Topics, A

a-theorem
* Idea: The statement (made in 1988 by John Cardy) that the number a of ways in which quantum fields can be energetically excited is always greater at high energies than at low energies; 2011, It seems to have been proved by Komargodski & Schwimmer.
@ References: Cardy PLB(88); Komargodski & Schwimmer JHEP(11)-a1107 + news nat(11)nov [proposed proof].

Abel's Theorem > see elementary algebra.

Abelian Group > see types of groups.

Abell Catalog > see galaxy distribution.

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".
@ General references: Timberlake TPT(13)-a1208 [history, and parallax].
@ Relativistic: Gjurchinovski EJP(06)phy/04; Beig & Heinzle AJP(08)jul-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.

ABJM Theory (Aharony, Bergman, Jafferis, and Maldacena) > see chern-simons theory.

Aboav-Weaire Law > see random tiling.

Abraham Tensor > see energy-momentum tensor.

Abraham-Lorentz Formula > see self-force.

Absolute Object > same as Ideal Element.

Absolute Parallelism > see teleparallel structures; types of manifolds [parallelizable].

Absolute Space > see models of spacetime.

Acceleration > s.a. acceleration radiation; anomalous acceleration [Pioneer effect]; Fermi Acceleration.

Accelerated Quantum Fields
@ References: Kiosses a1807 [and quantum gravity].

Accretion Process > s.a. astrophysics [accretion disk]; black-hole perturbations and matter near black holes [accretion onto black holes].
* Idea: The process whereby gas falls onto an astronomical body that has already been formed and is acquiring mass.
@ References: Shakura AR(18)-a1809 [Zeldovich and the foundation of accretion theory].

Accumulation Point
$ Def: Given a topological space \((X, {\cal T})\) and a sequence \(\{x_n\}\), a point \(x \in X\) is an accumulation point for the sequence if every neighborhood of x contains infinitely many elements of the sequence; Remark: This does not mean that the sequence converges to x (that would require all elements of the sequence to be inside that neighborhood from some n onwards), but if the space X is first countable, then it is possible to find a subsequence converging to x.

Accuracy > s.a. Precision.
* Idea: The offset of a measured quantity away from its true value.

ACES (Atomic Clock Ensemble in Space) > see gravitational redshift.

Achronal Set > see spacetime subsets.

Acoustic Equation > see types of wave equations.

Acoustic Geometry / Spacetime > see emergent gravity.

Acoustics > see sound [including thermoacoustics].

Action > see lagrangian dynamics.

Action at a Distance > s.a. causality; locality.
* Idea: The effect of a force on an object that acts between two bodies without any intermediary (such as a field), and usually without delay (such as that due to the causal propagation of a field); The main examples are Newtonian gravity and electrostatics.
* History in physics: The idea was favored in the 1930s by John Wheeler and Richard Feynman as an alternative to field theory.
@ General references: & Fokker; Hoyle & Narlikar 74, 96; Sidharth in(99)gq/98; Henry SHPSA(11), Ducheyne SHPSA(11) [Newton's views].
@ In electrodynamics: Wheeler & Feynman RMP(49); Hoyle & Narlikar RMP(95); Hollander & De Luca PRE(03)mp [2-body problem]; Ibison AP(06); De Luca JMP(09)-a0901 [variational principle]; Pietsch SHPMP(10) [how to allow for action at a distance]; Kastner IJQF-a1502 [Haag's theorem as a reason to reconsider]; Kastner Quanta-a1509 [and antimatter].
@ In quantum mechanics: Hardy CP(98); Boughn a1806 [no action at a distance]; > s.a. types of quantum interpretations [transactional].
@ And special relativity: Wigner &(71); Medvedev NTF(77); Louis-Martinez PLB(06)ht/05, PLA(07)ht/06 [relativistic], FP(12)-a1104 [and field theories].

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

Action-Angle Variables > s.a. oscillator.
* Idea: Pairs of phase-space variables for dynamical systems used to write the equations of motion of integrable systems explicitly as those of a free particle.
@ References: Bates PRSE(88) [obstructions]; Bates & Śniatycki ARMA(92); Khein & Nelson AJP(93)feb [persistent error in the literature]; Lahiri et al PLA(98) [in quantum mechanics]; Nguyen a1204 [on Dirac manifolds], a1706 [conceptual approach].
> Online resources: see Wikipedia page.

Action-Reaction Principle > s.a. Newton's Laws.
@ References: Brown & Lehmkuhl a1306-in [Einstein, and general relativity].

Active Dynamical System > see friedmann equation.

Active Learning > see physics teaching / or Machine Learning.

Active Matter > see non-equilibrium statistical mechanics.

Acyclic Complex > see Complex.

Adams Conjecture > see conjectures.

Adelic Numbers, Structures > see distributions; geometry; Non-Archimedean Structures.
@ In physics: Dragovich ITSF(98)mp/04, Djordjević & Nešić LNP(03)ht/04 [quantum mechanics]; Dragovich NPPS(01) [quantum mechanics and quantum field theory]; Dragovich AIP(06)ht [cosmology, dark matter + dark energy]; Dragovich a0707-proc [mathematical physics]; > s.a. modified uncertainty relations; quantum particles; quantum oscillators; quantum cosmology.

Adiabatic Approximation / Evolution
@ Quantum, conditions: MacKenzie et al PRA(07)-a0706; Comparat PRA(09)-a0906; Boixo & Somma PRA(10)-a0911.
@ Related topics: Brouder et al PRA(08)-a0807 [and Gell-Mann-Low theorem, degenerate Hamiltonian case].

Adiabatic Theorem
* Idea: An initial eigenstate of a slowly changing Hamiltonian evolves into an instantaneous eigenstate at a later time.
* Applications: It provides the basis for the adiabatic model of quantum computation.
@ Consistency: Marzlin & Sanders PRL(04)qp; Sarandy et al QIP(04)qp; Pati & Rajagopal qp/04; Du et al a0801; Amin PRL(09); Lobo et al EJP(12) [geometric pedagogical derivation].
> Online resources: see Wikipedia page.

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 \(pV^\gamma\) = constant, where \(\gamma\) = c_p/c_v is the ratio of specific heats.
* Examples: For a photon gas, \(\gamma\) = 4/3.
@ References: Oreshkov & Calsamiglia PRL(10)-a1002 [in quantum Markovian dynamics].

Adiabaticity > see QED phenomenology.

Adinkras
* Idea: Graphical tools created by Michael Faux and S J Gates for the study of representations in supersymmetry.
@ References: Faux & Gates PRD(05)ht/04; Zhang a1111 [rev].
> Online resources: see Wikipedia page; see also the Wikipedia page on the Akan visual symbols.

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, respectively, then an "adjoint" operator "A": W → V can be defined by "A^†":= g⁻¹A^†h; This is what happens for an operator on a Hilbert space (metric = inner product).
$ Usual definition: Given a densely defined A: \(\cal H\) → \(\cal H\), the adjoint A^†: \(\cal H\) → \(\cal H\) has domain

\(\cal D\)(A^†):= {y ∈ \(\cal H\) | ∃ z ∈ \(\cal H\): ∀x ∈ \(\cal H\), \(\langle\)y, Ax\(\rangle\) = \(\langle\)z, x\(\rangle\)},

and its action on each y ∈ \(\cal D\)(A^†) is defined by A^† y:= z.
@ References: Selby & Coecke a1606 [operational meaning].

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:
- Sⁿ⁻¹ ⊂ Eⁿ,  f : Sⁿ⁻¹ → p, constant function: Z_f = Eⁿ ∪_const p ≅ Sⁿ;
- Sⁿ⁻¹ ⊂ Eⁿ,  f : Sⁿ⁻¹ → Sⁿ⁻¹, identity: Z_f = Eⁿ ∪_id Sⁿ⁻¹ ≅ Eⁿ (as cell complex);
- Sⁿ⁻¹ ⊂ Eⁿ,  f : Sⁿ⁻¹ → Sⁿ⁻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 Function
@ References: Adler PRD(74); Nesterenko eConf-a0710 [in the analytic approach to QCD]; Horch PoS-a1311 [from the vacuum polarization function]; Shifman & Stepanyantz PRL(15)-a1412 [in supersymmetric QCD].

Adler-Bardeen Theorem > see chiral anomalies.

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 > s.a. time.
* Advanced time: The advanced time at a point x' with respect to (x, t) is the time at which a signal leaving (x, t) and traveling at the speed of light, arrives at x',

t_adv:= t + r/c ,   where   r:= ||x−x'|| .

Aether > see under Ether.

Affine Gravity > s.a. gravity / approaches to quantum gravity; Metric-Affine Theories.
* Idea: A theory of gravity with a connection field as the primary structure; Metric and matter are to be deduced from the connection.
@ General references: Petti GRG(01)-a1412 [affine defects]; von Borzeszkowski & Treder GRG(02); Popławski FP(09) [Ferraris-Kijowski theory and the cosmological constant]; Popławski GRG(14) [and dark energy]; Gültekin EPJC(16)-a1512 [with torsion]; Moti et al PDU(21)-a2101 [quantum corrections].
@ And cosmology: Filippov a1008; Azri a1805-PhD.

Affine Quantization
@ References: Klauder 00; Watson & Klauder JMP(00)qp; Klauder JPA(12)-a1108 [and affine coherent states]; Klauder JHEP(18)-a1803 [no zero-point energy].
> General theory: see canonical quantum theory [enhanced quantization]; formulations of quantum theory.
> Gravitational and cosmology: see approaches to canonical quantum gravity; brans-dicke theory; FLRW and minisuperspace quantum cosmology.
> Other applications: see quantum systems; types of quantum field theories.

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.

Agency > s.a. observers in physics.
@ References: Rovelli a2007 [agency as mechanism to transform low entropy into information].

Agravity > see gravitation; approaches to quantum gravity.

Aharonov-Bohm Effect

Aharonov-Casher Effect > s.a. geometric phase.
* Idea: The phase difference between wave functions of magnetic dipoles (neutrons) going different ways around a line of electric charges; In some cases it 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-en; Horsley & Babiker PRA(08) [role of internal degrees of freedom]; Eskin a1007 [simple proof]; Dulat & Ma PRL(12)-a1203; Vaidman in(14)-a1301-fs [paradoxes, locality and entanglement]; Kang PRA(15)-a1408 [locality]; Silva & Andrade a1412 [spin-1/2, Lorentz-violating effects].
> Online resources: see Wikipedia page.

Airy Beams / Packets > s.a. diffraction.
* Idea: Wave packets, initially predicted by Berry and Balazs in 1979, which remain diffraction-free over long distances and tend to freely accelerate sideways (paraxial Airy beams accelerate along a parabolic trajectory, non-paraxial beams accelerate in a circular trajectory).
@ References: news Phy(07) [paraxial, first observation]; Chen Phy(12) [non-paraxial]; Nassar & Miret-Artés AP(14)-a1404 [Bohmian trajectories].

Airy Functions
* Idea: The functions Ai(x) and Bi(x) are the two linearly independent solutions of the Airy or Stokes differential equation y"(x) − x y(x) = 0.
@ Applications: Rosu PS(02)qp/01 [role of Bi]; Vallée & Soares 10 [in physics].
@ Related topics: Nikishov & Ritus mp/05 [related functions and integrals]; Belloni & Robinett JPA(09)-a1007 [constraints on zeros from quantum mechanics]; Fernández a0911 [integrals of products]; Babusci et al OP(11)-a1002.
@ Generalizations: Fernández et al LMP(09)-a0901 [over non-archimedean local fields].
> Online resources: see Wikipedia page; MathWorld page.

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.

Alexandroff Space
$ Def: A topological space in which the intersection of every family [not just finite ones] of open sets is open.
* Examples: All finite topological spaces; Any set endowed with the discrete topology.
@ References: Kukieła Ord(10) [homotopy types].
> Online resources: see PlanetMath page.

Alexandrov Sets > s.a. causal structures in spacetime; time functions.
* Idea: Spacetime subsets defined as intersections of the future of a point with the past of another point; Also called "causal diamonds" and "intervals".
* Volume: In a 4D Lorentzian manifold the volume of the Alexandrov set defined by two events with timelike separation τ is

\[ V = {\pi\tau^4\over24}\Big[1+\Big({1\over180}\,R(0)-{1\over30}\,R_{00}(0)\Big)\,\tau^2+{\rm h.o.t.}\Big]\; .\]

@ General references: Solodukhin JHEP(09)-a0812 [in asymptotically de Sitter spacetimes, and irreversibility]; Khetrapal & Surya CQG(13)-a1212 [volume]; Roy et al PRD(13)-a1212 [number of k-chains in curved spacetime]; Berthiere et al PRD(15)-a1507 [comparison theorems, inequalities]; Jubb CQG(17)-a1611 [volume]; Jacobson et al CQG(18)-a1710 [area deficits and gravitational energy]; Wang PRD(19)-a1904 [geometry]; Krishnan SciP(19)-a1902 [asymptotic].
@ And gravity: Chandrasekaran & Prabhu JHEP(19)-a1908 [covariant phase space]; Visser a1908-PhD [thermodynamic, emergent and holographic aspects].
@ And quantum field theory: Su & Ralph PRD(16)-a1507 [thermal states for the Minkowski vacuum]; de Boer et al JHEP(16)-a1606 [interacting theories on the moduli space of causal diamonds]; Jacobson & Visser SciPost(19)-a1812 [gravitational thermodynamics]; Banks et al a2008 [Euclidean continuation, gravitational entropy].

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 \(\mathbb R\) or \(\mathbb C\), 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; Patil & Storch 10 [and commutative algebra]; Borceux 14; Tomašić in Bullett et al 17.
@ Applications: Roan in(02)mp/00 [in physics, rev]; in Bernardi & Carusotto JPA(12) [tools for the decomposition of tensors and polynomials].
@ Related topics: Douglas mp/05-en [random, attractors and flux vacua]; Laudal 11, Cornelissen & Landi JGP(13) [non-commutative]; > s.a. Arithmetic Geometry.

Algebraic Number > see numbers.

Algebraic Quantization / Quantum Theory > s.a. algebraic quantum field theory; canonical quantum theory.
* Idea: An approach to quantum theory in which the primary structure is a non-commutative C*-algebra of observables; Physical states are defined as linear functionals on this algebra; An equivalent abstract characterization of quantum theory is provided by Jordan-Lie-Banach algebras.
@ Intros: Bény & Richter a1505 [finite-dimensional, pedagogical, for quantum information]; Moretti IJGMP(16)-a1508-ln [advanced short course]; Drago a2102.
@ General references: Sudarshan et al AIHP(88); Rieffel CM-qp/97-proc [operator algebra]; Slavnov qp/01, qp/04; Accardi & Dhahri a1401 [C*-non-linear second quantization]; Cruz & Zilber PTRS(15)-a1410 [geometric semantics]; LaChapelle a1505 [functional integral representations of C*-algebras]; Hiley ch(16)-a1602 [Hans Primas' and David Bohm's approach]; Zalamea a1612-PhD [observables as Jordan-Lie algebras, etc].
@ Related topics: Roberts & Teh a1602 [representations of Jordan-Lie-Banach algebras]; Hiley ch(16)-a1602 [non-commutative symplectic algebra underlying quantum dynamics]; > s.a. entangled systems.

Algebraic Quantum Field Theory

Algebraic Topology

Algebraically Special Spacetimes > see petrov classification.

Algebroid > see Courant Algebroid; Leibniz Algebroid; lie algebras / Double Field Theory.

Algorithmic Complexity > see complexity.

Algorithmic Decidability > see 2-manifolds; 3-manifolds; 4-manifolds; types of manifolds.

Algorithmic Randomness > see quantum information theory; random processes.

Algorithmic Topology > see 3-manifolds.

Algorithms > s.a. computation; quantum computing.
@ References: Knuth 69, 73 [fundamental computing algorithms]; Mitzenmacher & Upfal 05 [randomized, probabilistic methods]; Soltys 12 [analysis of algorithms].
> Gravity-related algorithms: see Apparent Horizons; black-hole simulations; gravitational-radiation analysis; petrov classification; Ricci Tensor.
> Other physics-related algorithms: see computational physics; Domain Walls [PRS algorithm]; locality in quantum theory [algorithmic definition]; phenomenology of entanglement [measurement]; spin models [ground-state energy].
> Other specific algorithms: see 3-manifolds [classification]; brownian motion [random-search algorithms]; conservation laws [Van Holten's covariant algorithm]; differential equations; graph invariants [dimension] and graph types; Greechie Diagram; Grover Algorithm; knot invariants; operator theory [Lanczos algorithm]; lyapunov exponents; Metropolis Algorithm; montecarlo method; ordinary differential equations [genetic algorithm]; Polygons [Minkowski sums]; posets [linear-extension algorithm]; quantum information theory; ramsey theory; random walk; Shor's Algorithm; statistical geometry; voronoi tiling [Delaunay refinement algorithm]; Word [Deutsch-Josza algorithm].
> Algorithmic approaches to theories: see graph theory; information theory [algorithmic thermodynamics].

Alice Fields > see gauge theories; modified electromagnetism; monopoles.

Alien Calculus > s.a. quantum field theory techniques.
@ References: Dorigoni a1411 [and resurgence].

Alignment > see lorentzian geometry; Preorder.

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

Allan Factor > s.a. statistical geometry [Poisson point process].
* Idea: A statistic widely used to assess if the rate of occurrences of an event tends to cluster and show persistence in a range of space and/or time scales.

Almost Complex Structure > see complex structure.

Almost Hamiltonian Structure > see symplectic manifolds.

Almost Periodic Function > not to be confused with a Quasiperiodic Function; s.a. functions.
* Idea: A function of a real number that is periodic to within any desired level of accuracy for long enough intervals of the variable.
> Online resources: see Wikipedia page.

Alternating Group > see finite groups.

Alternating Tensor (Also called Levi-Civita tensor.)
$ Def: The volume element or n-form ε_{ab ... c} for an n-dimensional manifold.
* Useful expression: In terms of an orthonormal basis eⁱ_a of covectors, ε_{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].

AMANDA (Antarctic Muon and Neutrino Detector Array) > s.a. neutrino experiments.
@ References: Andres et al APP(00)ap/99; Desiati ap/03-conf; Halzen ap/03-conf; Silvestri MPLA(07); Ackermann et al ApJ(08)-a0711.

Amenable Group > see group action; Topological Group.

Amorphous Solids > see condensed matter.

Ampère's Law > see magnetism.

Amplitude Death
* Idea: An emergent phenomenon arising when non-linear dynamical systems are coupled, and consisting in the complete suppression of oscillations.
@ References: Saxena et al PRP(12).

Amplituhedron
* Idea: A mathematical object generalizing the positive Grassmannian, which looks like an intricate, multifaceted jewel in higher dimensions and allows one to calculate scattering amplitudes for maximally supersymmetric Yang-Mills theory in terms of its volume, much more efficiently than using Feynman diagram techniques; In this approach, locality and unitarity are derived concepts and not fundamental; The technique also suggests that thinking in terms of spacetime is not the right way of going about this.
* Precursors: A 1980s formula found by Stephen Parke and Tomasz Taylor of Fermilab, which reduced a 2-gluon to 4-gluon amplitude calculation from several billion terms using Feynman diagrams to a single term; The 2000s BCFW recursion relations using twistor diagrams, which gave instructions for calculating the volume of pieces of the positive Grassmannian; Twistor diagrams and the less efficient Feynman diagrams can now be seen as ways of calculating the volume of the amplituhedron piece by piece.
* Master amplituhedron: An amplituhedron with an infinite number of facets; Its volume represents, in theory, the total amplitude of all physical processes and lower-dimensional amplituhedra, which correspond to interactions between finite numbers of particles, live on the faces.
@ References: Arkani-Hamed & Trnka JHEP(14)-a1312 + news Quanta(13)sep; Ferro et al JHEP(16)-a1512 [volume]; Damgaard et al a1905 [momentum amplituhedron]; Ferro & Lukowski a2007 [rev].

Analog Geometry and Gravity > see emergent gravity [including acoustic].

Analog Transformations in Physics
@ References: García-Meca et al SciRep(13)-a1306 [and applications to acoustics].
> In gravitational physics: see black-hole analogs; emergent gravity; lorentzian geometry.

Analysis > s.a. functional analysis.

Analytic Continuation > see computational physics.

Analytic Functions, Mappings

Anderson Localization / Model > s.a. diffusion; locality in quantum mechanics; wave phenomena; types of graphs [quantum].
* Localization, idea: The localization of matter (electron) wave functions in a random medium; The origin of the localization is interference between multiple scatterings of the wave function by random defects in the potential, altering the eigenmodes from being extended (Bloch waves) to exponentially localized; In the case of electrons, the material can be transformed from a conductor to an insulator; The effect is commond in 2D disordered systems.
* Examples: Electron waves in condensed matter, electromagnetic and acoustic waves in disordered dielectric structures.
* Avoidance: A disordered system can avoid Anderson localization if waves cannot scatter, as in the quantum Hall effect, in which a strong magnetic field gives rise to topologically protected electron edge states, or some optical systems with synthetic magnetic fields.
* Anderson transitions: Phase transitions in disordered systems, involving isolated states (both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states).
@ 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)feb [1D]; Gavish & Castin PRL(05) [for atoms]; Kirsch a0704 [multiparticle systems on a lattice]; Brandenberger & Craig EPJC(12)-a0805 [towards a new proof]; Hamza et al JMAA(10)-a0907 [1D, proof based on fractional moments method]; Lagendijk et al PT(09)aug [history, overview]; Abrahams ed-10; Hartnett SA(17)aug.
@ 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]; Stolz CM-a1104 [mathematical theory]; Germinet & Klein JEMS(13)-a1105 [with singular random potentials, proof of localization]; Schenker LMP(15)-a1305 [estimating the critical disorder].
@ For photons: Chabanov et al nat(00)apr [indirect observation]; Schwartz et al nat(07)mar; news lfw(13)jan, pw(13)apr [observation]; Khanikaev & Genack Phy(14) [avoidance].
@ Metal-insulator transition: Chabé et al PRL(08), Sadgrove Phy(08) [in atomic matter waves]; Lemarié et al PRL(10) [critical point].
@ Other applications, experiments: Aspect & Inguscio PT(09)aug, White et al a1911 [of ultracold atoms]; > s.a. photons.
@ Related topics: Evers & Mirlin RMP(08) [Anderson transitions, review]; Strybulevych et al nPhys(08)oct [for ultrasound]; Damanik & Stolz JRAM-a0912 [1D localization, Kunz-Souillard approach]; Tautenhahn JSP(11)-a1008 [on locally finite graphs]; Izrailev et al PRP(12) [1D weakly disordered systems]; Yusipov et al PRL(17)-a1612 [in open systems].

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

Angular Diameter Distance
* Idea: A distance measure used in astronomy and cosmology, defined as the ratio of an object's physical transverse size to its angular size (in radians).
> Online resources: see Wikipedia page.

Angular Momentum

Anholonomy > see holonomy / types of constrained systems [non-holonomic]; geometric phase.

Anhomomorphic Logic > see logic.

Anisotropy of Spacetime > s.a. cosmological principle [(an)isotropy of spacetime on cosmological scales]; Isotropic Metric.
* Idea: A possible direction dependence of local properties of spacetime, such as the speed of light; A particular case of Lorentz invariance violation.
@ General references: Ruebenbauer IJTP(80); Müller et al PRL(03)phy; Ahmed et al IJP(12)-a1011; Castaño & Hawkins a1103-wd [theoretical argument for isotropy].
@ Tests: Mamone-Capria FP(11)-a1008 [special relativity and experimental tests of spatial isotropy]; Appleby & Shafieloo JCAP(14) [local anisotropy, method of smoothed residuals]; Ramazanov & Rubtsov PRD(14)-a1402 [with WMAP9 data]; Chang et al MPLA(14)-a1405 [constraints from supernovae and GRBs]; Bengaly et al ApJ(15)-a1503, Lin et al MNRAS(16)-a1504 [using type-Ia supernovae]; Saadeh et al MNRAS(16)-a1604, PRL(16)-a1605 [using the cmb]; Shiraishi et al a2009 [using galaxy surveys]; Mann et al a2011 [anisotropic quantum gravity corrections to low-energy physics]; > s.a. constants [speed of light]; phenomenology and tests of lorentz invariance; special relativity.

ANITA (Antarctic Impulsive Transient Antenna) > s.a. cosmic rays.
* Idea: A NASA-sponsored long-duration balloon payload used for cosmic-ray detection.

Annihilation Operator > s.a. fock space.

Annulus Conjecture > see spheres.

Anomalies in Quantum Theory > s.a. chiral and trace anomalies.

Anomalies in Scientific Data > see Discovery.

ANTARES > s.a. astronomy [multimessenger]; neutrino experiments.
* Idea: An astrophysical neutrino detector (Astronomy with a Neutrino Telescope and Abyss environmental RESearch).
@ References: Montaruli ap/02-conf; Katz EPJC(04)ap/03-proc; Korolkova et al NPPS(04)ap; Becherini JPCS(06)ap; Kouchner a0710-conf; Giacomelli in(09)-a0812; Carminati a0905-proc; Brown et ANTARES AIP(09)-a0908; Antares a0909-wd [search for point sources]; Coyle a1002 [status and first results]; Eberl eConf-a1205 [first results]; Zornoza & Zúñiga a1209-proc; Mangano AIP(14)-a1310 [results].

Anthropic Principle

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

Antieigenvalue Theory
* Idea: The antieigenvectors of a matrix or operator A are the vectors most turned by A.
@ References: Gustafson 11 [antieigenvalue analysis].
> Online resources: see Encyclopedia of Mathematics page; Wikipedia page.

Antichain > see posets.

Anticommutation Relations > s.a. Commutation Relations; Pseudoclassical Systems.
@ Canonical: Dereziński LNP(06)mp/05 [representations].
@ Anticommuting field variables: Jora a1602 [vacuum energy]; > s.a. supersymmetric field theories.

Antiferromagnetism > see coupled-spin models; ising models.

Antigravity > s.a. gravity theories; gravitational energy / massive gravity; test-particle motion.
* 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); Mannheim FP(00)gq; Matilsky ap/00/ApJL; Quirós gq/04 [and the cosmological constant]; Hossenfelder PLB(06)gq/05 [anti-gravitating fields], criticism Noldus & Van Esch PLB(06); Minkevich APPB(07)gq/05 [extreme conditions]; Perkowitz pw(09)jan; Hossenfelder AIP(10)-a0909 [possible extension of the classical theory]; Quirós a1409 [symmetry relating gravity and antigravity]; Chardin & Manfredi a1807-proc.
@ Specific systems: Hajdukovic gq/06 [proposed test with antiprotons]; Hossenfelder gq/06 [and cosmology]; Luongo & Quevedo a1005-MG12 [near naked singularities, and invariant definition]; Abramowicz & Lasota a1608 [not from gravitational wave emission]; > s.a. types of singularities.
@ In specific theories: Scherk PLB(79), in(79) [from supergravity]; Gershtein et al TMP(05) [in "relativistic theory of gravitation", and singularity avoidance]; Bars & James PRD(16)-a1511 [Weyl-invariant Standard Model coupled to General Relativity]; McGruder & VanDerMeer a1801 [Johannes Droste's 1916 PhD dissertation]; Klinkhamer & Queiruga PRD(18)-a1803 [from a spacetime defect].
@ Related topics: Felber gq/06-conf [propulsion and hyperdrive].

Antimatter > s.a. matter / early-universe cosmology.

Antirelativity > see special relativity.

Antisymmetrization > see tensors.

Antonov Instability > see gravitational thermodynamics.

Anyons > s.a. generalized particle statistics; quantum computing.
* Idea: A type of quasiparticle of arbitrary spin that may arise in 2D systems, for example as low-energy excitations of topologically ordered phases; Also, graphs are known to support anyons of an even richer variety than 2D space; > s.a. graphs in physics.
* Phenomenology: Abelian anyons have been detected and play a major role in the fractional quantum Hall effect; Anyons could also lead to technology for storing quantum information.
@ References: Buisseret EPJP(17)-a1605 [in quantum theories with a minimal length]; Burton a1610 [and modular functors]; Papić et al PRX(18) [imaging with scanning tunneling microscopy]; Esmaeilifar et al a1811 [relativistic quantum information].
@ In 3D: Teo & Kane PRL(10) + Stern & Levin Phy(10); Vijay & Fu a1706 [new approach].
@ Experiments: news sn(20)apr [evidence as quasiparticles in 2D materials]; news sn(20)jul [new evidence].
> Online resources: see Wikipedia page.

Apollonian Circles > see spheres.

Apollonian Gasket / Circle Packing
* Idea: A fractalconsisting of a set of circles tangent to each other and (almost) filling a larger circle.
> Online resources: see Wikipedia page.

Apparent Horizon > see horizons.

Approach Space
* Idea: A structure introduced by R Lowen as a natural generalisation of both topological and metric spaces; It is based on point-to-set distances, as opposed to point-to-point distances.
@ References: Lowen 97; Banaschewski T&A(06) [sober]; Brümmer & Sioen T&A(06) [asymmetry and bicompletion].
> Online resources: see Wikipedia page.

Approximation Methods
* Idea: An approximation is an inexact description of a physical system.
@ General references: Norton PhSc(12) [approximation vs idealization].
@ In mathematics: Villiers 12 [for continuous functions].
> In mathematics: see Diophantine Approximation; Factorial Function; functional analysis; Galerkin Approximation; Padé Approximant; Perturbation Methods; Polynomials [polynomial approximations of functions]; Steepest-Descent Approximation; Stirling Formula.
> Specific approximations in gravity: see Einstein-Infeld-Hoffmann Approximation; gravitational phenomenology [PN approximation]; Quadrupole Formula.
> Other approximations in physics: see Adiabatic Approximation; Bethe-Peierls Approximation; Born-Oppenheimer Approximation; Cluster Expansion; Eikonal Approximation; Geometric Optics; Hartree-Fock Approximation; Mean-Field Theory; One-Loop Approximation; Perturbation Methods; Proximity-Force Approximation; Stationary-Phase Approximation [or saddle-point]; WKB Approximation.

Apse, Apsidal Angle
@ References: Santos et al PRE(09).

Archimedean Lattices > see 2D ising model.

Archimedean Property > s.a. Non-Archimedean Structure.
* Idea: The property of having no infinitely large or infinitely small elements.
> Online resources: see Wikipedia page.

Arcwise (or Pathwise) Connected Space > see connectedness.

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

Area Law > s.a. laws of black-hole dynamics; Holographic Screen.
@ References: Mozgunov a1708 [entanglement in many-body localized systems].

Area Metric > s.a. differential geometry [generalizations].
@ References: Stritzelberger a1905-MG15 [electromagnetism, birefringence and corrections to the Schwarzschild metric].
> Phenomenology: see cosmology in modified gravity; cosmological acceleration in modified gravity; gravitational lensing; relativistic particles; unified theories.

Area-Preserving Map > see Poincaré Recurrence.

Argument Principle (a.k.a. Cauchy's argument principle) > see Wikipedia page; PlanetMath page; MathWorld page.

Aristotle / Aristotelian Physics > s.a. history of physics, classical mechanics, astronomy, and cosmological models; physical theories.
@ References: Rovelli JAPA(15) [as an approximation to Newtonian physics].
> Related topics: see Emergence; Explanation; logic; spacetime and spacetime structure; special relativity [Aristotelian universal friction]; vacuum.

Arithmetic > s.a. symmetries [relativity of arithmetics].
* Multiplication: 2019, The fastest method for multiplying integers predicted nearly 50 years ago is O(n log n), where n is the number of digits; Simple multiplication methods are O(n²), but a new method that is O(n log n) has just been announced; Multiplication of huge numbers is useful for certain detailed calculations, such as finding new prime numbers with millions of digits or calculating π to extreme precision.
* 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.
@ References: news sn(19)apr [faster algorithm for multiplying large integers].
@ Variations: Rotman ThSc(97)nov [non-Euclidean].

Arithmetic Geometry
* Idea: The part of Algebraic Geometry connected with the study of algebraic varieties over arbitrary rings, in particular over non-algebraically closed fields; It lies at the intersection between classical algebraic geometry and number theory.
@ References: Colliot-Thélène et al ed-10.

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; Cheng & Yang JDG(09); Efthymiopoulos & Harsoula PhyD(13)-a1302 [speed of diffusion].
@ Estimate of time of stability: Nekhorossiev RMS(77).

Arnold Transformation
@ References: Aldaya et al JPA(11)-a1010 [quantum version].

Arrangement Field Theory > see emergent gravity.

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; Li et al T&A(12).
@ Lorentzian version: Noldus CQG(04)gq/03.

Ashtekar Variables > see connection formulation of canonical gravity.

Ashtekar-Horowitz Model > see dirac quantization.

Associated Bundle > see fiber bundle.

Associative Operation > see sets.

Asteroids > see solar system objects.

ASTROD Mission > s.a. gravity tests; modified newtonian gravity; space-based gravitational-wave detectors.
* Idea: Astrodynamical Space Test of Relativity using Optical Devices, a series of missions for testing relativity in space.
@ References: Ni IJMPD(13)-a1212 [ASTROD GW, overview]; Selig et al IJMPD(13)-a1212 [ASTROD I].

Astrometry > see stars / astronomy; Gaia Mission; astrophysical tests of general relativity.

Astronomical Unit > see earth [orbit].

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; modified approaches to quantum gravity.
@ References: Asorey et al a2012 [in higher-derivative gauge theories].

Asymptotic Safety > see renormalization of gauge theories [QED] and other types of theories; asymptotic safety in quantum gravity.

Asymptotic Silence > see loop quantum cosmology.

Asymptotic Simplicity > similar to aymptotic flatness.

Atiyah-Singer Theorem > see Index Theorem.

Atomic Physics > s.a. atomic elements.

Attenuation > see wave phenomena.

Attractor
* 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; examples are found in the Lorenz system and the Rössler system).
@ General references: Milnor CMP(85); Gobbino Top(01) [topology]; Podvigina & Ashwin Nonlin(11)-a1008 [local attraction properties]; Komech & Kopylova a1907 [Hamiltonian non-linear partial differential equations].
@ Strange attractors: Sprott 93, PLA(94); > s.a. chaos; quantum computing [simulations].
@ 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-proc [for physicists].

Automorphism > see category.

Auxiliary Field
* Idea: In field theory, a field entering the action without any derivative, so that it does not propagate.

Auxiliary-Field Method > see schrödinger equation.

Averaging Physical Quantities > see tensor fields.

Averaging Problem in Cosmology

Avogadro's Number > see physical constants.

Axial Gauge > see gauge.

Axial Symmetry > see axisymmetry.

Axial Vector (a.k.a. pseudovector) > see vector.

Axino > s.a. axion.
* Idea: The fermionic superpartner of the axion; It is a well-motivated candidate for cold dark matter if it is the lightest supersymmetric particle.
@ References: Covi et al PRL(99) [as cold dark matter]; Freitas et al JHEP(11) [constraints from cosmology and tests at colliders].

Axioms for Physical Theories > see axiomatic quantum field theory; axioms for quantum mechanics; physical theories; special relativity.

Axiom of Choice
* Idea: Making an arbitrary choice of one element from each of an infinite collection of sets is a valid way to construct a set.
* Status: It is regarded by mathematicians with suspicion in a way that most mathematical axioms aren't; In 1938 Kurt Gödel proved that the axiom of choice can't be disproved from the other axioms of set theory, and in 1963 Paul Cohen proved that it can't be proved from the other axioms either; There are versions of set theory in which the axiom of choice holds and versions in which it doesn't, and they're equally consistent.
$ 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 θ: 2^X \ {∅}→ X, such that for all Y in 2^X \ {∅}, θ(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; > s.a. lorentzian geometry; Wikipedia page.
* Other statements: It is equivalent to the well-ordering principle (> see Well-Ordered Set).
@ References: Gödel 40; Moore 82; Rubin & Rubin 85; blog pt(20)may [nice intro].

Axion > s.a. dark matter types.

Axisymmetric Solutions, Spacetimes

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