**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].

**Racah Coefficients / Formula**
> s.a. SU(2).

* __Idea__: A formula
and coefficients used in the theory of coupling of quantum mechanical
angular momenta.

@ __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 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)** > light.

* __Remark__: It cannot
form in the middle of the day because the Sun has to be at most 40^{o}
above the horizon in order for its light to refract at the right angles.

@ __References__: Pincock SHPMP(11)
[explanations].

**Rainbow Gravity / Metric**
> s.a. friedmann equation; spacetime
models [emergent spacetime]; wormhole
solutions.

* __Idea__: (A.k.a.
doubly-general relativity) A class of theories generalizing doubly-special
relativity to incorporate curvature, which leads to an energy-dependent
metric, connection and curvature, and a simple modification to Einstein's
equation; Its phenomenlogy includes modified dispersion relations, with
field excitations propagating with different energy-momenta perceiving
different background geometries.

* __Cosmology__: 2015,
There have been claims that the initial singularity might be avoided, but
recent more careful results are inconclusive.

@ __General references__: Magueijo & Smolin CQG(04)gq/03
[proposal]; Garattini & Saridakis a1411
[and Hořava-Lifshitz gravity]; Loret & Barcaroli MG13(15)-a1501-MG13
[and relative locality]; Nilsson & Dąbrowski PDU(17)-a1701 [energy scale of Lorentz violation].

@ __And higher-order gravitational Lagrangians__: Olmo JCAP(11)-a1101; Hendi et al a1607 [*f*(*R*) gravity, and charged black-hole solutions].

@ __FLRW cosmology__: Ling JCAP(07)gq/06
[Friedmann equation]; Garattini & Sakellariadou PRD(14)-a1212
[and inflation without an inflaton]; Amelino-Camelia et al PRD(13)
[scale-invariant fluctuations]; Awad et al JCAP(13)-a1308
[modified dispersion relations and non-singular solutions]; Assanioussi et
al PLB(15)-a1412
[general mechanism for emergence from a quantum cosmological model]; Gorji
et al PLB(17)-a1606
[and lqc]; > s.a. electromagnetic fields in curved spacetime.

@ __Cosmology, initial singularity__: Majumder IJMPD(13)-a1305-GRF
[bouncing, non-singular universe]; dos Santos et al JCAP(15)-a1502;
Khodadi et al GRG(16)-a1602
[ideal fluid with a cosmological constant, Hamiltonian approach];
Assanioussi & Dapor PRD(17)-a1606
[effective cosmological models].

@ __Black holes__: Farag Ali PRD(14)-a1402
[thermodynamics and remnants]; Farag Ali et al EPL(15)-a1406
[absence of effective horizon]; Farag Ali et al PLB(15)-a1410
[no black holes at the LHC]; Farag Ali et al NPB(15)-a1410
[black-object remnants]; Farag Ali et al IJGMP(15)-a1506
[collapse]; Mu et al JCAP(15)-a1507
[thermodynamics and luminosity]; Hendi et al PLB(17)-a1602
[massive theory]; Rudra et al NPB(16)-a1606
[Vaidya spacetime in Galileon rainbow gravity]; Gangopadhyay & Dutta EPL(16)-a1606
[constraints on functions].

@ __Other phenomenology__: in Mattingly LRR(05)gq;
Farag Ali & Khalil EPL(15)-a1408
[and gravity tests]; Khodadi et al EPJC(15)-a1505
[stability of the Einstein static universe]; Garattini & Mandanici a1601-MG14, EPJC(17)-a1607 [compact stars equilibrium configurations]; Yadav et al a1605
[and the Unruh effect]; Bezerra et al a1704 [massive scalar fields]; Braidotti & Conti a1708 [simulation by non-local non-linearity]; Bezerra et al a1708 [vacuum fluctuations of a scalar feld and Casimir effect].

> __And particle
dynamics__: see relativistic
particles [propagation]; particles
in schwarzschild spacetime.

> __Online resources__:
see Wikipedia on Thomas Pynchon's novel.

**Raindrop**

* __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 **

* __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].

**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__: Ambjřrn in(96)ht/94
[and quantum gravity, lecture notes]; Wheater JPA(94)
[rev]; Ambjørn et al 97.

**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*+*p*_{z})/(*E*–*p*_{z})]
in high-energy physics, which corresponds to tanh(*y*) = *p*_{z}/*E*,
or *r* = cosh^{–1}*γ*, with
*p*_{z} 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 *ψ*_{a... c}, and
obey a generalized Dirac equation

(*γ*^{a} ∇_{a}
+ *κ*) *ψ*_{m... p}
= 0 ,

where *γ*^{a} *ψ*_{ab...
c} = 0.

* __k____ = 0__:
There is a gauge symmetry *ψ*_{a}
→ *ψ*_{a} + ∇_{a}
*φ*, with *γ*^{a} ∇_{a}
*φ* = 0.

* __k____ = 1__:
The Lagrangian is *L* = *ψ**_{a}(*γ*^{m}
∇_{m} + *κ*) *ψ*^{a}
– \(1\over3\)*ψ**_{a} (*γ*^{a}
∇_{b} + *γ*_{b}∇^{a})
*ψ*^{b} + \(1\over3\)*ψ**_{a}
*γ*^{a} (*γ*^{b
}∇_{b} – *κ*)
*γ*^{c} *ψ*_{c}
.

* __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 a1711 [equivalent to Einstein gravity].

**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 R ^{n}**
> see lines.

**Raychaudhuri Equation**

$ __Def__: Given a
timelike geodesic congruence parametrized by proper time, with unit
tangent vector *u*^{a},
the rate of change of its expansion *θ* is given by

d*θ*/d*τ* = –\(1\over3\)*θ*^{
2} – *σ*_{ab}
*σ*^{ab} + *ω*_{ab}
*ω*^{ab} – *R*_{ab}
*u*^{a}*u*^{b}
,

where *σ*_{ab} is the
shear and *ω*_{ab} the
vorticity of the congruence, and *R*_{ab}
the Ricci tensor ; For lines that are not geodesics, a term ∇_{a}(*u*^{b}
∇_{b} *u*^{a})
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 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].

@ __Qantum 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].

> __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].

> __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.

@ __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]; > 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*,
*B* ∈ *R*, *A*^{–1}
∈ *R* and *AB* ∈ *R*.

**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*:= (*λ*_{2}–*λ*_{1})/*λ*_{1}
= *ω*_{1}/*ω*_{2
}– 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.

**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 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**
(no, the spelling is not "Reissner-Nordstrřm")

**Rejection Sampling**

* __Idea__: A technique
for sampling from a target distribution, given the ability to sample from
another distribution; > used on the Monte
Carlo 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*]:=
{*y* ∈ *X* | *y* ~ *x*} whose union is the
whole set *X*; > s.a. Wikipedia page.

* __ k-homogeneous__:
A relational structure \(\cal A\) is called

@

@

@

>

**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; 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).

@ __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.

@ __Quantum theory__: Anderson & Kneller IJMPD(14)
[quadrilateralland]; Ipek & Caticha a1601-conf
[based on entropic dynamics]; Laudisa a1710; Rovelli a1712 [overview and update]; de Ronde & Fernández a1712 [history and present use].

@ __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 a1607
[and the cosmological singularity]; > 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.**
**@

@

@

@

@

@

**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]; Grřn 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 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].

> __Application to
specific theories__: see electrodynamics;
general relativity; special
relativity.

**Relator**

$ __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
, 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 *z*_{0},
*f*(*z*) = *F*(*z*)/(*z*–*z*_{0})^{n},
where *F*(*z*) is regular at *z*_{0},
the residue at *z*_{0} is *F*^{(n)}(*z*)/*n*!

**Resistance / Resistor** > see electricity.

**Resolution** > s.a. metric
spaces.

$ __Def__: A local
homotopy equivalence *M* → *X*, when *X* is a
manifold.

**Resolvent of a Matrix M** > see matrices.

**Resource Theory**

* __Idea__: Quantum
resource theory is an approach to the study of the connection between
quantum information theory and thermodynamics.

@ __References__: Brandăo & Gour PRL(15)-a1502;
Lostaglio et al NJP(17)-a1511
[non-commutativity and maximum entropy principles]; del Rio et al a1511
[resource theories of knowledge]; Sparaciari et al a1607
[without a thermal reservoir]; Ben Dana et al a1704 [of coherence]; Regula a1707 [geometry of quantum resource quantification].

**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.

**Resummation**

@ __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.

**Retraction**

$ __Def__: Given an
inclusion *i*: *A* → *X* between topological
spaces, a retraction is a continuous map *r*: *X* → *A*
such that *r* \(\circ\) *i* = id_{A},
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__: S^{1}
is a retract of the disk minus a point, D^{2}
\ {0}, but not of D^{2}.

**Retro-MACHOs** > see black-hole
phenomenology.

**Retrocausality / Retrocausation**

* __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 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]; Price & Wharton a1307
[and hidden variables]; 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; > s.a. modified quantum theory [time-symmetric].

> __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.

@ __In quantum mechanics__: Mermin PRL(95)
[problems]; Hartle Compl(97)gq,
PS(98)gq/97
[prediction vs retrodiction]; > 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 *L*^{3}.

$ __Def__: The number *R*
:= *UL*/*ν*_{mol} = (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, *ν*_{mol}
is usually of the order of 10^{–2}–10^{–1}
cm^{2}/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].

> __Online resources__:
see Wikipedia page.

**Rheology**

* __Idea__: The study
of deformations and flows of unusual materials.

@ __References__: TIP 2004 article;
Society of Rheology site.

**Ribbon**

* __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 }*g*_{ab}
= –*R*_{ab} + (2/*n*)
*r* *g*_{ab}, with
*g*_{ab}(0) = *g*_{ab},
for an *n*-dimensional manifold *M* (usually closed), with
*g*_{ab} 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; 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.

@ __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 *u*_{t}
+ *uu*_{x} = 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 Zeta Function** > see Zeta
Function.

**Riemann-Cartan Structure**
> s.a. affine connection
[structure equations]; differential geometry.

* __Idea__: A
differentiable manifold with a vielbein *e*_{a}^{i}
and a Lorentz connection *ω*^{ ij}_{a},
in terms of which the curvature and the torsion are

*R*^{ ij}
:= d*ω*^{ ij}, *R*
= *e*^{a}_{i}
*e*^{b}_{j}
* R*^{ ij}_{ab}
, and *T*^{
i} := d*e*^{i}
+ *ω*^{ i}_{j}
∧ *e*^{ 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

* __Idea__: A
boundary-value problem for partial differential equations.

**Riemann-Lovelock Curvature Tensor**
\ s.a. riemann tensor.

* __Idea__: The *k*th-order
Riemann-Lovelock tensor is a tensor with 4*k* indices, of *k*th
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 a1503
[*k*th-order tensor, two formulations].

**Riemann-Roch Theorem**

* __Relationships__:
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__: 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 R_{g}(*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].

**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}
= *k*^{a} + Ω_{H}
*m*^{a}, where Ω_{H}
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 a1608 [Hausdorff closed limits and rigidity for Lorentzian horospheres].

@ __In cosmology__: Lidsey GRG(93);
Aguirregabiria & Lazkoz MPLA(04)gq
[of tachyonic inflation].

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

**Ripple**

* __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.

**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.

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

**Rokhsar-Kivelson Point** \
s.a. Dimers.**
***

@

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

**Roton**

* __Idea__: A kind of
quasiparticle present, e.g., in liquid ^{4}He,
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 π_{1}(\(\cal
C\)) = *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].

**Runaway Solutions** > see self
force.

**Runge-Kutta Method** > see computational
physics; differential equations.

**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** × **L**
– *mk* (**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** /(2*m* |*E*|)^{1/2},
then

[*A*_{i}, *L*_{j}]
= ε_{ijk}* A*_{k}
, [*A*_{1},*
A*_{2}] = –[*p*^{2}
– (2*mk*/*r*)] *L*_{3}
, but [*D*_{i},
*D*_{j}] = *ε*_{ijk}
*L*_{k} ;

The *L*_{i} and *D*_{i}
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.

**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.

main page – abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 2 jan
2018