Topics, C

C Operator > see Charge Conjugation.

C-Metric

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-in [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]; Sekhon a2104 [Gel'fand-Naimark theorem and GNS theorem]; > 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 Materials > see condensed matter.

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.

Carbon

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 [\(\mathbb N\)] = ℵ₀; The first uncountable one is ℵ₁; Notice that [\(\mathbb R\)] = c > ℵ₁.

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 a1402v1-5, JPCS(15)-a1506 [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

C₀ = 1 ,     C_n = \((n + 1)^{-1} {2n \choose n}\)  for n ≥ 1 ,     S(x) = ∑_{k = 0}^∞ C_k x^k = [1 − (1 − 4x)^1/2] / 2x .

* Generalization: _pC₀ = 1; _pC_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; Abalos & Reula a1811 [constant-coefficient first order systems, nsc's].

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]; Minguzzi RVMP(20)-a1909 [regularity].

Cauchy Theorem > see analytic functions.

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

Cauchy-Schwarz Inequality > see inequalities; quantum correlations.

Causal Action Principle > see fermions.

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 Fermion Systems > see fermions.

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

Causal Model > s.a. causal sets; indefinite causal structures; 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]; Barrett et al a1906 [framework], a2002 [extension to cyclic causal structures]; Wolfe et al a1909 [characterization of correlations].

Causal Networks > see networks.

Causal Sets

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

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

Causality Violations

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 a Bethe lattice).

Cayley-Hamilton Theorem
> Online resources: see MathWorld page; Wikipedia page.

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 Complex

Cell-Like Map > same as Resolution.

Cellular Automaton

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 gs 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]; Kozameh et al PRD(20)-a1911 [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]; Roques-Carmes et al PRX(18) [and non-perturbative quantum electrodynamics].
@ Gravitational: Kostelecký & Tasson PLB(15)-a1508, Schreck Symm(18)-a1909 [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.
@ Related topics: Franca et al a1906 [reversed Vavilov-Čerenkov radiation].
> In modified theories: see modified electrodynamics; tests of lorentz invariance and phenomenology.

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]; Sabulsky et al a1812 [constraints from atom interferometry].
@ 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]; Naik et al MNRAS(19)-a1905 [chameleon-f(R) gravity, galaxy rotation curves]; > 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]; Kalita et al IJMPD-a1912 [violation in noncommutative geometry].
> 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 EPJC(17)-a1703 [vs scalar field]; Aurich & Lustig APP(18)-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 Equation; fluid dynamics.

Character of a Group G > s.a. group representation.
$ Def: A 1D complex representation of G, i.e., an element of Hom(G, U(1)).
@ References: Balantekin AIP(10)-a1011 [character expansions of invariant functions on a group].

Character of an Algebra > s.a. Spectrum.
$ Def: A non-zero algebra homomorphism χ: A → \(\mathbb C\).

Characteristic Classes

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.

Characteristic Polynomial

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 (∂lnZ/∂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; It vanishes for particles in an ideal phonon or photon gas.
* Quote: "A vague discomfort at the thought of the chemical potential is still characteristic of a physics education" (Kittel's Introduction to Solid State Physics, e7).
@ 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 Classes / Numbers

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 ∧ dA − \(2\over3\)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-Simons Theory

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]; Chowdhury et al a2009 [dismantling wave-particle duality].

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.

Chimera States
* Classical: States characterized by the spatial coexistence of synchronized and desynchronized dynamics; They are examples of partial synchronization patterns emerging in nonlinear dynamics.
@ References: Bastidas et al ch(16)-a1807 [in quantum mechanics].

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.
@ And phenomenology: Alexander et al PRD(20)-a2003 [(nearly vanishing) cosmological constant].

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 IJGMP(18)-a1805-proc [gravitational and electromagnetic interactions with matter]; Krasnov CQG+(16) [chiral modifications of general relativity].
@ Chiral bosons: Abreu & Dutra PRD(01)ht/00; Abreu & Wotzasek ht/04-ch; Townsend PRL(20)-a1912 [Lorentz-invariant action]; > s.a. self-dual gauge theories.
> Chiral fermions: see lattice theories; non-commutative gauge theories; particle models; spinors in field theory; types of spinors / cosmological 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; a href="t.html#TopologicallyMassiveGravity">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.

Christ-Lee Model
@ References: Anjali & Gupta a1908 [Faddeev-Jackiw quantization].

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.

CHSH Inequalities > see under Clauser-Horne-Shimony-Holt Inequalities.

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."
@ General references: Gurevich a1901 [generalization that cannot possibly be true].
@ 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 of a Vector Field > s.a. magnetohydrodynamics [circulation theorem].
$ Def: The circulation of a vector field A around a closed curve c is \(\oint_c {\bf A}\cdot{\rm d}{\bf s}\).
@ References: Redžić a2006 [time derivative of the circulation, and Faraday's law].

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 degree of classicality / decoherence; classical limit; 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 > s.a. quantum correlations / hidden variables; Fine's Theorem.
@ References: Khrennikov a1808-conf [and Bell inequalities].

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 dS_sys ≥ dQ/T, or \(\oint\) dQ/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 dp/dT 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 ΔS/ΔV between the change in entropy and the change in volume for some amount of substance crossing the line at that point; ΔS can be expressed as l/T in terms of the appropriate latent heat l.
* Remark: For second-order phase transitions in which ΔV = 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 S₄, 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 Algebra

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].

Clocks

Cloning > s.a. quantum technology [quantum cloning].
@ References: Fenyes JMP-a1010 [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]; Fialho a2001 [Fernández-Procacci convergence criterion].
@ 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]; Zhao a1903 [for field theories, translationally and rotationally invariant]; Scola a2005 [d-dimensional Ising model].
> In spin models: see coupled-spin models; ising model.
> Variations: see coupled-spin models [Cluster Mean-Field approach].
> Other applications: see gas [correlation functions]; 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]; Raju et al PRE(18) [and information loss, geometrical]; de Gosson a1901 [coarse-grained dynamics]; Duarte et al a2011 [emergent effective dynamics, mathematical tools].
@ 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 PRD(18)-a1412 [short-distance vs long-distance degrees of freedom]; Bény Quant(18)-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 PRA(19)-a1707; Schindler et al PRA(20)-a2005 [mixed states in multipartite systems].
@ 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]; Bodendorfer & Haneder PLB(19)-a1811 [as a representation change]; Eichhorn et al a1811-Univ [background-independent]; Eichhorn et al a2103 [in quantum gravity]; > s.a. averaging in cosmology; connections; covariant quantum gravity; dynamical triangulations; spin-foam models; spin networks.
> Other specific theories: see emergent theories; 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 M₁ and M₂ by an (n+1)-dimensional M, with ∂M = M₁ ∪ M₂; 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: M₁ and M₂ 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: It requires that M₁, M₂ 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: It is 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.
@ For photons: Picozzi & Haelterman PRL(02) [hidden coherence]; Wolf 07; Chatterjee et al a1812 [Sudarshan's contribution]; Berera PRD-a2009 [over interstellar distances].
@ 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 Quant(18)-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]; Kiukas et al a2011 [and measurement incompatibility]; > s.a. generalized thermodynamics.
@ As a physical resource: Streltsov et al RMP(17)-a1609 [rev]; Ben Dana PRA(17)-a1704; Saxena et al a1910 [dynamical resource theory].
@ 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]; Ringbauer et al PRX(18) [multilevel coherence]; Kendall & Kempf a2004 [n-fragility, and entanglement generation]; Theurer et al a2004 [and entanglement]; Qian et al PRA(20)-a2007 [genuine three-party pure-state coherence].

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: It prevents spacetime symmetries from being unified with internal ones, as some unification ideas would want; This 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]; Nigsch & Vickers a1910 [new approach, and differential geometry].
@ 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]; Gsponer EJP(09) [electrodynamics]; > 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 \(\sqrt3\), ...?); 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.

Combinatorics

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_Pc)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].
@ Representations: Mnatsakanova et al PoS-a1102 [regularity criterion]; Jorgensen & Tian a1601 [and stochastic calculus operators]; Arai 20.
@ 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]; Bosso a2005 [issues and misunderstandings]; > 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 Set / Compactness

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; Englert et al JMO(00)qp/99; 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)
* Idea: N-complexes are objects related to generalized cohomology and involve a linear operator d that satisfies d^N = 0.
@ References: Henneaux IJGMP(08) [and higher-spin gauge theories].

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]; Renou et al a2101, Chen et al a2103 [needed]; Finkelstein a2103 [not all real quantum theories are ruled out].
@ 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 Structure

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

Complexity = Action Proposal > s.a. complexity [laws of complexity].
@ References: Goto et al JHEP(19)-a1901 [for charged black holes].

Componendo & Dividendo
* Idea: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d).

Composite Models of Quarks

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]; Ciaglia et al LNCS(19)-a1908 [general notion].
> 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]; Durham a2011-FQXi [why is the universe comprehensible?].

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 effect: 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 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].

Condensed Matter

Condenser > see Capacitor.

Conduction / Conductors / Conductivity > see electricity; Heat Flow; Transport Phenomena.
* Thermal conduction, macroscopically: Governed by Fourier's law J = −κ ∇T, with J = heat flux, κ = coefficient of thermal conductivity, T = temperature.
* Thermal conduction, microscopically: Thermal conductivity relies mainly on free electrons in conductors, and phonons in insulators; The phonon contribution is a function of three parameters, the speed at which phonons propagate through the lattice (which depends on atomic vibrations at finite temperature), the phonon heat capacity, and how far phonons can travel before they are scattered by lattice imperfections, electrons, and other phonons.
@ General references: Bertola & Cafaro PLA(07) [in Liouvillean form, speed of propagation].
@ Fourier's law: Bonetto et al mp/00 [derivation]; Seligman & Weidenmüller JPA(11)-a1011 [in quantum mechanics]; Simoncelli et al PRX(20) [generalization].
@ Thermal conductivity models: Desloge AJP(62)dec [gas]; Komatsu et al PRL(08) [microscopic derivation]; Collet & Eckmann CMP(09) [model, and Boltzmann equation]; Wu & Segal PRA(11)-a1105 [role of quantum correlations]; Andersson Phys(19) [general discussion, and phonon contribution in metals].

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 PRD(17)-a1609 [gravitational spin connection]; Chaichian & Frasca PLB(18)-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(16)-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 book(17)-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 Gravity

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.

Conical Sections

Conjectures in Mathematics

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.

Connected Topological Space

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
* For spacetime geometry: The claim that it can be inferred from the properties of matter without recourse to spatiotemporal presumptions.
@ General references: Ozhigov a0811-ln [mathematical and physical constructivism].
@ And spacetime, gravity: Norton BJPS(08) [constructive relativity]; Schuller a2003-MG15 [physical and mathematical foundations].

Constructor Theory
* Idea: The theory about which physical transformations can be caused to happen and which cannot, and why; Counterfactuals are a key concept; It aims at changing the mode of explanation for physical phenomena.
@ References: Deutsch a1210 [motivation and principles].
> Online resources: see Quanta article (and YouTube video) on Chiara Marletto.

Contact Geometry / Manifold
* Idea: A generalization of symplectic geometry and symplectic manifolds.
$ Contact manifold: A (2n+1)-dimensional differentiable manifold M with a global 1-form ω such that ω ∧ (dω)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]; de León & Lainz a2011 [contact Hamiltonian and Lagrangian systems, rev].

Contextuality

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]; Martinec 19 [r PT(20)].
@ Problems and solutions: Eglit & Hodges 96; Vekstein 13.
@ Related topics: Gollub PT(03)jan [vs discrete description]; Pronko a0908, Pavelka et al a1907 [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. Infinitesimal; 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]; Bell 19.
@ 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 x₀ in X, or id_X ≅ x₀.
* 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 x₁ and x₂ in X,

d(f(x₁), f(x₂) ≤ c d(x₁, x₂) .

> 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]; Sanayei et al a1807 [trimer states].

Coordinates

Copernican Principle > s.a. cosmological principle [including constraints, tests]; cosmology; paradigms in physics; Typicality.
* 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]; Arjona & Nesseris a2105.
> Related topics: see cosmic microwave background.
> 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]; Hernández et al a1907 [in relativistic quantum mechanics].
@ 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 Censorship

Cosmic Coincidence Problem > see cosmology.

Cosmic Doomsday > see cosmological singularities.

Cosmic Explorer > see gravitational-wave interferometers.

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 Rays

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]; Bolotin a1812 [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.

Cosmological Principle

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]; Harada PRD-a2105 [and gravitational field equations].
@ 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].
@ Counterfactual communication: Salih et al a1806 [it is possible]; Hance a1909 [it is truly quantum].
@ 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(ρA) − tr(ρB).
@ 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].

Cover

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_*π₁(E, e₀) is a normal subgroup of π₁(X, x₀).
$ 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-a1003 [introductory text].

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]

CPT Symmetry, Theorem

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.

Creation Operator

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

Critical Phenomena

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.

Crystals

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.

Cut Metric > s.a. Graphons.
@ References: Bollobás et al JSP(10)-a0901; Coja-Oghlan & Hahn-Klimroth a1905 [for probability distributions].

CW-Complex > s.a. graphs.
* Idea: A space X with a decomposition \(X^0 \subset X^1 \subset \ldots \subset X^n = X\), 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 e₁, e₂, ..., 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.

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