Topics, N

N-Body Problem > see Many-Body Systems.

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 0| (x)(y) |0 = 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 a0704 [calculation, and non-perturbative renormalization group].

Nambu Algebras / Brackets / Mechanics > see poisson brackets; deformation quantization; modified classical mechanics.

Nambu Tensor > see killing fields.

Nambu-Goldstone Boson > see symmetry breaking.

Nambu-Jona-Lasinio Model
@ General references: Battistel et al a0803-PRD [strategy to handle divergences].
@ In curved spacetime: Elizalde et al PRD(94)ht/93 [phase structure and chiral symmetry breaking].

Nariai Metric > see schwarzschild-de sitter.

Navier-Stokes Equation > see fluid.

Nebulae > see interstellar matter.

Negative Probabilities > see probability in physics.

Nekhoroshev Theorem > see integrable systems.

Nernst Theorem > see thermodynamics [3rd law].

Nester Form > see Metric-Affine Gravity.

Net > s.a. 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 = N; Observable nets in quantum field theory (> see observable algebras).

Network

Neumann Functions > see bessel functions.

Neumann Problem
* Idea: A bdry value problem for second order elliptic pde's.
* Result: 1! solution to ² = –/₀ inside a region V, with /n fixed on V, up to an additive constant.

Neural Network > see network.

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

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

Neutron Stars

Newman-Penrose Formalism > see spin coefficients.

Newman-Tamburino Metrics
@ References: Steele gq/04 [Killing vector].

News Tensor > see asymptotic flatness at null infinity.

Newton-Cartan Theory

Newton-Wigner Localization > see localization; quantum particles.

Newton's Gravitational Constant

Newton's Laws of Dynamics > s.a. classical mechanics; modifications.
* First law: For a free particle in an inertial frame, du/dt = 0, where u = velocity and in the relativistic version t is usually proper time.
* Second law: In an inertial frame, m du/dt = f, where f is the force acting on the particle.
* Third law: The "principle of action and reaction," for objects A and B, the force F_AB of A on B is equal and opposite to the force 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.
* 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) [objections and formulation]; Brehme AJP(85) [laws as definitions]; Anderson AJP(90) [not definitions]; Grabowska & Urbanski mp/04 [frame-independent].
@ First law: Pfister FPL(04); Rabinowitz IJTP(08) [and quantum mechanics].
@ Third law: Anandan & Brown FP(95) [and pilot-wave theory]; Fraser SHPSA(05) [and argument for second law].
@ Bounds on violations: Abramovici & Vager PRD(86) [ok down to 3 10^–9 cm/s²]; Gundlach et al PRL(07) + pn(07)apr [ok down to 5 10^–14 cm/s²].
@ Modifications: Milton & Willis PRS(07) [second, continuum elastodynamics]; Unzicker gq/07 [and gravity]; > s.a. force, MOND.
> Related concepts: see force, inertia and inertial frame, mass.

Newton's Theorem
* Idea: The gravitational field outside a spherically symmmetric mass distribution depends only on its total mass.

Newtonian Gravitation > s.a. tests of newtonian gravitation.

Nijenhuis Torsion Tensor > see types of symplectic structures.

9j Symbol > see SU(2).

No-Cloning Theorem > see quantum technology.

No-Hair Theorems > see black hole hair, brans-dicke [cosmic].

Noether Symmetries, Theorem > s.a. [hamiltonian and lagrangian symmetries]; symmetries.
* Idea: Exploit a symmetry of a theory so as to reduce the number of variables needed to treat a problem.
* History: Soon after Hilbert's discovery of the variational principle for general relativity, people including Hilbert, Klein, and Einstein were concerned about the failure of local energy conservation in the theory; Noether's theorems solved the problem.
$ Def: To every continuous symmetry x^a = X^a_b ^b, = _a ^a of the Lagrangian for a field theory there corresponds a conserved current J^a_b with _a J^a_b = 0, and a conserved quantity, the charge Q_b:

J^a_b:= (/(_a)) _b – ^a_c X^c_b ,   Q_b:= _Sigma dS_a J^a_b.

@ General references: Noether NKGG(18) [translation TTSP(71)phy/05]; Govinder & Leach PLA(95) [integrals]; Byers phy/98 [historical].
@ Second theorem: Gogilidze & Surovtsev ht/96 [and constraints]; Bashkirov et al JPA(05)m.DG/04 [generalized setting], JMP(05)mp/04 [BRST symmetries]; Cariñena et al m.DG/05 [gauge symmetries in classical mechanics].
@ Related topics: García & Pons IJMPA(01)ht/00 [canonical realization]; Sanyal & Modak CQG(01)gq [and field couplings]; Brown & Holland AJP(04) [first theorem, in quantum mechanics and electromagnetism]; Butterfield phy/05-in; Albeverio et al JMP(06) [quantum]; Bokhari et al IJTP(06) [and spacetime isometries]; Bokhari & Kara GRG(07) [vs Killing vectors].
@ In classical mechanics: Desloge & Karch AJP(77); Marinho EJP(07).
@ For gauge theories/quantum field theories: Buchholz et al AP(86); Karatas & Kowalski AJP(90); Fatibene et al JMP(97); Julia & Silva CQG(98)gq; Gràcia & Pons JMP(00)mp; Bashkirov JPA(05) [reducible gauge symmetries].
@ Generalized: Rosen AP(72), AP(74), AP(74); Cariñena & Rañada LMP(88) [singular Lagrangians]; Lunev TMP(90) [non-local symmetries]; Gràcia & Pons JPA(95) [higher-order Lagrangians]; Govinder et al PLA(98) [approximate symmetries]; Magro et al AP(02)ht/01 [superfields]; Torres m.OC/03-in [non-smooth solutions]; Paal mp/06, mp/06-in [from Moufang transformations]; Agostini et al MPLA(07)ht/06, Arzano & Marcianò ht/07, Amelino-Camelia et al a0710-in [for Hopf algebra spacetime symmetries]; Cicogna & Gaeta JPA(07) [for -symmetries]; Agostini a0711 [in -Minkowski].
@ In gravitation: Sorkin PRS(91) [Noether operator, and electromagnetism];
@ In cosmological models: Vakili PLB(08)-a0804.
@ Other applications: Danos FP(97)ht [in quantum field theory]; García & Pons IJMPA(00)ht/99 [constrained systems]; Sardanashvily mp/03 [classical mechanics]; Hanc et al AJP(04) [examples and teaching]; > s.a. energy-momentum tensor.

Noise > s.a. partial differential equations [stochastic].
@ 1/f noise: in Kaplan & Glass 95 [II].
@ Shot noise: Beenakker & Schönenberger PT(03)may.

Noiseless Subsystems > see generalized coherent states.

Non-Archimedean Structures > see numbers; geometry.
@ And physics: Avinash & Rvachev FP(00) [cosmology]; El Naschie CSF(04) [fundamental length and all that]; > s.a. quantum particles.

Non-Associative Geometry > see modified quantum mechanics; proposals for quantum spacetime; types of gauge theories.

Non-Commutativity > s.a. non-commutative geometry; in physical theories, in field theory and gravitation.
@ References: MacKenzie ThSc(97)may [general notion].

Non-Conservative System > see classical systems.

Non-Degenerate Bilinear Form > see Bilinear Form.

Non-Equilibrium Systems > see modified sm and thermodynamics; states in quantum field theory; temperature.

Non-Euclidean Geometry > see geometry.

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-Linear Quantum Mechanics > s.a. brownian motion; causality violations; schrödinger equation [including WKB].
* Motivation: Include as part of the dynamical evolution the transformation associated with the wave function collapse.
* Feature: Superluminal propagation, a generic phenomenon in a large class on non-dissipative quantum theories.
@ Intros, reviews: Goss Levi PT(89)oct; news Nat(90)jul; Svetlichny qp/04 [arXiv bibliography]; Habib et al qp/05-in [intro].
@ General references: Bialynicki-Birula & Mycielski AP(76); Kibble CMP(78) [relativistic]; Giusto et al PhyD(84); Bialynicki-Birula in(86); Bon et al NCA(87) [solutions, thermodynamic description]; Weinberg AP(89), PRL(89) + comment Peres PRL(89); Castro JMP(90) [and geometric quantum mechanics]; Jordan PLA(90); Nattermann qp/97; Puszkarz qp/97, qp/97, qp/99, qp/99, qp/99; Strauch a0707-PRE [propagation scheme].
@ Derivations, motivation: Parwani qp/06-in, TMP(07) [information theory-motivated]; Adami et al JSP(07) [from many-body dynamics].
@ Non-polynomial: Parwani qp/04-in, AP(05) [info-theoretic argument]; Parwani & Tan qp/06 [solutions].
@ Special systems: Gattobigio et al JMP(99) [on half-line]; Gardiner et al PRA(00)qp/99 [with chaotic potential]; Teismann RPMP(05) [with L² solutions]; Shu & Zhang JMP(06) [harmonic potential]; Doikou et al NPB(08)-a0706 [on an interval].
@ And gravity: Soni Pra(02)gq/00; Singh gq/03; Svetlichny IJTP(05), qp/06-in [at Planck scale]; > s.a. quantum gravity phenomenology.
@ Difficulties: Gisin PLA(90) [causality]; Caticha PLA(98)qp, PRA(98)qp; Doebner qp/98-in; Puszkarz qp/98 [E non-uniqueness]; Lücke qp/99 [non-locality]; Mielnik PLA(01) [causality]; Kent PRA(05) [consistent with causality]; Burq & Zworski CMP(05) [semiclassical, instability].
@ Consequences: Cassidy PRD(95) [and closed timelike curves]; Czachor & Doebner PLA(02)qp/01 [multi-particle correlations]; > s.a. cosmic rays.
@ Tests: Bollinger et al PRL(89); Majumder et al PRL(90) [optically pumped ²⁰¹Hg].
@ Related topics: Minelli & Pascolini LNC(80), in(80) [solitons], NCB(85) [conservation laws]; Valentini PRA(90) [and complementarity]; Polchinski PRL(91) [causality and EPR, comment Mielnik qp/00]; Lücke qp/95 [observables]; Bona qp/99-in [for density matrices], qp/99-in [symmetries]; > s.a. Ehrenfest Theorem.

Non-Local Theories > see locality; types of quantum field theories.

Non-Metricity > see connection.

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

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

Non-Standard Analysis

Non-Symmetric Geometry and Gravity > s.a. unified theories.
* History: A non-symmetric theory of gravity was proposed by J Moffat.
@ 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).
@ 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].
@ 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-in [massive, as dark matter].
> Related topics: see anomalous acceleration [Pioneer 10/11]; bianchi models; equivalence principle.

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.

Norm / Normed Spaces

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 Product / Ordering > s.a. fock space.
$ Def: The normally-ordered product in the algebra of boson and fermion operators on Fock space is

:A₁ A₂ ... A_n: or N(A₁ A₂... A_n):= (–1)^p A_{p_1} A_{p_2} ... A_{p_n} ,

where {p₁, p₂, ..., 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 =  _k (a_k a_k + a_k a_k)    diverges,   H = : _k (a_k a_k + a_k a_k) : = _k a_k a_k    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) [basic ideas and results].
@ Combinatorics, formulas: Katriel LNC(74); Katriel JOB(02); Blasiak et al PLA(03), qp/05-in, JMP(05); Solomon et al qp/04-in; Horzela et al qp/04-in; Schork PLA(06) [q-deformed bosons]; Blasiak qp/05-PhD; Mansour et al qp/06 [non-crossing], PLA(07)qp/06 [generalization]; Blasiak et al AJP(07)-a0704 [introduction].
@ In curved spacetime: Nikolic GRG(05)ht/02 [generalization based on 2-point function].

Normal Space > see types of topologies.

Normal Subgroup of a Group > see group.

Normalizer of a Subset of a Group > see group.

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 1960's by Ogievetskii and Polubarinov, but the theory goes under the names of Kalb & Ramond.
@ References: Dvoeglazov PS(01)phy/98; Malik ht/03-in [as Hodge theory].

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 computation.

NP Formalism > stands for Newman-Penrose.

Nuclear Operator > same as Trace-Class.

Nuclear Physics

Nucleosynthesis > see early universe.

Null Cone > see Light Cone.

Null Coordinates > see coordinates.

Null Curve, Surface > see spacetime subsets.

Numbers > s.a. constants; number theory; types of numbers.
* Special numbers: All numbers are special, but some are more special than others; Other "special" numbers studied are the Sandreckoner's number 10⁶³, and Googolplex.
* Notation: We use a "geometric" notation, with a set of 10 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; Alternatives are the roman numerals, or the factorial number system [> MathPages page].
@ Special numbers: Flannery 06 [2^1/2].
> Examples: see Catalan Numbers; e; Euler-Mascheroni; Feigenbaum; Golden Ratio; i; Infinite; Omega; pi; 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 [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 nonvanishing 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]; Nouri-Zonoz et al CQG(99)gq/98 [dual]; Dadhich & Patel gq/02 [G → 0 limit]; Manko & Ruíz CQG(05)gq [interpretation].

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