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, \(\square\) = gabab .
* 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 \(\square\,\phi\) = gabab 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].

Deformation Quantization

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 xX 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.

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

$ 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 va, v(f) = vaa 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 u: AA such that there exists a usual derivation d of A satisfying the generalized Leibniz rule u(a b) = u(a) b + a d(b) for all a, b in A.
@ General references: Mukhopadhyay 12 [higher-order derivatives].
@ Special types: Hurley & Vandyck IJGMP(08) [D-differentiation].
@ Related topics: Gangopadhyaya & Ramsey AJP(13)apr [consequences of imprecise notation].
@ Generalizations: Penot 13 [non-smooth analysis and subdifferentials]; Heller et al CJP(13)-a1301 [generalized derivations and differential geometry, general relativity]; Moré & Wild JCP(14) [derivatives vs finite differences]; Marinho et al a2006 [Jackson and Hausdorff derivatives, and generalized statistical mechanics]; > s.a. analysis [Itō calculus].
> On a manifold: see connection [covariant derivative]; exterior derivative; lie derivative; Parallel Transport; Pfaff Derivative.
> Other types of situations: see Banach Space [Fréchet derivative]; fractals.

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, dPr/dt = 0, from the master equation, but is not a necessary condition; Related to reversibility.
@ General references: Evans JPA(05) [generalization for non-equilibrium states]; Colangeli et al JPA(11)-a1101 [systems driven away from detailed balance by a force].
@ Quantum detailed balance: Duvenhage & Snyman JPA(15)-a1407 [and entanglement].

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].

* 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 Forms

Differential Geometry

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.

* 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.

Dimension of a Space

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.

@ 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.

@ 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.

@ 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 Effect / Shift

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.

* 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 Horizon

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.

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