**Topics, N**

**N****-Body Problem**
> see Many-Body Systems.

**n****-Plectic Structures **
> see symplectic structures.

**N****-Point Correlation Functions**
> s.a. correlations; green functions,
in quantum field theory; Normal Order.

* __In quantum field theory__: They are
important in the study of the short-distance structure of quantum field theory;
For small |*x* − *y*| and the right function spaces one gets
behaviors like \(\langle\)0| *φ*(*x*) *φ*(*y*)
|0\(\rangle\) = *c* |*x *− *y*|^{−d};
> s.a. quantum field theory effects.

@ __In quantum field theory__:
Lu ht/05 [1-point functions, perturbative];
Mestre & Oeckl JMP(06) [relationships, Hopf-algebra approach];
Guerra et al EPJB(07)-a0704 [calculation, and non-perturbative renormalization group].

**n****-Poset**
> see generalizations of posets.

**Nagaoka Model**

* __Idea__: A model for ferromagnetism
based on the Hubbard model, popular with theorists for its simplicity.

@ __References__:
news pt(20)apr [observation in a quantum simulation with an array of quantum dots].

**Nambu Algebras / Brackets / Mechanics**
> s.a. poisson brackets; deformation quantization.

* __Idea__: A generalized Hamiltonian
dynamics characterized by an extended phase space and multiple Hamiltonians.

@ __References__: Nambu PRD(73);
Fecko JMP(13)-a1306 [symmetries and conserved quantities];
Müller & Névir JPA(14) [geometric application];
Horikoshi PTEP(19)-a1908 [hidden Nambu structures in quantum/semiclassical systems].

**Nambu Tensor**
> see killing fields.

**Nambu-Goldstone Boson**
> see symmetry breaking.

**Nambu-Jona-Lasinio Model**

@ __General references__: Battistel et al PRD(08)-a0803 [strategy to handle divergences];
Novello IJMPA(11)-a1004 [from gravitational interaction and extended Mach principle].

@ __In curved spacetime__: Elizalde et al PRD(94)ht/93 [phase structure and chiral symmetry breaking].

> __Online resources__:
see Wikipedia page.

**NANOGrav** (North American Nanohertz Observatory for Gravitational Waves)
> see gravitational-wave detection; gravitational-wave
background; gravitational waves from compact objects.

**Nanoscale Systems**
> see generalized thermodynamics.

**Narcowich-Wigner Spectrum**

@ __References__: Dias & Prata RPMP(09)-a0812 [of a pure state].

**Nariai Metric**
> see schwarzschild-de sitter spacetime.

**Narratability of a Theory**

* __Idea__: A theory is narratable
if specifying the state of the system on every spacelike hypersurface in a given
foliation of spacetime is sufficient to determine the states on other hypersurfaces,
so the history of the system can be told as a narration of states at successive times.

@ __References__: Judes a1002 [and cluster decomposition].

**Naturalness of a Physical Theory**
> s.a. Fine Tuning; Hierarchy
Problem; physical theories [criteria].

* __ Types__: Structural and
numerical naturalness (no fine tuning); The former is more subjective,
and has generated many wrong theories.

* __Rem__: This criterion
has been for many years a guiding principle in the search for physics
beyond the Standard Model, but it has been called into question; It might
be rejected at some deep scales (see, e.g. Linde's smörgåsbord
picture of many universes created after inflation), and needs to be treated
with care.

@ __References__: Dine a1501-ARNPS [rev, conjectures, challenges and alternatives];
Pivovarov a1508-proc
[formal criterion, and the Standard Model];
Hossenfelder Syn(19)-a1801 [critical analysis];
Williams FP(18)-a1812 [two notions];
Borrelli & Castellani FP(19)-a1904-conf [historical-philosophical perspective];
Dijkstra a1906.

**Navier-Stokes Equation**
> s.a. fluid; variational principles.

$ __Navier-Stokes equation__: The
non-linear set of pde's describing the time evolution of the fluid velocity
and pressure of an incompressible viscous homogeneous Newtonian fluid in terms
of given initial velocity and external body forces; It expresses momentum
conservation, and is obtained applying Newton's laws to the flow of the fluid,
and adding a term that accounts for energy lost through viscosity

∂_{t }*u*
+ (*u* · ∇) *u*
= −*ρ*^{−1} ∇*p*
+ *ν*∇^{2}*u* ,

where *u* satisfies ∇ · *u* = 0, *p* = pressure,
*ρ* = density (constant), and *ν* = (kinematic molecular)
viscosity; > s.a. mathematics.

@ __References__:
Succi 01 [lattice Boltzmann equation;
r PT(02)dec];
Streater mp/01 [corrections];
Brüger et al JCP(04) [high-order numerical solution];
Gill & Zachary NLA(10)mp/07,
mp/07 [initial data for global solutions];
Romeo a1007-conf [aspects of the initial-value problem];
news sf(14)feb [Terence Tao's new approach];
Dlotko a1501 [new approach];
news Quanta(15)jul [and the Boltzmann equation, incompleteness];
Gutiérrez-Santacreu a1508 [existence of smooth solutions];
Poliakovsky a2011 [variational approach];
> s.a. Boltzmann Equation.

@ __Variations__: Lichtenfelz a1502
[on negatively curved Riemannian manifolds, non-uniqueness of solutions].

> __Related topics__:
see solutions of einstein's equation.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Nebulae**
> see interstellar matter.

**Negative Probabilities**

> see probability in physics.

**Nekhoroshev Theorem**
> see types of integrable systems.

**Nernst Theorem**
> see thermodynamics [3rd law].

**Nerve**
> s.a. cover [nerve of a covering].

$ __Of a category__: The
simplicial set in which the simplices in degree *k* are the chains of morphisms
*a*_{0}, *a*_{0},
..., *a*_{k} in the category;
In other words: A vertex for each object, and a *k*-simplex for
each *k*-tuple of composable morphisms.

> __Online resources__:
see Webb ln(99).

**Nester Form**
> see Metric-Affine Gravity.

**Net** > s.a. Conformal
Net; Filter Base; geodesic net.

$ __Def__: A function from a
directed set *X* to some other set; A "sequence" whose
index set is not totally ordered.

* __Examples__: A sequence,
which is obtained for *X* = \(\mathbb N\); Observable nets in
quantum field theory (> see C*-Algebras;
observable algebras).

@ __References__: Mustafa & Ray CG(10)
[basis for weak *ε*-nets for finite sets of points in Euclidean space].

**Neumann Functions**
> see bessel functions.

**Neumann Problem**
> s.a. green functions.

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

* __Result__: ∃ 1!
solution to ∇^{2}*φ*
= −*ρ*/*ε*_{0} inside a region *V*,
with ∂*φ*/∂*n* fixed on ∂*V*, up to an additive constant.

> __Specific theories__:
see action for general relativity
[covariant notion of Neumann boundary conditions].

**Neural Network**
> see network.

**Neutral Manifold / Metric**
> see types of metrics.

**Neutralino**
> s.a. dark-matter phenomenology.

* __Idea__: A particle predicted by
supersymmetric extensions of the Standard Model that is a candidate for dark matter.

@ __References__:
Abbasi et IceCube PRL(09) [bounds from muon flux in Antarctic ice];
Grothaus et al JHEP(13)-a1207 [parameter space and naturalness];
Fan & Reese JHEP(13)-a1307 [searches for light, nonthermal wino dark matter].

**Neutrino**
> s.a. in cosmology and astrophysics;
mixing and oscillations.

**Neutrinoless Double-Beta Decay**
> see Beta Decay.

**Neutrix Calculus**
> s.a. regularization; renormalization.

* __Idea__: Neutrices are
additive groups of negligible functions that do not contain any constants
except 0; Their calculus was developed by van der Corput and Hadamard in
connection with asymptotic series and divergent integrals.

**Neutron Stars**
> s.a. pulsars.

**Newman-Janis Algorithm**
> see solution-generating methods for the einstein equation.

**Newman-Penrose Formalism**
> see spin coefficients.

**Newman-Tamburino Metrics**

@ __References__: Steele gq/04 [Killing vector];
De Groote & Van den Bergh CQG(08),
CQG(08) [with Maxwell field].

**News Tensor**
> see asymptotic flatness at null infinity.

**Newton-Cotes Methods**
> see integration.

**Newton-Schrödinger Equation**

* __Idea__: A type of non-linear
Schrödinger equation in which a term in the potential felt by a wave function
*ψ* is the Newtonian gravitational potential generated by *ψ*
itself; It plays a prominent role in alternative quantum theories, emergent quantum
mechanics, macroscopic quantum mechanics, gravitational decoherence and semiclassical
gravity.

@ __References__: Anastopoulos & Hu NJP(14)-a1403 [problems];
Maimone et al PS(19)-a1809 [cure for causality violations].

**Newton-Wigner Localization**
> see localization; quantum particles.

**Newton's Bucket Argument**
> s.a. mach's principle [rotation problem].

* __Idea__: The argument by Newton
for the existence of absolute space, based on the fact that one can tell whether
a reference frame is rotating with respect to absolute space by looking at whether
the surface of the water inside a bucket is flat or concave (once the water is
at rest with respect to the bucket); The argument was criticized by Mach, and
is considered an argument against relationalism.

@ __References__: in Barbour & Pfister 95;
Kubiak v1110,
Sekhar a1305 [and the liquid-mirror telescope];
Staley PT(13)dec [Mach's position];
Shuler PhysEd(15) [and Mach's principle].

> __Online resources__:
see Wikipedia page.

**Newton's Cradle**

* __Idea__: A device that demonstrates
conservation of momentum and energy using a set of swinging metal spheres.

@ __Quantum analog__:
news pw(14)may;
news cosmos(18)jun [and thermal equilibrium].

> __Online resources__:
see Wikipedia page.

**Newton's Gravitational Constant**

**Newton's Laws of Dynamics**
> s.a. classical mechanics [including variations and generalizations].

* __First law / Law of inertia__:
For a free particle in an inertial frame, d**u**/d*t* = 0,
where **u** = velocity and in the relativistic version *t*
is usually proper time; In words, unless a net force is acting on an object, the
object will keep moving with constant speed in the same direction; As opposed to
the Aristotelian view that any object moving at a constant speed must be continually
pushed if it is to maintain its motion.

* __Second law__: In an inertial
frame, *m* d**u**/d*t* = **f**, where
**f** is the force acting on the particle; Has been considered as a
definition rather than a true law of nature, based on the argument that force cannot
be measured directly but only through the acceleration it produces on a given mass.

* __Third law__: The "principle
of action and reaction," for objects *A* and *B*, the force
\({\bf F}_{AB}^~\) of \(A\) on \(B\) is equal and opposite to the force
\({\bf F}_{BA}^~\) of \(B\) on \(A\); It holds unless there are accelerated charges
(magnetic forces, or self-forces from radiation reaction); The laws of conservation
of energy-momentum and angular momentum are based on it; It can be violated when the
interacting particles are embedded in a non-equilibrium environment.

* __Limitations, tests__: Newton's
second law is expected to break down for subatomic scales; At macroscopic ones, one
way to test its validity is to check that the frequency of a pendulum is independent
of amplitude (if small).

@ __General references__:
Eisenbud AJP(58)mar [objections and formulation];
Brehme AJP(85)oct [laws as definitions];
Anderson AJP(90)dec [not definitions];
Grabowska & Urbański mp/04 [frame-independent];
Kokarev a0905 [three lectures];
Zimba 09 [illustrated guide];
Lee FP(11) [and infinite systems];
Stocklmayer et al TPT(12) [introduction of the laws in the classroom].

@ __First law__: Pfister FPL(04);
Rabinowitz IJTP(08) [and quantum mechanics];
Weatherall a1206-ch
[inertial motion and its explanation by the geodesic principle];
Hecht TPT(15)#2 [history, origins].

@ __Second law__: Hacyan AJP(09)jul [meaning of modifications or tests];
Pourciau AJP(11)oct
[what did Newton say in the *Principia*?];
Stefanini & Reali a1502 [new interpretation proposed].

@ __Third law__: Anandan & Brown FP(95) [and pilot-wave theory];
Fraser SHPSA(05) [and argument for second law];
Dykstra AJP(09)aug;
Pinheiro PS(11) [possible violations];
Ivlev et al PRX(15)
+ news PhysOrg(15) [violation, and statistical mechanics];
> s.a. Action-Reaction Principle;
photon phenomenology [negative effective mass].

@ __Bounds on violations__:
Abramovici & Vager PRD(86)
[ok down to 3 × 10^{−9} cm/s^{2}];
Gundlach et al PRL(07)
+ pn(07)apr
[ok down to 5 × 10^{−14} cm/s^{2}].

@ __Modifications__: Milton & Willis PRS(07) [second, continuum elastodynamics];
Unzicker gq/07 [and gravity];
Sprott AJP(09)sep [modified third law];
Alonso-Blanco & Muñoz-Díaz a1811 [for field theory];
> s.a. force; MOND.

> __Related concepts__:
see force; inertia
and inertial frame; mass.

**Newton's (Shell / "Superb") Theorem**

* __Idea__: A spherically symmetric mass
distribution attracts a body outside of it as if the entire mass were concentrated at
the center; In Newtonian physics it is also true that a hollow, spherically symmetric
mass distribution produces no gravitational effect inside it; In general relativity,
while a corresponding statement about points outside a spherically symmeric object
is still true (*mutatis mutandis*), a spherical shell does have an effect at
locations inside.

@ __References__: Schmid AJP(11)may [elementary geometric proof];
Zang & Yi IJMPCS(12)-a1203-conf [in general relativity];
> s.a. schwarzschild solution [interior metric].

**Newtonian Limit of General Relativity**
> see phenomenology of gravity.

**Newtonian Physics** > see cosmological models
[Newtonian cosmology]; newtonian gravity and tests of newtonian
gravity; physical theories [frameworks].

**Newtorites**

* __Idea__: Particles with only gravitational interactions.

@ __References__: Ronga JPCS(16)-a1511 [detection].

**Nicolai Map**

* __Idea__: A change of variables for
a gauge theory from the gauge connection in a fixed gauge to the anti-selfdual
part of the curvature.

@ __References__: Bochicchio & Pilloni JHEP(13)-a1304.

**Nieh-Yan Form**

@ __General references__: Guo et al CTP(99)ht/98;
Li JPA(99)ht.

@ __And physics__: Soo PRD(99)ht/98 [and the Adler-Bell-Jackiw anomaly];
Mielke PRD(09),
Banerjee CQG(10)-a1002 [and gravity];
> s.a. theories of gravity.

**Nightmare Scenario**

* __Idea__: The scenario
in which the Large Hadron Collider (LHC) fails to discover physics
Beyond the Standard Model (BSM).

@ __References__: Bertone et al JCAP(12) [consequences for dark matter, in cMSSM].

**Nijenhuis Torsion Tensor**
> see types of symplectic structures.

**Nilpotent Element of a Ring**

$ __Def__: An element *x*
of a ring *R* is called nilpotent if there exists some positive
integer *n*, called the index (or sometimes the degree), such that
*x*^{n} = 0.

> __Online resources__: see
Wikipedia page.

**9 j Symbol**
> see SU(2).

**No-Cloning Theorem**
> s.a. quantum technology.

@ __References__:
Hari Dass a1005 [for harmonic-oscillator coherent states];
Gauthier RPMP(13) [for an infinite multiverse];
Ortigoso a1707 [1970 proof];
Vagenas et al a1811 [and the GUP];
Kuzyk AJP(19)-a1903 [and entanglement].

**No-Hair Theorems**
> see black-hole hair; Cosmic No-Hair Conjecture.

**No-Signaling in Quantum Theory**
> see causality in quantum theory.

**Noise**
> s.a. partial differential equations [stochastic].

* __Shot noise__: Fluctuations in
a counting rate that are Poissonian, i.e., if the mean number of counts in
a certain time interval or spatial volume is *N*, then the variance
(δ*N*)^{2} of that number is also equal to *N*.

@ __1/ f noise__: in Kaplan & Glass 95 [II];
Kuzovlev PU(15)-a1504 [origin and significance].

@

@

**Noiseless Subsystems**
> see generalized coherent states.

**Non-Archimedean Structures / Numbers**
> s.a. Archimedean Property / Adelic and
*p*-Adic Structures; types of numbers;
geometry.

* __Idea__: A structure which has
a pair of non-zero elements, one of which is infinitesimal with respect to the other;
In other words, one with infinitely large and/or small elements; For example, hyperreal
numbers, surreal numbers, *p*-adic numbers.

@ __In physics__: El Naschie CSF(04) [fundamental length and all that];
Rosinger m.HO/05 [use in general];
Benci et al Quant(19)-a1901 [in quantum mechanics];
Sinclair a2002 [electrostatics].

@ __And elementary particles__:
Dimitrijevic et al FU(04)ht;
Varadarajan & Virtanen a1002.

@ __And cosmology__: Avinash & Rvachev FP(00);
Djordjević & Nešić PAOB-a1011 [*p*-adic and adelic quantum cosmology and tachyonic inflation];
> s.a. quantum particles.

> __Online resources__:
see Wikipedia page.

**Non-Associative Algebras**

* __Motivation__: They appear
in some exotic systems such as magnetic monopoles.

@ __In physics__: Kupriyanov a1603-PoS;
Liebmann et al a1909 [historical perspective];
> s.a. observable algebras [in quantum theory];
Supersymmetry.

@ __Non-associative quantum mechanics__:
Bojowald et al PRL(15)-a1510 [potentially testable effects];
Bojowald et al PRL(18)-a1810 [small magnetic charges and monopoles];
Bojowald et al a2011 [ground state of non-associative hydrogen];
> s.a. deformation quantization;
quantum mechanics in curved spacetime.

**Non-Associative Geometry**
> s.a. Octonions \ types of manifolds.

@ __ General references__: Wulkenhaar ht/96,
ht/96,
PLB(97)ht/96 [standard model],
ht/96 [grand unification];
Nesterov & Sabinin CMUC(00)ht-conf,
PRD(00)ht [and spacetime];
Beggs & Majid JPCS(10);
> s.a. particle physics.

@ __And gravity / spacetime__:
Nesterov & Sabinin ht/00-conf,
Sabinin IJTP(01) [and discrete spacetime];
Nesterov & Sabinin PRD(00)ht [de Sitter spacetime],
IJGMP(06)ht/04 [FLRW models];
Harikumar & Rivelles CQG(06)ht [and non-commutative gravity];
Farsnworth & Boyle JHEP-a1303 [spectral action];
Blumenhagen & Fuchs JHEP(16)-a1604;
Aschieri et al JHEP(18)-a1710.

@ __Other physics__:
Sbitneva IJTP(01) [special relativistic kinematics];
Blumenhagen & Plauschinn JPA(11)-a1010 [from string theory];
Mylonas & Szabo FdP(14)-a1404-proc [on non-geometric spaces];
Barnes et al PoS-a1601 [field theory];
Bakas a1605-proc [in Dirac's generalization of Maxwell theory];
Nesterov & Mata PRE(20)-a1812 [complex networks];
Szabo PoS-a1903 [pedagogical introduction].

> __ Other physics__: see modified quantum mechanics;
particle physics beyond the standard model; phase space [quantum];
Supersymmetry; types of gauge theories.

**Non-Commutativity**
> s.a. non-commutative geometry
and spacetime; in physical theories,
in field theory and gravitation.

@ __References__: MacKenzie ThSc(97)may [general notion];
Curcuraci a1803 [non-commutativity in quantum theory].

**Non-Conservative System**
> see classical systems.

**Non-Degenerate Bilinear Form**
> see Bilinear Form.

**Non-Demolition Principle**
> s.a. experiments in quantum mechanics.

@ __References__: Marcovitch & Reznik a1002-wd [and quantum predictions].

**Non-Equilibrium Systems**
> see in statistical mechanics and thermodynamics;
states in quantum field theory; temperature.

**Non-Euclidean Geometry**
> see geometry.

**Non-Extensive Statistical Mechanics**

**Non-Gaussianity**
> see under Gaussianity.

**Non-Hermitian Quantum Theory**
> see modified quantum mechanics [PT-symmetric]; quantum phase transitions.

**Non-Holonomic Systems**
> see types of constrained systems; quantum systems.

**Non-Imprisonment Conditions on Spacetime**

@ __References__: Minguzzi JMP(08) [and distinction property].

**Non-Linear Analysis** > see analysis.

**Non-Linear Systems / Field Theory** > see classical systems;
sigma-model; types of quantum field theories.

**Non-Local Systems**
> see locality; modified gravity theories;
non-local field theories; quantum oscillators;
types of quantum field theories.

**Non-Metricity**
> s.a. affine connection;
Metric-Affine Theories.

@ __General references__: Casanova et al MPLA(08)-a0712 [and torsion, Schouten classification];
Foster et al PRD(17)-a1612 [constraints from Lorentz violation].

@ __And gravity__:
Sobreiro & Vasquez Otoya BJP(10)-a0711;
Mol AACA(17)-a1406 [and general relativity];
Järv et al PRD(18)-a1802 [non-metricity formulation of general relativity and scalar-tensor theory];
Rünkla & Vilson PRD(18)-a1805 [scalar-nonmetricity theories of gravity];
Lazkoz et al PRD(19)-a1907 [observational constraints];
Delhom et al PLB(18)-a1709 [constraints];
Beltrán et al PRD(18)-a1710;
Delhom EPJC-a2002 [minimal coupling];
D'Ambrosio et al a2007 [coincident general relativity, ADM formulation];
> s.a. brans-dicke theory; torsion in physical theories.

@ __Cosmology__: D'Ambrosio et al a2004 [and MOND];
Anagnostopoulos et al a2104
[*f*(*Q*) gravity and ΛCDM].

**Non-Perturbative Features of Field Theory**
> see instantons; solitons.

**Non-Renormalization Theorems**
> see renormalization; supersymmetry in field theory.

**Non-Squeezing Theorem**
> see symplectic geometry.

**Non-Symmetric Geometry and Gravity**
> s.a. kaluza-klein theory; unified theories of gravity and electromagnetism.

* __Idea__: A geometry
and theory of gravity in which the metric tensor is not symmetric;
The vacuum field equations are

g_{μν}_{, σ}−gΓ_{ρν}^{ρ}_{μσ}−gΓ_{μρ}^{ρ}_{σν}= 0 , [(−g)^{1/2}g^{[μν]}]_{,ν}= 0 ,R_{(αβ)}= 0 ,R_{[αβ], γ}+R_{[βγ], α}+R_{[γα], β}= 0.

* __History__: The non-symmetric
theory of gravity was proposed by J Moffat; It attracted a lot of attention until
the mid-1990s, when it was thought that the theory does not include black holes,
but this claim was proved incorrect by work by Burko and Ori.

@ __General references__:
Kunstatter et al JMP(83) [geometrical structure];
Damour et al PRD(93),
gq/93 [problems];
Cornish & Moffat gq/94;
Moffat JMP(95),
JMP(95);
Ragusa PRD(97);
Jurco et al a1512-MG14 [and generalized geometry].

@ __Related topics__: Clayton IJMPA(97)gq/95 [Hamiltonian];
Mebarki et al PS(97) [quantization];
Wanas & Kahil GRG(99)gq [quantization of paths].

@ __Cosmology__:
Moffat ap/97 [birefringence];
Prokopec & Valkenburg PLB(06)ap/05 [inflation and cmb].

@ __Black holes__: Burko & Ori PRL(95)gq,
gq/95 [black holes do form].

@ __Other phenomenology__:
Woolgar PRD(90) [lunar orbit];
Legare & Moffat gq/95 [test particles];
Moffat & Sokolov PLB(96)ap/95 [galaxies];
Moffat gq/04,
JCAP(05)ap/04 [galaxy rotation curves];
Moffat gq/04 [Gravity Probe-B],
CQG(06) [time delay];
Janssen & Prokopec CQG(06)gq [instability];
Prokopec & Valkenburg ap/06,
Janssen & Prokopec JPA(07)gq/06-conf [massive, as dark matter];
Hammond IJMPD(13) [non-symmetric part as the potential for the spin field];
Pérez Teruel MPLA(14) [and particle interactions].

> __Related topics__:
see anomalous acceleration [Pioneer 10/11];
bianchi models; equivalence principle.

> __Online resources__:
see Wikipedia page.

**Nordström Theory of Gravity**
> see Scalar Theory of Gravity.

**Nordtvedt Effect**
> s.a. equivalence principle; tests of general relativity with orbits.

* __Idea__: A (possible) violation
of the strong equivalence principle in the Earth-Moon system; It would show up
in a departure from geodesic motion, e.g., with a "polarization" of
the Moon's orbit.

@ __References__: Dicke in(64);
Nordtvedt PR(68),
PR(68);
in Misner et al 73, p1128;
in De Rújula a2005.

> __Online resources__:
see Wikipedia page.

**Normal Coordinates on a Lie Group**
> see coordinates.

**Normal Coordinates on a Manifold**
> see coordinates.

**Normal Distribution**
> see gaussian.

**Normal Matrix / Operator**
> see operator theory.

**Normal Modes**
> see molecular physics.

**Normal Product / Ordering**
> s.a. fock space.

$ __Def__: The normally-ordered
product in the algebra of boson and fermion operators on Fock space is

**:***A*_{1}
*A*_{2} ...
*A*_{n}**:** or
N(*A*_{1} *A*_{2} ...
*A*_{n}):=
(−1)^{p}
*A*_{p1}
*A*_{p2} ...
*A*_{pn} ,

where {*p*_{1}, *p*_{2},
..., *p*_{n}} is a permutation of {1, 2, ..., *n*}
such that annihilation operators always appear to the right of creation operators, and *p* is
the number of times two fermion operators have been commuted; In addition, to make this operation
well-defined, we require linearity, **:***A*+*B***:**
= **:***A***:** + **:***B***:** and
**:***cA***:** = *c* **:***A***:**.

* __Applications__: Resolve operator-ordering
ambiguities, and regularize divergent quantities; e.g. for a scalar field

*H* = \(\frac12\)∑_{k}
(*a*_{k}^{†}
*a*_{k}
+ *a*_{k}
*a*_{k}^{†} )
*ω* diverges, *H*
= **:**\(\frac12\)∑_{k}
(*a*_{k}^{†}
*a*_{k}
+ *a*_{k}
*a*_{k}^{†} )
*ω***:**
= ∑_{k}
*a*_{k}^{†}
*a*_{k} *ω* does not.

* __Conditions__: It depends on
a choice of vacuum, so it is not obvious how to define it in curved spacetime.

@ __General references__: Wurm & Berg AJP(08)jan [basic ideas and results];
Plimak & Stenholm AP(12)-a1307 [generalization to interacting fields].

@ __Combinatorics, formulas__: Katriel LNC(74);
Katriel JOB(02);
Błasiak et al PLA(03),
CzJP(05)qp-conf,
JMP(05);
Solomon et al qp/04-conf;
Horzela et al qp/04-conf;
Schork PLA(06) [*q*-deformed bosons];
Błasiak PhD(05)qp;
Mansour et al IJTP(08)qp/06 [non-crossing],
PLA(07)qp/06 [generalization];
Błasiak et al AJP(07)jul-a0704 [introduction];
Błasiak & Flajolet SLC-a1010.

@ __In curved spacetime__: Nikolić GRG(05)ht/02 [generalization based on 2-point function].

> __Online Resources__:
see Wikipedia page.

**Normal Space**
> see types of topologies.

**Normal Subgroup of a Group**
> see group.

**Normalizer of a Subset of a Group**
> see group.

**Nothing**
> s.a. boundary conditions in quantum cosmology [universe-from-nothing proposals].

* __Idea__: The absence of spacetime.

@ __References__: Brown & Dahlen PRD(12)-a1111
[as an infinitely-negatively-curved spacetime, and tunneling in cosmology];
Šekrst a2006 [before the Big Bang].

**Notoph**

* __Idea__: A particle whose
helicity properties are opposite to those of the photon; A (3+1)-dimensional
free Abelian 2-form, described by a gauge theory; Proposed in the 1960s
by Ogievetskii and Polubarinov, but the theory goes under the names of Kalb
& Ramond, who developed the theoretical concept.

@ __References__: Dvoeglazov PS(01)phy/98;
Malik ht/03-proc [as Hodge theory],
PPNL(11)-a0912-conf [superfield formalism, BRST transformations];
Dvoeglazov JPCS(15)-a1308 [notoph-graviton-photon coupling];
Bandos & Ortín PRD(15)-a1502 [in *N* = 8 supergravity].

**Nova** > see star types.

**Nowhere Dense Subset**

* __Idea__: *A* ⊂ *X*
is nowhere dense if every ball *U* ⊂ *X* contains another ball
*V* ⊂ *X* which has no points in *A*.

**NP-Completeness**
> see computational complexity;
mathematics [Millennium Problems].

**NP Formalism**
> stands for Newman-Penrose.

**Nuclear Operator**
> same as Trace-Class.

**Nuclear Physics**
> s.a. nuclear technology.

**Nucleosynthesis**
> see nuclear physics; astrophysics;
early-universe nucleosynthesis.

**Null Cone**
> see Light Cone.

**Null Coordinates**
> see coordinates.

**Null Curve, Surface**
> see spacetime subsets.

**Nullity (Spinor)**
> see types of spinors.

**Numbers**
> s.a. Mathematical Constants; physical
constants; types of numbers.

* __Special numbers__: All numbers
are special, but some are more special than others; Large numbers studied are
the Sandreckoner's number 10^{63}, Googol and
Googolplex; Other special integers include 142,857 (see what happens to it when
multiplied by 2, 6, 4, 5, and by 7), 26 (the only one between a square and a
cube); > for more see number theory
[including prime numbers, factoring, ...].

* __Notation__: We use a "geometric"
modulus 10 notation, with a set of 10 "Arabic" symbols {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
that can be used to write more than 10 integers by position-dependent convention, in which the
one placed in the *n*-th position from the right is to be interpreted as multiplied by
10^{n−1}; Babylonians used
to count in modulus 60; Alternatives are the Roman numerals, or the factorial number system
[> see MathPages page].

@ __General references__: Johnson & Jedrzejewski 14 [patterns and drawings].

@ __Special numbers__: Adrian 06;
Flannery 06 [2^{1/2}];
Posamentier & Lehmann 09.

> __Examples__: see Catalan Numbers; e;
Euler(-Mascheroni) Constant; Feigenbaum Number;
Golden Ratio; Googol; Googolplex;
Googolplexian; i; Infinite;
Omega Number; π; Silver Mean;
Zero.

> __Online Resources__: see Internet Encyclopedia of Science
pages.

**Number Operator**
> see fock space.

**Numerical Methods**
> see computational physics; numerical general relativity.

**Numerical Relativity**
> s.a. models.

**NUT Space**
> s.a. maxwell fields in curved spacetime; null
infinity; schwarzschild spacetime [modified solution, and NUT parameter].

* __Idea__: A solution of Einstein's
equation that can be interpreted as describing the exterior field of two
counter-rotating semi-infinite sources possessing negative masses and infinite angular
momenta, which are attached to the poles of a static finite rod of positive mass.

* __NUT 4-momentum and charges__: The NUT
4-momentum is the magnetic dual of the Bondi-Sachs 4-momentum at null infinity, and
it is absolutely conserved, even if there is gravitational radiation; Gravitational
fields with non-vanishing NUT 4-momenta are not physically significant in classical
general relativity, but may play a role in quantum gravity [@ Ashtekar & Sen
JMP(82)].

@ __References__:
Krori & Bhattacharjee PLA(81) [in Brans-Dicke theory];
Nouri-Zonoz et al CQG(99)gq/98 [dual];
Dadhich & Patel gq/02 [*G* → 0 limit];
Manko & Ruíz CQG(05)gq [interpretation].

**Nyquist Theorem**

* __Idea__: One of the few general results
known in non-equilibrium thermodynamics; It relates tiny equilibrium voltage fluctuations
across a conductor with its resistance.

@ __References__: Singh a1409 [non-linear Nyquist theorem].

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

send feedback and suggestions to bombelli at olemiss.edu – modified 3 may 2021