**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. anomalous acceleration [Pioneer
effect]; Fermi
Acceleration.

**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.

**Accumulation Point**

$ __Def__: Given a
topological space (*X*, *τ*) and a sequence {*x _{n}*},
a point

**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.

**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; Hardy CP(98)
[in quantum mechanics]; Sidharth in(99)gq/98;
Henry SHPSA(11),
Ducheyne SHPSA(11)
[Newton's views].

@ __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].

@ __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 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* resp, 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.

@ __References__: Petti GRG(01)-a1412
[affine defects]; von Borzeszkowski & Treder GRG(02);
Popławski
FP(09)
[Ferraris-Kijowski
theory and the cosmological constant]; Filippov a1008
[and
cosmology]; Popławski GRG(14)
[and dark energy]; Gültekin EPJC(16)-a1512
[with torsion].

**Affine Quantization**

@ __ References__: Klauder 00;
Watson & Klauder JMP(00)qp;
Klauder JPA(12)-a1108
[and affine coherent states].

> __General theory__:
see canonical quantum theory [enhanced
quantization]; formulations of quantum theory.

> __Applications,
examples__: see approaches to
canonical quantum gravity; 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.

**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].

@ __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].

**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.

**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].

**Analog Gravity** > see in emergent
gravity.

**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.

**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.

**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].

@ __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].

@ __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].

@ 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 fractal
consisting 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** > see relativistic
particles; differential geometry
[generalizations]; cosmological
models; theories of cosmological
acceleration; 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].

* __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)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; 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.

**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].

@ __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}
\ {Ø}→ *X*, such that for all *Y* in 2^{X}
\ {Ø}, *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 2 sep
2017