* Idea: The Lorentzian geometry version of the Laplacian,
= gab 
ab .* On scalars: Can be written as
=
|g|–1/2 (|g|1/2 gab
b), a .* In harmonic coordinates: It simplifies to
= gab ab .Topics, D
D'Alambertian Operator > s.a. wave
equations.
* Idea: The Lorentzian
geometry version of the Laplacian, = gab ab .
* On scalars: Can be
written as =
|g|–1/2 (|g|1/2 gabb), a .
* In harmonic coordinates:
It simplifies to = gab ab .
Damped Systems > s.a. dissipation; oscillators.
@ Negative damping: Green & Unruh AJP(06) [and Tacoma Narrows bridge].
> Quantum: see quantum
oscillators; quantum systems; states
in quantum field theory; types of quantum
states.
Darboux Space > see 2D geometries, 3D geometries.
Darboux Transformation > s.a.
potential [quasi-exactly solvable]; toda lattice.
@ General references: Darboux CRAS(1882);
Rosu qp/98-in
[review].
@ Related topics: Bagrov et al mp/98-in
[of coherent states]; Samsomov JMP(98)qp/97 [and
phase space transformations]; Ustinov RPMP(00)mp [and
solutions of differential equations].
@ Generalized:
Morales et
al JMP(01);
Humi NCB(02)mp [fractional];
Song & Klauder JPA(03)
[t-dependent Hamiltonian systems].
Darboux's Theorem > see symplectic manifold.
Dark Energy > s.a. observational cosmology.
Dark Matter > s.a. matter contents of the universe; types of dark matter.
Davenport Constant > see finite groups.
de Broglie-Bohm Interpretation of Quantum Mechanics > see pilot wave.
De Donder Gauge > see gauge.
De Donder-Weyl Formalism > see types of symplectic structures.
de Rham Complex
de Rham Theorem > s.a. types
of cohomology theories.
$ Def:
The de Rham cohomology H*de Rham(M)
is the dual of the real singular homology H*(M;R).
@ References: in Warner 71.
de Sitter-Fokker Precession
* Idea: The precession
of the spin axis of an orbiting gyroscope; a.k.a. geodesic precession.
Debye Cutoff Length
@ References: Spiegel ap/98-in
[and gravity]; Rubab & Murtaza PS(06)
[non-Maxwellian plasmas].
Debye Model > see specific heat.
Debye-Waller Factor
Decay > see particles; quantum mechanical effects.
Decidability > see computation.
Decision Theory > see game.
Decoherence > s.a. black hole information and thermodynamics; quantum cosmology; semiclassical quantum mechanics; vacuum.
Decoherent Histories Formulation of Quantum Theory > see quantum histories.
Decomposition of Functions and Tensors
Dedekind Cut > see numbers.
Defects > s.a. particles [geometrical
models]; quantum field theory
effects; topological
defects.
@ General references: Mazenko 02 [fluctuations,
order]; Manko et al PhyA(04)
[local states]; Bazeia
ht/05-ln
[in scalar field theory]; Caudrelier a0704 [in
integrable field theories]; Afonso et al a0710 [building
networks of defects].
@ And condensed matter: Mermin RMP(79)
[and homotopy]; Nelson 02 [r PT(03)may];
Cancès et al CMP(08) [electrons, mean-field model].
@ Dislocations, disclinations: Katanaev cm/04-in
[Riemann-Cartan framework]; Comer & Sharipov mp/05 [differential
equations
and differential geometry]; Kleman & Friedel RMP(08)
[rev]; > s.a. spacetime
models.
@ And spacetime curvature / torsion: Maluf & Goya CQG(01)gq [and
teleparallelism];
Schmidt & Kohler GRG(01)gq [simplicial,
Regge calculus];
Tartaglia IJMPA(05)gq/04-in.
Deformations > see hamiltonian dynamics [phase space]; lie algebras; Logarithms; particle models; Planck Cube; quantization.
Degeneracy of Eigenvalues of the Hamiltonian > s.a. quantum
systems.
@ When: Fallieros & Hadjimichael AJP(95);
Chau AJP(95)
[from supersymmetric
quantum mechanics].
Degree Theory in Banach Spaces
* History: Developed by
Leray and Schauder in the 1930's.
@ References: Leray & Schauder AENS(34); Rothe 86.
Dehn's Lemma
Delaunay Triangulation > see voronoi tilings.
Delayed Choice Experiments > see experiments in quantum mechanics.
Delta Function > see non-standard analysis; distribution.
Dense Subset > see posets; [topology].
$ Def: A subset A of
a
topological space X such that every open neighborhood of x 
X contains
an element
of A.
Density, Tensor > see tensor fields.
Density Matrix > s.a. mixed
states.
* Uses: As "statistical
mixture", "reduced density matrix", and "conditional density
matrix".
@ References: Dürr et al FP(05)qp/03,
Maroney FP(05)
[Bohmian
mechanics]; Tulczyjew a0711 [non-normalized, and selective measurements].
Denumerability (R Smullyan, UM talk 2000)
* Analogy: Satan tells
a soul it will go free if he guesses (i) a positive integer, (ii) an integer,
(iii) two integers, (iv) a ratio, or (v) a real number; In which cases does
the soul know that he is not eternally damned?
Dependence > see Independence.
Derivation
$ Def: A linear mapping
between two vector spaces, satisfying the Leibniz rule.
Derivatives > see analysis [including
fractional];
tensor field.
* Directional derivative:
Depends on a vector field va, v(f)
= va a f,
and coincides with the Lie derivative wrt va.
@ Special types:
Hurley & Vandyck IJGMP(08) [D-differentiation].
> For functions on a manifold:
see connection [covariant
derivative]; exterior derivative; lie
derivative; Pfaff Derivative
> Other types of situations:
see Banach Space [Fréchet derivative].
Descriptors of a Mapping > see diffeomorphisms.
Design (Argument by) > see cosmology.
Designer Gravity
* Idea: Theories in which
gravity is coupled to a tachyonic scalar with anti-de Sitter boundary
conditions.
@ References: Hertog & Hollands CQG(05)ht,
Hertog CQG(05)
[stability].
Detailed Balance
* Idea: A joint condition
on the dynamics and a statistical state of a system described by a set
of states r, s, ...; States that the rate of occurrence of
any transition
is the same as the rate of occurrence of the inverse transition, or Pr Wrs =
Ps Wsr.
* Use: It implies that
the state is stationary, dPr/dt =
0, from the master equation, but is not necessary; Related to reversibility.
@ References: Evans JPA(05) [generalization for non-equilibrium states].
Detectors in Quantum Theory > s.a. experimental
particle physics; particle
effects; quantum field theory effects in curved
spacetime.
@ General references: Bloch PR(67);
Bloch & Burba PRD(74)
[and presence of particle]; Hinton JPA(83), CQG(84);
Marshall FP(91)
[efficiency and fluctuations of electromagnetic field]; Marolf PRA(94)gq/93;
Brunetti & Fredenhagen PRA(02)qp/01 [time
of occurrence]; Bondurant PRA(04)
[pointlike model]; Langlois AP(06)
[topologically non-trivial spacetime].
@ Accelerated: Klyshko PLA(91);
Sriramkumar & Padmanabhan CQG(96)
[finite-time]; Davies et al PRD(96)gq [rotating];
Kim PRD(99)
[accelerated oscillator]; Sriramkumar gq/01 [accelerated
D+1-dimensional]; Sonego & Westman
CQG(04)gq/03 [and
geodesic motion]; Lin & Hu PRD(06)
[vacuum fluctuations to radiation]; Louko & Satz gq/06-in
[with regularisation]; Costa & Piazza a0805 [and Unruh effect]; > s.a. mirrors.
Determinant > see matrices [including functional].
Determinism > s.a. causality;
paradigms in physics; Predictability;
reversibility.
* History: Introduced
for a system by Newton [@1687]; extended to the whole universe by Laplace [@1820];
However, Laplace thought that the
step from determinism to predictability was only a quantitative issue, of
having
enough
data; This we now know to be false, after quantum mechanics (and special relativity)
dealt a severe blow to this
view.
* In quantum mechanics:
The wave function evolves deterministically, not the results of single experiments.
@ General references: Earman 86; Ruelle 94 [chance and determinism,
I]; Bishop phy/05-in
[in physics, rev].
@ In classical physics: Stein PhSc(91),
Maxwell PhSc(93)
[in special relativity]; Bhat & Bernstein
IJTP(97),
Kosyakov FP(08)ht/07 [example
of non-unique evolution].
@ In quantum physics: Peres & Zurek AJP(82)
[unavoidable indeterminism];
Knill qp/96 [and
randomness]; 't Hooft ht/00-in
[and dissipation]; > s.a. bell's inequalities, experiments, hidden
variables, pilot
wave, time
in
quantum mechanics.
Diagram
* In category theory:
Any collection of objects connected by morphisms.
Diamagnetism > see magnetism.
Diameter > see metric spaces.
Dichroism > see polarization.
Dicke Model
* Idea: A collection
of two- and three-level atoms interacting with (a single quantized mode of)
the electromagnetic field and contained within a volume much smaller than the
smallest
resonance
wavelength.
@ General references: Buzek et al qp/05 [ground
state instabilities]; Dimer et al qp/06 [realization
in cavity QED].
@ Generalized: Aparicio et al a0706 [generalized fermion, phase transition].
Dickey Bracket > see lagrangian dynamics.
Dielectrics > see electricity.
Difference Equations
@ References: Lakshmikantham & Trigiante 02 [including numerical];
Legault & Senior
JMP(02)
[second-order, method]; Krichever mp/04 [rational
and elliptic coefficients]; Elaydi 05 [II/III, intro]; Sasaki a0708,
Odake & Sasaki a0708 [quasi-exactly
solvable].
Difference Operator > see sequences.
Differentiable Functions and Maps
Differentiable Manifolds > s.a. diffeomorphisms.
Differentiable Structure > see differentiable manifolds.
Differential Equations > s.a. ordinary differential equations; partial differential equations.
Differential Group
$ Def: An R-module
generated by 1 and d, such that d2 =
0, with R = {m + nd | m, n Z},
i.e., and abelian group A with a nilpotent homomorphism d: A → A.
Differential Operator > see under Derivative.
Differential Space
* History: Developed to
describe Brownian motion.
@ References: in Paley & Wiener 34, ch9; Wiener & Siegel PR(53), NC(55)
[in hidden
variable theory].
Differential Topology > see differentiable manifolds.
Digraph > see graph types.
Dilation of a Map between Metric Spaces > see distance.
Dilaton Field / Gravity > s.a. scalar-tensor.
Dimensional Analysis > see thermal radiation [example of use of pi-invariants and Buckingham's pi-theorem].
Dimensional Reduction > see gauge theories; spacetime models.
Dimensional Regularization Scheme > see regularization.
Diophantine Analysis / Equations
* Idea: Equations with
more than one independent variable and integer coefficients, for which integer
solutions are desired.
@ References: Pillay BAMS(97), erratum BAMS(98) [and model theory]; Shimura BAMS(06) [quadratic].
Diophantine Geometry > see geometry.
Dipoles, Dipole Moments > see atomic physics [electric]; electromagnetism; Magnetic Moment; multipoles.
Dirac Bracket > see constrained
systems.
@ General references: Bergmann & Goldberg PR(55)
[and phase space transformations].
@ Modifications: Krivoruchenko et al PRD(06)ht/05 [Moyal-like
quantum deformation].
Dirac Delta Function > see distribution.
Dirac Fields, Theory > s.a. in curved spacetime; quantum dirac fields.
Dirac Hole / Sea > s.a.
quantum field theory [pilot wave theory];
vacuum.
* Remark: Dirac's hole
theory and quantum field theory are usually considered to be equivalent.
@ For bosons: Nielsen & Ninomiya ht/98,
PTP(05)ht/04,
PTP(05)ht/04;
Habara et al ht/05,
ht/05-in
[and susy].
@ And quantum field theory: Jackiw ht/99-in
[physical consequences]; Coutinho et al CJP(02)qp/00;
Solomon CJP(03)qp/02, qp/03, ht/04-in,
CJP(05)qp;
Moffat PLB(05)ht [for
gravity, and the cosmological constant]; Esposito FP(06)
= FP(07)
[Majorana manuscript].
Dirac Matrices > see under Gamma Matrices.
Dirac Monopoles > see monopoles.
Dirac Quantization of Constrained Systems
Direct Limit > see limits.
Direct Product > see categories; manifolds.
Direct Sum > see categories; modules.
Directed Graph > see types of graphs.
Directed Set > see set theory.
Dirichlet Problem
* Idea: A boundary value
problem, in which one looks for a solution to an elliptic partial differential
equation, given the value on the boundary.
Disaster Scenarios > see black hole formation.
Disclination > see Defects.
Disconnected Set > see connectedness.
Discrete Groups > see finite groups.
Discrete Models in Physics > see Continuum; discrete geometry.
Disk > see electromagnetism [charged, rotating]; gravitating matter.
Dislocation > see Defects; geodesics.
Disorder > s.a. Order; QCD
phenomenology; quantum systems.
* Remark: In quantum statistics,
it is described in terms of entropy and algorithmic complexity, which is
not antithetical to the notion of order.
@ General references: Sewell a0711-in
[in quantum statistical mechanics, survey].
@ Strong disorder:
Iglói & Monthus PRP(05) [RG approach].
Dissipation, Dissipative System
Distance Function > s.a. special types and manifolds with metrics.
Distribution (Generalized function)
Distribution (On a manifold) > see tangent structures.
Distribution Function > see states in statistical mechanics; wigner.
Divergence of a Vector Field > see vector field.
Division Algebra
$ Def: An algebra without
zero divisors, i.e., such that there do not exist a, b 
0
with ab =
0.
Domain
* Idea: Domains are mathematical
structures for information and approximation; They combine order-theoretic,
logical, and topological ideas and provide a natural framework for modelling
and reasoning about computation; The theory of domains has proved to be a useful
tool for programming languages and other areas of computer science, and for
applications in mathematics.
Domain of Dependence, of Outer Communications > see spacetime subsets.
Domain Wall > see topological defect.
Donaldson-Witten Theory > see 4D manifolds.
Doomsday Argument > s.a. inflation;
mind.
* Doomsday argument:
(Carter, Leslie, Gott, Nielsen) The observation that we are among the first
1011 or so humans reduces the prior probability
that we find ourselves in a species whose total lifetime number of individuals
is much higher, and the chance of a disaster which would obliterate humanity
is much larger than usually thought.
@ References: Kopf et al gq/94;
Olum
PhilQ(02)gq/00;
Tegmark & Bostrom ap/05 [upper
bound 1 event/109 years,
99.9% c.l.].
Dot Product > see vectors.
Double Wieferich Primes > see number theory.
Drell-Yan Process
Drinfeld Doubles
@ References: Ballesteros et al JPA(07) [and Lie algebras].
Drum > see sound; spectral geometry.
Duality for Mathematical Structures > see cell complex; forms [Hodge dual]; functors; operator [spaces].
Dulong-Petit Law > see specific heat.
Dust > see fluid; interstellar; matter.
Dyad > see spheres [complex dyad on 2-sphere], or vielbein in general.
Dyadosphere
* In astrophysics:
A hypothetical region around a compact object where the electric field exceeds
the
critical
value
for
rapid Schwinger
pair production; Pair production is a self-regulating
process
that would discharge a growing electric field, in the example of a hypothetical
collapsing charged stellar core, before it reached 6% of the minimum dyadosphere
value.
@ References: Page ap/06,
ap/06-in,
ApJ(06)ap
[self-regulation].
Dynamical Horizon > see horizons.
Dynamical System > see formalism of classical mechanics.
Dynamics
* Idea: The study of
the evolution of a physical system, that can be a material object (mechanics
of particles or extended objects), a material medium (continuum
mechanics – fluid mechanics and condensed matter physics), a field (field
theory), or some more general structure.
Dynkin Diagram
@ Generalized: Zuber ht/97-in.
Dyon > s.a. black
hole entropy; monopole.
* Idea: A particle with both electric and magnetic charge.
* Result: In ordinary
4D field theory, it has to be structureless because there are no bound states
of an electric charge in the field of a magnetic monopole.
@ General references: Schwinger Sci(69)aug;
Bjoraker & Hosotani PRD(00)ht [in
4D Einstein-Yang-Mills theory]; Teh & Wong IJMPA(06)ht/05 [SU(2)
Yang-Mills-Higgs, 1/2 monopole charge]; Barnich & Gomberoff a0705 [duality-invariant
formulation, and black hole thermodynamics].
@ Spin and statistics: Friedman & Sorkin
PRD(79), CMP(80); Lechner & Marchetti
JHEP(00)ht.
@ From Kaluza-Klein theory: Davidson & Davidson PRD(86).
Dyson-Schwinger Equation > see under Schwinger-Dyson.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
13 jul 2008