Topics, D

D'Alembertian / D'Alembert Operator > s.a. causal sets; laplacian; types of wave equations.
* Idea: The Lorentzian geometry version of the Laplacian operator, \(\square\) = gabab .
* On scalars: Can be written as \(\square\)φ = |g|−1/2 (|g|1/2 gabbφ), a .
* In harmonic coordinates: It simplifies to \(\square\)φ = gabab φ .
> Online resources: see MathWorld page; Wikipedia page.

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.

Davenport Constant > see finite groups.

Davisson-Germer Experiment > see electron.

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 > see computation.

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.

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.

Defects > s.a. topological defects.
* Spacetime defects: A distribution of topological defects embedded in a classical spacetime is one possible way to model the effects of a quantum spacetime structure.
@ General references: Mazenko 02 [fluctuations, order]; Manko et al PhyA(04) [local states]; Afonso et al PLB(08)-a0710 [building networks of defects]; Grigorio et al PLB(10)-a0908 [dual approaches to effective theory of condensation]; Epstein & Segev a1305-conf [unified geometric treatment].
@ In various types of field theories: Bazeia ht/05-ln [scalar field theory]; Caudrelier IJGMP(08)-a0704 [integrable field theories]; Fuchs et al NPB(11)-a1007 [rational conformal field theory, classifying algebra for defects]; Klinkhamer & Rahmede PRD(14)-a1303 [in Skyrme model, non-singular, with non-trivial spacetime topology]; Balasubramanian JHEP(14)-a1404 [codimension-2 defects in 4D, N = 2 SCFTs].
@ And condensed matter: Mermin RMP(79) [and homotopy]; Nelson 02 [r PT(03)may]; Cancès et al CMP(08) [electrons, mean-field model]; Alexander et al RMP(12) [in nematic liquid crystals]; Tuomisto & Makkonen RMP(13) [identification in semiconductors with positron annihilation]; Freysoldt et al RMP(14) [point defects, first-principles calculations]; Schecter & Kamenev PRL(14) [phonon-mediated interactions between defects in quantum liquids]; Kamien & Mosna NJP(16)-a1510 [in smectic liquid crystals, topological structure of the defects]; > s.a. carbon [in graphene]; Elasticity; gauge theories; Impurities; ising models.
@ Dislocations, disclinations: Katanaev PU(05)cm/04-ln [Riemann-Cartan framework]; Comer & Sharipov mp/05 [differential equations and differential geometry]; Kleman & Friedel RMP(08) [rev]; Van Goethem & Dupret a1003 [mesoscale, geometric distributional approach]; Christodoulou & Kaelin ATMP-a1212 [dynamics of a crystalline solid with a continuous distribution of dislocations]; Malyshev a1612 [Einstein-like Lagrangian geometrical field theory]; > s.a. Extended Objects; Plasticity; types of lorentzian geometries.
@ And spacetime curvature / torsion: Maluf & Goya CQG(01)gq [and teleparallelism]; Schmidt & Kohler GRG(01)gq [simplicial, Regge calculus]; Kleinert BJP(04)-proc; Tartaglia IJMPA(05)gq/04-proc; Kleman a0905 [matter as condensed-matter-type defects]; Radicella & Tartaglia AIP(10)-a0911 ["cosmic defect theory"]; Randono & Hughes PRL(11)-a1010 [torsional monopoles]; Kleman a1204 [classification of 2D defects of a 4D maximally-symmetric spacetime]; Bennett et al IJMPA(13)-a1209 [and Plebański's theory of gravity]; Hossenfelder PRD(13)-a1309, PRD(13)-a1309, AHEP(14)-a1401 [and phenomenology]; Klinkhamer PRD(14)-a1402 [non-singular, Skyrmion-type defect]; Klinkhamer & Sorba JMP(14)-a1404 [defects which are homeomorphic but not diffeomorphic]; Brunner et al CMP(15)-a1404 [discrete torsion defects]; Arzano & Trzesniewski AHEP(17)-a1412 [energy-momentum and group momentum space]; Hossenfelder & Gallego a1709 [modification of general relativity with local space-time defects, and FLRW models]; > s.a. approaches to quantum gravity; einstein-cartan theory; geons; photon phenomenology in quantum gravity; types of quantum spacetime.
> Related effects: see examples of entangled systems; particle models; quantum-field-theory effects; spin; wave propagation.

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].
@ 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
> Online resources: see MathWorld page; see 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-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 a1709 [dipole moments of polar molecules].
@ And computation: Sellier & Dimov JCP(14) [Wigner Monte Carlo approach]; news pt(16)jul, Burke Phy(17)sep [simplifying the detailed computations].
> 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 atrix 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: Depends on a vector field va, v(f) = vaa f, and coincides with the Lie derivative with respect to va.
* 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]; > 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.

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 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 if 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]; Hümmer et al PRD(16)-a1506 [Unruh-DeWitt detectors for fermionic and bosonic fields, renormalized]; Martín-Martínez PRD(15)-a1509 [causality constraints]; Sriramkumar a1612-fs [review of concept and response to quantum field].
@ 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]; > 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].
> Related topics: see bell inequalities [detection loophole]; measurement in quantum theory; unruh effect.

Determinant > see operations on matrices [including functional].

Determinism > s.a. causality and causality violations; non-causal spacetimes; paradigms in physics.
* Idea: A property of the evolution of a system, by which complete knowledge of the state at one time determines uniquely the state at a future time.
* History: The concept was introduced for a system by Newton [@ 1687], then 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.
* Status: Our fundamental theories are detrministic, at least in the sense that the evolution of the variables in the theory (i.e., not necessarily the values obtained when measuring observables) is uniquely determined by the appropriate initial conditions; The possible exceptions are two gravity-related situations, the beginning of the universe (Strominger: "if you have nothing and then there's something, that's not deterministic") and black-hole evaporation; However, the creation of virtual black holes by quantum fuctuations would then lead to violations of determinism everywhere, and there are very strong bounds on that.
* In quantum mechanics: The theory is deterministic in that a wave function evolves deterministically, but results of single experiments are not predictable.
@ General references: Earman 86; Ruelle 94 [chance and determinism, I]; Bishop phy/05-en [in physics, rev]; Lapiedra & Montes a1006 [macroscopic, electrocardiogram test]; D'Ariano et al PS(15)-a1301 [without causality, toy theory]; Werndl SHPMP-a1310; Gilead a1510 [the twilight of determinism in biophysics]; Durham a1703 [emergent determinism from randomness].
@ And free will: Candales a1407; 't Hooft a1709-conf, Altaie a1802 [and the theory of everything].
@ In classical physics: Stein PhSc(91)jun, Maxwell PhSc(93)jun [in special relativity]; Bhat & Bernstein IJTP(97), Kosyakov FP(08)ht/07 [example of non-unique evolution]; Wilson BJPS(09) [and the problem of 'missing physics']; Norton PhSc(08)dec, Malament PhSc(08)dec [the dome issue]; Palmer CP(14) [and causality, in fundamental physics].
@ In quantum physics: Peres & Zurek AJP(82)sep [unavoidable indeterminism]; Knill qp/96 [and randomness]; 't Hooft ht/00-in [and dissipation]; Earman PhSc(08)dec [and cure for classical indeterminism]; Lapiedra & Pérez a1010 [proposed tests]; Paul a1011 [classical unpredictability and quantum indeterminism]; Reznikoff JPCS(12)-a1203 [deductive theories that cannot be deterministic]; Palmer CP(14)-a1309 [deterministic but non-computable theory of fundamental physics]; Spekkens FP(14)-a1312 [and proofs of the impossibility of a noncontextual model of quantum theory]; Vaidman QSMF(14)-a1405; Cator & Landsman FP(14) [relationship between the Bell and Conway-Kochen (free will) theorems]; Ionicioiu et al PRL(15)-a1406; Sudbery a1605 [and quantum mechanics, I]; > s.a. bell's inequalities; experiments in quantum theory; hidden variables [including superdeterministic hidden variables]; pilot-wave interpretation; time in quantum mechanics.
> Related topics: see chaos; Free Will; Predictability; random processes; reversibility; Superdeterminism.
> Online resources: see Wikipedia page.

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

Diagonalization > see matrices.

* 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 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].
@ 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 d2 = 0, with R = {m + nd | m, n ∈ \(\mathbb Z\)}, i.e., an abelian group A with a nilpotent homomorphism d: AA.

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 models.

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]; > 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 (2-form giving the) Poisson brackets 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 ofotherwise 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]; > 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].

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.

@ 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.

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].
@ 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.

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].
> 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.

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
$ 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 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 a1508 [canonical formulation and conserved charges].
@ 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 a1611.
@ Phenomenology: Wu & Yang JCAP(14)-a1307 [cosmology]; Wu & Yang a1312 [cosmological signatures]; Bekaert & Park JHEP(16)-a1605 [of higher-spin gravity].
> Videos: Zwiebach conf(12) [32'].

Double Layers > see gravitating matter fields.

Double Wieferich Primes > see number theory.

Double-Beta Decay > s.a. Beta Decay; 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].

Drum > see sound; spectral geometry.


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.

Dufour Effect > see dynamics of gravitating bodies.

Dulong-Petit Law > see specific heat.

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].
> 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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