**Topics, C**

**C Operator** > see Charge Conjugation.

**C-Theory** > see theories of gravity.

**C*-Algebra** > s.a. Grupoid;
Inductive System; lie group [groupoid];
operator theory; tiling.

$ __Def__: A
norm-closed subalgebra of the space \(\cal B\)(\(\cal H\)) of bounded
operators on a Hilbert space \(\cal H\), stable under the adjoint operation.

* __Representations__:
Any C*-algebra can be represented as the algebra of bounded operators on a Hilbert space.

* __Real C*-algebras__:
Used in the classification of manifolds of positive scalar curvature, in
representation theory, and in the study of orientifold string theories.

* __In physics__:
Causal reversibility is related to the fact that the observables of a
quantum theory form a real C*-algebra, which can be represented as an
algebra of operators on a real Hilbert space; Locality and separability
then impose the restriction to complex C*-algebras and complex Hilbert spaces.

* __Relationships__:
One is canonically defined by a Lie groupoid.

@ __General references__: Sakai 71;
Dixmier 77; Pedersen 79;
Douglas 80 [extensions];
Goodearl 82;
Ruzzi & Vasselli CMP(12)-a1005 [nets of C*-algebras, representations];
Rosenberg a1505 [real C*-algebras, structure and applications];
Lindenhovius IJTP(15)-a1501 [classification by posets of their commutative C*-subalgebras];
Chu in Bullett et al 17.

@ __Applications__:
Solovyov & Troitsky 00 [K-theory, and non-commutative differential geometry];
Odzijewicz mp/05 [polarized, and quantization].

@ __In physics__: Keyl IJTP(98) [and spacetime structure];
Landsman mp/98-ln [intro];
David PRL(11)-a1103 [and causal reversibility, etc];
Buchholz et al LMP(15)-a1506 [for the electromagnetic field];
> s.a. quantum
field theory and algebraic approach.

> __Online resources__:
see Wikipedia page.

**Cabibbo Angle** > see standard model.

**Cabibbo-Kobayashi-Maskawa Matrix** > see under CKM Matrix.

**Cahill-Glauber Formalism** > see quantum mechanics in phase space.

**Calabi-Yau Space** > s.a. solutions in general relativity.

$ __Def__: A compact,
3D complex manifold with a Ricci-flat Kähler metric.

* __Applications__:
They are used as (compact) internal spaces for 10D string theory, compactified to 4D.

@ __References__: Horowitz in(86);
Hübsch 92; Fré & Soriani 95;
Baez ht/05
[10D, and standard model]; De Bartolomeis & Tomassini IJGMP(06) [generalized].

**Calculating Theorem** > see fundamental group.

**Calculus** > s.a. analysis;
Derivatives; integration;
matrix calculus; series.

@ __References__: Moskowitz & Paliogiannis 11 [several variables].

@ __Generalizations__: Bazunova et al LMP(04) [ternary algebras];
Harrison mp/05-ln [geometric];
> s.a. fractional calculus;
Stochastic Calculus.

**Calculus of Variations** > see variational principles.

**Caldeira-Leggett Model**
> see Damped Systems.

* __Idea__: A simple
system-reservoir model that can explain the basic aspects of dissipation
in solid state physics; In the high-temperature and weak-coupling limit it
can also account for quantum Brownian motion.

@ __References__: Caldeira & Leggett PhyA(83);
Ramazanoglu JPA(09)-a0812 [approach to equilibrium];
Sels et al PhyA(13) [propagator for the reduced Wigner function];
Benderskii et al PLA(13) [revivals];
Kovács et al AP(17)-a1603 [quantum-classical transition].

**Calendars** > see clocks.

**Callan-Symanzik Equation** > see renormalization group.

**Calogero, Calogero-Moser, Calogero-Sutherland Model** > see types of integrable systems.

**Caloric Theory** > see heat.

**Calugareanu's Theorem** > see Ribbons.

**Campbell-Magaard Theorem** > see embeddings.

**Canonical Coordinates** > see symplectic manifold.

**Canonical Distribution / Ensemble**
> see states in statistical mechanics; quantum
statistical mechanics.

**Canonical Form on a Lie Group** > see differential forms.

**Canonical Formulation of Dynamics**

> __In general__:
see hamiltonian dynamics; hamiltonian
systems; canonical quantum mechanics.

> __Specific types of
theories__: see canonical general relativity;
canonical quantum gravity; Double Field Theory.

**Canonical Function Method** > see schrödinger
equation [solution of radial equation].

**Canonical Quantization** > see canonical
quantum mechanics; approaches to quantum field theory.

**Canonical Transformation**

> __In mathematics and classical theory__:
see hamiltonian dynamics; symplectic structure.

> __In quantum theory__:
see canonical approach to quantum theory; states in quantum mechanics.

**Cantor Dust / Set** > see fractal.

**Cap Product**

* __Idea__: A product
involving homology and cohomology classes.

@ __References__: in Maunder 72.

**Capacitor** > see electricity.

**Capacity, Tensor** > see tensor fields.

**Cardassian Expansion** > see cosmology
in modifed gravity theories.

**Cardinal Number / Cardinality**
> s.a. Continuum; Infinite.

$ __Def__: The
cardinality of a set *X* is the number of its elements, [*X*].

* __Examples__: The
first infinite cardinal is [\(\mathbb N\)] = ℵ_{0};
The first uncountable one is ℵ_{1}; Notice
that [\(\mathbb R\)] = *c* > ℵ_{1}.

**Cardy-Verlinde Formula** > see entropy bound.

**Carmeli Metric** > see gravitating matter.

**Carnot Cycle **> see thermodynamical systems.

**Carroll Group** > see asymptotic flatness at null infinity.

**Cartan Geometry / Space**
> s.a. geometry.

* __Idea__: A local
form of Klein geometry, in which the tangent space of (pseudo)-Riemannian
geometry is replaced by one of the homogeneous spaces of Klein geometry,
and the structure is characterized by a Cartan connection which tells us
how elements of that space are parallel transported along curves on the
base space; One can describe the geometry by a *G*-bundle, but in
reality one just needs a principal *H*-bundle, where *H*
is the stabilizer group.

* __Applications__:
Effectively used in the MacDowell-Mansouri approach to 4D gravity, and in
the approaches to 3D gravity as a Chern-Simons theory.

@ __General references__: Peyghan & Tayebi a1003 [metric and Kähler structure];
McLenaghan & Smirnov 12
[and Hamilton-Jacobi theory of separation of variables];
Westman & Zlosnik AP(15)-a1411
[the physics of Cartan gravity];
Hohmann JMP(16)-a1505
[symmetry-generating vector fields].

@ __Cartan formulation of general relativity__: Barnich et al a1611-proc.

**Cartan Structure Equation** > see affine connection.

**Cartan Subalgebra of a Lie Algebra**

$ __Def__: Its maximal commuting subalgebra.

**Cartan-Randers Systems**

@ __References__: Torromé a1402,
a1506-proc
[emergence of quantum mechanics, diffeomorphism invariance and the weak
equivalence principle].

**Carter Constant** > s.a. kerr
spacetime.

* __Idea__: A conserved
quantity for motion around stationary black holes in general relativity.

@ __General references__: Rosquist et al IJMPD(09)-a0710
[physical characterization]; Komorowski et al CQG(10),
CEJP(11)-a1101
[inclined orbits].

@ __Generalized__: Will PRL(09)-a0812
[in Newtonian gravity and electrodynamics]; Isoyama et al PTEP(13)-a1302
[evolution, for an inspiraling orbit].

> __Online resources__:
see Wikipedia page.

**Carter Spacetime** > see causality conditions.

**Cartesian Coordinates** > see coordinates.

**Cascading Gravity** > a realization of the idea of Degravitation.

**Casimir Effect** > s.a. various systems and geometries.

**Casimir Operator**

@ __Spectra, group invariants__:
Gruber & O'Raifeartaigh JMP(64);
Shirokov TMP(00)mp/01.

@ __Construction__: Karadayi & Gungormez JMP(97)ht/96,
JMP(97)ht/96;
Gladush & Konoplya JMP(00)mp/99.

> __Online resources__:
see Wikipedia page.

**Casimir-Lifshitz Force**

* __Idea__: A repulsive Casimir-type force.

@ __References__: Antezza et al PRA(08)
[out of equilibrium]; Munday et al Nat(09)jan
+ new pt(09)jan
[observation]; Markus et al a1011
[and modified Maxwell equations]; Noto et al PRA(14)-a1408
[out of thermal equilibrium between dielectric gratings].

**Casimir-Polder Forces** >
s.a. QED phenomenology / quantum
field theory in curved backgrounds.

* __Idea__:
Electromagnetic fluctuation-induced forces between atoms and surfaces; The
fluctuations give rise to energy-level shifts which are position-dependent
and therefore induce forces.

@ __General references__: Buhmann et al PRA(04)qp
[non-perturbative approach], OSID(06)ap
[microscopic origin]; Buhmann & Scheel PRL(08)-a0803
[vs thermal Casimir force]; Intravaia et al LNP(11)-a1010
[rev]; Milton et al NCC-a1301-proc [three-body interactions];
Cherroret et al EPL(17)-a1701 [and dissipation].

@ __Specific situations and experiments__: Sukenik et al PRL(93)
[measurement]; Dalvit et al PRL(08) [atom
+ corrugated surface]; Contreras & Eberlein PRA(09)-a0907
[atom + dielectric slab]; Sambale et al PRA(10)-a0907
[atom + small magnetodielectric sphere]; Bender et al PRX(14)
+ news at(14)mar
[measurement]; Zhou & Yu PRA(14)-a1408
[atom out of thermal equilibrium near a dielectric substrate].

> __Online resources__:
see Wikipedia page
on the Casimir effect.

**Cassini Oval**

* __Idea__: A figure in 2D Euclidean geometry
which is the locus of points such that the product of their distances from two fixed points is a
constant (as opposed to the sum an in an ellipse);
@ see Malin Christersson page.

**Cat States** > see types of quantum states.

**Catalan Numbers** > s.a. series;
stochastic processes.

$ __Def__: The numbers
*C*_{n}, with generating
function *S*(*x*), defined by

*C*_{0} = 1
, *C*_{n}
= \((n + 1)^{-1} {2n \choose n}\) for
*n* ≥ 1 , *S*(*x*)
= ∑_{k = 0}^{∞}
*C*_{k} *x*^{k}
= [1 − (1 − 4*x*)^{1/2}] / 2*x* .

* __Generalization__:
_{p}*C*_{0}
= 1; _{p}*C*_{n}
= (1/*n*) \({pn \choose n-1}\) for *n* ≥ 1.

@ __References__: Nilsson & Sundell JMP(95)
& refs therein; Cano & Díaz a1602
[continuous analog]; Stanley 15.

> __Related topics__:
see Triangulation.

**Catalan's Conjecture** > see conjectures.

**Catastrophe Theory** >
related to phase transitions.

* __Idea__: The
combination of singularity theory and its applications; Developed by René Thom
in his 1968 *Structural Stability and Morphogenesis*, and used by Syephen Smale
as the basis of his own work on chaos theory.

* __Catastrophe__: An abrupt change
in a system, as a sudden response to a smooth change in external conditions.

@ __General references__: Poston & Stewart 78;
Marmo & Vitale FS(80);
Arnold 86;
Tilley & Lovett AJP(96)may [soap films];
Castrigiano & Hayes 03.

@ __Related topics__: Baranov G&C(11)-a1112 [and the Petrov classification of gravitational fields].

**Category Theory** >
s.a. types of categories;
categories in physics.

**Catenary**

* __Idea__: The shape
taken by a hanging chain, suspended from its two ends.

@ __References__: Behroozi EJP(14)
[from equilibrium conditions, without calculus of variations].

**Cauchy's Argument Principle** > see under Argument Principle.

**Cauchy Horizon** > see horizons.

**Cauchy Principal Part / Value
**> see distribution.

**Cauchy Problem** > s.a. initial-value
problem for general relativity.

* __Idea__: A
boundary-value problem for partial differential equations in which one
specifies the solution and its normal derivative on the boundary of the
region of interest.

@ __References__: Fattorini 83.

**Cauchy Sequence** > see sequence.

**Cauchy Surface** > s.a. Hypersurface.

$ __Def__: A closed
achronal hypersurface Σ whose full domain of dependence is the whole
spacetime *M*, i.e., *D*(Σ) = *M*, where *D*(Σ):=
*D*^{−}(Σ) ∪ *D*^{+}(Σ).

@ __References__: Bernal & Sánchez LMP(06) [smoothability and time functions];
Kim CQG(08)-a0801 [encoding of causal structure];
Stoica AUOC(12)-a1108 [for spacetimes with black-hole singularities].

**Cauchy Theorem** > see analytic functions.

**Cauchy-Riemann Condition**
> see analytic functions [holomorphic].

**Cauchy-Schwarz Inequality** > see inequalities;
quantum correlations.

**Causal Action Principle / Causal Fermion System** > see Fermion Fields.

**Causal Continuity** > see causality conditions.

**Causal Diamond** > see under Alexandrov Set.

**Causal Entropic Principle**
> see cosmological-constant problem;
large-scale geometry of the universe.

**Causal Future or Past of a Subset of Spacetime** > see
spacetime subsets.

**Causal Model** > s.a. causal
sets; networks.

* __Idea__: An abstract
representation of a physical system as a directed acyclic graph.

@ __Quantum causal models__: Pienaar & Brukner NJP(15)-a1406 [generalised graph separation rule];
Allen et al PRX(17) [and quantum common causes].

**Causal Networks** > see networks.

**Causal Structure** > see causal structure in spacetime;
indefinite causality.

**Causality** > s.a. causality
conditions; causality in quantum theory.

**Causaloids** > see indefinite causality
in quantum theory.

**Caustics** > s.a. geodesics;
light [in cosmology].

* __Idea__: Places
where light rays from the same source intersect after passing through an
optical system that bends them.

**Cavendish Experiment** > see physics experiments.

**Cayley Determinant** > see simplices [volume of tetrahedron].

**Cayley Numbers** > see Octonions.

**Cayley Tree** (s.a. Bethe Lattice)

> __Online resources__:
see MathWorld page;
Wikipedia page
(on Bethe lattice).

**CDMS (Cryogenic Dark Matter Search)** > see dark-matter searches.

**Čech Cohomology / Complex** >
s.a. types of cohomology; types
of homology.

@ __References__: Attali et al CG(13)
[Čech complexes and shape reconstruction].

**Celestial Mechanics** > see orbits
in newtonian gravity.

**Celestial Navigation**

@ __References__: Van Allen AJP(04)nov
[basic principles].

**Cell-Like Map** > same as Resolution.

**Censorship** > see cosmic
censorship; models of topology
change [topological censorship].

**Center of a Group or of a Lie Algebra**

$ __Def__: The set of
all elements which commute with all other elements, i.e., the set of *g*s
such that: *gh* = *hg* for all *h* (group), or [*g*,
*h*] = 0 for all *h* (Lie algebra).

**Center of Mass**

* __Idea__: The mean or
first moment of the mass density of a system, considered as a
distribution.

@ __Relativistic particles__: Lehner & Moreschi JMP(95);
Alba et al JMP(02)
[and rotational dynamics].

@ __And curved spacetime__: Nester et al gq/04-MGX
[teleparallel gravity], JKPS-gq/04-conf
[in general relativity, quasilocal]; Huang a1101-proc
[in general relativity].

@ __In non-commutative theories__: Helling ht/05;
Chryssomalakos et al JPCS(09)-a0901
[and non-commutative effective description of spacetime].

@ __Related topics__: Paterson et al a0707
[solution of the overhang problem].

**Central Charge**

@ __In 2+1 general relativity__: Brown & Henneaux CMP(86);
Terashima PRD(01)
[path integral derivation].

**Central Extension of a Lie Algebra / Group**
> s.a. loop group; Virasoro Algebra.

* __Example__:
Bargmann's group is a central extension of the Galilei group, motivated by
quantum-theoretical considerations.

@ __General references__: de Saxcé & Valleé JGP(10) [construction from symplectic cohomology].

@ __And physics__: Marmo et al PRD(88);
> s.a. symmetries in quantum physics.

**Central-Limit Theorem** >
s.a. probability.

* __Idea__: The sum of
a large number of statistically independent random variables is a Gaussian
random variable, independent of the individual probability distributions.

@ __General references__: Dudley 14.

@ __Generalizations__: Vignat & Plastino JPA(07) [deformed, *q*-Gaussians],
PLA(09) [non-extensive case];
Jakšić et al JMP(10)
[sums of independent identically distributed non-commutative random variables];
Calvo et al JSP(10)
[renormalization-group approach];
Leggio et al JSP(12)
[central-limit-theorem-based approximation method in statistical physics];
Fedele a1212
[spin random variables with interactions described by multi-species mean-field Hamiltonians].

**Centralizer of a Subset of a Group** > see group theory.

**Centrifugal Force** > s.a. orbits
of test bodies in gravitation.

* __Idea__: A ficticious force seen by a rotating observer.

**Čerenkov Effect / Radiation** >
s.a. radiation.

* __Čerenkov radiation__:
Radiation emitted by a charged particle moving inside a medium at a speed
greater than the speed of light in that medium, the optical equivalent of
a sonic boom; __Remark__: In a photonic crystal, it is emitted without
a speed threshold, and in the backward direction.

* __Vacuum Čerenkov
radiation__: The emission of photons by charged particles moving in a
vacuum, which would signal a photon propagation speed smaller than the
invariant speed *c*, and thus possibly modified photon dispersion
relations; 2015, It has not been experimentally observed.

@ __General references__: Jelley TPT(63); Balakin
et al CQG(01)gq/00
[and gravitational waves]; Stevens et al Sci(01)jan [subluminal];
Rohrlich & Aharonov PRA(02)qp/01 [in vacuum];
Luo et al Sci(03)jan
+ pw(03)jan
[in photonic crystal];
Afanasiev 04 [Vavilov-Cherenkov and synchrotron radiation];
Casalderrey-Solana et al PRL(10) [of mesons by quarks in the quark-gluon plasma];
Razpet & Likar AJP(10)dec [Hamiltonian approach];
Watson NPPS(11)-a1101 [history and applications];
Kaminer et al PRX(16)-a1411 [quantum theory].

@ __ Gravitational__:
Kostelecký & Tasson PLB(15)-a1508
[and Lorentz symmetry violation]; > s.a. tachyons;
phenomenology of higher-order gravity.

@ __ Vacuum Čerenkov radiation__:
Hohensee et al PRL(09) [not seen at LEP];
> s.a. phenomenology of lorentz symmetry violation.

> __In modified
theories__: see modified electrodynamics;
tests of lorentz invariance.

**CGHS Model** > see 2D quantum
gravity; black holes in modified theories.

**Chain** > s.a. homology
[algebraic geometry notion]; posets.

$ __Idea__: A totally
ordered subset of a partially ordered set.

**Chain Complex** > see homology.

**Chameleon Gravity / Scalar Field**
> s.a. Screening.

* __Idea__: A
dark-energy motivated scalar field whose mass depends on the local matter
density; It is massive on Earth, where the density is high, but
essentially free in the solar system, where the density is low; All
existing tests of gravity are satisfied, but it could lead to a different
effective *G* in space than on Earth, and violations of the
equivalence principle.

@ __Reviews, intros__: Waterhouse ap/06 [pedagogical];
Brax et al a0706-proc [primer];
Khoury CQG(13)-a1306;
Lombriser AdP(14)-a1403;
Zanzi Univ(15)-a1602;
Burrage & Sakstein LRR(18)-a1709 [tests].

@ __General references__: Khoury & Weltman PRL(04)ap/03 [gravity in space];
Nojiri & Odintsov MPLA(04)ht/03 [instability];
Brax et al PRD(10)-a1006 [field-dependent couplings];
Kraiselburd a1310-proc [and the equivalence principle].

@ __Astrophysics__: Brax & Zioutas PRD(10)-a1004 [production deep inside the sun];
Chang & Hui ApJ(11)-a1011,
Dzhunushaliev et al PRD(11) [stellar structure];
Sakstein a1502-PhD.

@ __Cosmic acceleration__: Khoury & Weltman PRD(04)ap/03;
Brax et al PRD(04)ap;
Brax et al PRD(08)-a0806 [and *f*(*R*) gravity];
Bagchi Khatua & Debnath ApSS(10)-a1012;
Hees & Füzfa PRD(12)-a1111 [cosmological and solar-system constraints];
Farajollahi & Salehi PRD(12)-a1206;
Wang et al PRL(12)-a1208 [no-go results];
Ivanov & Wellenzohn a1607 [and quintessence].

@ __Other cosmology and galaxies__: Brax et al JCAP(13)-a1303 [structure formation];
Terukina et al JCAP(14)-a1312 [and the Coma cluster];
Khosravi Karchi & Shojaie IJMPD(16)-a1401 [FLRW cosmology];
Moran et al a1408/MNRAS [at the Virgo cluster scale].

@ __Earth-based experiments__: Burrage et al JCAP(15)-a1408,
Schlögel et al PRD(16)-a1507 [atom interferometry constraints];
news UCLA(15)aug,
Sci(15)aug [bounds];
Zhang a1702 [with two cold atom clouds].

@ __Other phenomenology__: Mota & Shaw PRL(06) [viability and possible experiments];
Upadhye et al PRD(06) [and inverse-square tests];
Burrage et al PRD(10)
+ news ns(10)aug [polarization of light];
Brax et al PRD(12) [detection prospects];
Sakstein et al IJMPD(14)-a1409 [constraints from stars];
Pignol IJMPA(15)-a1503 [neutron-interferometer constraints on the chameleon theory];
> s.a. tests of newtonian
gravity [constraints].

@ __Generalizations__: Noller JCAP(12)-a1203 [field-derivative-dependent couplings];
De Felice et al GRG(18)-a1702 [chameleon bigravity].

**Chance** > see probability in physics.

**Chandrasekhar Mass Limit**
> s.a. star types; supernovae; White Dwarf.

* __Idea__: The upper
bound on the mass of a white dwarf, beyond which it cannot exist as a
stable star; It is given by

\[M_{\rm limit} = {\omega_3^0\sqrt{3\pi}\over2}\,\bigg({\hbar c\over G}\bigg)^{\!3/2}\,{1\over(\mu_{\rm e}m_{\rm H})^2}\;,\]

with \(\mu_{\rm e}\) = average molecular weight per electron (which depends on the chemical composition of the star),
\(m_{\rm H}\) = mass of the hydrogen atom, and \(\omega_3^0 \approx 2.018236\) is a constant related to the solution to
the Lane-Emden equation; Its value is of order \(M_{\rm Pl}^3/m_{\rm H}^2\), numerically approximately equal to 1.4 \(M_\odot\).

@ __References__: Gregg & Major IJMPD(09)-a0806 [changes from modified dispersion relations];
Garfinkle AJP(09)aug [and the Planck mass];
Blackman a1103/PT [history];
Das & Mukhopadhyay IJMPD(12)-a1205-GRF,
PRL(13)-a1301
+ news pw(13)feb
[strongly magnetized white dwarves can exceed the Chandrasekhar limit].

> __Online resources__:
see Wikipedia page.

**Chandra X-Ray Observatory** >see astronomy in various wavelength ranges.

**Chaos** > s.a. chaotic systems;
mathematical description of chaos; quantum chaos.

> __Field theories and
gravity__: see chaos in field theories;
chaos for gravitating bodies;
chaos in gravitational-field dynamics.

**Chaplygin Gas**
> s.a. dark energy.

* __Idea__: A gas with
an exotic equation of state, *p*_{X}
= −*A* / *ρ*^{r}_{X}
(polytropic, with negative constant and exponent).

@ __General references__: Debnath & Chakraborty IJTP(08)gq/06
[and collapsing spherical cloud]; Giannantonio & Melchiorri CQG(06)gq
[and Sachs-Wolfe effect]; Banerjee & Ghosh MPLA(06) [gravity coupling];
Banerjee et al PRD(07)gq/06
[generalized, action]; Myung ApSS(11)-a0812
[thermodynamics]; Zschoche AHP(13)-a1303
[equation of state for a quantized free scalar field]; > s.a. wormholes.

@ __And cosmology__: Fabris et al PLB(11)-a1006 [ruling out modified Chaplygin gas cosmologies]; Pereira Campos et al EPJC(13)-a1212;
vom Marttens et al PDU-a1702 [and the dark sector];
del Campo et al a1703 [vs scalar field];
Aurich & Lustig a1704 [compatibility with recent data].

@ __In loop quantum cosmology__: Zhang et al MPLA(09)-a0902;
Chowdhury & Rudra IJTP(13)-a1204
[and the cosmic coincidence problem].

**Chapman-Enskog Method** > see Boltzmann
; fluid dynamics.

**Character of a Group G**
> s.a. group representation.

$

@

**Character of an Algebra**
> s.a. Spectrum.

$ __Def__: A non-zero
algebra homomorphism *χ*: *A* → \(\mathbb C\).

**Characteristic Equation**

> __For a differential
equation__: see Wikipedia page.

> __For a matrix__:
see Characteristic Polynomial below.

**Characteristic Evolution / Initial-Value Problem** >
see initial-value problem for general
relativity; types of wave equations.

**Charge** (Mathematical Notion) > s.a. charge
[physics].

$ __Def__: A finitely
additive, extended real-valued set function defined on a field of sets.

@ __References__: Bhaskara Rao & Bhaskara Rao 83.

**Charge Conjugation** >
s.a. CPT symmetry.

@ __References__: Ramsey PR(58);
Rosen AJP(73)apr
[form
electromagnetic quantities]; Nefkens et al PRL(05)
[test
of invariance with *η* decay]; Carballo Pérez & Socolovsky EPJP(11)-a1001
[non-relativistic limit].

**Charles' Law** > s.a. gas
[ideal-gas law].

* __Idea__: The volume
of a gas of fixed mass and pressure is proportional to the gas's
temperature, *V/T* = constant when *p* = constant.

> __Online resources__:
see Wikipedia page.

**Chasles' Theorem**

* __Idea__: Any rigid
body displacement can be produced by a translation along a line followed
(or preceded) by a rotation about that line.

@ __References__: Minguzzi JMP(13)-a1205
[relativistic generalization].

> __Online resources__:
see Wikipedia page.

**Chebyshev Polynomials**

@ __Generalizations__: Chen & Lawrence JPA(02).

**Cheeger-Gromov Theory** >
s.a. riemannian geometry.

* __Idea__: The study
of the convergence and degeneration of Riemannian metrics on a given
manifold *M*.

@ __References__: Cheeger & Gromov JDG(86),
JDG(90); Anderson gq/02-proc
[in general relativity].

**Cheerios Effect**

* __Idea__: The fact
that floating clumps of solid matter on the surface of liquid tend to
cluster together.

@ __References__: Vella & Mahadevan AJP(05)sep;
Berhanu & Kudrolli PRL(10).

**Chemical Potential** >
s.a. thermodynamics.

* __Idea__: The
thermodynamical variable *μ*:= (∂*F*/∂*N*)_{V,
T }, which is the important parameter determining
the equilibrium conditions between phases or chemical components; In a
canonical ensemble with partition function *Z*, *μ*:= −*kT*
(∂ln*Z*/∂*N*)_{V, T },
and if the particles are non-interacting, such as in an ideal gas, *μ*:=
−*kT* ln(*ζ*/*N*).

* __Specific systems__:
For conduction electrons in a metal, it coincides with the Fermi energy;
Vanishes for particles in an ideal phonon or photon gas.

@ __General references__:
Cook & Dickerson AJP(95)aug;
Baierlein AJP(01)apr [meaning];
Tobochnik et al AJP(05)aug
[understanding, Monte Carlo algorithms];
Kaplan JSP(06) [correct definition].

@ __Specific systems__: Herrmann & Würfel AJP(05)aug,
Hafezi et al PRB(15)-a1405 [for light, non-zero].

**Chemistry / Chemical Reactions** > see critical
phenomena; elements; galaxies,
galaxy evolution and the
milky way galaxy.

**Cherenkov Radiation** > see under Cerenkov.

**Chern-Gauss-Bonnet Theorem** > A generalization of the
gauss-bonnet theorem.

**Chern-Simons Function**

* __Idea__: A function
defined by a connection on a 3-manifold, which is topological in the sense
that it is invariant under gauge transformations in the connected
component of the identity; It can be used as (a contribution to) the
action for a topological gauge theory.

$ __Def__: For a 3D
manifold, the function

*Y*[*A*]:= \(k\over4\pi\) tr[*A* ∧ d*A*
− \(2\over3\)*A* ∧ *A* ∧ *A*] ,

where *k* is an integer.

* __In quantum field
theory__: Exponentiated, it is known as the Kodama state for quantum
gauge theories; > see gauge theories and
quantum gravity in the connection
representation.

@ __References__: Jackiw mp/04
[as a surface integral]; Szabados CQG(02)gq/01
[and Hamiltonian constraint]; Zanelli CQG(12)
[in gravitation theories, rev].

**Chern-Weil Theory** > see characteristic classes.

**Chernov-Sinai Ansatz** > see ergodic theory.

**Cheshire Cat Effect**

@ __References__: news dm(14)jul
[measurement of a neutron's magnetic moment independently of the
particle].

**Chevalley Groups** > see finite
groups.

**Chevreton Superenergy Tensor**
> s.a. stress-energy pseudotensors.

* __Idea__: Introduced
in 1964 as an electromagnetic counterpart of the gravitational
Bel-Robinson tensor.

@ __References__: Bergqvist et al CQG(03)gq,
Edgar CQG(04)
[conservation
laws]; Bergqvist & Eriksson CQG(07)gq
[traceless, in source-free electrovac spacetime].

**Chi-Squared Test** > see statistics.

**Chiral Symmetry / Phase Transition**

* __Chiral phase transition__:
The transition in QCD from hadronic matter to the quark-gluon plasma at high
temperatures and/or net-baryon densities; It is associated with the restoration
of chiral symmetry and can be investigated in the laboratory via heavy-ion collisions.

* __Analogy__: A simple
example can be obtained with a clamped string; If you pluck it within a
plane, and the amplitude of the vibrating string is large enough, the
plane in which the string moves will actually start to rotate; For QCD,
pions are considered to be the Goldstone bosons associated with the
spontaneously broken chiral symmetry.

@ __Breaking__: Giusti & Necco JHEP(07) [lattice QCD];
Zhang et al PRL(09) [coaxial nanotubes sliding against each other];
Jora PRD(10)-a1004 [for SU(*N*) gauge theories, and restoration];
Eser et al PRD(15)-a1508 [functional renormalization group study];
> s.a. mass [generation]; theta sectors.

**Chirality / Chiral Theories** >
s.a. differential forms [chiral forms].

* __Idea__: A theory is
chiral if solutions with different handedness have different properties,
i.e., if parity symmetry is violated.

@ __General references__: Tucker & Walton a1805-proc [gravitational and electromagnetic interactions with matter].

@ __Chiral bosons__: Abreu & Dutra PRD(01)ht/00;
Abreu & Wotzasek ht/04-ch;
> s.a. self-dual gauge theories.

> __Chiral fermions__:
see lattice theories; non-commutative
gauge theories; particle models; spinors
in field theory; types of spinors / cosmologica
constant; cpt violation; higher-dimensional
gravity; kaluza-klein theory; lorentz symmetry.

> __Chiral gauge
theory__: see renormalization;
Wikipedia page.

> __Related topics__:
see parity [violation]; particle phenomenology in quantum gravity;
Topologically Massive Gravity.

**Chisholm's Theorem** > see quantum field theory formalism.

**Choquet Space**

* __Idea__: A
generalization of the notion of topological space.

$ __Def__: A
convergence space in which a filter \(\cal F\) converges to *x*
whenever every ultrafilter finer than \(\cal F\) converges to *x*.

**Christodoulou Memory Effect** > see Gravitational Memory.

**Christoffel Symbols **> see affine connection.

**Chromatic Number**

@ __References__: Soifer JCTA(05)
[dependence upon axioms for set theory].

**Chronogeometry** > s.a. models of spacetime.

* __Idea__: The
determination of spacetime geometry just using clocks and the exchange of
light signals.

@ __References__: Lusanna a0708-conf
[in general relativity].

**Chronological Future / Past of a Subset of Spacetime**
> see spacetime subsets.

**Chronological Space**

$ __Def__: A pair (*M*,
<), with < a relation obeying (i) Transitivity; and (ii) If *x*
< *x*, then there exists *y* in *M*, with *y*
≠ *x*, such that *x* < *y* < *x*.

* __Full chronological
space__: One in which, in addition, (iii) For all *p*, *q*,
*x*, *u*, and *v* in *M*, with *p*,
*q* < *x* < *u*, *v*, there exist *y*,
*z* in *M* such that *p*, *q* < *y*
< *x* < *z* < *u*, *v*; and (iv)
For all *x* in *M*, there exist *p*, *q*
in *M* such that *p* <* x* < *q*;
> s.a. Alexandrov Topology
[base].

@ __References__: Kronheimer & Penrose PCPS(67);
Carter GRG(71);
Kronheimer GRG(71);
Harris CQG(00)gq/99 [topology].

**Chronology** > see cosmological history.

**Chronology Protection** > see non-causal spacetimes.

**Chronon** > s.a. dirac fields;
spinning particles; time in quantum theory.

* __Idea__: A unit of discrete time.

@ __References__: Jackson JPA(77)-a1208.

**Chu Space** > see formulations of quantum theory.

**Church-Turing Thesis (a.k.a. Turing's Thesis)**

* __Idea__: The only
computable functions are the partial recursive ones, and they are also the
ones computable by Turing machines, or "everything algorithmically
computable is computable by a Turing machine."

@ __And physics__: Deutsch PRS(85) [quantum theory and computers];
Svozil in(98)qp/97;
Etesi & Németi IJTP(02)gq/01
[general relativity]; Szudzik LNCS(12)-a1201;
Wüthrich Syn-a1405 [and quantum gravity].

> __Online resources__:
see Stanford Encyclopedia of Philosophy page;
Wikipedia page.

**CIBER (Cosmic Infrared Background Experiment)** > see observational
cosmology.

**Circles** > see spheres.

**Cirel'son's Bound** > see bell's
inequality; quantum
correlations.

**Circularity Condition**

$ __Def__: A stationary
axisymmetric spacetime satisfies the circularity condition if the action
of the 2-parameter isometry group is orthogonally transitive.

**Circulation Theorem** > see magnetism.

**Civilizations**
(Cosmic Civilizations)

**C ^{k} Open Topology**
> see lorentzian geometries.

**CKM Matrix** (Cabibbo-Kobayashi-Maskawa)
> see CP violation; standard
model.

**Class** > see set.

**Classical Mechanics** > see approaches
and formalism; systems.

**Classicality in Quantum Theory** > see decoherence;
classical limit of quantum mechanics; relationship
between quantum and classical mechanics.

**Classicalization**

* __Idea__: The phenomenon
in which a theory prevents itself from entering into a strong-coupling regime,
by redistributing the energy among many weakly-interacting soft quanta;
In a scattering process of initial hard quanta split into a large number
of soft elementary processes, and at very high energies, the outcome is
the production of soft states of high occupation number that are approximately
classical.

@ __References__: Dvali proc(17)-a1607-ln.

**Classicalons**

* __Idea__: Classical
solutions produced in high-energy scattering processes for some theories
that would otherwise be non-renormalizable but are protected against
non-unitarity by their appearance, and may thus not need an ultraviolet
completion.

@ __References__: Bajc et al a1102
[path integral discussion].

**Classifying Space**

* __Example__: B(U(1))
= \(\mathbb C\)P^{∞}.

> __Online resources__:
see John Baez page;
Wikipedia page.

**Clauser-Horne-Shimony-Holt (CHSH) Inequalities**
> see quantum correlations / hidden
variables; Fine's Theorem.

**Clausius Inequality / Relation **
> s.a. thermodynamics [Clausius
formulation of the second law].

* __Idea__: The
inequality for a system in a bath or surrounding environment that d*S*_{sys}
≥ d*Q*/*T*, or \(\oint\) d*Q*/*T*_{bath}
≤ 0 for a cyclic process.

@ __Generalizations__: Deffner & Lutz PRL(10)-a1005 [for non-equilibrium quantum processes];
Maes & Netočný JSP(14) [overdamped mesoscopic and macroscopic diffusions].

> __Online resources__:
see everyscience page.

**Clausius-Clapeyron Equation**

* __Idea__: An equation
giving the slope d*p*/d*T* of the phase equilibrium line at
a point in the *p*-*T* plane for a substance that can exist
in different phases, as equal to the ratio Δ*S*/Δ*V* between
the change in entropy and the change in volume for some amount of
substance crossing the line at that point; Δ*S* can be expressed as
*l*/*T* in terms of the appropriate latent heat *l*.

* __Remark__: For
second-order phase transitions in which Δ*V* = 0, one uses instead
the Ehrenfest Equations.

@ __References__: Krafchik & Sánchez AJP(14)apr [second derivatives of the phase transition line, pedagogical derivations].

**Claw Graph**

* __Idea__: The
complete bipartite graph K_{(1,3)}, a tree
that is isomorphic to the star graph S_{4},
and is sometimes also known as the Y graph.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Clay Institute** > see Millennium
Problems.

**Clebsch Potential**

@ __And electromagnetism__: Wagner PLA(02)
[problems].

**Clebsch Variables** > see perfect
fluids.

**Clebsch-Gordan Theory / Coefficients**

**Clifford Group**

@ __References__: Bengtsson a1202-conf
[representation].

**Clifford Operators**

* __Idea__: A subset of
quantum operations, well studied for both the qubit and higher-dimensional
qudit systems.

@ __References__: Farinholt JPA(14)-a1307 [characterization].

**Cloak** > s.a. metamaterials
[electromagnetic "invisibility" cloak]; sound [acoustic cloak].

* __Idea (in physics)__:
Something put in place to hide or disguise the presence an object by
modifying the object's effect on radiation or its environment.

@ __General references__: Matson SA(09)aug [without metamaterials];
Pendry Phy(09) [rev].

@ __Different types__: news ns(11)jul,
wp(12)jan [time cloaking];
news sci(12)mar [for water waves];
Alù Phy(14) [thermal cloaks];
Brûlé et al PRL(14)
+ Sheng Phy(14) [seismic cloak].

**Clock Effect** > see gravitomagnetism.

**Clock Hypothesis** > see time [in special relativity].

**Cloning** > s.a. quantum technology [quantum cloning].

@ __References__: Fenyes a1010/JMP [classical].

**Closed Set** > s.a. topological space.

$ __Def__: The
complement of an open set in a topological space.

* __Rem__: The empty
set and the whole topological space are always (open and) closed.

**Closed Timelike Curves (CTCs)** > see causality
conditions; causality violations.

**Closure of a Subset**

* __Idea__: The
sequential closure of a subset *A* ⊂ *X* is the set of all
points *x* ∈ *X* for which there is a sequence in *A*
that converges to *x*.

@ __References__: Borodulin-Nadzieja & Selim T&IA(12) [sequential closure in the space of measures].

**Cloud Chamber** > see physics experiments.

**Cluster Algebra**

* __Idea__: A structure
introduced by Fomin & Zelevinsky in 2002; A cluster algebra of rank *n*
is an integral domain *A*, together with some subsets of size *n*
called clusters whose union generates the algebra *A* and which
satisfy various conditions.

@ __References__: Williams BAMS(14)-a1212.

> __Online resources__:
see Wikipedia page.

**Cluster Expansion / Variation
Method** > s.a. Mayer Series;
renormalization group.

* __Idea__: A hierarchy
of approximate variational techniques for discrete (Ising-like) models in
equilibrium statistical mechanics.

@ __General references__: Pelizzola JPA(05) [rev];
Poghosyan & Ueltschi JMP(09)-a0811 [general setting];
Miracle-Solé MPRF(10)-a1206 [short exposition with complete proofs].

@ __Special systems__: Bissacot et al JSP(10)-a1002 [for polymer gases];
Pulvirenti & Tsagkarogiannis CMP(12)-a1105
[canonical partition function for particles in a box, alternative more direct derivation];
Jansen JSP(15)-a1503 [multi-species Tonks gas];
Bastianello & Sotiriadis NPB(16)-a1601
[for ground states of local Hamiltonians].

> __In spin models__:
see coupled-spin models; ising model.

> __Variations__:
see coupled-spin models [Cluster
Mean-Field approach].

> __Other applications__:
see gravitational statistical mechanics; lattice
field theory and lattice gauge theory;
networks [cluster growth].

**Cluster Separability**

@ __References__: Damjanović & Marić FBS-nt/95 [in scattering theory];
Hájíček FP(11)-a1001,
a1003, JPCS(11)-a1011 [and the quantum measurement problem].

**Clustering** > s.a. star
clusters; gas [including cluster
expansion]; gravitating matter.

@ __References__: Janowitz 10 [modeling with posets, + CD-ROM].

**CO Space** > see types of topological spaces.

**Coalgebra**

* __Idea__: A structure that is dual
(in the category-theoretic sense of reversing arrows) to unital associative algebras.

@ __References__: Brouder mp/05-em [use in quantization];

Jacobs 16 [intro].

> __Online resources__:
see Wikipedia page.

**Coarse Structures in Geometry**

@ __References__: Roe 03;
Dydak & Hoffland T&IA(08).

**Coarse-Graining** > s.a. information;
network; renormalization;
Smoothing; thermodynamic
concepts.

* __Example__: Examples
of physical coarse-graining processes are the coalescence of small bubbles
into large ones in beer, or stars into galaxies.

* __Rem__: A typical
feature of coarsening is that the system "forgets" its initial state,
developing a statistical steady state at large time.

@ __General references__: Ridderbos SHPMP(02) [inadequate approach to statistical mechanics];
Dvurečenskij et al RPMP(05)qp/04 [of observables];
Kawasaki JSP(06) [maximum entropy and reduced dynamics];
Korzyński CQG(10)-a0908 [covariant, in curved spacetime].

@ __In quantum theory__: Anastopoulos PRD(97)ht/96 [quantum field theory, in terms of open systems];
Wetterich a1005
[and non-commutativity of position and momentum];
Radonjić et al PRA(11)-a1105 [system of non-linear oscillators];
Raeisi et al PRL(11) [and micro-macro entanglement];
Dittrich NJP(12)-a1205 [and cylindrically consistent dynamics];
Agon et al a1412 [short-distance vs long-distance degrees of freedom];
Bény a1509
[and distinguishability of field interactions];
Singh & Carroll PRA(18)-a1709
[states in a finite-dimensional Hilbert space];
> s.a. decoherence
in quantum field theory.

@ __And quantum entropy__: Gell-Mann & Hartle PRA(07)qp/06; Šafránek et al a1707.

@ __In statistical physics__: Rodríguez & Santamaría-Holek PhyA(07) [and non-extensive effects in gas of Brownian particles];
Barmak et al PRB(11)
+ Kohn Phy(11)
[statistics of grain boundaries in a polycrystalline material, irreversibility and critical phenomena];
Noorbala AP(14)-a1407 [entropy evolution and decoherence];
Kalogeropoulos Ent(15)-a1507 [behavior of different entropies];
Alonso-Serrano & Visser Ent(17)-a1704 [simple conceptual models, and entropy].

@ __Gravity-related examples__: Dittrich & Steinhaus NJP(14)-a1311 [time evolution as refining, coarse graining and entangling];
Rastgoo & Requardt PRD(16)-a1606
[geometric renormalization scheme for metric spaces and emergent spacetime];
Charles PhD-a1705 [in lqg];
Eichhorn CQG(18)-a1709 [in causal set theory];
> s.a. averaging in cosmology;
connections; covariant quantum gravity; dynamical triangulations; spin-foam models; spin networks.

> __Other specific
theories__: see lattice field theory;
lattice gauge theory.

**COBE Mission / Satellite** > see cosmic microwave background.

**Cobordism** > s.a. 2D,
3D, 4D manifolds;
discrete geometries [graph cobordism]; morse theory;
Surgery; types of categories.

* __Idea__: The study
of the interpolation between *n*-dimensional manifolds *M*_{1}
and *M*_{2} by an (*n*+1)-dimensional
*M*, with ∂*M* = *M*_{1}
∪ *M*_{2}; It can be traced to
H Poincaré, and in its modern form to L Pontrjagin.

* __Terminology__: The
field is now usually referred to as "bordism", and the term "cobordism" is
reserved for a cohomology-type theory introduced by Atiyah from a
homology-type theory constructed using bordism; Here we will still call
cobordism the equivalence relation among manifolds defined by the
condition that there be a bordism between them.

* __Equivalent condition__:
*M*_{1} and *M*_{2}
are cobordant by a smooth compact manifold iff all their Stiefel-Whitney numbers agree.

* __Properties__: Any
smooth cobordism admits a Morse function; Any smooth cobordism can be
decomposed into a union of cobordisms each of which has Morse number
(minimal number of critical points) equal to 1.

* __Oriented cobordism__:
Requires that *M*_{1}, *M*_{2}
and *M* be oriented, with compatible orientations, and the
equivalent condition in terms of characteristic classes includes equality
of their Pontrjagin numbers.

* __Cobordism classes__:
The equivalence classes of *n*-dimensional cobordant manifolds is
a \(\mathbb Z\)_{2}-algebra, in which the
addition is disjoint union and the product is Cartesian product.

* __In physics__:
Important for the study of possible topology changes in spacetime; >
s.a. topology change; topological
field theories.

@ __General references__: Stong 58;
Milnor 65; Peterson 68;
Landweber MPCPS(86);
Vershinin 93;
Atiyah BAMS(04) [Thom's cobordism theory];
in Wall 16.

@ __Lorentzian__: Larsson MS(14)-a1406;
Smirnov & Torres a1804 [specific types, necessary and sufficient conditions].

> __Online resources__:
see Wikipedia page;
Encyclopaedia of Mathematics page.

**Cochain** > a concept in cohomology
theory; s.a. Triangulations.

**Coefficient of Restitution**

@ __References__: Ferreira da Silva EJP(07) [concept].

**Coevent** > see histories
formulations of quantum theory; measures [quantum measures].

**Cogravity**

@ __And perihelion precession__: de Matos & Tajmar gq/00.

**Coherence**
> s.a. Interference [measuring coherence].

* __Idea__: Two
(particles or) waves are coherent when the phase difference between
them is constant, over a certain time and/or spatial region.

* __In classical
mechanics__: For a wave, Glauber defined a notion of degree of
coherence based on whether certain *n*-th order correlation
functions vanish.

* __In quantum mechanics__:
For a wave function *ψ* and points *x* and *y*,
the mutual coherence is the 2-point function Γ(**x**, **y**;
*τ*):= \(\langle\)*ψ**(**x**, *t*)
*ψ*(**y**, *t*+*τ*)\(\rangle\)*;
> s.a. coherent states.

@ __In optics__: Picozzi & Haelterman PRL(02) [hidden coherence];
Wolf 07.

@ __In quantum mechanics__: Slosser & Meystre AJP(97)apr [quantum optics, RL];
Ponomarenko et al PLA(05) [optical field, significance];
Cavalcanti & Reid PRL(06)qp/07 [criteria for macroscopic coherence];
Sewell a0711-en [in quantum statistical mechanics, survey];
Silverman 08;
Pei et al a1011,
Yu et al QIP(14)-a1402 [relationship with quantum correlations];
Winter & Yang PRL(16)-a1506,
Singh et al a1506 [operational theory];
Xu PRA(16)-a1510 [of Gaussian states];
Giorgi & Zambrini a1706 [unified framework for coherence and correlations];
Sperling et al PRA(17)-a1707 [for indistinguishable particles];
Tan et al PRL(17)-a1703
+ news PhysOrg(17)nov [and non-classicality of light];
> s.a. generalized thermodynamics.

@ __As a physical resource__: Streltsov et al RMP(17)-a1609 [rev];
Ben Dana PRA(17)-a1704.

@ __Evolution, loss of coherence__: Hu & Fan SRep(16)-a1512 [evolution equation];
Bu et al PRA(16)-a1608;
> s.a. decoherence.

@ __Measures of quantum coherence__: Baumgratz et al PRL(14)-a1311;
Streltsov et al PRL(15)-a1502
+ news PhysOrg(15)jun [measured with entanglement];
Frérot & Roscilde PRA(16)-a1509 [quantum variance, for many-body systems];
Rastegin PRA(16)-a1512 [based on Tsallis relative entropies];
Napoli et al PRL(16)-a1601 [robustness];
Chen et al PRA(16)-a1601 [pure quantum states];
Marvian & Spekkens PRA(16)-a1602;
Yue et al a1605 [for a superposition of two pure states];
Jin & Fei PRA(18)-a1806 [based on the Hellinger distance].

**Coherent States** > s.a.
generalized and modified states; types of coherent states.

**Cohomology** > s.a. types of cohomology.

**Coincidences in Physics**
> see cosmology [coincidence problem].

@ __References__: Frampton & Nielsen a1704
[two unexplained coincidences in particle physics and gravity].

**Coisotropic Submanifold** > see symplectic structure.

**Cokernel**

$ __Def__: The cokernel of a (group) homomorphism *f* :
*G* → *H* is Cok(*f*):= *H* / *f*(*G*).

**Colbeck-Renner Theorem**

* __Idea__: "No
alternative theory compatible with quantum theory and satisfying the
freedom of choice assumption can give improved predictions".

@ __References__: Landsman JMP(15)-a1509
[more precise version of the formulation and proof].

**Cold Fusion** > see nuclear technology.

**Cold Spot** > see cmb anisotropy.

**Coleman-Mandula Theorem**

* __Idea__: If the *S*-matrix
is based on a local 4D non-relativistic quantum field theory, there is
only a finite number of particles of a given mass, and there is an energy
gap between vacuum and the 1-particle states, then the most general
connected group of symmetries of the *S*-matrix is locally a
direct product of an internal symmetry group and the Poincaré group.

* __Remark__: Prevents
spacetime symmetries from being unified with internal ones, as some
unification ideas would want; Can be circumvented in the presence of a
cosmological costant, as in some proposals for unified theories (Lisi's
E8, Smolin), or replacing the Lie algebra of symmetries by a
supersymmetric or graded one, as in the Wess-Zumino model.

@ __General references__: Coleman & Mandula PR(67).

@ __Variations, generalizations__: Pelc
& Horwitz JMP(97) [higher-dimensional];
Lovelady & Wheeler PRD(16)-a1512 [alternative gauging of a simple group];
Fewster CMP(17)-a1609 [for quantum field theory in curved spacetimes].

**Coleman-Weinberg Effect**

@ __References__: Floreanini et al CQG(91) [in quantum gravity].

**Collapse** > see gravitational
collapse; wave-function collapse.

**Collineations** > s.a. affine
structures; FLRW models; symmetries.

* __Curvature
collineation__: A vector field on a manifold such that the Lie
derivative of the Riemann tensor along it vanishes.

* __Projective
collineation__: A vector field generating a local group of
geodesic-preserving diffeomorphisms.

@ __Curvature collineation__: Katzin et al JMP(69),
JMP(70);
Hall & Shabbir G&C(03) [spacetime examples];
Shabbir G&C(03) [Bianchi I];
Kashif & Saifullah a1005-MG12
[and Weyl collineations].

@ __Projective collineation__: Hall & Lonie CQG(95) [on spacetime].

@ __Matter collineations__: Sharif NCB(01)gq/05 [Bianchi I, II, III, VIII, IX, Kantowski-Sachs];
Qadir & Saifullah MPLA(09)-a1005;
> s.a. bianchi models.

**Collisions** > see scattering.

**Colloids** > see condensed
matter [soft matter]; entropy.

**Colombeau Algebra**

* __Idea__: A space of
generalized functions, more general than distributions, for which a
multiplication is defined.

@ __General references__: Gsponer a0807 [intro];
Gsponer EJP(09)
[and electrodynamics].

@ __Diffeomorphism-invariant__: Steinbauer in(04)m.FA/01;
Grosser in(04)m.FA/01;
Kunzinger in(04)m.FA/01.

@ __Applications__: Kamleh gq/00 [and signature change];
Gsponer JMP(08)-a0806
[and pointlike electrons]; Colombeau et al a0705,
Colombeau & Gsponer a0807
[quantum field theory]; > s.a. general
relativity solutions with matter; representations
in quantum theory; types of metrics.

**Color** > see light; QCD
[as a quantum number]; QCD effects [confinement].

**Colored Tensor Models**

@ __References__: Gurau & Ryan Sigma(12) [rev].

**Coloring Problems** > s.a. Four-Color Theorem.

* __Problem (Halmos)__:
Given any coloring of the plane by *n* colors, in which each point
is colored independently, can one always find two points exactly 1 cm
apart (say) with the same color?

* __Answer__: For *n*
= 2, yes (just consider the vertices of an equilateral triangle of edge length = 1); For *n* =
3, yes (circle of radius √3,
...?); For *n* = 7, no (can tile the plane with hexagons of
diameter 0.9, colored so that no two adjacent ones have the same color);
Unknown for *n* = 4, 5, 6 (as of 1986).

@ __References__: Di Francesco BAMS(00).

**Comb Space** > see types
of topological spaces.

**Combinatorial Geometry**
> see combinatorics.

**Combinatorial Group Theory**

* __Idea__: A group
theory based on words, generators and presentations.

* __History__: It
emerged in the 1880s from complex function theory with Klein, Fricke and
Poincaré.

@ __References__: Stillwell 80;
Cohen 89; Johnson 89.

**Combinatorial PDEs**

* __Idea__: Cochains
defined on chains.

@ __References__: in Grady & Polimeni 10.

**Combinatorial Topology**

* __Idea__: A type of
algebraic topology that uses combinatorial methods; It includes simplicial
homology.

@ __References__: Pontrjagin 52;
Aleksandrov 56; Prasolov 06;
Kozlov 08.

**Common Cause Principle** >
see causality; causality
in quantum theory / algebraic quantum
field theory.

**Communication** > see quantum communication.

**Commutant of a Group** > see group.

**Commutation Relations,
Commutators** > s.a. matrices.

* __Useful relationships
for matrices__: For powers of matrices/operators, [*M*^{
n}, *A*] = ∑_{i
= 1}^{n} *M*^{
i−1} [*M*,* A*] *M*^{
n−i}.

* __In quantum mechanics__:
The standard commutation relations are the ones defining the Heisenberg
Algebra; Quantum gravity considerations motivate a modified form for the
basic commutation relations that depends on a parameter *β*, in
terms of which

[*x*_{i}, *p*_{j}]
= i ℏ [1 + *β* (*p*/*m*_{P}*c*)^{2}]
δ_{ij} .

@ __In quantum mechanics__: Luis JPA(01)
[as a geometric phase]; Sergi PRE(05)qp
[non-Hamiltonian]; Tangherlini PS(08)
[covariant]; Ercolessi et al RNC(10)-a1005
[and equations of motion]; Pain JPA(13)-a1211
[commutators of operator monomials].

@ __In quantum mechanics, representations__: Mnatsakanova et al PoS-a1102
[regularity criterion]; Jorgensen & Tian a1601
[and stochastic calculus operators].

@ __In quantum mechanics, modified__: Pikovski et al nPhys(12)mar
+ news pt(12)mar [test using quantum optics];
Aste & Chung ASTP(16)-a1312
[Klein transformations changing anticommutation relations into commutation
relations]; D'Andrea et al Sigma(14)-a1406
[and metric structures]; > s.a. non-commutative
geometry.

> __In quantum
mechanics__: see annihilation and creation
operators; computer languages
[with Mathematica]; Heisenberg Algebra;
observable algebras; uncertainty
relations; Weyl Algebra.

> __In quantum
mechanics, modified__: see annihilation
operators; deformation quantization; modified
quantum theory; modified
uncertainty relations.

> __In quantum field
theory__: see annihilation and creation
operators; approaches to quantum
field theory [covariant]; photons.

**Compact Astrophysical Objects** > see star
formation and evolution.

**Compact-Open Topology**

* __Idea__: A topology
defined on the set of continuous maps between two topological spaces; It
is one of the commonly used topologies on function spaces.

@ __References__: Kundu & Garg T&A(09)
[on a Tychonoff space, properties].

> __Online resources__:
see Wikipedia page.

**Compactification of Extra Dimensions** > see strings.

**Compactification of Spacetime**
> see spacetime boundaries.

**Compactification of a Topological Space** > see Bohr
Compactification; compactness.

**Compacton**

* __Idea__: A solution
of the field equations of a theory that behave trivially outside a compact region.

@ __References__: Bazeia & Vassilevich a1501 [singularities and quantum theory].

**Complementarity** > s.a.
particles; quantum theory.

* __Idea__: Bohr's view
that microscopic objects can behave as particles or waves in different
situations, and no experiment can measure both the wave and the particle
behaviors simultaneously; For example, an object can have either a sharply
defined position or a sharply defined momentum, but not both;
Experimentally, no matter how a system is prepared for each degree of
freedom there is always a measurement whose outcome is totally
unpredictable; Conceptually, a statement about the relationship between
the mechanics and the field theory descriptions of matter dynamics.

* __In quantum mechanics__:
To some extent, it is incorporated in the uncertainty principle, although
the latter is a statement about spreads of values of measured quantities,
not of actual values of system properties; Demonstrated by Young's double
slit experiment with one particle going through the apparatus at a time.

@ __General references__: Rosenfeld Nat(61)apr;
Wootters & Zurek PRD(79) [and the double-slit experiment];
Folse 85;
Vol'kenshtein SPU(88);
Scully et al Nat(91)may;
Mermin PT(93)jan;
Cormier-Delanoue FP(95) [for light];
Holladay AJP(98)jan;
Ghose a0906;
Fedrizzi et al NJP(11)-a1002 [information complementarity];
Heunen FP(12)-a1009 [in categorical quantum mechanics];
Vaccaro a1012-proc [group-theoretic formulation].

@ __Conceptual__: Saunders FP(05)qp/04 [and Bohr];
de Ronde qp/05,
a0705 [and interpretations];
Camilleri SHPMP(07) [Bohr and Heisenberg];
Cuffaro SHPMP(10)-a1008 [views of Immanuel Kant and Niels Bohr];
Plotnitsky 12 [and Niels Bohr];
De Gregorio SHPMP(14)-a1212-conf [Bohr's views];
Kastner in(17)-a1601 [beyond Bohr's complementarity];
Qian et al a1803 [and entanglement];
> s.a. physical theories [generalized].

@ __Experimental analysis__: Auccaise et al PRA(12)-a1201 [interferometer in a closed-open quantum superposition];
Dieks & Lam AJP(08)sep [and the Einstein-Bohr photon box].

@ __Afshar's experiment__: Afshar SPIE(05)qp/07,
AIP(06)qp/07 [violation?];
Srinivasan IJQI(10)qp/05;
Qureshi qp/07;
Reitzner qp/07;
Steuernagel FP(07);
Flores SPIE(09)-a0803 [modified version];
Flores FP(08)-a0802 [reply to comments];
Flores & De Tata FP(10)-a1001;
Drezet a1008.

@ __Related topics__: Ross NCB(93) [???];
Roll-Hansen HSPBS(00) [and biology];
Luís PRA(01) [2D systems];
beim Graben & Atmanspacher FP(06) [in classical mechanics];
Arcioni & Suarez a0901 [slightly modified, and black holes];
Maccone et al PRL(15)-a1408 [and correlations].

> __Related topics__:
see Einstein Boxes; interference;
quantum representations; uncertainty;
Wave-Particle Duality.

**Complete Manifold** > see differential geometry.

**Complete Normed Space** > see Banach Space.

**Completely Regular Topological Space** > s.a. uniformity.

**Completeness** > s.a. Geodesic Completeness;
Incompleteness Theorem; NP-Completeness.

@ __Quantum completeness__: Hofmann & Schneider PRD(15)-a1504 [in curved spacetimes];
Jurić a1802.

**Completeness of Quantum Theory** > see foundations of quantum mechanics;
*ψ*-Ontic Theories.

**Complex (in Topology)**
> s.a. cell complex; CW-complex.

* __Idea__: A finite
family of polytopes such that (i) Every face of every polytope is itself
in the family, and (ii) The intersection between any two polytopes is a
face if each of them; In homological algebra, a sequence of modules.

* __Examples__: Chain
complex, Cochain complex; > s.a. graph invariants.

* __Acyclic complex__:
One without cycles, H_{q}(*X*)
= 0 for *q* = 0, and H_{red, 0}(*X*) = 0.

> __Related topics__:
see euler characteristic; homology
and cohomology [chain complex and dual operator complex].

**Complex ( N-Complex)**

*

@

**Complex Analysis** > see analysis.

**Complex Ginzburg-Landau Equation** > see under Ginzburg-Landau Equation.

**Complex Numbers** > s.a. analysis;
analytic functions; i.

* __Möbius transformation__:
The map *z* \(\mapsto\) (*az *+ *b*) (*cz *+
*d*)^{−1}, where the matrix {*a*,
*b* // *c*, *d*} is in SL(2,\(\mathbb C\)).

@ __General references__: Ahlfors 81 [Möbius transformation].

@ __In quantum mechanics__: Dirac PRS(37);
Accardi & Fedullo LNC(82);
Anastopoulos IJTP(03)gq/02-conf;
Lev FFTA(06)ht/03;
Bracken RPMP(06)qp/05 [Hilbert space quantum mechanics];
Anastopoulos IJTP(06);
Davis IJTP(06);
Goyal et al PRA(10)-a0907;
Sivakumar a1207
[motivation using the results of tandem Stern-Gerlach experiments].

@ __And physics__: Burko TPT(96) [meaning];
Benioff IJPAM(07)qp/05 [Fock-type representation];
> s.a. complex structure.

> __Physical systems__:
see hamiltonian systems; lagrangian dynamics.

**Complex Systems / Complexity**
> s.a. mathematics and posets.

**Componendo & Dividendo**

* __Idea__: If *a*/*b*
= *c*/*d*, then (*a*+*b*)/(*a*−*b*)
= (*c*+*d*)/(*c*−*d*).

**Composite Models of Spacetime / Gravity** > see quantum
spacetime proposals.

**Composite Systems** >
s.a. composite quantum systems; composite
particle models; Mereology.

@ __References__: Hardy a1303-fs
[fundamental axioms for any theory of composition].

> __Related topics__:
see mass [coupling of internal and center-of-mass dynamics].

**Comprehensibility** > s.a. mathematical
physics; Physical Laws.

@ __References__: de Waal a1610
[Charles Sanders Peirce and the logic of abduction].

**Compressibility**

$ __Def__: The
isothermal compressibility is \(\kappa_T^~\) := −(1/*V*) ∂*V*/∂*p*|_{T,N}
, and the adiabatic compressibility \(\kappa_T^~\) := −(1/*V*) ∂*V*/∂*p*|_{S,N} .

* __Example__: The
isothermal compressibility of an ideal gas is \(\kappa_T^~\) = 1/*p*.

@ __References__: Bragg & Coleman JMP(63)
[thermodynamic inequality]; Calvo & Velasco AJP(98)oct
[positive and negative]; Villamaina & Trizac EJP(14)
[fluctuations and finite-size effects].

@ __Metamaterials with negative compressibility__: Nicolaou & Motter NatMat(12)may
+ news ns(12)may;
news sn(17)nov.

**Compton Effect / Scattering**
>s.a. photon phenomenology.

* __Idea__: The
phenomenon in which a photons scatters off a free charged particle, to
which it transfers part of its energy; The particle is often an electron,
which may be bound inside an atom, in which case one uses the
approximation that the binding energy is much less than the photon energy.

* __History__: The
decrease in wavelength of the scattered photon was explained in a 1923
paper by Compton treating light quanta as particles.

* __Inverse Compton
__: The effect in which a charged particle transfers part of its energy to a photon.

@ __References__: Schrödinger AdP(27) [without quantum field theory];
Welton PR(48)
[and quantum fluctuations]; Di Mauro et al a1501
[Majorana's contributions]; > s.a. quantum-gravity
phenomenology.

> __Online resources__:
see Wikipedia page.

**Compton Wavelength**
> s.a. dirac fields coupled to gravity [Compton-Schwarzschild length].

* __Idea__: The
wavelength of a photon with energy equal to the rest energy of a particle;
It can be thought of as an indicative value for the smallest possible
uncertainty on the position of the particle, since localizing the particle
to within a smaller uncertainty would require a photon with a smaller
wavelength, which could lead to the creation of a particle pair instead.

$ __Def__: For a particle
of mass *m*, *λ*:= *h*/*mc*, and the reduced
Compton wavelength is *λ*_{r}:= ℏ/*mc*.

* __Values__: For an electron,
*λ* = 2.43 × 10^{−12} m and
*λ*_{r} = 3.86 × 10^{−13} m.

> __Online resources__:
see Wikipedia page.

**Computation** [including computability] > s.a. computer
languages; computational physics (and specific areas);
quantum computation.

**Comultiplication on a Manifold** > see manifolds.

**Concavity** > see functions.

**Concepts** > see philosophy of science.

**Concomitant**

* __Idea__: A differential
operator on a manifold that doesn't depend on a choice of connection.

**Concordance Cosmology** > s.a. cosmology;
cosmological models.

> __Online resources__:
see J Ostriker's page.

**Condensates** > see bose-einstein
condensation; phase transitions; semiclassical
quantum gravity; types of dark energy.

**Condensation** > see phase
transitions; quantum phase transitions [including fermion condensation].

**Condenser** > see Capacitor.

**Conduction / Conductors / Conductivity**
> see electricity; Heat
Flow; Transport Phenomena.

**Cone on a Space** > see topology.

**Configuration Space (in Physics)**

* __Idea__: A space *C* whose
elements represent possible configurations (instantaneous states) of a physical system.

* __Classical vs quantum__:
For systems with finitely many degrees of freedom, the classical and
quantum configuration spaces can be chosen to coincide; For inifinitely
many degrees of freedom (field theories), one normally has to extend *C*
to include distributional fields of some sort.

* __For field theories__:
It has the structure of a configuration bundle (*Y*, Σ, *π*)
over the space manifold Σ.

@ __General references__: Anderson a1412
[generalized configuration spaces, structures and examples].

@ __For point particle systems__: McGlinn et al IJMPA(96)ht/95;
> s.a. particle descriptions and effects.

> __Quantum__:
see particle statistics; quantum geometrodynamics;
quantum gauge theories.

> __Online resources__:
see Less Wrong page;
Wikipedia page.

**Confinement**

> __In QCD__: see QCD effects.

@ __In other theories__: Donoghue a1609 [gravitational spin connection]; Chaichian & Frasca a1801 [condition for confinement].

**Confirmation of a Theory** > see criteria for physical theories [verifiability].

**Confluent Hypergeometric Functions** > see Hypergeometric Functions.

**Conformal Completions / Extensions of Spacetime** > see
asymptotic flatness, at spatial
infinity and null infinity; conformal transformations.

**Conformal Cyclic Cosmology (CCC)** >
s.a. cmb anisotropy.

@ __References__: Gurzadyan & Penrose EPJP-a1512 [and the Fermi paradox].

**Conformal Field Theory** > s.a. conformal
structures; conformal invariance in
physics; Scale Invariance.

* __Idea__: Conformal
field theories are quantum field theories, generally defined in two
dimensions, that are invariant under conformal transformations; They are
solvable because conformal invariance in two spacetime dimensions implies
an infinite number of symmetries, and they have applications in critical
phenomena, black holes, and string theory.

@ __2D__: Friedan & Schenker NPB(87);
Giddings PRP(88);
Jain IJMPA(88)
[and strings in general backgrounds];
Segal in(88);
Moore & Seiberg CMP(89);
Cardy pw(93)jun;
Zuber Rech(93)feb;
Halpern et al PRP(96)
[irrational]; > s.a. Percolation;
supersymmetric field theories.

@ __2D, reviews__: Furlan et al RNC(89);
Kaku 91; Ketov 95;
Fuchs ht/97-ln;
Gaberdiel RPP(00)ht/99;
Efthimiou & Spector ht/00-ln;
Nagi IJMPA(06) [operator algebra];
Cardy a0807-ln [and 2D critical phenomena];
Fuchs et al JMP(10); Ribault a1609-ln.

@ __4D__: Caracciolo & Rychkov PRD(10)-a0912 [limits on interaction strength];
Giardino AACA(17)-a1509 [using quaternions].

@ __Higher-dimensional__: Anselmi PLB(00)ht/99
[classification, even dimensions]; Petkova & Zuber ht/01-in
[rational, rev]; Castro-Alvarado & Fring NPB(04)
[vacuum energies]; Bischoff et al BulgJP(09)-a0908
[various dimensions]; Qualls a1511-ln
[various dimensions, rev]; Rychkov a1601-ln [\(D \ge 3\)].

@ __Related topics__: Bartels et al a1001
[conformal nets]; > s.a. types
of entropy [Rényi entropy].

> __And bulk/boundary
duality__: see AdS-cft correspondence
[including dS-cft]; kerr spacetime
[Kerr-cft correspondence]; isolated horizons.

**Conformal Geometry**

@ __References__: Curry & Gover a1412
[and tractor calculus, and applications in general relativity].

**Conformal Infinity** > see asymptotic
flatness at null infinity.

**Conformal Invariance and Structures in
Physics**

**Conformal Structure and Transformations**

**Conformal Nets** > see Conformal
Field Theory.

**Congruence of Lines in a Manifold**
> see lines; Expansion;
Shear; Vorticity;
Raychaudhuri Equation.

**Conjugate Elements / Subgroups of a Group** > see group
theory.

**Conjugate Points in a Manifold** > see geodesics.

**Conjugate Representations** > see group
representations.

**Conjugate Variables** > see hamiltonian
dynamics.

**Connected Sum of Manifolds**
> see manifolds.

**Connection** > s.a. affine
connection.

**Connes Distance** > see distances
on manifolds; Spectral Distance.

**Consciousness** > see mind.

**Conservation Laws, Conserved
Quantities**

**Consistency of a Theory** >
see interactions; physical
theories; quantum field theory formalism
/ electromagnetism; types
of field theories; types of gauge theories.

**Consistent Histories Formulation of Quantum Theory** >
see histories.

**Constants, Mathematical**
> s.a. Numbers.

@ __References__:
Finch 03.

> __Special constants__:
see Catalan Numbers; e,
Euler-Mascheroni Constant; Feigenbaum
Number; Golden Ratio; Madelung
Constant; Omega Number; π;
Silver Mean.

**Constants,
Physical** > s.a. approximate
values; fine-structure and gravitational constant;
variation of constants.

**Constants of Motion** > see conservation laws.

**Constituent Models** (for quarks) > see composite models.

**Constraints** >
s.a. constraints in general relativity; quantization
of first-class systems and second-class
systems.

**Constructivism**

@ __References__: Ozhigov a0811-ln
[mathematical and physical constructivism]; Norton BJPS(08)
[constructive relativity].

**Constructor Theory**

* __Idea__: The theory
of which physical transformations can be caused to happen and which
cannot, and why.

@ __References__: Deutsch a1210
[motivation and principles].

**Contact Geometry / Manifold**

* __Idea__: A
generalization of symplectic geometry and symplectic manifolds.

$ __Contact manifold__:
A (2*n*+1)-dimensional differentiable manifold *M* with a
global 1-form *ω* such that *ω* ∧ (d*ω*)^{n}
≠ 0, for all *p* in *M*.

@ __General references__: Hurtado DG&A(08)
[stability numbers]; Kholodenko 13 [and topology, applications to physics];
Banyaga & Houenou 16.

@ __Contact geometry and physics__: Rajeev AP(08)mp/07 [thermodyamics, geometrical optics, and quantization];
Bravetti & Tapias JPA(15)-a1412 [dynamics of non-conservative systems];
Fernández de Córdoba & Isidro a1802 [and the ideal gas].

**Contextuality**
> s.a. foundations of quantum mechanics / Entropic Dynamics;
Hardy's Thought Experiment.

* __Idea__: The
impossibility of a hidden-variable model of quantum theory wherein the
representation of measurements does not depend on the context of the
measurement (see the Bell-Kochen-Specker theorem); The simplest proof is
considered to be Hardy's.

@ __General references__: Clifton AJP(93)may;
Svozil qp/99 [model],
AIP(05)qp/04 [and counterfactuals];
Appleby PRA(02)
[existential]; Steane JPA(07)qp/06
[and unitarity and time reversal]; Bishop & Atmanspacher FP(06);
Kupczynski
AIP(07)-a0710;
Karakostas JGPS(07)-a0811
[and non-separability]; Kurzyński et al PLA-a1111 [emergence of non-contextuality in macroscopic systems];
Hermens SHPMP(11);
Liang et al PRP(11)
[Specker's parable of the overprotective seer];
Griffiths SHPMP(13)-a1201 [Hilbert-space quantum mechanics is non-contextual];
Fields IJGS(13)-a1201 [in classical physics];
Grudka et al PRL(14)-a1209 [contextuality measures];
Acín et al CMP(15)-a1212 [general combinatorial formalism];
Durham a1307-FQXi [as Wheeler's universal regulating principle];
Asadian et al PRL(15)-a1502 [in phase space];
Dzhafarov & Kujala in(16)-a1508 [dialog];
Kocia & Love a1711
[and orders of \(\hbar\) as a resource in the Wigner-Weyl-Moyal formalism]; > s.a. logic.

@ __Mathematical formulation__: Abramsky et al a1502 [cohomology]; Carù EPTCS(17)-a1701.

@ __Proofs__: Ghirardi & Wienand FP(09)-a0904;
Vidick & Wehner PRL(11)
[example of large violation of non-contextuality in quantum mechanics];
Cabello et al PRL(13)-a1310 [simple Hardy-like proof];
de Ronde a1606 [physical and philosophical meaning].

@ __Quantification__: Durham Info(14)-a1409 [order-theoretic];
Amaral et al PRA(15)-a1507 [and absolute maximal contextuality];
Simmons a1712
[quantum mechanics is not as contextual as other possible theories].

@ __And non-locality__: Kurzyński et al PRL(14);
Abramsky EATCS-a1406;
Zhan et al PRL(16)
+ news PhysOrg(16)mar
[non-locality and contextuality never appear together]; Liu et al PRL(16)-a1603 [non-locality from local contextuality].

@ __State-independent contextuality__: Cabello a1201;
Cabello & Terra Cunha PRA(13)-a1212
[with identical particles].

@ __Non-contextual inequalities__: Bub & Stairs a1006
[use of the Klyachko inequality]; Cabello et al a1010
[as an axiom, and other theories]; Amselem et al PRL(12)-a1111
[and test].

@ __Theoretical models__: Chen et al PRA(13)
[for a relativistic spin-1/2 particle, an electron in a Coulomb
potential]; Laversanne-Finot et al JPA(17)-a1512
[Hilbert spaces of arbitrary dimensions]; Acacio de Barros et al a1707 [and indistinguishability].

@ __Experiments__: Svozil PRA(09)qp/04;
Cabello et al PRL(08)-a0804
[with neutrons]; Nambu a0805
[non-entangled photons]; Cabello PRL(08)-a0808;
Bartosik et al PRL(09)-a0904
[in neutron interferometry]; Pan & Home PLA(09)-a0912
[and subensemble mean values]; Cabello PRA(10)-a1002
[inequality]; Kurzyński et al PRL(12)-a1201
[simplest contextual inequality]; Zu et al PRL(12)
+ Paris & Paternostro Phy(12)
[state-independent experimental test on a single photonic qutrit]; Soeda
et al PLA(13)
[in macroscopic magnetization measurements]; Winter JPA(14)-a1408;
Cabello PRA(16)-a1603;
> s.a. quantum foundations.

@ __Related topics__: Raussendorf PRA(13)-a0907
[and quantum computation]; Dressel et al PRL(10)-a0911,
Dressel & Jordan PRA(12)-a1110 [contextual values of observables];
Dzhafarov et al LNCS-a1504 [contextuality by default];
Acacio de Barros et al a1511 [and negative probabilities];
Lostaglio a1705 [and quantum fluctuation theorems];
Shrapnel & Costa a1708 [and causal structure].

**Continued Fractions** > see types of numbers.

**Continuity Classes of Functions** > see analysis.

**Continuity Equation**
> see conservation laws.

**Continuous Matter Creation**
> s.a. matter distribution in cosmology;
Steady-State Cosmology.

@ __References__: Ramos et al PRD(14)
[creation of cold dark matter cosmology].

**Continuous Media / Continuum Mechanics**

* __Areas__: The main areas of continuum mechanics are condensed and solid matter,
fluid mechanics, thermodynamics, elasticity, electricity, field theory.

@ __Textbooks__: Roberts 94 [1D introduction];
Lautrup 11;
Tadmor et al 11
[mechanics and thermodynamics, r CP(12)#5];
Tadmor & Miller 11 [multiscale techniques];
Clayton 14 [focus on finite deformation kinematics and classical differential geometry].

@ __Problems and solutions__:
Eglit & Hodges 96;
Vekstein 13.

@ __Related topics__: Gollub PT(03)jan [vs discrete description];
Pronko a0908 [Hamiltonian description].

> __Material media__:
see condensed matter; Extended Objects; fluids [smoothed
particle hydrodynamics]; gravitating
matter; solid matter.

> __Related topics__:
see field theory; Stress Tensor; thermodynamics.

**Continuum (Mathematics) **> s.a. non-standard
analysis.

* __Remark__: Our view
of nature is based on the usual notion of continuum; but this may be a
historical accident.

* __Continuum problem__:
Are all infinite subsets of \(\mathbb R\) conumerous with either \(\mathbb
Z\) or \(\mathbb R\)?

@ __General references__: Ingram T&A(06)
[historical, indecomposable continua]; Prajs & Whittington T&A(07),
T&A(07)
[homogeneous, decompositions].

@ __Continuum hypothesis__: Yaremchuk qp/01
[intermediate cardinality], qp/01
[consequences of negation], qp/02
[and physics]; Czajko CSF(04)
[argument
against].

@ __And physical theories__: White 92
[history]; Baez a1609 [our struggles with it]; > s.a. Discrete Models and Discretization.

**Contorsion** > see torsion.

**Contractible Topological Space**

$ __Def__: *X*
is contractible if the identity map on it is homotopic to the constant map
on some *x*_{0} in *X*, or
id_{X} ≅ *x*_{0}.

* __Properties__: A
Contractible space has the same homotopy type as a point.

* __Relationships__:
Contractibility implies simple connectedness.

**Contraction Mapping**
> s.a. Lipschitz Condition.

* __Idea__: A mapping *f*
: *X* → *X* from a metric space to itself is an expansion
if there is a positive constant *c* < 1 such that for all *x*_{1}
and *x*_{2} in *X*,

*d*(*f*(*x*_{1}),
*f*(*x*_{2}) ≤ *c* *d*(*x*_{1},
*x*_{2}) .

> __Online resources__:
see Wikipedia page.

**Contraction of a Lie Algebra** > see lie
algebras.

**Contraction of Operators (Dyson-Wick)
or Chronological Pairing** > see fock
space.

**Contragradient**

* __Idea__: A gradient
with the index raised by a metric.

**Convection** > s.a. Heat
Flow; Lorentz Equations.

* __Idea__: The bulk
motion of particles within a fluid induced by temperature differences,
which results from the combination of diffusion (random motion of
individual particles) and advection (large-scale organized motion); It is
one of the major heat-transfer and mass-transfer mechanisms.

* __Rayleigh-Bénard
convection__: The kind that arises in a fluid heated from below; It is
turbulent / chaotic; Example: The solar convection zone.

@ __References__: Ahlers Phy(09)
[turbulent].

> __Online resources__:
see Wikipedia page.

**Conventionalism** > see spacetime
[conventionalism in geometry]; cosmology references; symmetries
[invariance, convention and objectivity].

**Convergence** > see limit; Filter; sequence; series.

**Convex Functions** > s.a.
functions; analysis.

@ __References__: Gibbons & Ishibashi CQG(01)gq/00, proc(01)-a1702 [and spacetime geometry].

**Convex Geometry**

@ __References__: Porta Mana a1105
[conjectures and open problems, and physics].

**Convex Normal Neighborhood**

$ __Def__: A convex
normal neighborhood is a subset *U* of spacetime such that for any
two points in it there is a unique geodesic connecting them, and contained
entirely within *U*.

@ __References__: in Hicks 65; in
Penrose 72; in Wald 84.

**Convex Sets / Spaces **> see affine
space [convex subsets]; euclidean
geometry; vector space [locally
convex].

**Convolution** > see functions.

**Conway Polynomial** > see knot
invariants.

**Conway-Kochen Theorem** > see Determinism;
Free Will.

**Cooper Pairs** > s.a. superconductivity.

* __For photons__: A pairing between photons that can occur in water and other transparent media, identified through correlations in light that is scattered inelastically; Data matches a model in which the photons interact by an exchange of virtual molecular vibrations.

@ __References__: news pn(07)dec
[in insulators]; Saraiva et al PRL(17)-a1709 [for photons].

**Copernican Principle**
> s.a. cosmological principle
[including constraints, tests]; cosmology; cosmic
microwave background; paradigms in
physics.

* __Idea__: In the
original version, the Earth is not the center of the Solar System; The
current version is that our location in the universe is not a special one
in any way, and is incorporated into the cosmological principle; As for
the latter, the question of its validity is a scale-dependent one.

* __History__: Hermann
Bondi coined the phrase in the 1950s.

* __Recent history__: A
violation of the Copernican Principle, in the sense that we live near the
middle of a void, has been proposed as an explanation for the apparent
cosmological acceleration, as an alternative to the existence of dark
energy.

@ __General references__: Nutku gq/05
[modern, multiverse version]; Graney BA-a0901
[and Galileo]; Danielson AS(09)jan
[history].

@ __Tests__: Clarkson CRAS(12)-a1204;
Sapone et al PRD(14)-a1402
[using Hubble and Supernova Ia data].

> __Online resources__:
see the story of the film The
Principle.

**Corepresentations** > see group
representations.

**Coriolis Force / Effect** >
s.a. force [in general relativity].

* __Consequences__: If
you flush a toilet in the Northern Hemisphere, the water will usually
spiral down in a counterclockwise direction.

@ __History__: Graney a1012
[early description by G B Riccioli];
Graney PT(17)jul-a1611 [Claude Francis Milliet Dechales, 1674].

@ __Related topics__: blog sa(01) [water down the drain]; Lan et al PRL(12)
+ Close & Robins Phy(12)
[and atom interferometry].

> __Online resources__:
see UIUC page.

**Corona (in a Tiling)**

$ __Def__: The first
corona of a tile is the set of all tiles that have a common boundary point
with that tile (including the original tile itself); The second corona is
the set of tiles that share a point with something in the first corona,
and so on [from Weisstein's Encyclopedia].

**Correlations** > s.a. quantum correlations;
types and bounds.

**Correlation Length**

* __Idea__: The spatial
distance between locations in an extended system over which the
fluctuations of microscopic degrees of freedom are significantly
correlated to each other (in a material this is usually a few interatomic
spacings).

$ __Def__: The constant
*ξ* such that the correlation function *C*(*x*_{i},
*x*_{j}) between the
variable *x* at locations *i* and *j* can be
expressed as exp{−|*i*−*j*|/*ξ*}.

**Correspondence Principle**
> s.a. classical limit of quantum mechanics.

* __Idea__: The
predictions of a quantum theory must agree with those of the corresponding
classical theory in the limit of large occupation numbers.

@ __General references__: Heller & Tomsovic PT(93)jul;
Makowski
EJP(06)
[formulations];
Goyal a0910
[evolution of expectation values and correspondence rules]; Gómez & Castagnino CSF(14)-a1009
[threats from fundamental graininess and chaos].

@ __Types of systems__: Karkuszewski et al PRA(02) [breakdown in chaos];
Makowski & Górska PRA(02) [exact cases];
Henner et al a1507
[for spin systems].

@ __In quantum field theory__: Kawai & Stapp PRD(95)qp [QED and S-matrix];
Kazakov NPPS(02)ht/01,
IJMPD(03)ht [quantum gravity];
Manjavidze a1106.

**Coset**

* __Left coset__: An
equivalence class of elements of a group *G* under the equivalence
relation *y* = *xh*, for some *h* in a given
subgroup *H*, i.e., a subset of *G* of the form *xH*;
A subset of the form *Hx* is a right coset.

* __Coset space__: The
set *G*/*H* of cosets of a group *G* with respect
to a subgroup *H*; __In physics__: > see, e.g., geometric
quantization; homotopy groups.

> __Online resources__:
see MathWorld page.

**Cosmic Balloons**

* __Idea__: Spherical
domain walls containing trapped relativistic particles; They may be
created in the early Universe

@ __References__: Holdom PRD(94).

**Cosmic Coincidence Problem** > see cosmology.

**Cosmic Doomsday** > see cosmological singularities.

**Cosmic Microwave Background**
> s.a. cmb anisotropy and polarization.

**Cosmic No-Hair Conjecture**
> see brans-dicke theory; scalar-tensor
theories; schwarzschild-de sitter solutions.

**Cosmic Strings** >
s.a. cosmic-string phenomenology.

**Cosmic Web** > see matter
distribution in cosmology.

**Cosmography**

* __Idea__: The branch
of cosmology which aims to describe the universe without the need of
postulating a priori any particular cosmological model.

@ __References__: Piazza & Schücker GRG(16)-a1511 [minimal requirement];
Dunsby & Luongo IJGMP(16)-a1511 [rev].

**Cosmological Argument**

* __Idea__: An argument for
the existence of a First Cause (or instead, an Uncaused cause) to the universe.

@ __References__: Romero & Pérez IJPR(12)-a1202 [remarks on versions of the argument].

> __Online resources__:
see Wikipedia page.

**Cosmological Constant** >
see also cosmological constant problem.

**Cosmological Models**
> see also general relativistic
models.

**Cosmology** > s.a. acceleration;
cosmological parameters; expansion;
geometry; history;
observational cosmology; perturbations;
references.

**Cosmon** > see gravitational
constant [variable-*G* theories].

**Cosserat Theory of Elasticity** > see Elasticity; 2-spinors.

**Cosymplectic Structures** > see hamiltonian systems.

**Cotangent Bundle, Vector** > see tangent structures to a manifold.

**Cotton Tensor** > s.a. riemannian
geometry [Cotton flow]; Topologically Massive Gravity.

* __Idea__: A tensor constructed
out of the curvature, which arises in the context of the Bianchi identities.

* __In 3D__: The
conformally invariant tensor, whose vanishing is equivalent to conformal
flatness (replaces the Weyl tensor) defined by

*C*^{a}_{b}
:= *ε*^{amn} ∇_{m}(*R*_{nb}
− \(1\over4\)*R* *g*_{nb}) .

@ __General references__: García et al CQG(04)gq/03 [properties].

@ __Cotton-York tensor__: Bini et al CQG(01)gq [stationary vacuum spacetime, congruence approach];
Valiente Kroon CQG(04)gq [asymptotic expansion];
Osano a1309 [and gravitational waves].

**Coulomb Gauge** > see gauge choice.

**Coulomb's Law** > see electricity.

**Coulomb Potential / Systems**
> see electromagnetism; quantum
systems; scattering.

**Council of Giants** > see milky way galaxy [neighborhood].

**Counterfactuality, Counterfactuals**

@ __In quantum mechanics__: Finkelstein Syn(99)qp/98 [and spacelike separated points];
Choy & Ziegeler qp/99/AJP [meaning, and non-locality];
Bigaj Syn(04) [and spacetime events];
Tresser qp/05 [weak realism];
Vaidman a0709-en;
Vaidman a1401
[counterfactuals don't have to be time-asymmetric];
Salih et al a1806 [counterfactual communication is possible].

@ __In general relativity__: Curiel a1509 [and dynamical geometry].

**Counting Function** > see Enumeration.

**Coupling Constant** >
s.a. charge; renormalization
theory and applications.

* __Idea__: Any
constant *g* appearing in the Lagrangian for a field theory in a
term containing different fields; For example, the electric charge *e*,
the gravitational constant *G*, or *g φ ψ***ψ*.

@ __References__: Besprosvany MPLA(03) [and particle compositeness].

> __For specific
theories__: see fine-structure
constant; gravitational constant;
GUTs.

**Coupon Collector's Problem**

* __Idea__: How often do you have to press
"Random Article" on Wikipedia until you have visited every page at least once?
The number of times you have to click is random; The average number of clicks it takes
if there are *n* articles is approximately *n* log(*n*) + *γn*,
where *γ* is the Euler-Mascheroni constant;
For *n* = 5,252,120 articles, this comes out to around 84,303,659 clicks
[from Sam Watson on Quora].

**Courant Algebroid** > Fluxes.

**Courant-Friedrichs-Lewy
Condition** > s.a. causality.

* __Idea__: A necessary
condition for convergence in the numerical solution of partial
differential equations representing time evolution of systems; It puts an
upper bound on the size of the time steps, depending on the size of the
spatial discretization, and can be viewed as a discrete "light cone" condition.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Covariance of a Physical Theory**

**Covariance of Random Variables**
> s.a. correlations.

$ __Def__: For two
random variables *X* and *Y* the covariance with respect
to a given probability distribution *p* is Cov_{p}(*X,Y*):=
\(\langle\)*X Y*\(\rangle\)_{p}
\(-\langle X \rangle_p\,\langle Y \rangle_p\) .

* __Covariance matrix__:
For a set of variables \(x_i\), the matrix of correlation functions
between pairs of variables \(x_i\) and \(x_j\), \(C_{ij}\)
:= \(\langle\)*x*_{i}* x*_{j}\(\rangle\)
− \(\langle\)*x*_{i}\(\rangle
\langle\)*x*_{j}\(\rangle\)
; The determinant of the covariance matrix is known as the generalized variance.

* __In quantum mechanics__:
Given a state *ρ*, one can define a symmetrized quantum analog of
covariance, Cov_{ρ}(*A*,
*B*):= \(1\over2\)tr[*ρ*(*AB*+*BA*)] − tr(*ρ**A*)
− tr(*ρ**B*).

@ __References__: Gibilisco & Isola JMAA(11) [quantum covariance and uncertainty relations].

> __Online resources__:
see Wikipedia page.

**Covariant Derivative**
> see tensor fields; Fermions / Parallel Transport.

**Covariant Quantization **> see approaches
to quantum field theory; covariant quantum gravity.

**Covariant Regularization Scheme** > see regularization [Pauli-Villars].

**Covector** > see differential forms [1-form].

**Covering Dimension** (Of a topological space) > see dimension.

**Covering Group** > see Universal Covering Group.

**Covering Number** > see cover.

**Covering Relation** > see posets.

**Covering Space** > s.a. lorentzian
manifolds; Universal Covering Space.

$ __Def__: The pair (*E*,*
p*: *E* → *X*) is a covering space of *X* if
for all *x* in *X*, there is a neighborhood *U* of
*x*, such that *p*^{−1}(*U*)
is a disjoint union of open sets in *E*, each mapped
homeomorphically onto *U* by *p*.

* __Example__: The
covering space of SO(3,1) is SL(2,\(\mathbb C\)); Covering groups of
special (pseudo)orthogonal groups are often called spin groups.

* __Remark__: *E*
and *X* have the same properties locally.

$ __Normal covering space__:
One in which *p*_{*}π_{1}(*E*,
*e*_{0}) is a normal subgroup of π_{1}(*X*,
*x*_{0}).

$ __Covering
transformations__: Given a covering space (*E*, *p*) of
*X*, the group *G* of covering transformations is the group
of all homeomorphisms of *E* which preserve the fibers, i.e., *φ*
∈ *G* implies that \(p\cdot \phi = p\).

@ __References__: Brown AMM(74).

**COW (Colella-Overhauser-Werner) Experiment** > see tests of the equivalence principle.

**Cox Rings**

@ __References__: Arzhantsev et al
14.

**Coxeter Graphs / Groups** > s.a.
group types / types of spacetime singularities.

* __Result__: Finite
Coxeter groups coincide with the finite reflection groups of Euclidean
spaces; Coxeter groups coincide with cocompact discrete reflection groups of geodesic spaces.

@ __General references__: Hiller 82;
Björner & Brenti 05
[combinatorics; r BAMS(08)];
Davis 08;
Sirag 16 [ADE Coxeter graphs].

@ __And Clifford algebra__: Dechant AACA(13)-a1205 [quaternionic representations];
Dechant AACA(13)-a1207.

@ __Related topics__: Hosaka T&A(06)
[and geodesic spaces]; Henneaux et al JMP(07)ht/06
[rank-10 and 11, special class]; Marietti EJC(08) [identities-dualities].

**CP Violation** [includes the strong CP problem]

**Crane-Yetter State-Sum Model**
> s.a. spin-foam models.

* __Idea__: A 4D spin-coupling theory.

@ __References__: Crane & Yetter gq/03;
Barrett et al JMP(07)m.QA/04 [observables].

**Creation of Matter** > see Continuous Matter Creation;
Steady-State Cosmology.

**Cremmer-Scherk Theory** > see spin-1 field theories.

**Critical Points** > s.a. phase transitions.

* __Idea__: Locations
on a phase diagram where the boundary between phases disappears.

> __Examples__: see QCD phenomenology.

**Cross Product** > see vectors.

**Cross Section in Scattering Theory** > see scattering; units [barn].

**Cross Section of a Bundle** > see bundle.

**Crossing Property** > see quantum field theory.

**Crum's Theorem**

* __Idea__: A result in
1-dimensional quantum mechanics, stating the existence, for any given
Hamiltonian system, of an associated Hamiltonian system with the same
energy spectrum except for the lowest energy state, which is deleted.

@ __References__: García-Gutiérrez et al PTP(10)-a1004
[for discrete quantum mechanics].

**Crumpling** > s.a. quantum regge calculus.

* __Idea__: A type of phase transition.

@ __References__: Foltin JPA(01) [in fluid membranes].

**Cryptology / Cryptography**
> s.a. quantum technology [quantum cryptography].

* __Idea__: Cryptology
is the science that makes secure communications possible; Its two
complementary aspects are cryptography (the art of making secure building
blocks) and cryptanalysis (the art of breaking them).

* __Tools__: In
cryptanalysis, frequency analysis is a code breaker's fundamental tool.

@ __Cryptology__: Beutelspacher 94;
Singh 99 [I]; Klima & Sigmon 12;
von zur Gathen 15.

@ __General references__: Kippenhahn 99 [cryptanalysis, I];
Stinson 05
[cryptography]; McAndrew 11
[cryptography, with open-source algebraic software].

@ __Related topics__: Baptista PLA(98)
[with chaos]; Smithline AS(09)mar
[breaking of a 200-year-old code].

**Crystallographic Groups** > see finite groups.

**Cubic Equations** > see elementary algebra.

**Cuntz Algebra**

@ __References__: in Coquereaux JGP(89),
JGP(93);
Jorgensen in(01)m.FA/00 [representations, and loop group/wavelets];
Abe & Kawamura mp/01 [and fermions];
Kozyrev mp/02 [*p*-adic representations].

**Cup Product** > see cohomology.

**Curl of a Vector Field** > see vector calculus.

**Current in Electricity** > see electricity.

**Current in Dynamical Theories** > see conservation
laws; field theory.

**Current in Quantum Mechanics** > see quantum mechanics.

**Current Algebra** > see history of physics.

**Curvaton** > s.a. inflationary
phenomenology [structure formation]; types
of inflationary scenarios.

* __Idea__: A light
scalar field present during inflation (in additional to the inflaton
itself) that is responsible for the observed inhomogeneities, in the sense
that its quantum fluctuations produce the primordial density perturbations
in a proposal for the origin of structure formation; Spatial variations in
the curvaton density are then transferred to the radiation density when
the curvaton decays some time after inflation.

@ __References__: Wands LNP(08)ap/07 [rev];
Chingangbam & Huang PRD(11)-a1006;
Mazumdar & Rocher PRP(11) [rev].

**Curvature** > s.a. line;
riemann tensor.

**Curve / Line**
[including quantum curve] > s.a. Fitting; geodesic;
loop; spacetime subsets; Timelike Curve;
vector field [integral curve].

**Cuscuton Model** > s.a. hořava-lifshitz gravity.

* __Idea__: A field with infinite speed of propagation, introduced in the context of cosmology.

@ __References__: Gomes & Guariento PRD(17)-a1703 [Hamiltonian analysis].

**Cusp**

* __Idea__: One of the
two generic singularities that occur in mappings from a 2-surface to a plane.

**CW-Complex** > s.a. graphs.

* __Idea__: A space *X*
with a decomposition *X*^{ 0} ⊂ *X*^{
1} ⊂ ... ⊂ *X*^{ n}
= *X*, where *X*^{ 0} is a
finite set of points, and *X*^{ k}
is obtained from *X*^{ k−1}
by attaching a finite number of *k*-cells.

$ __Def__: A Hausdorff
space *K* (underlying space) and a partition {*e*_{i}}
of *K*, such that *e*_{i}
is homeomorphic to an open *n*_{i}-cell,
and each point in the boundary of *e*_{i}
is in some other *e*_{j}
(with *n*_{j} < *n*_{i});
In addition, if *K* is not finite, each *p* in *K*
is contained in a finite subcomplex, and *K* has the direct limit
topology of its finite subcomplexes.

* __Properties__: It is always paracompact.

@ __References__: Whitehead BAMS(49);
Lundell & Weingram 69; in Banyaga &
Hurtubise 04;
Minian & Ottina math/06
[generalization, CW(A)-complexes].

**Cycle**

$ __In homology__: A
chain *c* whose boundary is zero, ∂(*c*) = 0.

$ __In graph theory__:
A closed chain (set of consecutive edges); > s.a. graph theory.

**Cyclic Cosmologies** >
s.a. Bounce.

> __Classical models__:
see brane cosmology; cosmological
models and early-universe models.

> __In quantum gravity__:
see cosmological-constant problem;
loop quantum cosmology.

**Cyclic Representation of a Group** > see group representation.

**Cylindrical Function**

$ __Def__: A function *f*
on an infinite-dimensional vector space is cylindrical with respect to a
finite-dimensional subspace *V*_{n}
of (the dual of) *V* spanned by *e*_{1},
*e*_{2}, ..., *e*_{n}
if *f*(*φ*) depends only on the components *φ*_{i}
= *e*_{i}(*φ*) of *φ*
in *V*_{n}.

**Cylindrical Symmetry** > see types of spacetimes.

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

send feedback and suggestions to bombelli at olemiss.edu – modified 29 jun 2018