Topics, R

Rabi Model
* Idea: The simplest model describing the coupling of light and matter; In its semiclassical form, the model describes the coupling of a two-level system and a classical monochromatic field, and it is the basis for understanding nuclear magnetic resonance; The fully quantum model considers the same situation with the light field quantized, and has been solved in 2011; 2012, There is renewed theoretical interest in the model due to Braak's analytical solution and introduction of a new criterion for integrability.
@ References: Braak PRL(11), Solano Phy(11) [analytical solution to the quantum model]; Travěnec PRA(12)-a1201 [two-photon Rabi Hamiltonian]; Moolekamp a1201 [understanding its solvability]; Ziegler JPA(12) [spectral density, continued-fraction approach]; Batchelor & Li a1510 [re Braak's solution]; Savasta et al a2006 [gauge invariance].

Racah Algebra
@ References: De Bie et al a2001-conf [overview].

Racah Coefficients / Formula > s.a. SU(2).
* Idea: A formula and coefficients used in the theory of coupling of quantum mechanical angular momenta; Also known as 6j-symbols.
@ References: Krasnov CQG(05)gq/04 [for the Lorentz group]; Krasnov & Louko JMP(06)mp/05 [for SO(1, d+1), AdS-cft methods].
> Online resources: see Wikipedia page.

Radial Gauge > see gauge.

Radiation > s.a. thermal radiation.

Radiation Memory > see Gravitational Memory.

Radiation Reaction > see self-force.

Radiative Corrections > see quantum field theory techniques; quantum-gravity phenomenology.

Radioactivity > see nuclear physics.

Radion Field > s.a. tests of newtonian gravity [constraints]; types of dark matter.
* Idea: A field that stabilizes an extra spatial dimension in a higher-dimensional theory of gravity/spacetime.
@ References: Chakraborty & SenGupta EPJC(17)-a1701 [stabilization from the higher-curvature gravitational action].

Radon Transform
* Applications: CAT, PET and NMR scanning, found by A Cormack (Nobel prize in medicine 1979).
@ References: Facchi et al JMO(09)-a0905.

Radon-Nikodym Derivative

Rainbow (Optics) > s.a. light.
* Remark: It cannot form in the middle of the day because the Sun has to be at most 40° above the horizon in order for its light to refract at the right angles.
@ References: Pincock SHPMP(11) [explanations]; Ford TPT(20)feb [graphical approach].

Rainbow Gravity / Metric

* Idea: An object which plunges radially toward a black hole from rest at infinity.

Raising Operator > s.a. Ladder Operators; spherical harmonics.
* Idea: An operator acting on a set of states or functions labelled by a discrete parameter; > Similar to a creation operator, but without the particle interpretation, and the opposite of a Lowering Operator.
@ References: Boyer and Miller JMP(74) [second-order, for two-variable Hamiltonians].

Raman Scattering
* Idea: Inelastic scattering of photons off molecules, in which the internal state of the molecules change.
* Applications: Detecting explosive powders from a distance [@ news sn(14)aug].

Ramsey Numbers / Theory

Randall-Sundrum Model > see branes; brane cosmology.

Randers Space > see finsler geometry; Cartan-Randers Systems / pioneer effect.

Random Dynamics
@ References: Nielsen & Kleppe a1403-proc [derivation of space]; Kleppe & Nielsen a1412-conf [derivation of diffeomorphism invariance].

Random Matrix (and Tensor) Models > see Matrix Models.

Random Medium > see Disordered System; electromagnetism in matter; light [propagation]; scattering; solid matter [amorphous solids, glass].

Random Process > s.a. random walk.

Random Surfaces > s.a. dynamical triangulations.
@ References: Ambjørn in(96)ht/94 [and quantum gravity, lecture notes]; Wheater JPA(94) [rev]; Ambjørn et al 97.

Random Walk (including quantum walk)

Rank of a Group > s.a. Group Presentation.
* Idea: For a discrete group, the number of generators in a presentation; For a Lie group, the maximum number of generators that can be simultaneously diagonalized; It is meaningful (invariant) for Abelian and free groups.

Rapidity > s.a. kinematics of special relativity / velocity.
* Idea: An angle-like quantity representing boosts in special relativity, with the additivity property that speed doesn't have.
$ Def: Defined as y = (1/2) ln[(E+pz) / (Epz)] in high-energy physics, which corresponds to tanh(y) = pz/E, or r = cosh−1γ, with pz the longitudinal momentum.
$ Pseudorapidity: The quantity −ln[tan(θ/2)], where θ is the angle with respect to the beam line in high-energy collisions; For approximately massless particles, it is approximately equal to the rapidity.
@ References: Lévy-Leblond & Provost AJP(79)dec; Rhodes & Semon AJP(04)jul [description of Wigner rotation and Thomas precession].
> Online resources: see Wikipedia page.

Rarita-Schwinger Equation / Lagrangian / Theory
* Idea: A theory of particles with spin k + \(1\over2\), with k an integer, that are described by a spinor field with k spacetime indices ψa... c, and obey a generalized Dirac equation

(γaa + κ) ψm... p = 0 ,

where γa ψab... c = 0.
* k = 0: There is a gauge symmetry ψaψa + ∇a φ, with γaa φ = 0.
* k = 1: The Lagrangian is L = ψ*a (γmm + κ) ψa − \(1\over3\)ψ*a (γab + γba) ψb + \(1\over3\)ψ*a γa (γbbκ) γc ψc .
* Applications: Its main application is in supergravity to describe gravitinos.
@ References: Rarita & Schwinger PR(41); Rashkov MPLA(99) [and AdS/conformal field theory]; Szereszewski & Tafel CQG(01) [stationary axisymmetric spacetime]; Pilling IJMPA(05) [massive, symmetries]; Kaloshin et al ht/05 [Lagrangian]; Bizdadea et al AdP(06)-a0704 [and Weyl graviton, no cross-couplings]; Carballo Pérez & García-Compeán FP(12)-a1104 [and deformation quantization]; Torres-Gómez & Krasnov IJMPA(13)-a1212 [and propagating spin-1/2 particles]; Adler PRD-a1508 [classical massless, gauged], PRD-a1508 [quantized massless, gauged].

Rastall Gravity > s.a. brans-dicke theory.
@ References: Visser PLB(18)-a1711 [equivalent to Einstein gravity]; Hansraj & Banerjee a1807 [equilibrium stellar configurations]; Cruz et al CQG(19)-a1904 [thermodynamics revision]; Bronnikov et al a2007 [static, spherically symmetric solutions with scalar field, stability].

Rational Function > s.a. Padé Approximant.
* Idea: A function that can be written as the quotient of two polynomials.
> Online resources: see MathWorld page; Wikipedia page.

Rational Numbers > see numbers.

Raumproblem (Problem of Space)
@ References: Loinger RNC(88); Urbantke IJTP(89); Rau a1009-proc; Scholz a1310-in [the views of H Weyl and É Cartan]; Scholz a1608 [hist].

Ray Emanating from a Point in Rn > see lines.

Raychaudhuri Equation
$ Def: Given a timelike geodesic congruence parametrized by proper time, with unit tangent vector ua, the rate of change of its expansion θ is given by

dθ/dτ = \(-{1\over3}\)θ 2σab σab + ωab ωabRab uaub ,

where σab is the shear and ωab the vorticity of the congruence, and Rab the Ricci tensor ; For lines that are not geodesics, a term ∇a(ubb ua) is added to the right-hand side.
@ General references: Raychaudhuri PR(55); in Hawking & Ellis 73; in Stephani et al 03; Dadhich gq/05 [derivation], comment Mitra gq/05; Kar & SenGupta Pra(07)gq/06 [rev]; in Witten a1905-ln [intro].
@ In specific spacetimes: Albareti et al JCAP(14) [FLRW and Bianchi-I models].
@ Variations: issue Pra(07)jul, including Dadhich Pra(07)gq [analog for quantum gravity?]; Abreu & Visser PRD(11)-a1012 [generalizations]; Mohseni GRG(15)-a1502 [for spinning test particles]; Burger et al PRD(18)-a1802 [beyond general relativity]; Iosifidis et al a1809 [with torsion and non-metricity]; Pesci a1809 [for a null congruence]; Namboothiri a1911 [in Kaluza-Klein spacetime].
@ Quantum version, based on Bohmian trajectories: Das PRD(14)-a1311; Farag Ali & Das PLB(15)-a1404; Lashin & Dou PRD(17)-a1606, response Das PRD(17)-a1702 [problematic points]; Vagenas et al NPB(18)-a1706 [and the GUP]; Chakraborty et al PLB(19)-a1904 [with zero-point length].
> Related topics: see Expansion, Shear and Vorticity of a Congruence.
> Online resources: see Wikipedia page.

Rayleigh Jets
* Idea: The phenomenon by which a charged droplet of liquid becomes unstable and explodes, ejecting a microscopic jet of liquid from each end before returning to equilibrium; This limit relates the maximum charge the droplet can bear to its surface tension and radius; Suggested by Rayleigh, visualized by Thomas Leisner and colleagues at Ilmenau Technical University [@ Duft et al Nat(03)jan].

Rayleigh Limit / Criterion > see diffraction; optical technology.

Rayleigh Scattering > s.a. interaction.
* Idea: Elastic scattering of photons; It is responsible for the blue color of the sky.
@ References: Chakraborti AJP(07)sep [simple experiment]; Vinogradov et al a2010 [quantum model].
> Online resources: see Wikipedia page.

Reaction Networks > see networks.

Real Numbers > see numbers.


Reality > s.a. realism.
* Quantum reality problem: Finding a mathematically precise definition of a sample space of configurations of beables, events, histories, paths, or other mathematical objects, and a corresponding probability distribution, for any given closed quantum system.
@ General references: Stern a2001 [Objective Cognitive Constructivism and recursive production of objective realities].
@ In quantum theory: Sudbery a1205 [quantum theory and dual attitudes to reality (realist vs idealist, material vs mental) and knowledge (objective vs subjective, scientific vs romantic)]; Kent PRA(14) [quantum reality problem, outline of solution]; Krizek a1708 [overview of conceptions, structural realism]; Bertlmann a2005 [John Bell's ideas]; > s.a. foundations of quantum theory; ψ-Ontic Theories.

Realization of a Group > see group action.

Rebit > s.a. quantum computation.
* Idea: A state of a quantum system with a 2D Hilbert space described by a real density matrix.

Reciprocity > see Born's Reciprocity Hypothesis; number theory [quadratic reciprocity theorem].

Reconnection / Recoupling Theory > see angular momentum.

Recueil (Souriau)
$ Pre-recueil: A set R whose elements are regular operators such that for all A, BR, A−1R and ABR.

Recurrence (for a Dynamical System)
* Idea: The return of a dynamical system to (a state arbitrarily close to) its initial state.
* Poincaré's recurrence theorem: If V is a space with measure \(\mu\), and \(T: V\to V\) a measure-preserving flow map, then for any open set \(A\subset V\) there exists an orbit of T that intersects A infinitely many times; Used with V the phase space for a physical system.
* Kac's lemma: For a discrete-time evolution, the average recurrence time for a point in a space V to return to a subset A is \(T = \mu(V)/\mu(A)\); For example, for all gas particles in a 1-liter container at room temperature and atmospheric pressure to be in one half of the container this time is many times larger than the current age of the universe, and, practically speaking, one can ignore the possibility [> see Wikipedia page].
@ General references: Wallace a1306 [unified account of classical and quantum recurrence theorems].
@ Poincaré recurrence: Burić et al JPA(03) [area-preserving maps]; Saussol RVMP(09) [quantitative results]; Varandas SD(16)-a1411 [Kac's lemma for suspension flows]; Dong & Stojković PRD(16)-a1611 [the effect of gravity].
@ Recurrence paradox: Steckline AJP(83)oct [Boltzmann & Zermelo].
@ In quantum theory: Campos Venuti a1509 [recurrence time].
> Online resources: see Wikipedia page.

Recursion Relations > see scattering [for quantum field theory scattering amplitudes].

Recursion Theory
@ References: Smullyan 93.

Redshift / Blueshift > s.a. doppler effect; tests of gravity with light [gravitational redshift].
* Idea: A frequency / wavelength change for electromagnetic waves / photons; It may be due to the relative velocity between source and observer, expansion of the background spacetime, or a gravitational field.
* Quantitatively: The redshift is defined as the quantity z:= (λ2λ1) / λ1 = ω1/ω2 − 1.
> And cosmology: see cosmological expansion; geometry of the universe [distance-redshift relation].

Reduce > see programming languages.

Reduced Density Matrix > see Density Matrix.

Reduced Mass > see classical systems [two-body problem].

Reduction of a Dynamical System > s.a. constrained systems; Emergence; Momentum Map; types of symplectic structures [polysymplectic].
* Idea: The process by which from a dynamical system with symmetries one obtains another one with a smaller number of degrees of freedom, by dividing out by the action of the symmetry group; The reduction procedure can be based on the standard momentum map (symplectic or Marsden-Weinstein reduction) or on generalized distributions (the optimal momentum map and optimal reduction); In general it yields a stratified structure.
* History: It was first formulated by Marsden and Weinstein for a phase space described by a symplectic manifold, and then generalized to a Poisson manifold by Marsden and Ratiu.
@ References: Ortega & Ratiu LMP(04) [rev], RPMP(06) [stratified structure]; Butterfield phy/05-ch; Laudato a1703-MS [classical and quantum interacting systems, and non-commutative geometry]; > s.a. symmetries in quantum theory.
> Specific systems: see klein-gordon quantum field theory.

Reduction of a Fiber Bundle > see fiber bundles.

Reduction Rule in Quantum Theory > same as Projection Postulate.

Reductionism > see paradigms in physics.

Reeh-Schlieder Property / Theorem > s.a. locality in quantum field theory.
* Idea: A result of relativistic local quantum field theory, stating that the vacuum is a cyclic vector for the field algebra of any open set in Minkowski space, or that it is spacelike superentangled relative to local fields, meaning that any part of the Hilbert space can be reached by operations on the vacuum state with operators constructed within any, arbitrarily small space-time region.
@ General references: Reeh & Schlieder NC(61); Schlieder CMP(65); Clifton et al PRA(98) [superentangled states]; Jaekel AdP(03)ht/00 [for ground states]; Sanders CMP(09)-a0801 [existence of Reeh-Schlieder].
@ In curved spacetime: Strohmaier CMP(00)mp [on a stationary spacetime]; Dappiaggi RVMP(11)-a1102 [for higher-spin free fields].
> Online resources: see Wikipedia page.

Reference Frame

Refined Algebraic Quantization > see dirac quantization of constrained systems.

Refinement of an Open Cover > see cover.

Reflection > see groups [reflection groups]; mirrors [moving mirrors]; quantum systems [potential step]; wave phenomena; Wigner Delay.

Reflexive Banach Space > see Banach Space.


Refrigerators > see technology.

Regge Calculus > s.a. quantum version.

Regge Poles / Theory
@ References: Bottino AAST-a1807, Del Duca & Magnea a1812-in [retrospective look].

Regge Trajectory
* Idea: Regge trajectories are straight lines connecting groups of particles in the J 2-m plane.
* Applications: They were important for the formulation of the phenomenological dual string models.

Regge-Wheeler Equation > see perturbations of schwarzschild; models in numerical general relativity.

Regular Topological Space > see types of topologies.

Regularization in Quantum Field Theory

Reidemeister Moves > s.a. knot theory.
* Idea: Elementary, local changes to knot or link diagrams preserving their topology.
* Types: There are three, called the (I) twist/untwist move, (II) poke/unpoke move, and (III) slide move.
> Online resources: see nLab page; MathWorld page; Wikipedia page.

Reissner-Nordström Solution (the spelling is not "Reissner-Nordstrøm")

Rejection Sampling
* Idea: A technique for sampling numerically from a target distribution, given the ability to sample from another distribution; > s.a. montecarlo method.
@ References: Ozols et al a1103 [quantum rejection sampling].

Relation > for examples, see graphs and posets.
$ Def: A relation in a set X is a subset R of X × X.
* Transitive relation: A binary relation R on a set X such that for all x, y, z in X, x R y & y R z implies x R z.
* Equivalence relation: One which is reflexive, symmetric, and transitive; An equivalence relation ~ partitions a set X into disjoint equivalence classes [x]:= {yX | y ~ x} whose union is the whole set X; > s.a. Wikipedia page.
* k-homogeneous: A relational structure \(\cal A\) is called k-homogeneous if each isomorphism between two k-element substructures of \(\cal A\) extends to an automorphism of \(\cal A\).
@ General references: Fraïssé 86; Kornyak mp/05-proc [compatibility of systems of relations].
@ Equivalence relation: Ellerman a1701 [two notions of quotient set, and superposition states in quantum mechanics].
@ Other types: Droste & Macpherson JCTA(91) [k-homogeneous]; Droste & Kuske JCTA(03) [random].
> Online resources: see Wikipedia on finitary relations.

Relational Blockworld > s.a. classical limit; relativistic quantum mechanics; time [block universe vs evolution].
@ References: Stuckey et al qp/06, FP(08); Stuckey & Silberstein a0712 [and quantum gravity]; Silberstein et al SHPMP(08) [and interpretation of quantum mechanics]; Miller SHPMP(08) [and quantum mechanics]; Stuckey et al a0903 [discrete path integral over graphs]; Stuckey et al a0908 [and interpretation of quantum field theory].

Relationalism / Relational Theories > s.a. parametrized theories; Newton's Bucket; time; spacetime.
* Idea: Relationalism can be thought of in two different ways, in terms of background independence (as opposed to absolutism) or in terms of processes (as opposed to substantialism).
* In quantum theory: In the late 1990s Mermin proposed an interpretation in which quantum correlations were considered as elements of physical reality; The proposal encountered difficulties after the no-go theorems proved by Cabello and Seevinck in the following years; > s.a. quantum correlations.
@ General references: Martin-Dussaud a2012 [and fundamental physics].
@ Relational mechanics: Barbour CQG(03)gq/02; Anderson a1001, CQG(11)-a1003 [of shape and scale], a1111 [problem of time and quantum cosmology]; Anderson IJMPD(14)-a1202 [quadrilateralland]; Ferraro GRG(16)-a1410 [as a gauge theory, and dragging effects]; Ferraro FP(17)-a1801 [and the frame of fixed stars]; > s.a. modified quantum mechanics; Shape Dynamics.
@ Quantum theory: Anderson & Kneller IJMPD(14) [quadrilateralland]; Ipek & Caticha a1601-conf [based on entropic dynamics]; Laudisa a1710; Rovelli PTRS(18)-a1712 [overview and update]; de Ronde & Fernández a1712 [history and present use]; Trassinelli FP(18)-a1803 [and probability]; de Ronde et al a1810 [evading Cabello's and Seevinck's no-go-theorems in Mermin's relational interpretation]; Höhn et al a1912, a2007 [three approaches]; Oldofredi FP-a2011 [mereological bundle theory approach]; Jennings a2011 [modified version]; Muciño et al a2105 [critique].
@ Gravity: Anderson a0908-MG12 [quantum cosmology]; Anderson a1205-talks [quantum gravity]; Anderson a1411 [and the problem of time]; Vidotto a1508-ch [relational quantum cosmology]; Koslowski et al PLB(18)-a1607 [and the cosmological singularity]; Marchetti & Oriti a2008 [cosmology, from group field theory]; > s.a. canonical general relativity; quantum gravity.

Relative Field Theory >s.a. anomalies.

Relative Locality > see locality.

Relativistic Theory of Gravitation > s.a. gravity theories.
@ General references: Logunov & Loskutov TMP(86); Vlasov & Logunov TMP(86) [cosmology]; Logunov et al IJMPA(88), SPU(88); Logunov TMP(89), SPU(90); Logunov TMP(90); Logunov & Mestvirishvili TMP(91), et al TMP(91), TMP(94); Logunov et al TMP(94) [equivalence principle]; Logunov TMP(95); Logunov TMP(95) [post-newtonian approximation]; Logunov SPU(95), 01-gq/02-book; Gershtein et al TMP(09)-a0810 [gravitational waves]; Soloviev & Tchichikina a0812 [canonical formalism].
@ Cosmology: Chugreev TMP(09) [vacuum cosmological solutions]; Zhelnorovich G&C(10) [positive cosmological constant], G&C(10).
@ Gravitational collapse: Vlasov & Logunov TMP(86), TMP(87); Gershtein et al TMP(09) [impossible for dust ball]; Logunov & Mestvirishvili TMP(13) [impossible].
@ Other phenomenology: Logunov & Loskutov TMP(87) [test bodies].
@ Criticism of general relativity: Logunov et al TMP(86); Loskutov TMP(90).
@ Support for general relativity: Zel'dovich & Grishchuk SPU(86), SPU(88); Chermyanin SPU(90); Ferrari TMP(90); Grishchuk SPU(90); Ohta & Kimura NCB(91).

Relativity Principle > s.a. Covariance; reference frames.
* Idea: The statement that physical laws should be invariant under a change of coordinate system; Different versions differ by which laws are included, and which coordinate systems (e.g., all smoothly related ones for general relativity); The latter basically amounts to specifying which group of transformations transforms one frame into another (e.g., the Galilean Group for Newtonian mechanics).
* And transformations between reference frames: In 1910 von Ignatowsky showed that the relativity principle, combined with linearity and isotropy, leads almost uniquely to either the Lorentz transformations or Galileo's transformations.
* Issue: Kretschmann's well-known criticism stated that general covariance of the Einstein equation is not sufficient to express the principle of general relativity, the failure being rooted in the fact that the metric in the Einstein equation is not uniquely determined by the matter distribution.
* Issue: One would expect the principle to be valid if the dynamical laws in question include all the dynamical quantities they should include; It might by violated, for example, at scales at which quantum gravity starts becoming important, if all the relevant degrees of freedom (or dimensions) are not properly taken into account.
@ General references: Budden BJPS(97) [Galilean vs modern definition]; Barbour gq/03-proc [new formulation]; Szabó FPL(04) [vs Lorentz covariance]; Leston & Ferraro gq/04 [and the Einstein equation]; Grøn AJP(09)apr-a0708, reply Hartman & Nissim-Sabat AJP(09)apr [with rotation and inertial dragging]; Bandyopadhyay EJP(09) [meaning, students' ideas]; De Angelis & Espirito Santo JAHH-a1504 [Giordano Bruno as precursor].
@ Einstein-Kretschmann debate: Sorkin PSA(02); Antoci & Liebscher a0910 [general coordinate transformations and isometries]; Pitts a0911 [adding artificial gauge freedom to a theory].
@ Generalizations: Delbourgo & Stack IJMPA(14)-a1401 [general relativity of space-time-property, and unification]; Hetzroni FP(20)-a2001 [in abstract spaces].
> Application to specific theories: see electrodynamics; general relativity; special relativity.

$ Def: One of the words that define a group presentation.

Relaxation Phenomena
* Relaxation time: The time taken by a system away from equilibrium to approach the equilibrium state; For molecules in a gas, it amounts to another name for the mean collision time, and can be estimated in the framework of the Boltzmann equation.
@ Systems: Chimonidou & Sudarshan PRA(08)-a0705 [two interacting harmonic oscillators]; > s.a. gas [bosons]; magnetism.
@ Relaxation time: Lemos & Pereira PhyA(07) [temperature effects]; Maiocchi & Carati CMP(10) [for a Hamiltonian system, lower bound].
@ Relaxation time, universal bound: Hod PRD(07)gq/06 [τ > \(\hbar\)/(π T)]; Pesci IJMPD(09)-a0807 [and covariant entropy bound].
@ In quantum theory: Tsekov IJMS(01)-a1505 [Markov processes, phase-space description of a mechanical system and extension of the Gibbs canonical distribution]; Santos et al PRL(12)-a1110, García-Mata et al PRE(15)-a1408 [and transition to chaos]; > s.a. pilot-wave quantum theory.
@ Related topics: Shapiro PLA(08) [energy duality]; Kochubei IEOT(11)-a1105 [and fractional calculus]; Ginoux & Letellier Chaos(12)-a1408 [relaxation oscillations, history].
> Specific types of systems: see spin models.
> Online resources: see Wikipedia page.

Relaxion Field > see Electroweak Hierarchy Problem.
* Idea: An axion-like field whose expectation value determines the electroweak hierarchy as well as the QCD strong CP violating \(\bar{\theta}\) parameter.
@ References: Nelson & Prescod-Weinstein a1708 [and the strong CP problem].

Relaxon > s.a. Heat Flow.
* Idea: A collective phonon excitation responsible for thermal conductivity in electrical insulators.

Renormalization > s.a. renormalization group; for quantum gravity, gauge theories, and other theories.

Rényi Entropy > see types of entropies.

Reparametrization-Invariant Systems > see parametrized theories.

Replica Symmetry / Theory > s.a. ising model.
* Idea: The "replica method" proposed by Mark Kac is an approach to the study of quenched disordered systems.
@ General references: in Dotsenko 95; Dotsenko 01 [r PT(02)jan].
@ Dynamical replica theory: Coolen et al PRB(96)cm/95 [for disordered spin systems]; Sakata JPA(13) [evolution of the autocorrelation function].
@ Related topics: Arenzon JPA(99)cm/98 [for granular systems]; Campellone et al JSP(10) [breaking, and finite-volume corrections to mean-field theory]; news gizm(17)jun [replica symmetry breaking in glasses].
> Online resources: see Wikipedia page on Replica Trick.

Representation > s.a. in quantum theory.

Residue of a Function > s.a. analytic function [Cauchy theorem].
$ Def: For a function with a pole of order n at z0, f(z) = F(z)/(zz0)n, where F(z) is regular at z0, the residue at z0 is F(n)(z)/n!

Resistance / Resistor > see electricity.

Resolution > s.a. metric spaces.
$ Def: A local homotopy equivalence MX, when X is a manifold.

Resolvent of a Matrix M > see matrices.


Resource Theory > s.a. generalized and non-equilibrium thermodynamics; probability theory in physics [in general probabilistic theories].
* Idea: Quantum resource theory is an approach to the study of the connection between quantum information theory and thermodynamics.
@ General framework: Brandão & Gour PRL(15)-a1502 [general structure]; Girolami a1709 [information geometry]; Bromley et al JPA(18)-a1802 [lower bounds for resource quantifiers]; Sparaciari et al Quant(20)-a1806 [first law]; Chitambar & Gour RMP(19)-a1806 [rev]; Egloff et al PRX(18) [focus on local operations and communication channels]; Takagi & Regula a1901 [operational]; Gour & Scandolo a2101 [dynamical resource theory]; Dey a2103 [introductory review].
@ Related topics: Lostaglio et al NJP(17)-a1511 [non-commutativity and maximum entropy principles]; del Rio et al a1511 [resource theories of knowledge]; Sparaciari et al PRA(17)-a1607 [without a thermal reservoir]; Ben Dana et al PRA(17)-a1704 [of coherence]; Regula JPA(18)-a1707 [geometry of quantum resource quantification]; Alicki a1801 [information is not a thermodynamic resource]; Morris et al a1908 [entanglement between identical particles]; Martins et al a2004 [quantum incompatibility].

Response Functions / Theory
@ References: in Alastuey et al 16.
> Examples, applications: see Compressibility; heat [thermal expansion]; fluctuations [fluctuation-dissipation theorem].

Rest Frame > see reference frames.

Rest of the Universe > see Feynman's Rest of the Universe.

@ References: DeWitt PRL(64); Isham, Salam & Strathdee PRD(71).

Resurgence / Resurgent Functions > see Alien Calculus / quantum field theory techniques.

Retarded Green Function > see green functions.

Retarded Potential > see Potentials in Physics.

$ Def: Given an inclusion i: AX between topological spaces, a retraction is a continuous map r: XA such that r \(\circ\) i = idA, i.e., a continuous deformation of X onto one of its subsets; A is called a retract of X.
* Deformation retract: A retraction such that r ≅ id rel A.
* Example: S1 is a retract of the disk minus a point, D2 \ {0}, but not of D2.

Retro-MACHOs > see black-hole phenomenology.

Retrocausality / Retrocausation (Backwards in Time Signaling)
* Idea: A phenomenon by which an effect can precede its cause; It arises for example in time-symmetric approaches to quantum mechanics, as the possibility of backward influences from measurement events to the state of systems at an earlier time; It gives rise to the "retrocausal loophole" in Bell's Theorem.
@ General references: Nielsen a1503-proc [arguments, including cosmological constant]; Guryanova et al Quant(19)-a1708 [no-backwards-in-time signaling conditions].
@ In quantum mechanics: Suarez qp/98, qp/98, qp/98/PLA; Costa de Beauregard qp/98; Wharton qp/98|AJP; Pegg FP(08) [and quantum measurement]; Berkovitz SHPMP(08); Price SHPMP(08) [toy models]; Hokkyo SHPMP(08) [and single-electron double slit]; Schulman SHPMP(08); Pegg SHPMP(08); Price SHPSB(12)-a1002 [and time symmetry]; Stapp AIP(11)-a1105, a1111 [and the principle of sufficient reason]; Aharonov et al a1206-conf [using weak + strong measurements]; Lewis SHPMP(13); Tan et al PRL(15) [retrodiction, for a qubit]; Aiello & Woerdman a1506/PS [in classical and quantum optics]; Evans a1506; Price & Wharton Ent(15)-a1508 [and Costa de Beauregard's explanation for quantum correlations], a1510 [as an alternative to quantum spooks]; Aharonov et al Quanta(16)-a1512 [and free will]; Sutherland AIP(17)-a1706; Carmi et al a1903 [and hidden variables]; Zwirn a2009 [no retrocausation, but not entirely determined past]; Norsen & Price a2102 [dialogue]; > s.a. modified quantum theory [time-symmetric].
@ And hidden variables: Price & Wharton a1307; Drezet FP(19)-a1904 [Lorentz-invariant, deterministic framework].
@ Models, systems: Wharton Ent(18)-a1805 [retrocausal but not retro-signaling classical fields]; Argaman Ent(18)-a1806 [toy models with 'lenient' arrow of time].
> And theory: see interpretations of quantum mechanics [transactional]; pilot-wave quantum theory.
> And experiment: see experiments in quantum mechanics [delayed-choice]; quantum effects.
> Online resources: see Wikipedia page.

Retrodiction > s.a. histories formulations of quantum theory.
* Idea: The making of statements about past events based on current information.
@ In quantum mechanics: Mermin PRL(95) [problems]; Hartle Compl(97)gq, PS(98)gq/97 [prediction vs retrodiction]; Barnett et al Symm-a2103; > s.a. time in quantum theory.

Reversibility > see arrow of time; quantum foundations; quantum measurements and types of quantum measurements.

Revival > see Quantum Revival.

Reynolds Number > s.a. turbulence.
* Idea: A dimensionless parameter for liquid flow in a pipe, that measures the importance of non-linearity in the Navier-Stokes equation and characterizes the conditions for laminar flow or turbulence in terms of speed of flow, density, viscosity and diameter of the pipe; It can be thought of as giving the ratio of inertial forces to viscous forces in the liquid; The number of active degrees of freedom in turbulent flow is about R 9/4 per L3.
$ Def: The number R := UL/νmol = (typical flow velocity) (typical length scale) / (kinematic molecular viscosity).
* Values: For small R, the flow is laminar; As R increases, instabilities set in; For large R, there is full turbulence (in real life, νmol is usually of the order of 10−2–10−1 cm2/sec).
@ References: Reynolds PTRS(1895), reprint PRS(95).

Reynolds Transport Theorem
* Idea: A result giving an expression for the rate of change of an integral over an evolving domain.
@ References: Falach & Segev a1312 [generalization]; Reddiger & Poirier a1906 [for manifolds with corners].
> Online resources: see Wikipedia page.

* Idea: The study of deformations and flows of unusual materials.
@ References: TIP 2004 article; Society of Rheology site.

* Calugareanu's theorem: A result which enables the integer linking number of the two edges of a ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe.
@ General references: Dennis & Hannay PRS(05)mp [Calugareanu's theorem].
@ Braided ribbon networks: Hackett a1106 [definition, evolution and invariants]; > s.a. spin networks.

Riccati Equation > see ordinary differential equations.

Ricci Collineations
@ References: Flores et al JMP(04)gq [type-B warped spacetimes].

Ricci Flow > s.a. Geometric Flow; riemannian geometry.
* Idea: The study of the initial-value problem ∂t gab = −Rab + (2/n) r gab, with gab(0) = gab, for an n-dimensional manifold M (usually closed), with gab a Riemannian metric, and r its averaged scalar curvature over M.
* Results: Solutions always exist for finite t, and continue while the curvature is bounded; Isometries and volume are preserved; Under certain circumstances, the flow converges to a sphere.
@ General references: Hamilton JDG(82), JDG(85), in(88); Carfora & Marzuoli CQG(88); Carfora et al JDG(90); Cao & Chow BAMS(99); Samuel & Chowdhury CQG(07), CQG(08) [gradient formulation in terms of entropy and general relativity].
@ Applications and special cases: Holzegel et al a0706 [with surgery, for biaxial Bianchi IX metrics]; Woolgar CJP(08)-a0708-proc [applications in physics]; Carroll a0710 [and quantum theory]; Abraham et al IJGMP(09)-a0810 [corresponding classical mechanical system]; Carfora & Romano RPMP(09)-a0902, Carfora a1001-conf [and renormalization-group flow for non-linear σ-models]; > s.a. averaging in relativistic cosmology; origin of quantum theory.
@ Related topics: Isidro et al IJMPA(09) [quantum-mechanical]; > s.a. cell complexes [simplicial]; types of manifolds [PL manifolds].

Ricci Rotation Coefficients > s.a. tetrads.
* Idea: The connection coefficients defined by an orthonormal frame field; It is named after Gregorio Ricci-Curbastro.
@ References: Ricci MRAL(1896); Levy BAMS(25).

Ricci Tensor > s.a. cell complex [simplicial]; riemann tensor; types of manifolds [PL manifolds].
* Idea: The once-contracted Riemann curvature tensor; Named after Gregorio Ricci-Curbastro.
@ General references: Carfora & Familiari IJGMP(20)-a2102 [relationships in physics and interpretation].
@ Classification: Cormack & Hall IJTP(79); Zachary & Carminati GRG(04) [Segre classification, algorithm].

Richardson Extrapolation (> n630)
> Online resources: see Wikipedia page.

Riemann Equation
* Idea: The first-order, non-linear partial differential equation ut + uux = 0, which describes a one-dimensional accelerationless perfect fluid; It is PT-symmetric, and possesses solutions that typically develop shocks in a finite time.
@ References: Bender & Feinberg JPA(08)-a0709 [complex deformation].

Riemann Hypothesis / Conjecture > see conjectures.

Riemann Integral > see integration.

Riemann Mapping Theorem
* Idea: If U is a non-empty simply connected open subset of the complex plane \(\mathbb C\) which is not all of \(\mathbb C\), then there exists a Riemann mapping (a biholomorphic mapping, i.e., a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk.

Riemann Normal Coordinates > see coordinates.

Riemann Surface > see 2D manifold.

Riemann Tensor

Riemann Zeta Function > see Zeta Function.

Riemann-Cartan Structure > s.a. affine connection [structure equations]; differential geometry.
* Idea: A differentiable manifold with a vielbein eai and a Lorentz connection ω ija, in terms of which the curvature and the torsion are

R ij := dω ij,   R = eai ebj R ijab ,   and   T i := dei + ω ije j .

@ And gravity theories: Vacaru et al gq/05-book [Riemann-Finsler]; Romero et al a0903 [higher-dimensional theories]; Sobreiro & Vasquez JGP(11)-a1003 [from metric-affine theories]; > s.a. gödel spacetime; Metric-Affine Gravity Theories; teleparallel gravity.
@ And cosmology: Minkevich gq/07-proc; Minkevich MPLA(13)-a1309 [regular, accelerating FLRW models].
> Related topics: see Defects; electroweak theory; thermodynamics; unified theories.

Riemann-Hilbert Problem > s.a. axisymmetry [Ernst equation].
* Idea: A boundary-value problem for partial differential equations.
@ References: Deift a1903 [method of non-linear steepest descent].

Riemann-Lovelock Curvature Tensor \ s.a. riemann tensor.
* Idea: The kth-order Riemann-Lovelock tensor is a tensor with 4k indices, of kth order in the Riemann curvature tensor, with which it shares its basic algebraic and differential properties; It was introduced to study the properties of Lovelock gravity theories in low dimensions.
@ References: Kastor CQG(12)-a1202, Camanho & Dadhich EPJC(16)-a1503 [kth-order tensor, two formulations].

Riemann-Roch Theorem
* Relationships: It can be considered as a special case of the Atiyah-Singer index theorem.

Riemann-Silberstein Vector
* Idea: A tool that simplifies the description of the electromagnetic field both in the classical and the quantum domain; It has also been considered as the best possible choice for the photon wave function.
@ References: Cheremisin ht/03-conf; Białynicki-Birula & Białynicka-Birula JPA(13)-a1211 [rev].

Riemannian Connection > see affine connection.

Riemannian Geometry / Manifold / Metric / Structure > see riemannian geometry; metric tensor.

Riesz Space
* Idea: A vector lattice, i.e., an ordered vector space which is also a lattice with respect to a partial order <.
@ References: Zaanen & Luxemburg 71; Zaanen 83.

Rigged Hilbert Space > see hilbert space.

Right Action of a Group > s.a. group action.
$ Right translation in a Lie group: The right action of G on itself given by Rg(h):= hg.

Rigid Body > s.a. reference frames; Chasles' Theorem; Screw Theory.
* In classical mechanics: Its dynamics is described by Euler's equations; Any rigid body can be replaced by an equivalent system of exactly four masses, located at the vertices of an irregular tetrahedron.
* And special relativity: An ideal rigid body cannot exist, since otherwise its ends would move simultaneously in all frames, and could be used to establish a "universal time".
@ In classical mechanics: Modugno & Vitolo mp/05 [geometrical]; Bender et al JPA(07)-a0705 [complex and PT-symmetric solutions]; Rajeev a0906 [geometry of motion]; Gil et al EJP(14) [equivalent system of four masses]; Wilkinson a1506-wd [no physical classical solutions to Euler's equations].
@ In quantum mechanics: Modugno et al mp/05 [covariant approach].
@ In special relativity: Martínez & Gambí IJTP(81) [definitions]; Eriksen et al PS(82); Kim & Jo JPA(04) [rigid rotation impossible]; Soler FP(06)gq/05; Sauer a0704-MGXI [Einstein-Varicak letters]; Brotas a0810 [history]; Butler et al a1005 [geometric, and Clifford algebras]; Llosa et al GRG(12)-a1103 [radar-holonomic congruences of wordlines as weaker substitutes]; Bażański GRG(11) [rigidly rotating circle]; Franklin FP(13) [dynamics].
@ In curved spacetime: Combi & Romero GRG(20) [in an expanding universe].

Rigidity > s.a. differential geometry [rigidity of a geometrical structure]; horizons; classical particles.
* Idea: For a theory, stability is a way of expressing its structural stability.
* Rigidity theorem: For a stationary black-hole solution or one with a Cauchy horizon, it establishes, under weak assumptions, the existence of a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon; If (M, g) is a pseudostationary, axisymmetric asymptotically flat spacetime, with a simply connected domain of outer communications, where the causality and circularity conditions are satisfied, then one can normalize the null tangent l to the event horizon such that l a = ka + ΩH ma, where ΩH is a constant.
@ Rigidity theorems: Hawking CMP(72) [black hole]; Isenberg & Moncrief [Cauchy horizon]; Chruściel CMP(97)gq/96; Friedrich et al CMP(99)gq/98, Rácz CQG(00)gq/99 [general], gq/07-wd [generalization]; Chruściel & Tod CMP(09)-a0712; Alexakis et al a0902 [without analyticity assumption]; Yu AHP(10)-a0903 [charged black holes]; Costa e Silva & Flores a1408 [null geodesic incompleteness]; Ionescu & Klainerman a1501 [mathematical survey]; Galloway & Vega AHP(17)-a1608 [Hausdorff closed limits and rigidity for Lorentzian horospheres].
@ In cosmology: Lidsey GRG(93); Aguirregabiria & Lazkoz MPLA(04)gq [of tachyonic inflation].

Rindler Space

Ring, Ring of Subsets, Ring Space > see ring.

* Idea: A connection at spatial infinity, induced by the conformally rescaled metric on an asymptotically flat spacetime used for compactification; Equivalently, an equivalence class of connections on spacetime, two of them being equivalent if they induce the same structure at spatial infinity.

Rips Complex > see under Vietoris-Rips Complex.

Rishons > see composite models.

Ritz Theory of Electrodynamics > see alternative formulations of electrodynamics.

Robbins Conjecture > see conjectures.

Robertson Uncertainty Principle > see modified uncertainty relations.

Robertson-Walker Universe

Robin Boundary Conditions / Problem > s.a. boundaries in field theory [edge states in quantum field theory].
* Idea: A boundary-value problem for partial differential equations.
@ General references: Bondurant & Fulling JPA(05)mp/04 [and Dirichlet problem solutions]; Belchev & Walton JPA(10) [reflecting wall].
@ Specific theories: Krishnan et al a1702-proc [general relativity].

Robinson-Bertotti Solution > see kantowski-sachs metrics.

Robinson-Trautman Spacetimes > s.a. schwarzschild spacetime.
* Idea: Metrics that describe gravitational radiation in the exterior of bounded sources.
@ General references: Bičák & Podolský PRD(97)gq/99 [Λ ≠ 0]; Moreschi & Pérez CQG(01)gq [perturbations with angular momentum]; Griffiths et al CQG(02)gq [type N]; Natorf gq/05-wd [solutions, type II = c-metric]; Bakas gq/05 [vacuum, integrability and analog of Liénard-Wiechert fields]; Natorf & Tafel JMP(06)gq [symmetries and reductions], CQG(08) [past quasilocal horizons]; Kozameh et al CQG(06)gq [Robinson-Trautman-Maxwell, meaning]; Podolský & Svítek PRD(09)-a0911 [with cosmological constant, past horizons]; de Oliveira et al PRD(11)-a1107 [numerical evolution]; Nashed IJTP(14) [energy and momentum in teleparallel gravity]; Davidson GRG(17).
@ In higher dimensions: Podolský & Ortaggio CQG(06)gq; Ortaggio et al CQG(08)-a0708 [with electromagnetic field]; Svítek MG12(12)-a1001 [existence of horizons]; Podolský & Švarc CQG(15)-a1406 [Weyl tensor algebraic structure].

Roche Lobe
* Idea: An hourglass-shaped surface surrounding the stars in a binary system, beyond which matter is not in the gravitational sway of either star.

Rodrigues Formula > see legendre polynomials.

Rogue Waves > see types of waves.

Rømer Time-Delay Measurement > see tests of general relativity [gravitational-wave speed].

Rokhsar-Kivelson Point \ s.a. Dimers.
* Idea: A construction initially developed in condensed matter for the quantum dimer model, in which a quantum Hamiltonian is constructed, on the same state space as that of a classical statistical-mechanics model (with a discrete state space, and endowed with a dynamics satisfying detailed balance), such that the ground state wavefunction coincides with the classical equilibrium distribution; The word "point" refers to the fine tuning of a parameter in the quantum Hamiltonian.
@ References: Henley JPCM(04)cm/01-proc; Fradkin et al PRB(04)cm/03 [bipartite]; Castelnovo et al AP(05)cm; Syljuåsen IJMPB(05)cm [diffusion Montecarlo method].

Root Lattice
@ References: Baake et al JPA(90)cm/00 [quasicrystals].

Rosen's Bimetric Theory > see bimetric.

Rössler System
* Idea: A system of three non-linear ordinary differential equations, originally studied by Otto Rössler, that exhibits chaotic dynamics associated with the fractal properties of its strange attractor.
> Online resources: see MathWorld page; Scholarpedia page; Wikipedia page.

Rotating Discs Argument
* Idea: An argument against perdurantism.
@ References: Butterfield BJPS(06) [defeat].

Rotation > s.a. mach's principle [rotation problem].

Rotation Curves > see dark matter; galaxies; milky way galaxy; unified theories.

* Idea: A kind of quasiparticle present, e.g., in liquid 4He, which is the quantum "ghost" of a vanishingly small vortex ring, or an extra He atom moving and leaving a swirling disturbance; Minimum energy 9 K.

Rotor \ s.a. constrained systems.
* Configuration space: Given by \(\cal C\) = SO(3) / H, where H is the symmetry group of the shape, with π1(\(\cal C\)) = H*.
@ Second-class constraints: Falck & Hirshfeld EJP(83); Foerster et al PLA(94); Kleinert & Shabanov PLA(97); Saa CQG(97); Bratek JPA(10)-a1004 [relativistic].
@ Quantum, δ-kicked: Ammann et al PRL(98) [decoherence]; Brouard & Plata JPA(03).
@ Quantum, other: DeWitt PR(52) [path integral]; Falck & Hirshfeld EJP(83); Abdalla & Banerjee BJP(01)qp/98.

Rotosurface > see spacetime subsets.

Routh Reduction Procedure > see lagrangian systems [singular]; noether's theorem.

Runaway Solutions > see self force.

Runge-Kutta Method > see differential equations / computational physics.

Runge-Lenz (Laplace-Runge-Lenz) Vector > s.a. orbits in newtonian gravity.
* History: Invented by Hamilton; or by Laplace?
* Idea: A conserved quantity in classical mechanics for a potential V = −a/r, related to the fact that there is no orbit precession.
$ Def: For the Kepler problem, the vector A := p × Lmk (r/r) (where k is the constant appearing in V(r) = −k/r), whose direction is parallel to the vector from the center of attraction to the perihelion, and whose magnitude is proportional to the orbital eccentricity.
* Poisson brackets: If we modify the vector A and define D := A /(2m |E|)1/2, then

[Ai, Lj] = εijk Ak ,      [A1, A2] = −[p2 − (2mk/r)] L3 ,      but      [Di, Dj] = εijk Lk ;

The Li and Di are generators of an SO(4) algebra, possibly related to transformations in the space one obtains by mapping the 3D Kepler problem to a 4D harmonic oscillator problem; > see the Kustaanheimo-Stiefel Transformation.
* Generalization: There probably are Runge-Lenz type vectors for other types of central potentials.
@ General references: in Goldstein 80; Kaplan AJP(86)feb [as a constant of the motion]; Dahl JPA(97) [physical origin]; O'Connell & Jagannathan AJP(03)mar; Morehead AJP(05)mar [corresponding symmetry].
@ In general relativity: Brill & Goel AJP(99)apr-gq/97 [deriving light deflection and perihelion precession].
@ In other theories: Hakobyan & Nersessian PRA(15)-a1504 [Clogero-Coulomb problem].
@ Generalizations: Leach & Flessas JNMP(03)mp/04; Kamath JMP(02) [planar, Chern-Simons electrodynamics]; Nikitin JMP(13)-a1308 [for arbitrary spin, and superintegrable quantum mechanical systems with so(4) dynamical symmetry]; Nikitin JPA(14)-a1403 [with spin, in any dimension].

Ruppeiner Metric / Theory > s.a. thermodynamics.
* Idea: A metric on the state space of a thermodynamical system, determined by the Hessian of the Gibbs surface; It provides a geometric description of thermodynamic systems in equilibrium, and is conformally related to the Weinhold metric.
@ References: Ruppeiner PRA(79) [introduction], RMP(95) [rev]; Ruppeiner JPCS(13)-a1210 [physical interpretation of the scalar curvature].
> And black holes: see 3D black holes; black-hole thermodynamics.

Russell Paradox
* Idea: Partition the set of all sets into two subsets, those that contain themselves as elements, and those that don't; Does the set of all sets that don't contain themselves contain itself? If it does, then it's one of those sets that don't contain themselves, so it doesn't contain itself; Likewise, if it doesn't, then it does.
* Rem: The upshot is that one has to be careful when defining what a set is, if one wants to avoid logical troubles.

Rutherford Scattering
@ References: in Das & Ferbel 03 [II]; Lemasson et al PRL(09) + Ackermann & Simon Phy(09) [21st-century version, and nuclear structure].

Rydberg Atoms > see atomic physics.

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