spacetime subsets.

Acoustic Equation > see wave equation.

Acoustics > see sound [including thermoacoustics].

Action > see lagrangian.

Action at a Distance > see causality.

Action of a Group on a Set > see group action.

Action-Angle Variables > s.a. oscillator.
@ References: Bates PRSE(88) [obstructions]; Khein & Nelson AJP(93) [persistent error in the literature]; Lahiri et al PLA(98) [in quantum mechanics].

Action-Reaction Principle > see formalism of classical mechanics.

Acyclic Complex > see Complex.

Adams Conjecture > see conjectures.

Adelic Numbers, Structures > see distributions; geometry.
@ In physics: Dragovich ht/06-in [cosmology, dark matter + dark energy]; Dragovich a0707-in [mathematical physics]; > s.a. modified uncertainty, oscillator. quantum cosmology.

Adiabatic Approximation
@ References: MacKenzie et al PRA(07)-a0706 [quantum, conditions for validity].

Adiabatic Theorem
* Idea: An initial eigenstate of a slowly changing Hamiltonian evolves into an instantaneous eigenstate at a later time.
@ Consistency: Marzlin & Sanders PRL(04)qp; Sarandy et al QIP(04)qp; Pati & Rajagopal qp/04; Du et al a0801.

Adiabatic Transformation
* Idea: A change in a fluid during which no heat is exchanged with the environment; In microscopic terms, the number of states available to the system remains constant; For a perfect fluid, the thermodynamic quantities satisfy p^{–} = constant, where = c_p/c_v is the ratio of specific heats.
* Examples: For a photon gas, = 4/3.

Adiabaticity > see qed phenomenology.

Adjacency Matrix > see graph functions.

Adjoint of an Operator
$ General definition: Given an operator A: V → W between two vector spaces, we can define the adjoint A: W* → V* (without the need of any metric) by

A w*(v):= w*(Av),    for all w* W*, v V .

* With metrics: If V and W have metrics g and h resp, then an "adjoint" operator "A": W → V can be defined by "A":= g^–1Ah; This is what happens for an operator on a Hilbert space (metric = inner product).
$ Usual definition: If A: → , the adjoint A: → is defined by

(A):= {y   | z  : x    y, Ax = z, x },      A y:= z .

It cannot be defined if (A) is not dense.

Adjoint Representation > see representations of a lie group or algebra.

Adjunction Space
$ Def: Given A X and a map f : A → Y, the adjunction space Z_f X _f Y is defined by X _f Y:= (X Y)/, where the equivalence relation identifies any point x in A with f(x) in Y.
* Idea: Intuitively, X _f Y is X and Y glued together along A f(A).
* Examples:
- S^n–1 Eⁿ,  f : S^n–1 → p, constant function: Z_f = Eⁿ _const p Sⁿ;
- S^n–1 Eⁿ,  f : S^n–1 → S^n–1, identity: Z_f = Eⁿ _id S^n–1 Eⁿ (as cell complex);
- S^n–1 Eⁿ,  f : S^n–1 → S^n–1, any homeomorphism; Same as second example.
* Remark: To prove things about adjunction spaces, use the map Y → X Y → Z_f, one-to-one.

Adler-Bardeen Theorem
@ References: Mastropietro ht/06 [non-perturbative version].

Adler-Kostant-Syms Theorem > see integrable system.

ADM Energy / Mass / Momentum > see ADM formulation of general relativity.

AdS-CFT Correspondence

Advanced Green Function > see green functions.

Advanced Time > see time.

Aether > see under Ether.

Affine Gravity > s.a. [gravity]; approaches to quantum gravity; Metric-Affine Theories.
* Idea: The connection field is the primary structure; Metric and matter are to be deduced from the connection.
@ References: von Borzeszkowski & Treder GRG(02).

Affine Structures, Collineation > s.a. affine connection.

Afshar's Experiment > a version of the quantum-mechanical two-slit Interference experiment.

Age of the Universe > see observational cosmology.

Aharonov-Bohm Effect

Aharonov-Casher Effect > s.a. phase.
* Idea: The phase difference between wave functions of magnetic dipoles (neutrons) going different ways around a line of electric charges; In some cases may be considered the electromagnetic dual of the Aharonov-Bohm effect.
@ References: Aitchison Nat(89)sep; Azimov & Ryndin hp/97 [particle motion], PAN(98)hp/97 [arbitrary spin]; Pati PRA(98) [Bell's inequality]; Bruce PS(01)qp; He & McKellar PRA(01) [and 2+1 electromagnetic dual]; Persson EJDE(05)mp [different self-adjoint extensions of Pauli operator]; Rohrlich a0708-in.

Airy Functions
@ Applications: Rosu PS(02)qp/01 [role of Bi]; Vallée & Soares 04.
@ Related topics: Nikishov & Ritus mp/05 [related functions and integrals].

Airy Process > see stochastic processes.

Akivis Algebras > see geodesics.

Alexander Polynomial > see knot invariants.

Alexander-Lefschetz Duality Relationships
* Examples: The most famous one is the Jordan Curve Theorem.

Alexandroff Line > same as Long Line.

Alexandrov Set > see causal structures in spacetime.

Alexandrov Topology > see spacetime topology.

Alfvén Waves

Algebra (Abstract)

Algebra, Algebraic Equation, Algebraic Function > see elementary algebra.

Algebraic Geometry
* Idea: A study of algebraic varieties in affine and projective spaces over R or C, loci defined by polynomial equations, called schemes.
* Techniques: Modern ones use Schemes and cohomology.
* And physics: Gained prominence with the development of string theory.
@ General texts: Zariski 35; Lefschetz 56; Dieudonné 74; Griffiths & Harris 78; Hirzebruch 78; Hartshorne 90.
@ And physics: Roan mp/00-in [rev].
@ Related topics: Douglas mp/05-in [random, attractors and flux vacua].

Algebraic Number > see numbers.

Algebraic Topology

Algebroid > see lie algebras.

Alignment > see lorentzian geometry.

Allais Effect
* Idea: An observed puzzle or anomaly of gravity, a phenomenon that occurs during solar eclipses.
@ References: Amador JPCS(05)gq/06 [measurements].

Almost Complex Structure > see complex structure.

Almost Hamiltonian Structure > see symplectic manifolds.

Almost Periodic Function > see functions.

Alternating Group > see finite groups.

Alternating Tensor (Also called Levi-Civita tensor.)
$ Def: The volume element or n-form _{ab ... c} for an n-dimensional manifold.
* Useful expression: In terms of an orthonormal basis eⁱ_a of covectors, _{ab ... c} = n! e¹_a e²_b ... eⁿ_c] (but add an i for a null tetrad).

Amalgamated Sum > see fundamental group [Seifert-Van Kampen theorem].

Amenable Group > see group action; Topological Group.

Ampère's Law > see magnetism.

Analog Gravity > see black hole analogs; lorentzian geometry.

Analysis > s.a. functional analysis.

Analytic Functions, Mappings

Anderson Localization and Model > s.a. diffusion; locality in quantum mechanics; photon; wave phenomena; types of graphs [quantum].
* Localization, idea: The localization of matter (electron) wave functions in a random medium.
@ General references: Anderson PR(58); Abrahams et al PRL(79) [scaling]; Ye & Gupta PLA(03), Ye PLA(03) [2D]; Aizenman et al IM(05)mp/03; Domínguez-Adame & Malyshev AJP(04) [1D]; Gavish & Castin PRL(05) [for atoms]; Kirsch a0704 [multiparticle systems on a lattice]; Brandenberger & Craig a0805 [towards a new proof].
@ Anderson model: Chen JSP(05) [3D localization length, small disorder]; García PRE(06)cm/05 [transition, spectral characterization]; Nakano JSP(06) [repulsion between localization centers], JMP(07) [finite-volume approximation].

Angles > see canonical quantum mechanics; geometrical operators in quantum gravity; trigonometry.

Angular Momentum

Anholonomy > see [holonomy]; constraints [non-holonomic]; geometric phase.

Anisotropy > see bianchi models; relativistic cosmology; minisuperspace quantum cosmology.

Annihilation Operator > s.a. fock space.

Annulus Conjecture > see spheres.

Anomalies in Quantum Theory

Anthropic Principle

Anti-de Sitter Spacetime > s.a. AdS-cft.

Antichain > see posets.

Anticommutation Relations
@ Canonical: Derezinski LNP(06)mp/05 [representations].

Antigravity > s.a. [gravitational energy and theories]; motion of test particles.
* Idea: In general, the suggestion that gravity can be a repulsive force in certain situations; One specific suggestion is that there is gravitational repulsion between matter and antimatter.
* With charged black holes: A phenomenon by which a system of non-rotating black holes can be in static equilibrium because of the balance between gravitational attraction and electromagnetic repulsion; The condition is that each black hole satisfy M = G^–1/2 (Q² + P²)^1/2, where Q is the electric charge and P the magnetic charge.
@ General references: Nieto & Goldman PRP(91); Scherk PLB(79), in(79) [from supergravity]; Quirós gq/04 [and the cosmological constant]; Hossenfelder PLB(06)gq/05 [anti-gravitating fields], criticism Noldus & Van Esch PLB(06); Gershtein et al TMP(05) [in "relativistic theory of gravitation", and singularity avoidance]; Minkevich APPB(07)gq/05 [extreme conditions]; Hajdukovic gq/06 [proposed test with antiprotons].
@ Related topics: Felber gq/06-in [propulsion and hyperdrive].

Antimatter > see matter; early universe.

Antirelativity > see special relativity.

Antisymmetrization > see tensors.

Anyons > see particle statistics.

Apparent Horizon > see horizons.

Approach Space
@ References: Banaschewski T&A(06) [sober]; Brümmer & Sioen T&A(06) [asymmetry and bicompletion].

Arcwise (or Pathwise) Connected Space > see connectedness.

Area of a Surface > see geometrical operators in quantum gravity.

Area Metric > see differential geometry [generalizations]; cosmological models.

Area-Preserving Map > see Poincaré Recurrence.

Arithmetic
* Modular arithmetic: The version in which two integers are said to be equal (or "congruent") modulo a particular, fixed integer N if they differ by a multiple of N; Applied in Shor's algorithm for factoring large numbers by quantum computers.
@ Variations: Rotman ThSc(97) [non-Euclidean].

Arnold Cat > see chaotic systems.

Arnold Conjecture > see Gromov-Witten Invariants.

Arnold Diffusion > s.a. quantum chaos.
* Idea: A phenomenon appearing in (soft) chaotic systems with at least 2 degrees of freedom; The non-resonant tori which are not broken by a small perturbation away from an integrable system do not foliate the energy surfaces in phase space, so the stochastic regions near resonant tori can join and cause trajectories in them to wander around.
* Consequences: Although most tori are not destroyed, a finite measure set of trajectories departs arbitrarily far from the unperturbed motion.
@ General references: In almost any book on chaos, for example Zaslavskii et al 91.
@ Estimate of time of stability: Nekhorossiev RMS(77).

Arrow of Time

Artificial Intelligence > see computation.

Aschenbach Effect > see kerr spacetime.

Ascoli-Arzela Theorem
* Idea: A useful criterion for compactness on function spaces.
$ Def: If X is a compact metric space, a bounded and equicontinuous subset K of the space C(X) of continuous functions on X with the uniform norm is compact.
@ General references: in Yosida 78; in Choquet-Bruhat et al 82, p61.
@ Lorentzian version: Noldus CQG(04)gq/03.

Ashtekar Variables > see connection formuulation of canonical gravity.

Ashtekar-Horowitz Model > see dirac quantization.

Associated Bundle > see fiber bundle.

Associative Operation > see sets.

Asteroids > see solar system objects.

Astrometry > see stars.

Astronomy > s.a. astronomical objects; extrasolar systems; history of astronomy; solar system.

Astrophysics

Asymptote (in General Relativity) > see asymptotic flatness at null infinity.

Asymptotic Analysis > see analysis.

Asymptotic Anti-de Sitter Spacetime > see Anti-de Sitter.

Asymptotic Expansion > s.a. series.
@ References: Copson 71.

Asymptotic Flatness > s.a. at null infinity and at spatial infinity.

Asymptotic Freedom > see QCD.

Asymptotic Safety > see approaches to quantum gravity.

Asymptotic Simplicity > similar to aymptotic flatness.

Atiyah-Singer Theorem > see Index Theorem.

Atomic Physics > s.a. atomic elements.

Attractor [> chaos].
* Idea: A set of phase space points that a dynamical system approaches as t → .
* Condition: Only dissipative systems can have attractors.
* Types: Point; Limit cycle; Strange attractor (a fractal).
@ General references: Milnor CMP(85); Gobbino Top(01) [topology].
@ Strange attractors: Sprott 93, PLA(94).
@ Strange, non-chaotic: Romeiras & Ott PRA(87); El Naschie & Kapitaniak PLA(90); Kapitaniak CSF(91).

Autocorrelation Function > see stochastic process.

Automaton > see computation.

Automorphic Form
@ References: Pioline & Waldron ht/03-in [for physicists].

Automorphism > see category.

Avogadro's Number > see constants.

Axial Gauge > see gauge.

Axiom of Choice

Axioms for Physical Theories > see axioms for quantum mechanics; physical theories.

Axino
@ References: Covi et al PRL(99) [as cold dark matter].

Axiom of Choice
$ Def 1: Given any non-empty set of disjoint non-empty sets X_i, with i in I, a set can be formed which contains exactly one element x_i from each X_i.
$ Def 2: Given any set X, there exists a choice function : 2^X \ {Ø}→ X, such that for all Y in 2^X \ {Ø}, (Y) is in Y.
$ Kuratowski lemma: (An equivalent statement) Each chain in a poset is contained in a maximal chain.
$ Zorn's lemma: (An equivalent statement) If each chain in a poset has an upper bound, then the set contains a maximal element.
* Other statements: It is equivalent to the well-ordering principle (> see Well-Ordered Set).
@ References: Gödel 40; Moore 82; Rubin & Rubin 85.

Axion > s.a. dark matter types.

Axisymmetric Solutions, Spacetimes

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