Topics, C
C Operator > see Charge Conjugation.
C*-Algebra > s.a. Grupoid; Inductive
System; lie group [groupoid];
operator theory.
$ Def: A norm-closed
subalgebra of a B(
),
stable under the adjoint operation.
* Relationships: One is
canonically defined by a Lie groupoid.
@ General references: Sakai 71; Dixmier 77; Pedersen 79; Douglas 80
[extensions]; Goodearl 82; Odzijewicz mp/05 [polarized,
and quantization].
@ In physics: Keyl IJTP(98)
[and spacetime structure]; Landsman mp/98-ln
[intro]; > s.a. quantum field theory and algebraic
approach.
Cabibbo Angle > see standard model.
Cabibbo-Kobayashi-Maskawa Matrix > see CP violation.
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 > see analysis;
Derivatives; integration; series.
@ Generalizations: Bazunova et al LMP(04) [ternary algebras]; Harrison
mp/05-ln
[geometric].
Calculus of Variations > see variational principles.
Calendars > see clocks.
Callan-Symanzik Equation > see renormalization group.
Calogero, Calogero-Moser, Calogero-Sutherland Model > see integrable systems.
Calugareanu's Theorem > see Ribbons.
Campbell-Magaard Theorem > see embeddings.
Canonical Distribution / Ensemble > see states in statistical mechanics.
Canonical Form on a Lie Group > see differential forms.
Canonical Formulation of Dynamics > see hamiltonian dynamics and systems; also canonical general relativity, quantum gravity, quantum mechanics.
Canonical Transformation > see states in quantum mechanics; symplectic structure.
Cantor Dust / Set > see fractal.
Cap Product
* Idea: A product involving
homology and cohomology classes.
@ References: in Maunder 72.
Capacitor > see electricity.
Cardassian Expansion > see cosmological models in general relativity.
Cardinal Number / Cardinality > s.a. Continuum.
$ Def: The cardinality
of a set X is the number of its elements, [X].
* Examples: The first
infinite cardinal is [N] = 
0;
The first uncountable one is 1;
Notice that [R] = c > 1.
Cardy-Verlinde Formula > see entropy bound.
Carmeli Metric > see gravitating matter.
Carnot Cycle > see thermodynamical systems.
Cartan Geometry > 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.
Cartan Structure Equation > see affine connection.
Cartan Subalgebra of a Lie Algebra
$ Def: Its maximal commuting
subalgebra.
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.
Casimir-Polder Force > see qed phenomenology.
Catalan Numbers > s.a. series;
stochastic processes.
$ Def: The numbers Cn,
with generating function S(x), defined by
* Generalization:
pC0 =
1; pCn =
(1/n) {pn \choose n–1} for n 
1.
@ References: Nilsson & Sundell JMP(95) & refs
therein.
Catalan's Conjecture > see conjectures.
Catastrophe Theory > related
to phase transitions.
* Idea: The combination
of singularity theory and its applications.
* Catastrophe: An abrupt
change in a system, as a sudden response to a smooth change in external conditions.
@ References: Poston & Stewart 78; Marmo & Vitale FS(80); Arnold 86; Tilley &
Lovett AJP(96) [soap films]; Castrigiano & Hayes 03.
Category Theory > s.a. types of categories.
Cauchy Horizon > see horizons.
Cauchy Problem > s.a. initial
value problem for general relativity.
* Idea: A boundary value
problem in pde's in which one specifies the solution and its normal derivative
on the boundary.
@ References: Fattorini 83.
Cauchy Sequence > see sequence.
Cauchy Surface > s.a. Hypersurface.
$ Def: A closed achronal
hypersurface 
whose
full domain of dependence is the whole spacetime M, i.e., D()
= M, where
D():=
D–() 
D+().
@ References: Bernal & Sánchez LMP(06)
[smoothability and time functions].
Cauchy Theorem > see analytic functions.
Cauchy-Riemann Condition > see analytic functions [holomorphic].
Cauchy-Schwarz Inequality > see inequalities.
Causal Continuity > see causality conditions.
Causal Future or Past of a Subset of Spacetime > see spacetime subsets.
Causality > s.a. causality conditions; causal structure in spacetime.
Caustics > see geodesics.
Cavendish Experiment > see physics experiments.
Cayley Numbers > see Octonions.
Cech Cohomology > see cohomology; types of homology.
Celestial Mechanics > see orbits in newtonian gravity.
Celestial Navigation
@ References: Van Allen AJP(04)
[basic principles].
Cell-Like Map > same as Resolution.
Cellular Automaton > s.a.
computation; game theory [life]; quantum
computation.
@ General references: Wolfram CMP(84),
86.
@ Related topics: Tisseur Nonlin(00)m.DS/03 [Lyapunov
exponents, entropy]; Alonso-Sanz PhyA(05)
[with memory, phase transitions].
@ Quantum: Svozil PLA(86)
[for quantum field theory]; Meyer JSP(96)qp, PLA(96)qp [no
homogeneous, scalar, unitary ones on Euclidean lattices], qp/96;
Gudder IJTP(99)
[overview]; Svozil qp/02-in
[as models]; McGuigan qp/03 [and
lattice field theory]; Andrecut & Ali PLA(04)
[entanglement dynamics]; Schumacher & Werner
qp/04 [reversible];
Pérez-Delgado & Cheung qp/05,
a0709-PRA.
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
@ Relativistic particles: Lehner & Moreschi JMP(95); Alba
et al JMP(02) [and rotational dynamics].
@ Other systems:
Nester et al gq/04-in
[teleparallel gravity], gq/04-in
[in general relativity, quasilocal]; Helling ht/05 [non-commutative
theories].
@ 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.
@ 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.
@ References: Vignat & Plastino JPA(07) [generalization to deformed, q-Gaussians].
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.
Cerenkov Radiation > see radiation.
Chain > s.a. homology [algebraic geometry
notion]; posets.
$ Idea: A totally ordered
subset of a partially ordered set.
Chain Complex > see homology.
Chameleon Scalar Field
* Idea: A 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.
@ General references: Khoury & Weltman PRL(04)ap/03 [gravity
in space];
Nojiri & Odintsov MPLA(04)ht/03 [instability];
Waterhouse ap/06 [pedagogical
intro]; Brax et al a0706-in
[primer].
@ And
cosmic acceleration:
Khoury & Weltman PRD(04)ap/03;
Brax et al PRD(04)ap;
Brax et al a0806 [and f(R) gravity].
@ Other phenomenology: Mota & Shaw PRL(06)
[viability and possible experiments]; Upadhye et al PRD(06)
[and inverse-square
tests]; > s.a. tests of newtonian gravity [constraints].
Chance > see probability in physics.
Chandrasekhar Limit
* Idea: The upper bound
on the mass of a white dwarf.
@ References: Gregg & Major a0806 [changes from modified dispersion relations].
Chaos > s.a. chaos in gravitation; chaos in the metric; chaotic systems; mathematical description; quantum chaos
Chaplygin Gas > s.a. dark
energy.
* Idea: A gas
with an exotic equation of state, pX = –A / 
rX (polytropic,
with negative constant and exponent).
@ References: Debnath & Chakraborty 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]; > s.a. wormholes.
Chapman-Enskog Method > see Boltzmann Equation.
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)).
Character of an Algebra > s.a.
Spectrum.
$ Def: A non-zero algebra
homomorphism 
: A → C.
Characteristic Polynomial / Equation
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
@ References: Rosen AJP(73)
[form electromagnetic quantities]; Nefkens et al PRL(05)
[test of invariance with 
decay].
Chasles' Theorem
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-in
[in general relativity].
Chemical Potential > s.a. thermodynamics.
* Idea: The thermodynamical
variable 
:=
(
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, :=
–kT (lnZ/N)V,T ,
and if the particles are non-interacting, such as in an ideal gas, :=
–kT ln(
/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);
Baierlein AJP(01)
[meaning]; Tobochnik et al AJP(05)
[understanding, Monte Carlo algorithms]; Kaplan JSP(06)
[correct definition].
@ Specific systems: Herrmann & Würfel AJP(05) [for light, non-zero].
Cherenkov Radiation > see under Cerenkov.
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 functon
Y[A]:= (k/4
)
tr[A 
dA –
(2/3) 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].
Chern-Weil Theory > see characteristic classes.
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].
Chiral Symmetry
@ Breaking: Giusti & Necco JHEP(07) [lattice QCD].
Chirality > s.a. spinors
in field theory.
@ Chiral bosons: Abreu & Dutra PRD(01)ht/00; Abreu & Wotzasek ht/04-in.
Chisholm's Theorem > see quantum field theory effects.
Choquet Space
* Idea: A generalization
of the notion of topological space.
$ Def: A convergence
space in which a filter 
coverges
to x whenever every ultrafilter finer than converges
to x.
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-in [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.
@ References: Kronheimer & Penrose PCPS(67); Carter GRG(71); Kronheimer GRG(71);
Harris CQG(00)gq/99 [topology].
Chronology > see cosmological history.
Chronology Protection > see causality violations.
Chronon > s.a. dirac fields; spinning
particles; time in quantum theory.
* Idea: A unit of discrete
time.
Church-Turing Thesis
* Idea:
The only computable functions are the partial recursive ones, and they are
also the ones computable by Turing machines.
@ And physics: Deutsch PRS(85)
[quantum theory and computers]; Svozil qp/97;
Etesi & Németi IJTP(02)gq/01 [general
relativity].
> Online resources:
Stanford Encyclopedia of Philosophy page;
Wikipedia page.
Cirel'son's Bound > see bell's inequality; 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 > s.a. anthropic
principle; Doomsday Argument.
* Fermi paradox: If
there had ever been a single advanced civilization in the cosmological history
of our galaxy, dedicated to expansion, it would have had plenty of time to
colonize the entire galaxy via exponential growth; No evidence of present
or past alien visits to Earth is known to us, leading to the conclusion that
no advanced expanding civilization has ever existed in the Milky Way.
@ General references: Kuiper & Brin AJP(89).
@ Types of civilizations: Kardashev SovAstr(64)www [Kardashev scale];
in Sagan 73, pp233-234; Hartle & Srednicki PRD(07)-a0704 [are
we typical?]; Cirkovic a0805-JBIS
[optimization-driven development].
@ Inflation, black holes, and civilizations: Linde PLB(89);
Tipler PLB(92),
rebuttal Ellis & Coule GRG(94)
[ultimate fate of life]; Garriga et al IJTP(00)ap/99-in;
Olum Anal(04)gq/03.
@ And future of the universe: Krauss & Starkman ApJ(00)ap/99;
Freese & Kinney
PLB(03)ap/02 [acceleration];
Cirkovic FP(04)ap/02 [long-term]; Page
ht/06 [universe rate of decay]; > s.a. cosmology.
@ Fermi
paradox:
Gros JBIS(05)ap;
Gato-Rivera phy/05-in;
Gato-Rivera phy/06-in
[inflation and brane world].
@ Communication, observation: Gato-Rivera phy/03 [undetectability];
Loeb & Zaldarriaga JCAP(07)ap/06 [radio
eavesdropping]; Learned et al a0805 +
pw(08)may [using
neutrinos].
CKM Matrix (Cabibbo-Kobayashi-Maskawa) > see CP violation; standard model.
Class > see set.
Classical Mechanics > see approaches and formalism; systems.
Classifying Space
* Example: B(U(1))
= CPinfty.
Clauser-Horne-Shimony-Holt Inequalities > see correlations; hidden variables.
Clausius Inequality / Formulation of Second Law > see thermodynamics.
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 
S/V between
the change in entropy and the change in volume for some amount of substance crossing
the line at that point; S can
be expressed as l/T in terms of the appropriate latent
heat l.
Clebsch Potential
@ And electromagnetism: Wagner PLA(02) [problems].
Clebsch Variables > see perfect fluids.
Clebsch-Gordan Theory / Coefficients
Cloak > see metamaterials [electromagnetic "invisibility" cloak]; sound [acoustic cloak]
Cloud Chamber > see physics experiments.
Cluster Expansion / Variation Method > s.a. gravitational
statistical mechanics; ising
model; lattice
field theory and lattice gauge theory.
* Idea: A hierarchy
of approximate variational techniques for discrete (Ising-like) models in
equilibrium statistical mechanics.
@ References: Pelizzola JPA(05) [rev].
Clustering > see astronomical objects [star clusters]; gas [cluster expansion]; gravitating matter.
CO Space > see types of topological spaces.
Coalgebra
@ References: Brouder mp/05-in
[use in quantization].
Coarse Structures in Geometry
@ References: Roe 03; Dydak & Hoffland T&IA(08)
[definition via uniformly bounded families].
Coarse-Graining > s.a. lattice
field theory; network; renormalization; thermodynamic
concepts.
@ General references: Anastopoulos PRD(97)ht/96 [in
quantum field theory, ito open systems]; Ridderbos SHPMP(02)
[inadequate approach to statistical mechanics]; Dvurecenskij
et
al RPMP(05)qp/04 [of
observables]; Kawasaki JSP(06)
[maximum entropy and reduced dynamics]; Gell-Mann & Hartle PRA(07)qp/06 [in
quantum
theory, and entropy].
@ Related topics: Rodríguez & Santamaría-Holek PhyA(07) [and
non-extensive effects in gas of Brownian particles].
Cobordism > s.a. 2D, 3D,
4D manifolds; discrete
geometries [graph cobordism]; morse
theory; Surgery.
* Idea: The study of
the interpolation between n-dimensional manifolds M1 and
M2 by an (n+1)-dimensional M,
with M = M1 M2;
Can be traced to H Poincaré, and in its modern form to L
Pontrjagin.
* 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.
* Cobordism classes: The
equivalence classes of n-dimensional cobordant manifolds is a Z2-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.
@ References: Stong 58; Milnor 65; Peterson 68; Landweber PCPS(86); Vershinin 93; Atiyah
BAMS(04) [Thom's cobordism theory].
Cochain > a concept in cohomology theory; s.a. Triangulations.
Coefficient of Restitution
@ References: Ferreira da Silva EJP(07) [concept].
Cogravity
@ And perihelion precession: de Matos & Tajmar gq/00.
Coherence > s.a. Interference [measuring
coherence].
* In classical mechanics:
For a wave, Glauber defined a notion of degree of coherence based on whether
certain
n-th order correlation functions vanish; > s.a. wave
equations.
* In quantum mechanics:
For a wave function 
and
points x and y, the mutual coherence is the 2-point
function 
(x, y; 
):= 
*(x, t) (y, t+)
;
> s.a. coherent states.
@ In quantum mechanics: Slosser & Meystre AJP(97)RL
[quantum optics]; Ponomarenko et al PLA(05)
[optical field, significance]; Cavalcanti & Reid PRL(06)qp/07 [criteria
for macroscopic coherence]; Sewell a0711-in
[in quantum statistical mechanics, survey].
Coherent States > s.a. generalized and modified states; types of coherent states.
Cohomology > s.a. types of cohomology.
Coincidence Problem > see cosmology.
Coisotropic Submanifold > see symplectic structure.
Cokernel
$ Def: The cokernel
of a (group) homomorphism f : G → H is Cok(f):= H / f(G).
Cold Fusion > see nuclear technology.
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.
@ References: Coleman & Mandula PR(67);
Pelc & Horwitz JMP(97)
[higher-dimensional generalization].
Coleman-Weinberg Effect
@ References: Floreanini et al CQG(91)
[in quantum gravity].
Collapse > see gravitational collapse; wave function collapse.
Collineations > s.a. affine
structures; FRW
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].
@ Projective collineation: Hall & Lonie CQG(95) [on spacetime].
@ Matter collineations: Sharif NCB(01)gq/05 [Bianchi
I, II, III, VIII, IX, Kantowski-Sachs]; > s.a. bianchi
models.
Collisions > see scattering.
Colloids > s.a. [entropy].
Colombeau Algebra
* Idea: A space of generalized
functions, more general than distributions, for which a multiplication is defined.
@ General references: Gsponer a0807 [intro].
@ 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 a0806 [and
pointlike electrons]; Colombeau et al a0705, Colombeau & Gsponer a0807 [quantum
field theory]; > s.a. general
relativity solutions with
matter, types of metrics.
Color > see light; QCD [as a quantum number].
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 (equilateral triangle of edge length = 1, ...); For n = 3,
yes (circle of radius 
,
...?); 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 1880's from complex function theory with Klein, Fricke and Poincaré.
@ References: Stillwell 80; Cohen 89; Johnson 89.
Combinatorial Topology
* Idea: A type of algebraic
topology that uses combinatorial methods; Includes simplicial homology.
@ References: Pontrjagin 52; Aleksandrov 56.
Common Cause > see causality.
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 =
1n M i–1
[M, A] M n-i.
@ In quantum mechanics: Luis JPA(01)
[as a geometric phase]; Sergi qp/05/PRE
[non-Hamiltonian]; Tangherlini PS(08) [covariant].
> In quantum mechanics:
see computer
languages [with Mathematica]; observable algebras; uncertainty
relations, modified
uncertainty relations; types of groups [Heisenberg
algebra].
Compactification of Extra Dimensions > see strings.
Compactification of Spacetime > see spacetime boundaries.
Compactification of a Topological Space > see Bohr Compactification; compactness.
Complementarity > s.a. [quantum
theory]; Interference; quantum
representations; uncertainty; Wave-Particle
Duality.
* Idea: Bohr's view
that microscopic objects can behave as particles or waves in different situations;
For example, an object can have either a sharply defined position or a sharply
defined momentum, but not both; No
matter how a system is prepared, for each degree of freedom there is always
a measurement whose outcome is totally unpredictable.
* 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.
@ General references: Rosenfeld Nat(61)apr;
Wootters & Zurek PRD(79)
[and 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); Saunders qp/04 [and
Bohr]; de Ronde qp/05,
a0705 [and
interpretations]; Camilleri SHPMP(07)
[Bohr and Heisenberg].
@ Afshar's experiment:
Afshar SPIE(05)qp/07, AIP(06)qp/07 [violation?];
Qureshi qp/07;
Reitzner qp/07;
Steuernagel FP(07);
Flores a0802.
@ Related topics: Ross NCB(93) [???]; Roll-Hansen HSPBS(00)
[and biology]; Luís
PRA(01)
[2D
systems];
beim Graben & Atmanspacher FP(06)
[in classical mechanics].
Complete Manifold > see differential geometry.
Complete Normed Space > see Banach Space.
Completely Regular Topological Space > s.a. uniformity.
Completeness of Quantum Theory > see foundations of quantum mechanics.
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.
* Acyclic complex: One
without cycles, Hq(X) =
0 for q = 0, and Hred, 0(X) = 0.
> Related topics:
see euler characteristic; homology and cohomology [chain complex and dual operatior complex].
Complex Numbers > s.a. analysis;
analytic functions; i.
* Möbius transformation: The map z 
(az + b)
(cz + d)–1, where
the matrix {a, b //
c, d} is in SL(2,C).
@ General references: Ahlfors 81 [Möbius transformation].
@ In quantum mechanics: Dirac PRS(37);
Accardi & Fedullo LNC(82); Anastopoulos IJTP(03)gq/02-in;
Lev
FFTA(06)ht/03;
Bracken RPMP(06)qp/05 [Hilbert
space quantum mechanics]; Anastopoulos IJTP(06);
Davis IJTP(06).
@ And physics: Burko TPT(96) [meaning]; Benioff qp/05 [Fock-type
representation]; > s.a. complex
structure.
Complexity > s.a. mathematics and posets.
Componendo & Dividendo
* Idea: If a/b = c/d, then (a+b)/(a–b) = (c+d)/(c–d).
Composite Systems > see composite quantum systems; composite particle models.
Compressibility
@ References: Bragg & Coleman JMP(63)
[thermodynamic inequality]; Hernández & Velasco AJP(98)
[positive and negative].
Compton Effect / Scattering > see photon.
Compton Wavelength
* Idea: The wavelength
below which kinetic energy can be used to produce an extra pair particle-antiparticle;
For a particle of mass m, 
C:= 
/ mc;
For an electron, C =
3.86 
10–13 m.
Computation > s.a. computer languages; computational physics; 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.
Condensation > see phase transition.
Conductors / Conductivity > see electricity; heat; Transport.
Cone on a Space > see topology.
Configuration Space
* 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 
to
include distributional fields of some sort.
* For field theories:
It has the structure of a configuration bundle (Y, , )
over the space manifold .
@ 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.
Confinement > s.a. QCD; QCD
phenomenology.
@ Models: Delfino & Grinza a0706 [in q-state
Potts field theory]; > s.a. Bag Model.
Confluent Hypergeometric Functions > see Hypergeometric Functions.
Conformal
Field Theory > s.a. conformal structures, conformal
structures in physics.
@
2D: Friedan & Schenker NPB(87);
Giddings PRP(88);
Jain IJMPA(88)
[and strings in general backgrounds]; Segal in(88); Moore & Seiberg
CMP(89); 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].
@ Higher-dimensional: Anselmi PLB(00)ht/99 [classification,
even D];
Petkova & Zuber
ht/01-in
[rational, rev]; Castro-Alvarado & Fring NPB(04)
[vacuum energies]; > s.a. AdS-cft [including
de Sitter-conformal field theory].
Conformal Gravity > s.a.
3D gravity; bianchi
I; gravity theories;
covariant quantum gravity;
schwarzschild; unified
theories [Weyl].
* Idea: A theory of gravity
that is invariant under conformal (local scale) transformations; There are
several versions in the literature, a popular one being the higher-derivative
theory with the Bach equation as the vacuum field
equation, and
action
S = 
d4x |g|1/2 Cabcd Cabcd.
* Motivation: Initially,
the dimensionless coupling constant ,
for quantization; Later, used to explain flat galactic rotation curves without
dark matter.
* And general relativity:
One can obtain Einstein gravity from conformal gravity in 4D by introducing
a scalar
compensator
with a vacuum expectation value that spontaneously breaks the conformal invariance
and generates the Planck mass, or by compactifying extra dimensions in
a higher-dimensional conformal theory of gravity (without the need to introduce
the scalar compensator).
* Solutions, and phenomenology:
All vacuum solutions of general relativity are solutions of conformal gravity
(e.g., Schwarzschild), but
not the other way around, and not with matter; Linearized theory gives a 4th-order
wave equation, (t2+
2)2 
=
0 around Minkowski.
* Results: Get extra
attractive effect on matter (from motion in Schwarzschild-like solutions, the
Newtonian potential is modified to V(r) = –b/r + cr),
but also an additional repulsive term for light, affecting light deflection
(and the latter does not fit observed data); The cosmological Geff is
smaller than the Cavendish one; > s.a. dark
matter [alternatives].
@ General references: Boulware et al PRL(83)
[zero-energy theorem]; Gorbatenko et al GRG(02)gq/01 [and
geometrodynamics]; Gorbatenko & Pushkin GRG(02)
[and causality]; Gorbatenko GRG(05)
[properties]; Carroll a0705 [and
quantum theory].
@ Quantum:
Wang JPCS(06)gq/05-in, PTRS(06)gq [canonical,
new variables and Immirzi parameter]; Mannheim a0707-in
[no ghosts].
@ Barbour's version:
Barbour CQG(03)gq/02 [particle
motion], Anderson et al CQG(03)gq/02 [geometrodynamics].
@ Cosmology: Mannheim GRG(90),
ap/96 [age
of universe], ap/98, ap/98-in, gq/99-in, ApJ(01)ap/99 [cosmic
acceleration]; > s.a. cosmological constant
problem.
@ Other phenomenology: Barabash & Shtanov PRD(99)ap [Newtonian
limit]; Navarro & Van Acoleyen JHEP(05)ht [compactification
and general relativity]
@ Solutions: Schmidt AdP(84)gq/01, AN(85)gq/01 [of
Bach equation]; Le Brun CMP(91);
Edery PRL(99)gq;
Dzhunushaliev & Schmidt
JMP(00)gq/99 [vacuum].
Conformal Invariance and Structures in Physics
Conformal Structure and Transformations
Congruence of Lines in a Manifold > see Expansion; Shear; Vorticity.
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.
Consciousness > see mind.
Conservation Laws, Conserved Quantities
Consistency of a Theory > see for example electromagnetism.
Consistent Histories Formulation of Quantum Theory > see histories.
Constants > 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.
Contact Geometry / Manifold
$ Contact manifold:
A (2n+1)-dimensional
differentiable manifold M with a global 1-form 
such
that
(d)n 0,
for all p in M.
@ General references: Hurtado DG&A(08) [stability numbers].
@ Contact geometry and physics: Rajeev AP(08)mp/07 [thermodyamics,
geometrical optics, and quantization].
Contextuality > see foundations of quantum mechanics; experiments in quantum mechanics.
Continued Fractions > see numbers.
Continuity Equation > see conservation laws.
Continuous Media > s.a. Extended
Objects; fluids;
field theory; gravitating
matter.
@ References: Gollub PT(03)jan
[vs discrete description].
Continuum > 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 R conumerous with either Z or R?
Cohen: This cannot be decided based on the Zermelo-Frankel axioms.
@ General references: White 92 [and physical theories, history];
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].
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 idX 
x0.
* Properties:
A Contractible space has the same homotopy type as a point.
* Relationships:
Contractibility implies simple connectedness.
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 > see Lorentz Equations.
Conventionalism > see spacetime.
Convergence > see sequence; series.
Convex Functions > s.a. functions; analysis.
@ References: Gibbons & Ishibashi CQG(01)gq/00 [and
spacetime geometry].
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 [convex subsets]; vector space [locally convex].
Convolution > see functions.
Conway Polynomial > see knot invariants.
Cooper Pairs > s.a. superconductivity.
@ References: news pn(07)dec [in insulators].
Copernican Principle > s.a. cosmology;
microwave background.
* Idea:
The Earth is not the center of the Solar System / Our location in the universe
is not a special one in any way.
* 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].
@ Tests:
Clarkson et al PRL(08)-a0712 [model-independent,
and
acceleration]; Uzan et al PRL(08)
[time-drift of cosmological redshift]; Clifton et al a0807 [redshift
dependence
of luminosity distance]; Bolejko & Wyithe a0807 [supernovas and cosmic flow]; > s.a. observation [homogeneity].
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 ccw direction.
> Online resources: 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 (including correlation length)
Correspondence Principle > s.a. classical
limit of quantum mechanics.
@ References: Heller & Tomsovic PT(93)jul;
Kawai & Stapp PRD(95)qp [QED
and S-matrix]; Kazakov NPPS(02)ht/01,
IJMPD(03)ht [in
quantum gravity];
Karkuszewski
et al PRA(02)
[breakdown in chaos]; Makowski & Górska PRA(02)
[exact cases]; Makowski EJP(06)
[formulations].
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 wrt a subgroup H; In
physics:
> see, e.g., geometric
quantization.
> Online resources:
see MathWorld page.
Cosmic Coincidence Problem > see cosmology.
Cosmic Microwave Background > s.a. cmb anisotropy.
Cosmological Constant > see also cosmological constant problem.
Cosmological Models > see also general relativistic models.
Cosmological Principle > see cosmology.
Cosmology > s.a. acceleration; history; observation; perturbations.
Cotton Tensor > s.a. riemannian
geometry [Cotton flow].
* 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
Cab := 
amn m(Rnb – 
R gnb )
.
@ 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].
Coulomb Gauge > see gauge.
Coulomb's Law > see electricity.
Coulomb Potential > see scattering.
Counterfactuality, Counterfactuals
@ In quantum mechanics: Finkelstein Syn(99)qp/98 [and
spacelike separated points]; Choy & Ziegeler qp/99/AJP;
Bigaj Syn(04)
[and spacetime events]; Tresser qp/05 [weak
realism]; Vaidman a0709-in.
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 
.
@ References: Besprosvany MPLA(03)
[and particle compositeness].
> For specific theories:
see fine structure constant; gravitational
constant; GUTs.
Covariance > s.a. Coordinates; Event;
Hole Argument;
Reference Frame; Relativity
Principle.
* General covariance:
A theory is generally covariant iff it is (a) Invariant under all changes of
coordinate
system, similar to saying that it is diffeomorphism-invariant, or (b) Expressed
in terms of only the metric (and other dynamical fields), with no background
geometry.
* Issue: Any theory can be reformulated
(by putting enough structure
among
the "variables") so as to satisfy the definition.
* Remark: This is not
always the same as saying that no preferred observer
is selected (e.g., see cobordisms).
@ In general relativity: Norton FP(89)
[Einstein's view and modern view]; Ellis
and Matravers GRG(95)
[questioning]; Zalaletdinov et al
GRG(96);
Guo et al PRD(03)
[and Noether charges]; Lusanna gq/05-in
[rev]; Dieks SHPMP(06)
[vs equivalence of reference frames]; Giulini gq/06-in
[issues + historical]; Mashkevich gq/06 ["geometricity"];
Gao & Zhang PRD(07)gq,
Sotiriou & Liberati PRD(07)gq [relationship
with gravitational dynamics].
@ Violations of general covariance: Pirogov gq/06-in [and extra particles].
@ In quantum field theory: Brunetti et al CMP(03)mp/01 [algebraic], mp/05 [rev];
> s.a. types
of quantum field theories [diffeo-invariant].
@ In quantum gravity: Padmanabhan MPLA(88);
Kazakov CQG(02);
Christodoulakis & Papadopoulos gq/04 [and
observables].
@ Related topics: 't Hooft pr(89) [2D, discrete model]; Mack gq/97;
Bing gq/98 [??];
Francis gq/02 [quantum
proposal]; Lusanna & Pauri gq/03 [and
gauge]; Wu & Ruan ht/03 [and
general relativity, ??]; Mekhitarian & Mkrtchian mp/04 [applications];
Colosi et al CQG(05)gq/04 [model,
info and evolution]; Treder & von Borzeszkowski FP(06)
[and spacetime structure].
Covariant Derivative > see tensor fields.
Covariant Regularization Scheme > see regularization [Pauli-Villars].
Covector > see differential forms [1-form].
Covering Dimension (Of a topological space) > see dimension.
Covering Number > see cover.
Covering Space > s.a. 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,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*1(E, e0)
is
a normal subgroup
of 1(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: 
G implies
that p = p.
@ References: Brown AMM(74).
Coxeter Groups [> s.a. group
types.]
* 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)].
@ 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].
Crane-Yetter State Sum > see spin foam models.
Cremmer-Scherk Theory > see spin-1 field theories.
Cross Product > see vectors.
Cross Section in Scattering Theory > see scattering, units [barn].
Cross Section of a Bundle > see bundle.
Crumpling > s.a. quantum
regge calculus.
* Idea: A type of phase
transition.
@ References: Foltin JPA(01) [in fluid membranes].
Cryptology
@ References: Beutelspacher 94.
Cryptography > s.a. quantum technology.
* Tools: In cryptography,
frequency analysis is a code breaker's fundamental tool.
@ References: Kippenhahn 99, Singh 99 [I].
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.
Curvaton > s.a. inflationary phenomenology [structure formation].
* Idea: A light scalar
field during inflation whose 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.
Curvature > s.a. line; riemann.
Curve > s.a. Fitting; geodesic; loop; spacetime subsets; Timelike Curve; vector field [integral curve].
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 {ei}
of K,
such that ei is homeomorphic
to an open ni-cell, and
each point in the boundary of ei is
in some other ej (with
nj < ni);
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.
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 > see cosmological models and general relativistic cosmologies.
Cyclic Representation of a Group > see group representation.
Cylindrical Function
$ Def: A function f on
an infinite-dimensional vector space is cylindrical wrt a finite-dimensional
subspace Vn of (the dual
of) V spanned by e1, e2,
..., en if f()
depends only on the components i = ei()
of in Vn.
Cylindrical Symmetry > see types of spacetimes.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
20 jul 2008