**Topics, D**

**D'Alembertian / D'Alembert Operator**
> s.a. causal sets [discretized]; laplacian;
types of wave equations.

* __Idea__: The Lorentzian geometry
version of the Laplacian operator, \(\square\) = *g*^{ab}
∇_{a}∇_{b} .

* __On scalars__: Can be written
as \(\square\,\phi = |g|^{-1/2} (|g|^{1/2} g^{ab}\partial_b \phi)_{,a}\) .

* __In harmonic coordinates__:
It simplifies to \(\square\,\phi\) = *g*^{ab}
∂_{a}∂_{b}
*φ* .

> __Online resources__:
see MathWorld page;
Wikipedia page.

**DAMA and DAMA/LIBRA Experiments**
> see dark matter detection.

**Damped Systems**
> s.a. dissipation; oscillators.

* __In quantum theory__:
Damped systems give rise to complex spectra and corresponding resonant states.

@ __Negative damping__: Green & Unruh AJP(06)aug [and the Tacoma Narrows bridge].

@ __In quantum theory__: Caldeira & Leggett PRA(85) [effect on interference];
Chruściński JMP(03)
[resonant states and irreversibility].

> __In quantum theory__:
see Lindblad Equation; quantum oscillators;
states in quantum field theory; types of quantum states.

**Darboux Space**
> see 2D geometries; 3D geometries.

**Darboux Transformation**
> s.a. quantum systems with special
potentials [quasi-exactly solvable]; toda lattice.

@ __General references__: Darboux CRAS(1882);
Rosu in(99)qp/98 [review].

@ __Related topics__: Bagrov et al mp/98-conf [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) [time-dependent Hamiltonian systems];
Hill et al RMS(15)-a1505 [for differential operators on the superline].

> __Online resources__:
see Encyclopedia of Mathematics page.

**Darboux's Theorem**
> see symplectic manifold.

**Dark Energy**
> s.a. dark-energy equation
of state; dark-energy
models; observational cosmology.

**Dark Matter**
> s.a. dark-matter
detection, distribution
and phenomenology, theoretical
models, types;
matter contents of the universe.

**Dark Photons**
> see dark-matter detection;
dark-matter types.

**Data Analysis**
> see statistics and data analysis in physics.

**Davenport Constant**
> see finite groups.

**Davisson-Germer Experiment**
> see electron.

**de Broglie Relation**
> see photons.

**de Broglie-Bohm Interpretation of Quantum Mechanics**
> see pilot-wave interpretation.

**De Donder Gauge**
> see gauge.

**De Donder-Weyl Formalism**
> see types of symplectic structures.

**de Finetti Theorem**

@ __References__: Barrett & Leifer NJP(09) [for test spaces];
Christandl & Toner JMP(09);
Leverrier & Cerf PRA(09)-a0904 [quantum, phase-space representation];
Rougerie a1409-ln [and mean-field theory and Bose-Einstein condensation].

> __Online resources__:
see Wikipedia page.

**de Groot Dual of a Topology**
> s.a. causal structures in spacetime; spacetime topology.

$ __Def__: The de Groot dual of
a topology τ on a set *X* is the topology τ* whose closed
sets are generated by compact saturated subsets of (*X*, τ).

> __Online resources__:
see Wikipedia page.

**de Rham Cohomology / Complex**
> see types of cohomology theories.

**de Rham Theorem**
> s.a. types of cohomology theories.

$ __Def__: The de Rham cohomology
H^{*}_{dR}(*M*)
is the dual of the real singular homology H_{*}(*M*;
\(\mathbb R\)).

@ __References__: in Warner 71.

**de Rham-Gabadadze-Tolley Theory**

* __Idea__: A non-linear
massive gravity theory in which the ghost present in the Pauli-Fierz
theory is eliminated by using a special form of potential to recover
the Hamiltonian constraint; Some desired solutions of the theory
however are unstable.

@ __References__: de Rham et al PRL(10)-a1011;
Kodama & Arraut PTEP(14)-a1312 [stability of the Schwarzschild-de Sitter black hole];
Bernard et al PRD(15)-a1410 [linearized, massive graviton field equations on an arbitrary background].

**de Sitter Spacetime**
> s.a. fields and particles in de sitter spacetimes.

**de Sitter-Fokker Precession**
> see Geodetic Precession.

**Debye Cutoff Length**

@ __References__: Spiegel ap/98-fs [and gravity];
Rubab & Murtaza PS(06) [non-Maxwellian plasmas].

**Debye Model**
> see specific heat.

**Debye-Waller Factor**

**Decay**
> see particles; quantum
state evolution; resonances.

**Decidability**
> s.a. computation [algorithmic decidability].

@ __References__: Paillusson & Booth a2005-FQXi [and science, historical];
Müller a2008-FQXi
[undecidability, unpredictability, and what we can know].

**DECIGO (Deci-Hertz Interferometer Gravitational-wave Observer)**
> see space-based gravitational-wave detectors.

**Decision Theory**
> see game.

**Decoherence**
> s.a. decoherence in specific systems; semiclassical
quantum mechanics; vacuum.

**Decoherence Functional**
> see quantum histories.

**Decoherent Histories Formulation of Quantum Theory**
> see quantum histories.

**Decomposition of Functions and Tensors**

**Deconfinement**
> see QCD effects.

**Dedekind Cut**
> see numbers.

**Deep Inelastic Scattering**
> see scattering; qcd and qcd phenomenology.

**Deep Learning**

@ __References__: Garg & Ramakrishnan a2005 [quantum].

**Defects** (in condensed matter physics, and spacetime)

**Deficit Angle** > s.a. cosmic strings;
magnetic monopoles; regge calculus.

@ __In spacetime__:
Clifton & Barrow PRD(10)-a1001 [effects, and constraints in the Solar System].

**Definitions**
> see mathematics.

**Deformation**
> s.a. Elasticity; hamiltonian dynamics [phase space];
lie algebras; Logarithms;
particle models; Planck Cube.

@ __Of varieties, schemes and manifolds__: Glazunov a1601 [elementary introduction, and applications];
Guan et al a1911,
a1912 [of algebraic structures].

@ __And gravity / spacetime__: Maia et al GRG(11) [of FLRW models];
> s.a. lorentzian metrics.

> __And emergent fields / gravity__:
see emergent gravity; formulations of general relativity;
gauge theories [origin]; spacetime structure [gravity as distortion].

**Deformed Special Relativity**
> see DSR.

**Degeneracy of Eigenvalues of the Hamiltonian**
> s.a. quantum systems.

@ __When__: Fallieros & Hadjimichael AJP(95)nov;
Chau AJP(95)nov
[from supersymmetric quantum mechanics].

**Degenerate Metrics**
> see gravity theories with extended signatures;
types of metrics.

**Degravitation**
> see brane-world gravitation.

**Degrees of Freedom of a Dynamical System**

@ __References__: Burić FP(15)-a1411 [relations between different notions].

**Degree Theory in Banach Spaces**

* __History__: Developed by Leray and Schauder in the 1930s.

@ __References__: Leray & Schauder AENS(34);
Rothe 86.

**Dehn's Lemma**

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Dehn Surfaces**
> see 3-manifolds.

**Dehn Surgery**

* __Idea__: An operation on3-manifolds.

@ __References__: Gang a1803 [algorithm].

> __Online resources__: see
MathWorld page;
Wikipedia page.

**Delaunay Graph / Triangulation**
> see voronoi tilings.

**Delayed-Choice Experiments**
> see experiments in quantum mechanics.

**Delone Set**

* __Idea__: A type of
well-spaced set of points in Euclidean space.

$ __Def__: A point set
*S* in \(\mathbb R\)^{d}
is called a Delone set if it is uniformly discrete and relatively dense;
i.e., if there are numbers *R* > *r* > 0, such that
each ball of radius *r* contains at most one point of *S*,
and every ball of radius *R* contains at least one point of *S*
[from Tilings Encyclopedia page].

@ __References__: Nagai a1702
[general framework for tilings, Delone sets, functions and measures].

> __Online resources__:
see Wikipedia page.

**Delta Function**
> see non-standard analysis; distributions.

**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, of a Graph**
> see graph invariants.

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

**Density Functional Method / Theory**

* __Idea__: An approach to
the study of properties of materials (many-particle systems) based on
the idea that they can all be derived from knowledge of the electron
density *n*(**r**) in the material, using an
appropriate functional *F*[*n*] of this density.

@ __References__: Kohn & Sham PR(65);
Kohn RMP(99) [Nobel lecture];
Prodan Phy(10) [use at finite temperature];
Blanchard et al IJQC(12)-a1011 [on phase space];
Capelle & Campo PRP(13) [and model Hamiltonians];
Hofer JPCS(14)-a1311 [and the future of physics];
Kvaal et al JChemP(14)-a1312 [Moreau-Yosida regularization and differentiable formulation of density-functional theory];
Zangwill a1403,
PT(15)jul [history];
Banks a1503 [for field theorists].

@ __Books__:
Engel & Dreizler 11;
Giustino 14.

@ __Types of systems__:
Koshelev a0812 [relativistic];
Meng ed-16 [for nuclear structure];
Hait & Head-Gordon JCTC(18)-a1709 [dipole moments of polar molecules];
Sanna et al PRL(20) [conventional superconductors];
Hait et al a2011 [small chemical systems].

@ __And computation__: Sellier & Dimov JCP(14) [Wigner Monte Carlo approach];
news pt(16)jul,
Burke Phy(17)sep [simplifying the detailed computations];
Godby Phys(20)
[extending simulations to larger length scales].

> __Online resources__:
see Wikipedia page.

**Density Matrix**
> s.a. mixed quantum states.

* __Idea__: An operator
*ρ* on the Hilbert space for a quantum system satisfying
*ρ*^{†} = *ρ* and tr *ρ* = 1.

* __Uses__: As "statistical
mixture", "reduced density matrix", and "conditional density matrix".

* __Reduced density matrix__:
Given a density matrix *ρ* for a system, and a subsystem
identified with a subspace of the Hilbert space, the reduced density
matrix for the subsystem is the trace of *ρ* over the orthogonal
subspace of the Hlbert space; It can be considered the density-matrix
equivalent of the concept of marginal distribution for probabilities;
> It is used to define entanglement entropy.

@ __General references__: Dürr et al FP(05)qp/03,
Maroney FP(05) [Bohmian mechanics];
Tulczyjew a0711 [non-normalized, and selective measurements];
Weinberg PRA(14)-a1405 [as basis for quantum theory];
Budich & Diehl PRB(15)-a1501 [topology].

@ __Conceptual__: Anandan & Aharonov FPL(99) [meaning of density matrix];
Lobo et al a1110 [ontological status].

@ __Measurement__: Thekkadath et al PRL(16).

> __Online resources__:
see Wikipedia page.

**Density of States**

@ __References__: Wörner & Muñoz EJP(12) [finite-size corrections];
Mulhall & Moelter AJP(14)jul-a1406 [calculation and visualization, for simple quantum systems].

> __Online resources__:
see Wikipedia page.

**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.

**Dequantization**

@ __References__: Cordero et al a1507 [for Born-Jordan quantization].

**Derivation**

$ __Def__: A linear
mapping between two vector spaces, satisfying the Leibniz rule.

**Derivatives**
> s.a. analysis [continuity classes]; fractional
derivatives; operators; tensor field.

* __Directional derivative__: It
depends on a vector field *v*^{a},
*v*(*f*) = *v*^{a}
∂_{a} *f*,
and coincides with the Lie derivative with respect to \(v^a\).

* __Generalized derivation
of an algebra A__: (Introduced by Bresar in 1991) A
linear mapping

@

@

@

@

>

>

**Descriptors of a Mapping**
> see diffeomorphisms.

**DESI (Dark Energy Spectroscopy Instrument)**
> see acceleration of the cosmological expansion.

**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 \(P_r W_{rs} = P_s W_{sr}\).

* __Use__: It implies that the
state is stationary, d*P _{r}*/d

@

@

**Detectors in Quantum Theory**
> s.a. experimental particle physics; particle effects.

* __Idea__: A model for a detector
is often a point particle with internal energy levels, which can get excited
due to its interaction with a quantum field.

@ __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;
Bondurant PRA(04) [pointlike model];
Buscemi & Compagno PRA(09)-a0904 [in quantum field theory, and non-local correlations];
D'Auria et al PRL(11) [quantum decoherence of single-photon counters];
Brown et al PRD(13)-a1212 [beyond perturbation theory];
Bruschi et al JPA(13)-a1212;
Martín-Martínez & Louko PRD(14) [and the zero mode of a quantum field];
Martín-Martínez PRD(15)-a1509 [causality constraints];
Sriramkumar a1612-fs [review of concept and response to quantum field];
Luis & Ares a1707 [and non-classicality];
de Ramón et al a2102 [and causality];
Tjoa et al a2102.

@ __Unruh-DeWitt detectors__: Hümmer et al PRD(16)-a1506 [for fermionic and bosonic fields, renormalized];
Cong et al a2009 [inside rotating shells];
Burbano et al JHEP(21)-a2012 [path integral formalism].

@ __Other models, examples__:
Wick a1901 [model for real position measurements];
Yang & Jacob JAP(19)-a1905 [using first-order quantum phase transitions];
Nehra & Jacob a1909 [Wigner functions];
Teufel & Tumulka a1912 [detectors as absorbing boundary conditions];
Ballesteros et al CMP(21)-a2007 [appearance of particle tracks];
Adjei et al PRA(20)-a2001 [simulation with non-linear optics];
Iyer et al a2104
[unified formalism for spacelike and timelike events, correlations].

@ __Time of detection__: Brunetti & Fredenhagen PRA(02)qp/01;
Tumulka a1601,
a1601,
a1601 [time distribution of clicks].

@ __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 JPCS(07)gq/06 [with regularisation];
Costa & Piazza NJP(09)-a0805 [and Unruh effect];
Kothawala & Padmanabhan PLB(10)-a0911 [time-dependent acceleration];
Thoma a1305
[quantum-field-theoretical model, for Unruh effect];
Anastopolos & Savvidou GRG(14)-a1403 [detection rates along non-inertial trajectories];
Doria & Muñoz a1503
[non-uniformly accelerating observers do not see a thermal state];
Costa a2008 [finite time interval, decoherence];
> s.a. mirrors.

@ __In non-trivial spacetimes__: Langlois AP(06) [topologically non-trivial];
Hodgkinson PhD(13)-a1309 [curved-spacetime quantum field theory];
Ng et al PRD(16)-a1606,
a1706 [and the non-local structure of spacetime];
Martín-Martínez et al PRD(20)-a2001 [fully covariant smeared particle detectors in curved spacetimes].

> __Related topics__:
see bell inequalities [detection loophole]; measurement
in quantum theory; unruh effect.

**Determinant**
> see operations on matrices [including functional].

**DGP (Dvali-Gabadadze-Porrati) Models**
>see brane cosmology.

**Diagonalization**
> see operations on matrices; matrices [Jordan normal form].

**Diagram**

* __In category theory__:
Any collection of objects connected by morphisms.

**Diagrammatic Methods in Mathematics**

> __Lie group / Lie algebra theory__:
see Dynkin Diagram; Young Tableau.

> __Combinatorics /
discrete structures__: see Hasse
Diagram [poset theory]; Schlegel
Diagram; Venn Diagram [set theory].

> __Other mathematical
areas__: see characteristic
polynomials; embedding; exact
sequence; Greechie Diagram; knot
theory; voronoi tiling.

**Diagrammatic Methods in Physics and Related Areas**

> __Quantum field
theory__: see quantum field theory formalism
(and Feynman Diagram); fermions
[fermion algebra]; generalized field theories.

> __Other quantum theory__:
see axioms for quantum theory; path integrals;
quantum information.

> __Gravitational theories__:
see einstein's equation [perturbative method];
lovelock gravity; Penrose Diagram;
Spacetime Diagram.

> __Other physics,
specific diagrams__: see Free-Body
Diagram; Krajewski Diagram
[standard model]; Phase Diagram.

> __Other physics, techniques__:
see heat kernel; non-commutative
gauge theories; scalar fields
[perturbative expansion of path integrals].

> __Astronomy__: see
HR Diagram, Hubble
Diagram [these are actually plots rather than diagrams].

**Diamagnetism**
> see magnetism.

**Diameter**
> see metric spaces.

**Diamond-Shaped Regions**
> see under Alexandrov Sets.

**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; It has a
phase transition with the atom-field coupling as control parameter.

@ __General references__:
Buzek et al PRL(05)qp [ground-state instabilities];
Dimer et al PRA(07)qp/06 [realization in cavity QED];
Garraway PTRS(11);
Bastarrachea-Magnani & Hirsch RMF-a1108 [numerical solutions];
Bhaseen et al PRA(12)-a1110 [dynamics of non-equilibrium Dicke models];
Hirsch et al AIP(12)-a1110 [mean-field description];
Braak JPB(13)-a1304 [*N* = 3, solution];
Kirton et al a1805-AQT [intro].

@ __Critical behavior__: Castaños et al PRA(12)-a1206;
Bastidas et al PRL(12) [non-equilibrium quantum phase transitions];
Dey et al PRE(12)-a1208 [information geometry, quantum phase transitions];
Nahmad-Achar et al PS(13) [catastrophe formalism and group theory];
Bastarrachea-Magnani et al PRA(14) [density of states and excited-state quantum phase transitions],
PRA(14) [chaos and regularity, quantum and semiclassical];
del Real et al PS(13)-a1409 [Husimi distribution and Wehrl entropy];
Bhattacherjee PLA(14) [non-equilibrium dynamical phases];
Bastarrachea-Magnani et al PRE(16)-a1509 [regular and chaotic regions in phase space].

@ __Generalized__: Aparicio et al a0706 [generalized fermion, phase transition];
Grinberg AP(11) [non-classical effects].

> __Properties,
related concepts__: see Fisher Information.

> __Related models__:
see Tavis-Cummings Model.

**Dicke States** > s.a. entanglement measures.

* __Idea__: Multi-particle
states of spin-1/2 particles with the maximal value of the total
angular momentum; They were proposed by Dicke in 1954 and have
become important more recently in quantum information theory.

@ __References__: Dicke PR(54);
Liu & Hu a1511
[in high spin multi-particle systems].

**Dickey Bracket**
> see lagrangian dynamics.

**Dielectrics / Dielectric Constant**
> see electricity [conductivity];
electromagnetic fields in matter.

**Difference Equations**

@ __General references__: Lakshmikantham & Trigiante
02 [including numerical];
Elaydi 05 [II/III, introduction];
Zharinov TMP(11) [symmetries and conservation laws].

@ __Techniques__: Legault & Senior JMP(02) [second-order];
Ablinger et al a1601 [coupled systems].

@ __Special types__: Krichever mp/04 [rational and elliptic coefficients];
Sasaki JMP(07)-a0708,
Odake & Sasaki JMP(07)-a0708 [quasi-exactly solvable];
Ramani et al JPA(09) [integrable];
Levi & Rodríguez JPA(10) [*λ*-symmetries];
Iglesias et al a1011 [in implicit form].

**Difference Operator**
> see sequences.

**Differentiable Functions and Maps**

**Differentiable Manifolds**
> s.a. diffeomorphisms.

**Differentiable Structure**
> see differentiable manifolds.

**Differential Algebra**

@ __References__: Pommaret a1707 [and mathematical physics].

**Differential Equations**
> s.a. ordinary differential equations;
partial differential equations.

**Differential Group**

$ __Def__: An \(R\)-module
generated by the elements 1 and \(d\), such that \(d^2 = 0\) with
\(R = \{m + nd \mid m,\, n \in {\mathbb Z}\}\), i.e., an abelian group
\(A\) with a nilpotent homomorphism \(d: A \to A\).

**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.

**Diffiety**

* __Idea__: Diffieties
formalize geometrically the concept of differential equation.

@ __References__: Vitagliano JGP(11)-a1104 [Hamilton-Jacobi diffieties].

**Diffraction**
> s.a. radiation [diffraction radiation].

**Digamma Function**

@ __References__: Coffey a1008 [series and integral representations].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Digraph**
> see graph types.

**Dilation of a Map between Metric Spaces**
> see distance.

**Dilaton Field / Gravity**
> s.a. scalar-tensor gravity.

**Dilogarithm Function** (a.k.a. Spence's Function)

> __Online resources__:
MathWorld page;
Wikipedia page.

**Dimensional Analysis**
> s.a. thermal radiation [example of
use of pi-invariants and Buckingham's theorem].

@ __References__: Misic et al EJP(10) [and the Buckingham theorem];
Bolster et al PT(11)sep;
Jonsson a1408 [theoretical framework and practical algorithm];
Robinett AJP(15)apr [methodology, examples, power and limitations];
Lemons 17.

**Dimensional Reduction**
> see gauge theories; spacetime dimensionality.

**Dimensional Regularization Scheme**
> see regularization.

**Dimer Models**

* __Dimer__: In chemistry, a dimer
is a structure formed from two similar sub-units (monomers), for example a
diatomic molecule; Formally, a dimer is an edge in a perfect matching of edges
and vertices in a finite, connected graph, i.e., a set of edges such that
each vertex is adjacent to exactly one one of those edges (not all graphs
have perfect matchings).

* __Applications__: Dimer models were
introduced to model the physics of resonating valence bond states in lattice
spin systems.

* __And integrable systems__:
A correspondence between dimer models and integrable systems was introduced
by Goncharov and Kenyon; Dimer models give rise to relativistic integrable
systems that match those arising from 5-dimensional *N* = 1 gauge
theories studied by Nekrasov.

@ __ General references__: Kenyon math/03-ln [intro];
Moessner & Raman a0809-ln [intro];
Cimasoni a1409-ln [geometry];
Bocklandt BLMS(16)-a1510 [recent developments];
Nash & O'Connor a1612 [geometrical approach].

@ __ Related topics__: Cislo PhyA(08) [and the Ising model];
Eager et al JHEP(12)-a1107 [and integrable systems];
Ambjørn et al JPA(14) [on a 2D random causal triangulation];
Flicker et al PRX(20)-a1902 [on rhombic Penrose tilings];
> s.a. Rokhsar-Kivelson Point.

**Diophantine Analysis / Equations**
> s.a. number theory.

* __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];
Andreescu et al 10 [II].

**Diophantine Approximation**

* __Idea__: The problem
of approximating a real number by rational numbers.

**Diophantine Geometry**
> see geometry.

**Dipoles, Dipole Moments**
> see atomic physics [electric]; electromagnetism
with matter; gas [dipole gas]; Magnetic Dipole
Moment; multipoles.

**Dirac Bracket** > s.a. constrained
systems and types of constrained systems [second-class].

* __Idea__: The pullback of the
Poisson brackets (symplectic form) to the constraint surface in phase space.

@ __General references__:
Bergmann & Goldberg PR(55) [and phase space transformations].

@ __Modifications__:
Krivoruchenko et al PRD(06)ht/05 [Moyal-like quantum deformation];
Kanatchikov a0807-proc
[generalization in the De Donder-Weyl Hamiltonian formalism].

> __Online resources__:
see Wikipedia page.

**Dirac Cone**

* __Idea__: A characteristic
feature in the electronic band structure of graphene.

**Dirac Conjecture**
> see types of constrained systems [1st-class].

**Dirac Delta Function**
> see distribution.

**Dirac Equation / Fields / Theory**
> s.a. dirac equation in curved spacetime; generalized
dirac fields; quantum dirac fields.

**Dirac Hole / Sea**
> s.a. quantum field theory [pilot-wave theory]; vacuum.

* __Idea__: A model for
the vacuum in which a positron is seen as a hole in an infinite set
of otherwise filled states of negative energy.

* __Remark__: Dirac's hole theory
and quantum field theory are usually considered to be equivalent.

@ __For bosons__: Finster ATMP(98)ht/97 [with external fields];
Nielsen & Ninomiya ht/98,
PTP(05)ht/04,
PTP(05)ht/04;
Habara et al ht/05,
PTPS(07)ht/05 [and supersymmetry];
Habara et al IJMPA(08)ht/06 [new formulation of quantum field theory],
IJMPA(08)ht/06 [renormalization method].

@ __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-ch,
CJP(05)qp;
Moffat PLB(05)ht [for gravity, and the cosmological constant];
Esposito FP(06)
= FP(07) [Majorana manuscript];
Finster & Grotz JMP(10) [and causal perturbation expansion];
Dimock LMP(11)-a1011 [alternative construction].

> __Online resources__:
see Wikipedia page.

**Dirac Manifolds**

@ __References__: Bursztyn a1112-ln.

**Dirac Matrices**
> see under Gamma Matrices.

**Dirac Monopoles**
> see monopoles.

**Dirac Oscillator**

* __Idea__: An interacting system
of a relativistic massive fermion under the action of a linear potential.

@ __References__: Martínez-y-Romero et al EJP(95)qp/99;
Alhaidari IJTP(04)ht [Green function];
de Lima PLA(08)-a0707;
Sadurní et al JPA(10)-a0902 [coupled to an external field];
Quimbay et al EJTP(14)-a1201 [canonical quantization, in 1+1 and 3+1 dimensions];
Franco-Villafañe et al PRL(13)-a1306
[experimental realization];
de Castro a1906
[as a spin-1/2 fermion in a transverse homogeneous magnetic field];
Montañez & Quimbay a2005 [different spatial dimensionalities];
> s.a. green function.

@ __In 2+1 dimensions__: Andrade & Silva EPL(14)-a1406;
Menculini et al PRD(15)-a1411 [with minimal length, quantum phase transitions].

@ __And minimal length__: Benzair et al JMP(12) [with GUP, path integral];
Boumali et al APPB(16)-a1511 [thermal properties];
Valtancoli JMP(17)-a1611.

**Dirac Quantization of Constrained Systems**

**Direct-Action Theories**
> see under Action at a Distance.

**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 Eta Function**

* __Idea__: A special
function, a.k.a. alternating zeta function.

@ __References__: Milgram JoM(13)-a1208 [integral and series representations].

**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.

**Dirichlet Space**

* __Idea__: One of the three
fundamental Hilbert spaces of holomorphic functions on the unit disk.

@ __References__: El-Fallah et al 14.

**Disaster Scenarios**
> see black-hole formation.

**Discernibility of Particles**
> see particle descriptions.

**Disclination**
> see defects.

**Disconnected Set**
> see connectedness.

**Discord**
> see quantum discord.

**Discovery**

@ __References__: Loeb a1207 [nurturing scientific discoveries];
Gilead a1402 [discovery of actual vs possible entities];
Peiris a1410-IAU [anomalies and discoveries in cosmology];
Wells a1904 [in high energy physics].

**Discrete Geometry**
> see geometry;
combinatorial geometry; discrete spacetimes.

**Discrete Groups**
> see finite groups.

**Discrete Mathematics**
> s.a. combinatorics;
computation; number theory;
proof theory; set theory.

@ __References__: Penner 99 [II].

**Discrete Models / Systems in Physics**
> s.a. Continuum; discrete geometries;
time in physical theories.

@ __General references__: Easton 98 [geometric methods];
Kornyak in(09)-a0906 [gauge invariance and quantization],
in(10)-a1006 [structure and symmetries];
Khare et al Pra(12)-a1111 [solutions in terms of Lamé polynomials];
Kornyak PPN(13)-a1208 [discrete gauge connections, origin of quantum behavior];
Navascués et al JPA(13)-a1110
[spectra of coarse-grained variables based on a collection of microscopic variables];
Marrero et al a1303 [local description];
Kornyak MMG-a1501
[combinatorics, statistics and continuum approximations].

@ __Matter fields__: in da Paz et al PLA(14)-a1406 [granularity of the electromagnetic field].

@ __Condensed-matter-inspired models__: Tahim et al MPLA(09)-a0705 [deformable solid];
't Hooft IJMPA(09) [4D crystal with defects].

@ __Continuum limit__:
Bergman & Inan ed-04 [continuum models];
Tarasov JPA(06) [with long-range interactions].

@ __Minisuperspace models__:
Gambini & Pullin PRL(03)gq/02,
CQG(03)gq/02;
Baytaş & Bojowald PRD(17)-a1611.

> __Gauge theories__:
see chern-simons theory; gauge
theories; lattice gauge theories;
self-dual solutions; types
of gauge theories; types of yang-mills
theories [on a complex].

> __Other examples__:
see Bernoulli Map; cellular automata; dirac
fields; generalized quantum field theories;
hamiltonian systems; lagrangian
systems; integrable systems; quantum
systems; Sequential Dynamical Systems; spin models;
types of wave equations.

**Discrete Topology** > see types of topologies.

**Discretization**

@ __General references__: Tonti JCP(14) [purely algebraic formulation of physical laws, without discretization].

@ __Techniques__: Seslija et al JGP(12)-a1111
[discrete exterior geometry, Dirac structures and finite-dimensional port-Hamiltonian systems];
Palha et al JCP(14) [basic concepts];
Höhn JMP(14)-a1401 [systems with temporally varying discretization, quantization];
Levi & Rodriguez a1407
[discrete variables and invariant schemes when the discrete Schwarz theorem is satisfied];
> s.a. Finite-Element Method.

> __Mathematical__:
see Continuum; Derivatives;
differential equations; discrete
spacetimes; distributions [Dirac delta];
laplace equation; riemannian geometry.

> __Gravity-related systems__:
see approaches to quantum gravity; Barrett-Crane Model
[discretized BF theory]; BF theory; bianchi models;
brane world [Randall-Sundrum models];
canonical quantum gravity models;
constraints in general relativity;
formulations of general relativity;
FLRW spacetimes; gowdy spacetimes;
lattice gravity; loop quantum gravity;
perturbations in general relativity;
riemannian geometry.

> __Quantum systems__:
see canonical quantum theory; formulations of quantum theory;
modified quantum mechanics; path-integral quantum mechanics;
path-integral quantum field theory; QED;
quantum chaos; types of quantum field theories.

> __Other physical systems__:
see computational physics; constrained systems;
Continuous Media; field theory; fluids;
graph theory in physics; modified electromagnetism;
heat equation; klein-gordon fields;
Kolmogorov System; lattice field theories;
regge calculus; types of field theories;
types of yang-mills theories; wave equations.

**Disentropy**

@ __References__: Ramos a1901 [information theory].

**Disformal Interactions / Transformations**
> s.a. Horndeski Action; Mimetic
Gravity; Vainshtein Mechanism.

@ __General references__: Brax & Burrage PRD(15)-a1407 [disformal scalars, and atomic and particle physics];
Bittencourt et al CQG(15)-a1505 [and the Dirac equation];
Fumagalli et al a1610 [as a change of units].

@ __Disformal gravity__: Ip et al JCAP(15)-a1507 [solar system constraints];
Sakstein & Verner PRD(15)-a1509 [Jordan-frame analysis].

@ __And cosmology__: Minamitsuji PLB(14) [cosmological perturbations in scalar-tensor theory];
Sakstein JCAP(14)-a1409;
Sakstein PRD(15)-a1409 [cosmological solutions];
Motohashi & White JCAP(16)-a1504 [invariance of curvature perturbations];
Domènech et al JCAP(15)-a1505;
Alinea & Kubota a2005 [primordial perturbations].

@ __Other spacetimes__: Anson et al a2006 [disformal versions of Kerr metric + scalar field].

**Disk**
> see electromagnetism [charged, rotating]; gravitating matter.

**Dislocation**
> see defects; geodesics.

**Disordered Systems**
> s.a. Order; quantum systems; Random
Medium; solid matter [amorphous solids, glass].

* __In a solid__:
Disorder has a strong influence on the solid's elastic properties;
In terms of electronic properties, disorder in a crystal tends
to localize electrons and drive a transition from a metallic to
an insulating state (Anderson localization transition).

* __Remark__: In quantum
statistics, disorder is described in terms of entropy and algorithmic
complexity, which is not antithetical to the notion of order.

@ __General references__: Binder & Kob 05,
Bovier 06 [statistical mechanics, r JSP(08)];
Sewell a0711-en [in quantum statistical mechanics, survey];
Brody et al JPCS(09)-a0901 [in thermal equilibrium];
Giacomin et al a0906 [and critical behavior];
Wreszinski JMP(12)-a1208-ln [quantum, rev].

@ __Strong disorder__:
Iglói & Monthus PRP(05) [RG approach];
Monthus & Garel JPA(08) [equilibrium properties and phases];
Vojta et al PRB(09)
+ Refael Phy(09)jan [RG approach, universal behavior];
Goldsborough & Evenbly PRB(17)-a1708 [entanglement renormalization].

@ __In condensed matter__:
Foster et al PRB(09)
+ Vojta Phy(09) [typical electron wave function];
Pollet et al PRL(09)
+ Weichman Phy(09)
[patches of order in disordered boson systems and superfluid-insulator transition];
Blundell & Terentjev PRS(11) [influence on deformations in semiflexible networks];
Briet & Savoie RVMP(12) [magnetic response];
Chern et al NJP(14) [disorder-induced criticality in artificial spin ices];
Ashhab PRA(15)-a1510 [effect on the transfer of quantum states];
Kurečić & Osborne a1809 [interacting quantum systems, stochastic integral representation];
Skinner et al PRL(21)
+ news Phys(21) [detecting hidden order].

> __Related concepts / tools__:
see Anderson Localization [random media];
Replica Symmetry; QCD phenomenology;
wave phenomena [propagation].

> __Related phenomena__:
see bose-einstein condensates; casimir
effect; localization.

**Dispersion, Dispersion Relation**

**Dissipation, Dissipative System**

**Distance Function**
> s.a. special types and manifolds with metrics.

**Distance Measurements**
> see Parallax; spatial geometry of the universe [in cosmology].

**Distance-Redshift Relation**
> see geometry of the universe.

**Distinguishable Particles**
> s.a. Identity of Indiscernibles; Indistinguishability;
particle statistics.

* __Idea__: Two
particles are distinguishable if their quantum state changes under
exchange of the spatial locations of the two particles.

@ __References__: Marletto a2009 [and thermodynamic work extraction].

**Distinguished Curves**
> same as unparametrized geodesics.

**Distinguishing Spacetime**
> see causality conditions.

**Distorsion / Distortion**
> see formulations of general relativity; spacetime
structure; s.a. Deformation.

**Distribution** (Generalized function)

**Distribution** (On a manifold)
> see tangent structures.

**Distribution Function**
> see states in statistical mechanics; wigner function.

**Disturbance**
> see uncertainty [error-disturbance relations].

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

**Division Algebra**
> s.a. Tenfold Way [real super division algebras].

$ __Def__: An algebra
without zero divisors, i.e., such that there do not exist *a*,
*b* ≠ 0 with *ab* = 0.

* __Finite-dimensional
real division algebras__: The Frobenius theorem states that up to
isomorphism there are exactly three such algebras, the reals themselves
(dimension 1), the complex numbers (dimension 2), and the quaternions
(dimension 4).

@ __References__:
Baez & Huerta in(10)-a0909 [and supersymmetry];
Wills-Toro a1007 [graded, not necessarily associative];
Baez FP(12)-a1101 [and quantum mechanics].

> __Online resources__:
see Wikipedia page.

**Domain Theory** > s.a. posets.

* __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 formalizes the intuitive ideas of approximation and convergence in
a very general way, and 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 defects.

**Donaldson-Thomas Theory**

@ __References__: Meinhardt a1601 [gentle introduction].

**Donaldson-Witten Theory**
> see 4D manifolds.

**Doomsday Argument**
> see civilizations;
cosmological singularities [cosmic doomsday].

**Doppler Lensing**

* __Idea__: The apparent
change in object size and magnitude due to peculiar velocities.

@ __References__: Bacon et al MNRAS(14)-a1401 [and cosmology].

**Dot Product** > see vectors.

**Double Copy**

* __Idea__: 2010, A correspondence
between scattering amplitudes in gravity and their gauge theory counterpart,
subsequently extended to other quantities, providing gauge theory analogues,
for example, of black holes.

@ __References__: Bern et al PRL(10)-a1004;
Bern et al PRD(10)-a1004;
White CP(18)-a1708 [rev].

**Double Field Theory**
> s.a. types of field theories.

* __Idea__: A concept
developed in order to make manifest the hidden O\((d,d;{\mathbb Z}\))
T-duality symmetry of string theory, and used asan effective field theory
capturing the low energy dynamics of closed strings; It is based on a
doubled spacetime with generalized coordinate transformations, which
unify diffeomorphisms and *b*-field gauge transformations.

@ __ General references__: Hull & Zwiebach JHEP(09)-a0904;
Hohm & Kwak JPA(11)-a1101;
Kan et al a1201-proc [particle equations of motion];
Aldazabal et al CQG(13)-a1305 [rev];
Naseer JHEP(15)-a1508 [canonical formulation and conserved charges];
Chatzistavrakidis et al a1903-proc [algebroid structure];
Lescano & Mirón-Granese a2003 [phase space];
Alfonsi & Berman a2101 [and geometric quantisation].

@ __Flux formulation__: Geissbühler et al JHEP(13)-a1304;
du Bosque et al JHEP(16)-a1509.

@ __ Geometry__: Vaisman JMP(12)-a1203;
Hohm & Zwiebach JHEP(12) [Riemann tensor],
JMP(13)-a1212 [invariant geometry];
Park JHEP(13)-a1304 [and diffeomorphisms];
Hohm et al FdP(13)-a1309 [spacetime, rev];
Blumenhagen et al JHEP(14)-a1312 [non-associative deformations];
Berman et al JHEP(14)-a1401 [global aspects];
Cederwall JHEP(14)-a1402 [metric on doubled space],
JHEP(16)-a1603 [geometry of superspace];
Hassler JHEP-a1611;
Penas FdP(19)-a1807 [generalized connection];
Berman a1903-proc [Kaluza-Klein approach].

@ __Phenomenology__: Wu & Yang JCAP(14)-a1307 [cosmology];
Wu & Yang a1312 [cosmological signatures];
Bekaert & Park JHEP(16)-a1605 [of higher-spin gravity];
Krasnov NPB(18)-a1803 [and the Standard Model fermions].

> __Videos__:
Zwiebach conf(12) [32'].

**Double Layers**
> see gravitating matter fields.

**Double Wieferich Primes**
> see number theory.

**Double-Beta Decay**
> s.a. Beta Decay [including neutrinoless]; neutrino;
types of particles [lepton number].

@ __References__:
Klapdor-Kleingrothaus 10;
Klapdor-Kleingrothaus & Krivosheina in(09)-a1006 [fundamental physics and cosmology].

**Double-Slit Experiment**
> see interference.

**Doubly General Relativity**
> see under rainbow gravity.

**Doubly Special Relativity**
> see DSR.

**Drell-Yan Process**

* __Idea__: A high
energy hadron-hadron scattering process in which a pair of
oppositely-charged leptons is produced out of the annihilation
of a quark-antiquark pair from the two hadrons.

> __Online resources__:
see Wikipedia page.

**dRGT Gravity Theory**
> see under de Rham-Gabadadze-Tolley.

**Drinfel'd Doubles**

@ __References__: Ballesteros et al JPA(07) [and Lie algebras];
Ballesteros et al CQG(13)-a1303 [for 2+1 gravity];
Ballesteros et al CQG(18)-a1809 [for the Poincaré group].

**Drum**
> see sound; spectral geometry.

**Dual Charge** > see Charge.

**Dualities in Field Theory**
> s.a. Triality.

**Duality for Mathematical Structures**
> see cell complex; forms [Hodge dual];
functors; operator [spaces]; posets.

> __Online resources__:
see Wikipedia page.

**Duffin-Kemmer-Petiau Theory**
> see modified QED [SDKP4].

**Dufour Effect**
> see dynamics of gravitating bodies.

**Duhem-Quine Problem**
> see statistics.

**Dulong-Petit Law**
> see specific heat; history of physics.

**Dust**
> see fluid; interstellar; matter.

**Dutch Book Argument**
> s.a. hidden variable theory.

* __Idea__: An argument in the theory of probability.

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

**Dvali-Gabadadze-Porrati Models**
> see DGP Models.

**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-proc,
ApJ(06)ap [self-regulation];
Cherubini et al PRD(09)-a0905 [Reissner-Nordström, "dyadotorus"];
Raychaudhuri et al MPLA(09) [test-particle motion in dyadosphere geometry].

**Dynamical System**
> see formalism of classical mechanics.

**Dynamical Triangulations**
> s.a. causal dynamical triangulations.

**Dynamically Assisted Sauter-Schwinger Effect**
> see particle effects.

**Dynamics**
> s.a. physical theories.

* __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.

* __Structure__: It is
described in terms of physical laws and initial conditions; This dichotomy
appeared with Newton, and modern physics has extended the notion of
initial conditions to internal degrees of freedom and fields; Some
quantization methods try to overcome the distinction.

@ __References__: in Janssen SHPMP(09) [vs kinematics];
Spekkens a1209-FQXi
[kinematics and dynamics must yield to causal structure];
Gogioso a1501
[monadic framework, and shift from histories to dynamics];
Gallego Torromé a2007 [non-reversible].

> __Related topics__:
see Kinematics; Motion;
Symbolic Dynamics.

**Dynkin Diagram**

* __Idea__: A type
of diagram used to classify semisimple Lie algebras.

@ __Generalized__: Zuber ht/97-proc;
Keller AM-a1103
[proof of the periodicity conjecture for pairs].

> __Online resources__:
see Wikipedia page.

**Dyon**
> s.a. black-hole entropy; black-hole
solutions [diholes]; 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;
Teh & Wong IJMPA(06)ht/05 [SU(2) Yang-Mills-Higgs theory, 1/2 monopole charge];
Barnich & Gomberoff PRD(08)-a0705 [duality-invariant formulation, and black-hole thermodynamics];
Singh & Tripathy IJTP(13) [non-abelian, topological].

@ __In Einstein-Yang-Mills theory__: Bjoraker & Hosotani PRD(00)ht [4D];
Nolan & Winstanley CQG(12)-a1208 [and dyonic black holes, in asymptotically anti-de Sitter spacetime].

@ __Spin and statistics__:
Brandt & Primack IJTP(78);
Friedman & Sorkin PRD(79),
CMP(80);
Lechner & Marchetti JHEP(00)ht.

@ __From Kaluza-Klein theory__: Davidson & Davidson PRD(86).

**Dyson Gas**

* __Idea__:
A 2D gas of Coulomb charges in a background potential.

@ __References__: Zabrodin CAOT(10)-a1002 [canonical and grand canonical partition functions].

**Dyson Spheres** > see civilizations.

**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 21 apr 2021