**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, \(\square\) = *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.

**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** (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].

@ __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;
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 *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-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.

**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__: 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.

**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];
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];
Wick a1901 [model for real position measurements].

@ __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];
Esfeld a1812 [no conflict].

@ __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];
Gisin a1803-conf [finite-information physics and non-determinism].

@ __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];
Del Santo a1807 [Einstein and Bohm];
> 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.

**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; quantum 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 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];
> 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].

**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.

**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.

**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].

> __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 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 [algebroid structure].

@ __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 a1807 [generalized connection].

@ __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.

**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].

> __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 25 mar 2019