**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].

@ __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*^{−1}*A*^{†}*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__:

- S^{n−1}
⊂ E^{n},* f* :
S^{n−1} → *p*, constant
function: *Z*_{f}
= E^{n} ∪_{const}
*p* ≅ S^{n};

- S^{n−1}
⊂ E^{n}, *f*
: S^{n−1}
→ S^{n−1}, identity:
*Z*_{f}
= E^{n} ∪_{id}
S^{n−1} ≅ E^{n}
(as cell complex);

- S^{n−1}
⊂ E^{n}, *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 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.

**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].

@ __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.

> __Applications,
examples__: see approaches to
canonical quantum gravity; brans-dicke theory;
FLRW and minisuperspace
quantum cosmology; quantum systems.

**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.

**Agravity** > see gravitation; approaches to quantum gravity.

**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 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 a1904 [geometry].

@ __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 a1812 [gravitational thermodynamics].

**Alexandrov Topology**
> see spacetime topology.

**Alfvén Waves**

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

**Algebraic 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].

@ __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**

**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; Monte
Carlo 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 *e ^{i}*

**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].

**Analog 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.

**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.

@ __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 [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.

**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];
> 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 ap/04-proc;
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].

**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}
(*Q*^{2}
+ *P*^{2})^{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.

* __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 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].

> __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*(*n*^{2}), 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.

**Artificial Intelligence** > see computation.

**Aschenbach Effect** > see kerr spacetime.

**Ascoli-Arzela Theorem
***

$

@

@

**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.

**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.

**Asymptotic Safety** > see renormalization of
specific 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**

$ __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.

**Axion**
> s.a. dark matter types.

**Axisymmetric Solutions, Spacetimes**

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 18 jul 2019