Topics, M

M-Theory

Macdonald Polynomials
@ References: Gadde et al CMP(13)-a1211 [and gauge theories]; Blondeau-Fournier et al a1211 [Macdonald superpolynomials].
> Online resources: see Wikipedia page.

MacDowell-Mansouri Formalism / Theory > s.a. actions for general relativity; Gauge Theory of Gravity; self-dual fields; supergravity.
* Idea: An approach to gravity treated as a gauge theory, in which the basic variable is a connection-like quantity A = ω + (1/l) e that combines an actual connection ω valued in the Lie algebra of a group G with a cotetrad e; Used for gravity with a cosmological constant; For Λ > 0 the group G = SO(4, 1), while for Λ < 0, the group G = SO(3, 2); The "trick" can be geometrically understood in terms of Cartan geometry, where the tangent space at each point is replaced by a tangent sphere or hyperboloid.
* Remark: It can be considered an extension to 4D of Witten's 3D approach to gravity based on the Chern-Simons action (but it preceded the latter), and has also been considered as a perturbed BF theory [& Freidel, Smolin & Starodubtsev]; It can be generalized to other Cartan geometries with dim(G/H) = 4.
* Important quantities: The connection A = ω + (1/l) e has values in the vector space so(3,1) + \(\mathbb R\)3,1; Its curvature can be written as

F[A] = R ± (1/l 2) ee + (1/l) dωe ,

where the second term is a correction to the curvature of the Lorentz connections, and the third one is a torsion term.
* Action: The Einstein-Hilbert action plus a topological term; With Fcorr = R ± (1/l 2) ee, and l 2 = 3/Λ, then

Scorr = # tr(FcorrFcorr) ;

Notice that at the action level, the Λ = 0 case gives just RR, the Chern form, so one obtains a topological theory and not general relativity.
@ References: MacDowell & Mansouri PRL(77) [with Λ > 0]; Wise CQG(10)gq/06 [and Cartan geometry]; Anabalón JHEP(08)-a0805 [and gauged Wess-Zumino-Witten term]; López-Domínguez et al a1401 [generalization]; Reid & Wang JMP(14) [conformal holonomy]; Berra-Montiel et al a1703 [covariant Hamiltonian formulation].

Mach's Principle

MACHOs > see types of dark matter.

Macrorealism > see realism.

Macroscopic Systems > see macroscopic quantum systems; classical systems; s.a. cosmological general relativity [macroscopic, averaged gravity].

Macrostate > s.a. Boltzmann Principle; Microstate; types of entropy.
* Idea: In statistical mechanics, a macrostate is the specification of a particular set of values for the variables characterizing the state of a thermdynamical system, e.g., the energy, number of particles and volume for a gas; The same macrostate, in this sense, can be represented by different ensembles, depending on whether the system is considered to be isolated with a well-defined value of its energy (microcanonical ensemble), in thermal equilibrium with a heat bath (canonical ensemble), etc.
> Online resources: see Wikipedia Microstate page.

Macsyma > see programming languages.

Madelung Constant
* Idea: A constant characterizing a periodic crystal of N positive and negative point charges, used to calculate its electrostatic energy.
@ References: Tyagi PTP(05)cm/04 [fast-converging series representation]; Baker & Baker AJP(10)jan [finite and bulk materials, calculation]; Mamode a1511 [hypercubic crystal structures in any dimension].

Madelung Equations / Quantum Hydrodynamics > see interpretations and origin of quantum theory.

Magic Numbers > see nuclear physics.

Magic Squares
* Idea: Magic squares are arrays of distinct numbers whose rows, columns and diagonals add to the same total.
@ References: news SA(14)sep [do 3 × 3 magic squares of squares exist?].

Magic States > see types of quantum states.

Magnetars > see astronomical objects.

Magnetic Dipole Moment > see magnetism / Gyromagnetic Ratio / neutrinos; particle types [electron, muon]; photon.

Magnetic Mass > see duality.

Magnetic Part of the Weyl Curvature > see weyl tensor.

Magnetic Permeability > see magnetism.

Magnetic Susceptibility > see Susceptibility.

Magnetism > s.a. magnetic effects and phenomenology [including Magnetohydrodynamics].

Magnetohydrodynamics > see phenomenology of magnetism.

Magnon
* Idea: The Nambu-Goldstone boson of (anti-)ferromagnets; A particle-like excitation in a solid arising from a moving magnetic-spin disturbance; In the presence of a magnetic field strength larger than a certain value, atoms with an intrinsic magnetic moment can be oriented all in one direction; In this configuration a small input of energy can tilt some of the spins out of the general formation; The successive tilting of spins can take the form of a wave moving through the sample; If also the temperature of the sample is extremely low, the moving wave can be considered as a particle-like (or quasiparticle) entity, like mechanical vibrations in a solid can be construed as sound waves or as phonons.
* Giant magnons: Classical solitons of the O(N) σ-model, which play an important role in the AdS-CFT correspondence.
@ General references: Kämpfer et al NPB(05) [low-energy effective theory].
@ Giant magnons: Zarembo JHEP(08)-a0802.
@ Related topics: Kenzelmann Phy(11) [electromagnons].

Magnus Effect > s.a. turbulence.
* Idea: The effect in which a spinning object curves away from its flight path; It is important in many ball sports, and some areas of engineering.
* Reason: It is a manifestation of Bernoulli's theorem (fluid pressure decreases where the fluid speed is high), together with air drag by the object.
@ Optical: Bliokh & Bliokh PLA(04) [and Berry phase].
> Online resources: see Britannica page; Wikipedia page; YouTube video (see also the Bouncing Bomb).

Majorana Equation > see arbitrary-spin field theories.

Majorana Particles / Spinors > see 4-spinors.

Majorana's Stellar Representation > see quantum systems.

Majumdar-Papapetrou Solutions > see under Papapetrou-Majumdar.

Makeenko-Migdal Equation
@ References: Driver et al a1601 [for Yang-Mills theory on the plane, three proofs], a1602 [for Yang-Mills theory on compact surfaces].

Malament-Hawking Theorem > see causal structures.

Malament-Hogarth Spacetimes
@ References: Welch BJPS(08) [extent of possible computations].

Malcev Algebra > see abstract algebra.

Maldacena Conjecture > see AdS–conformal field theory correspondence.

Malus' Law > see polarization.

Mandelbrot Set > see fractals.

Mandelstam Identities
* Idea: SL(2, \(\mathbb C\)) identities, like the spinor identity (tr A) (tr B) – tr AB – tr AB–1 = 0, for all A, B ∈ SL(2, \(\mathbb C\)).

Manifold > s.a. 2D, 3D, 4D manifolds; types of manifolds.

Manko-Novikov Solutions
* Idea: A family of stationary axisymmetric solutions of the vacuum Einstein equation; They represent compact objects that are not black holes, since they do not have event horizons, and they evade the uniqueness theorems by having ring singularities on their surfaces; They are parametrized by the anomalous magnetic moment q, representing the deviation from the value of the quadrupole moment of the Kerr solution with the same mass and spin (depending on the sign of q they are more prolate or more oblate than the corresponding Kerr solution); They are useful as models for exteriors (not too close to the horizon) of non-Kerr black spacetimes.
@ References: Bambi JCAP(11)-a1103, a1104-proc; Contopoulos et al IJBC(11)-a1108 [orbits].

Mansouri-Sexl Theory > see tests of lorentz invariance.

Mansuripur's Paradox > see Lorentz Force.

Many-Body Systems > see classical systems; Fermions; gravitating many-body systems; many-particle quantum systems.

Many-Minds Interpretation > see many-worlds.

Many-Worlds Interpretation

Map > s.a. maps between differentiable manifolds.
@ Positive maps: Majewski OSID(04)qp [quantization of classical Banach spaces], qp/04 [classification], & Marciniak qp/04 [decomposability].

Maple > see programming languages.

Mapping Class Group of a Manifold M > see group types.

Marginal Distribution > see probability theory.

Marginally Trapped Surface > see Trapped Surface.

Margolus-Levitin Theorem > s.a. evolution of quantum states; quantum information / black holes [complexity evolution].
* Idea: A quantum system of energy E needs at least a time h/4E to go from one state to an orthogonal state.
@ References: Margolus & Levitin PhyD(98)qp/97.
> Online resources: see Wikipedia page.

Markov Chain / Process

Martingale > s.a. diffusion; markov processes.
@ References: Revuz & Yor 91; Baldi et al 02 [and exercises].
> Online resources: see MathWorld page.

Maslov Index > s.a. geodesics.
* Idea: A number defined for each path of symplectomorphisms of a symplectic vector space (an integer if the path is a loop, and a half-integer in general).
@ References: Pletyukhov & Brack JPA(03) [canonically invariant calculation]; de Gosson & de Gosson JPA(03) [Hamiltonian periodic orbits]; in de Gosson 17.

Mass

Mass Inflation > s.a. brans-dicke gravity; reissner-nordström black holes.
* Idea: The exponentially growing, relativistic counter-streaming instability at the inner horizon of a two-horizon black hole, first pointed out by Poisson and Israel, causing an apparent increase of the mass of the black hole for a traveler moving toward it.
@ References: Poisson & Israel PRD(90); Bonanno PRD(96)gq/95; Oda gq/97 [for Reissner-Nordström black holes, in quantum gravity]; Chan PhD(98); Hamilton & Avelino PRP(10) [physical cause and consequences]; Brown et al PRD(11) [for loop black holes]; Hwang et al JCAP(11)-a1110 [in f(R) gravity]; Dokuchaev CQG(14).

Massive Gravity

Massless Particles > see field theory.

Master Equation > s.a. non-equilibrium statistical mechanics; open systems; statistical-mechanical systems.
* Idea: An equation describing a classical stochastic process, of the form dPc/dt = ∑c' (Wc'c Pc'Wcc' Pc) in its Markov version, where Wcc' is the transition rate from state c to state c'.
@ General references: Alicki IJTP(77) [and the Fermi golden rule]; Joos qp/05-talk [from strong decoherence]; Sun PRL(06) [path summation formulation]; Kryszewski & Czechowska-Kryszk a0801 [pedagogical]; Sano JPA(08) [steady-state distribution]; Lafuerza & Toral JSP(10) [Gaussian approximation]; Hall et al PRA(14)-a1009 [Lindblad-like canonical form]; Kamiya PTEP(15)-a1409 [quantum-to-classical reduction].
@ Applications: Mendes & Farina qp/06 [atomic energy level corrections].
@ Special types: Belavkin TMP(97)qp/05 [quantum, irreversible]; Öttinger a1002 [dissipative, non-linear]; Rivas et al NJP(10) [Markovian, model derivation].
@ Non-Markovian: Maniscalco PRA(07) [spin-1/2, with exponential memory]; Krovi et al PRA(07)-a0707 [qubit + Ising spin bath]; Vacchini PRA(08) [bipartite system, generalized Lindblad equation]; Breuer & Vacchini PRL(08)-a0809 [quantum semi-Markov process]; Kossakowski & Rebolledo OSID-a0902 [positivity-preserving, characterization]; Chruściński & Kossakowski PRL(10); Bellomo et al JPA(10)-a1005 [tomographic approach]; Chruściński PS(13) [criteria, mathematical aspects]; Chruściński & Maniscalco PRL(14) [degree of non-Markovianity of quantum evolution]; Pagnini PhyA(14) [emergence of fractional non-Markovian master equations]; Ribeiro & Vieira PRB(15)-a1412 [in electronic and spin transport].
@ Specific types of systems: Bhattacharya et al a1610 [spin and spin bath].
> Related topics: see brownian motion; Lindblad Equation.
> Online resources: see Wikipedia page.

Matching of Metrics

Material Cause > see causality.

Materials > see condensed-matter physics.

Mathematica > see programming languages; BRST symmetry; heat kernel; partial differential equations.

Mathematical Physics

Mathematics

Mathieu Equation, Function > s.a. oscillator.
@ References: Frenkel & Portugal JPA(01) [algebraic methods].

Mathisson-Papapetrou-Dixon Equations > see spinning particle.

Matlab > see programming languages.

Matrix > s.a. characteristic polynomial; operations on matrices [determinant, inverse, etc].

Matrix Mechanics > see history of quantum physics.

Matrix Models / Theories in Physics > s.a. dynamical triangulations; entropy; Tensor Models.
* Idea: (Probably) the simplest non-commutative geometries; Instead of a Riemannian metric, a matrix model is described by a matrix-valued self-adjoint symmetric two-tensor that plays a role of a "non-commutative'' metric.
@ General references: Di Francesco et al PRP(95) [2D gravity]; Starodubtsev NPB(03)ht/02 [quantization]; Avramidi IJGMP(05)mp [Dirac operator]; Smolin a0803 [universality, gauge theory and gravity]; Steinacker CQG(10)-a1003 [emergent geometry and non-commutative gravity].
@ Random matrix models: Guhr et al PRP(98); Bleher a0801-in [Riemann-Hilbert approach]; Zirnbauer a1001-en [and symmetry classes]; Marchal PhD(10)-a1012 [geometric and integrable aspects]; Small a1503-PhD [unified framework]; Eynard et al a1510-ln [self-contained introduction] > s.a. matrices.

Matroid Theory
* Idea: A branch of combinatorics started by Whitney (1935), which captures and generalizes the notion of linear independence in vector spaces; The theory also goes by the name of (combinatorial) pregeometry, and matroids are also called independence structures.
$ Matroid: A pair M = (S, I), with S a finite set and I a non-empty family of ("independent") subsets of S, satisfying (1) (AB) ∧ (BI) implies AI; (2) (A, BI) ∧ (|A| < |B|) implies that there exists bB \ A, such that A ∪ {b} ∈ I (there is also an alternative definition in terms of circuits rather than independent sets).
* Remark: Most are not representable in vector spaces.
$ Free matroid: M = F n, k has as independent sets all subsets of k or less out of n points; k = 0 gives a trivial matroid; k = n gives a free geometry (Boolean algebra).
* Examples: Finite sets of vectors in a vector space V and linearly independent ones; Finite sets of points in an affine space.
> Other examples: see Geometrically Independent Points and Combinatorial Geometries.
@ General references: Whitney AJM(35); Tutte 80; White 86; Crapo & Schmitt EJC(05) [free product]; Gordon & McNulty 12.
@ Related topics: Cameron et al JCTA(13) [combinatorial representations].
@ And physics: Nieto & Marín IJMPA(03) [gravity]; Brunnemann & Rideout CQG(10)-a1003, Nieto RMF-a1003 [loop quantum gravity]; > s.a. hilbert space.
> Online resources: see MathWorld page; Wikipedia page.

Matter > s.a. condensed matter; matter content of the universe.

Matter-Gravity Entanglement Hypothesis > see types of entropy.

Maupertuis Principle > see hamiltonian dynamics; quantum mechanics formalism; variational principles.

Maurer-Cartan Form, Structure Equation > see forms [canonical].

MaxEnt > see Maximum Entropy Estimation below.

Maximal Acceleration > see acceleration.

Maximal Surface > see extrinsic [extremal surface].

Maximal Variety Principle > see origin of quantum mechanics.

Maximally Symmetric Geometry
$ Def: An n-dimensional manifold M with metric gab is maximally symmetric if it has n(n+1)/2 independent Killing vector fields.
* Curvature: The full Riemann tensor can then be expressed in terms of the Ricci scalar R, as Rabcd = [2/(n–1)(n–2)] R gc[a gb]d.

Maximum Entropy Estimation (MaxEnt) > s.a. entropy; quantum states / non-equilibrium statistical mechanics; Thermodynamic Limit.
* Idea: A formalism developed by Jaynes that determines the relevant ensemble in statistical mechanics by maximizing the entropy functional subject to the constraints imposed by the available information.
* Principle of Maximum Caliber: A non-equilibrium generalization of the MaxEnt Principle.
@ General references: Jaynes PR(57), PR(57); Gzyl 95 [lecture notes]; Meléndez & Español JSP(14)-a1402 [alternative derivation]; van Enk SHPMP-a1408 [and statistical mechanics, and the Brandeis dice problem]; Holik et al a1501 [generalization, introduction of symmetry constraints and group theory].
@ Maximum caliber: Pressé et al RMP(13); González et al FP(14) [and Newtonian dynamics].

Maximum Entropy Principle > see entropy; molecular physics [molecular gas].

Maximum Power / Force Principle > see force.

Maximum Tension Principle > see matter phenomenology in gravity.

Maxwell's Demon > s.a. thermodynamics / computation; heat; Szilard's Engine.
* Idea: A creature used by Maxwell in a thought experiment about a possible way to violate the second law of thermodynamics.
@ General references: Bennett SA(87)nov; Leff & Rex AJP(90)mar [RL]; Maddox Nat(90)may; letters Nat(90)347, p24; Von Baeyer 98; Maruyama et al RMP(09)-a0707 [and information]; Ainsworth PhSc(11)jan [(few) restrictions from Liouville's theorem]; Hemmo & Shenker PhSc(11)apr [Szilard's perpetuum mobile and a detailed phase-space analysis of the compatibility of Maxwell's demon with statistical mechanics]; Hosoya et al PRE(11)-a1110 [equivalence of information-theoretical and thermodynamic entropies]; van Hateren a1407 [Maxwell's demon's goal, and information].
@ Implementations: Serreli Nat(07)feb + pw(07)jan [molecular information ratchet]; Schaller et al PRB(11)-a1106 [electronic]; Mandal & Jarzynski PNAS(12)-a1206 [solvable model, and the thermodynamics of information processing]; Strasberg et al PRL(13) + news sn(13)mar [implementation using quantum dots]; news sn(15)dec [new experiment]; Vidrighin et al PRL(16)-a1510 + news PhysOrg(16)feb [photonic]; > s.a. non-equilibrium thermodynamics.
@ Quantum version: Kim et al PRL(11)-a1006 [complete quantum analysis of Szilard's engine]; news at(12)jul [it can do work, write and erase data]; Cottet et al a1702; news sn(17)jul [testing the role of information].
@ Variations: Mandal et al PRL(13) [Maxwell's refrigerator, powered by information].
> Online resources: see Wikipedia page.

Maxwell's Equations > see electromagnetic field equations.

Maxwell Relations
* Idea: Relations obtained by expressing the integrability of the first law of thermodynamics, as expressed in terms of different potentials.

Maxwell Stress Tensor > see energy-momentum tensor.

Maxwell-Boltzmann Distribution
* Idea: The distribution of speeds/momenta in a dilute hard-sphere gas in a box with hard walls at equilibrium,

fMB(v) = 4πn(m/2πkT)3/2 v2 exp{–mv2/kT},   fMB(p) = (1/2πmkT)3/2 exp{–p2/2mkT},

where T is defined by U = (3/2) NkT; Can be derived from statistical mechanics if we use Sinai's theorem.
* Origin: Particles acquire this f starting from Brownian motion effectively by viscosity.
@ General references: in Srednicki cm/94-conf; Mohamed JSP(11) [fast, efficient sampling algorithm]; Shivanian & López PhyA(12) [model of decay to the Maxwell distribution].
@ Generalization: Cubero et al PRL(07)-a0705 [in special relativity]; Maslov TMP(06) [correction]; Rajeev a0907 [not the equilibrium solution to the Fokker-Planck equation on a hyperboloid]; Shim a1211 [non-Maxwellian molecular velocity distribution at large Knudsen numbers].

Maxwell-Lorentz Equations > s.a. fluid dynamics.
* Idea: The equation of motion of charged dust coupled to Maxwell's equations on a fixed, possibly curved spacetime.
@ References: Perlick & Carr JPA(10)-a1007 [the initial-value problem is well posed]; Aharonovich & Horwitz EPL(12)-a1203 [refutation of a non-existence theorem].

Mayer Cluster / Series Expansion > s.a. Cluster Expansion; Virial Expansion / QCD phenomenology [confinement].
* Idea: A series expansion used to evaluate grand canonical partition functions, or the pressure equation of state for a gas in powers of the activity.
@ General references: Jansen JSP(12) [at low temperatures, interpretation of the radius of convergence]; Bourgine NPB(14)-a1310 [and matrix models].
@ Bounds for the convergence radius: Morais et al JSP(14)-a1407; Procacci & Yuhjtman a1508.

Mayer-Vietoris Sequence / Theorem > s.a. ER = EPR Conjecture.
* Idea: It can be regarded as a generalization of the finite-set formula card(AB) = card A + card B – card(AB).

McDonald Functions > see bessel functions.

McVittie Metric > s.a. schwarzschild solution; relativistic cosmology.
* Idea: A type of metric used to model isolated objects in FLRW cosmological backgrounds.
* Line element: The k = 0 line element in Hogan's form is

\[ \def\dd{{\rm d}}\def\ee{{\rm e}} \dd s^2
= -\left({1-GM\ee^{-\beta(t)/2}/2c^2r\over1+GM\ee^{-\beta(t)/2}/2c^2r}\right)^{\!2} c^2\dd t^2
+ \ee^{\beta(t)} \left(1+{GM\ee^{-\beta(t)/2}\over2c^2r}\right)^{\!4}(\dd r^2 + r^2\dd\Omega^2)\;, \]

where M ~ "mass at the singularity," and β(t) ~ (asymptotic) expansion rate; It can also be written in the "Painlevé-Gullstrand form"

\[ \dd s^2 = -\big(f(r)-H^2(t)\,r^2\big)\,\dd t^2 - {2\,H(t)\,r\over\sqrt{f(r)}}\,\dd r\,\dd t + {\dd r^2\over f(r)} + r^2\dd\Omega^2\;,\ \ f(r):=1-{2m\over r}\;.\]

* Relationships: The metric becomes a FLRW metric for r → ∞, Schwarzschild for dβ/dt = 0, and Schwarzschild-de Sitter for d2β/dt2 = 0, or dH/dt = 0.
@ General references: McVittie MNRAS(33); Newman & McVittie GRG(82) [many particles]; Hogan ApJ(90); Nolan PRD(98)gq, CQG(99), CQG(99)gq; Carrera & Giulini PRD(10)-a0908 [re generalizations]; Kaloper et al PRD(10)-a1003 [with and without cosmological constant]; Lake & Abdelqader PRD(11)-a1106 [black hole in an asymptotically ΛCDM cosmology]; Faraoni et al PRD(12) [behavior of horizons]; Landry et al PRD(12)-a1207 [with Λ < 0]; da Silva et al PRD(13)-a1212 [causal structure analysis]; Nolan a1707 [with non-flat FLRW backgrounds].
@ Matter in McVittie spacetimes: Bolen et al CQG(01)gq/00 [orbit precession]; Arakida NA(09)-a0808, GRG(11)-a1103 [gravitational time delay]; Nolan CQG(14)-a1408 [particle and photon orbits].
@ Generalizations: Patel et al G&C(00) [higher-dimensional]; Faraoni et al PRD(14)-a1404 [charged]; Bejarano et al a1707 [in f(T) gravity].

Mean of a Collection of Numbers / Probability Distribution > see statistics.

Mean Curvature > see riemann tensor.

Mean-Field Approximation / Theory > s.a. history of physics.
* Idea: An approximation method used in statistical calculations for many-degree-of-freedom systems, by which one focuses on one of the degrees of freedom and replaces the values of the variables associated with the other ones by their mean values; The approximation reduces the many-body system to an effective single-body one, and neglects correlations; It sometimes produces qualitatively incorrect results, for example near critical points.
* Entanglement mean-field theory: A variant in which a many-body quantum system is reduced to a two-body system.
@ Books: Suzuki 15 [mathematical structure].
@ General references: Caracciolo et al AP(98) [statistical theory]; Pluchino et al PhyA(05) [Monte Carlo study]; Ponomarenko et al JPA(06) [finite quantum system, canonical ensemble]; Kiessling AIP(08)-a0711 [and thermodynamical equilibrium]; Genovese & Barra JMP(09)-a0812 [mechanical-system approach]; Pickl a0907 [quantum, new strategy]; Salcedo PRA(12)-a1201 [and consistency requirement]; Lacroix et al PRC(12)-a1203 [stochastic mean-field approach]; Singh a1402 [and phase transitions and renormalization, pedagogical]; Ammari et al a1411 [rate of convergence to the mean-field limit].
@ Dynamical mean-field theory: Aoki et al RMP(14) [generalization to non-equilibrium situations]; Benedikter et al a1411 [fermionic mixed states].
@ Ising model: Yapage & Nagaoka JPA(08) [information-theoretic approach]; Barra JSP(08) [interpolation techniques].
@ N-particle bosonic quantum theory: Ammari & Nier JMP(09); Erdős & Schlein JSP(09); Lewin a1510-proc; Liard a1609; > s.a. Hartree Equation.
@ Entanglement mean-field theory: De & Sen JPCS(11)-a1105 [properties of time-evolved states].
@ Related topics: Chayes CMP(09) [1D and 2D systems]; Boers & Pickl JSP(15)-a1307 [propagation of molecular chaos in mean-field situations and Vlasov-type limits]; Dai Pra et al JSP(13) [Curie-Weiss model with dissipation]; Petrat PhD-a1403 [mean-field dynamics for fermions]; Dudek et al JSP(14) [space-dependent mean-field approximation]; Paul et al a1708 [and chaos].
> Other systems: see composite quantum systems; condensed-matter physics; entanglement; Hubbard Model; modified formulations of QED; QCD phenomenology; spin models [spin glasses]; Vlasov-Poisson System.
> Variations: see coupled-spin models [Cluster Mean-Field approach].
> Related topics: see Defects; Replica Symmetry [finite-volume corrections].

Mean Free Path > see scattering.

Meander > see molecular physics [polymers].

Meaning > see mathematics [conceptual aspects].
@ In mathematics and physics: Polkinghorne 11; Sánchez a1604; Rovelli a1611 [meaningful information].

Measure Theory

Measurement > s.a. experiments in physics; measurement in quantum physics, types and effects; units.
@ Metrology: Crease PT(09)dec [Peirce and the first absolute measurement standard].
@ Quantum metrology: Giovannetti et al nPhot(11)-a1102 [reducing the statistical error by repeating the measurement].
@ Related topics: Ridgeway a0707 [measurements in infinite lattices]; Sassoli de Bianchi FS(15)-a1208 [indeterminism in physical observations].
> Specific physical quantities: see Frequency.

Mechanical Similarity > see conformal and scale symmetry.

Mechanism > see Physical Laws and, for examples, higgs mechanism, Vainshtein Mechanism.

Median
* Idea: In a set of data consisting of values of a variable x, the value m of x such that half of the data have values lower than m, and half have values higher than m.
* History: The concept was introduced into the formal analysis of data by Gustav Fechner.
> Online resources: see Wikipedia page.

Mediocrity Principle > see civilizations.

Meissner Effect > see superconductivity / fields near black holes.

Mellin Transform
@ References: Oberhettinger 74.

Melnikov Integral / Method > see description of chaos.

Melvin Solution > s.a. Robinson-Bertotti Spacetimes; kerr-newman spacetimes [Melvin-Kerr-Newman black holes].
* Idea: A family of solutions of the Einstein-Maxwell equations representing spacetimes with large-scale electric or magnetic fields; There are two seemingly different types, a dynamical one found in 1962 by Gerald Rosen in which the field is homogeneous, with magnetic stress-energy driving an anisotropic cosmic expansion, and one describing a static bundle of magnetic flux-lines bound together by self-gravity, originally discovered in 1963 by Melvin; Melvin solutions are typically discussed in the context of black hole physics (they were an important clue leading to the formulation of Thorne's) hoop conjecture, while Rosen cosmologies interpolate between different anisotropic Kasner vacuum solutions at early and late times; Kastor and Traschen showed that the Rosen and Melvin solutions are essentially the same, with a Wick rotation of coordinates converting the time evolution of a Rosen cosmology into the radial evolution of a Melvin fluxtube.
@ References: Melvin PL(64); Havrdová & Krtouš GRG(07)gq/06 [as limit of C-metric]; Kastor & Traschen CQG(14)-a1312 [in Einstein-Maxwell-dilaton theory]; Kastor & Traschen CQG(15)-a1507 + CQG+(16) [Melvin magnetic fluxtube/cosmology correspondence].

Membrane > s.a. black-hole geometry [membrane paradigm].

Memory Effects > s.a. electromagnetism and matter.
@ Specific systems: Lahini et al PRL(17) + Keim Phy(17) [crumpled sheets].
> Gravitational systems: see Gravitational Memory; gravitational-wave propagation; loop quantum cosmology [quantum memory].

Memristor > see electronic technology.

Menger's Sponge > the 3D analog of a Sierpinski Carpet.

Mereology
* Idea: The part of philosophy and mathematical logic that studies the relationship between parts and the wholes they form.
> Online resources: see Wikipedia page.

Mermin-Wagner Theorem
* Idea: The statement that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2; Also known as Mermin-Wagner-Hohenberg theorem or Coleman theorem.
@ References: Drexler blog(09)nov [and graphene]; Kelbert & Suhov AMP(13)-a1210 [for quantum Gibbs states on 2D graphs]; Kelbert et al BJPStat(14)-a1211 [Lorentzian triangulations with quantum spins], JSP(13) [Gibbs states on Lorentzian triangulations].
> Online resources: see Scholarpedia page; Wikipedia page.

Meromorphic Function > s.a. Argument Principle.
* Idea: A complex function whose only singularities are poles.
@ References: Yang & Yi 04 [uniqueness theory].

Meron > see solutions of gauge theories.

Mersenne Primes > see number theory.

Mesons > see hadrons.

Mesoscopic Systems > see classical-quantum relationship.

Metals > s.a. hydrogen; metamaterials [metal foams]; Mott Transition; solid matter [metallic glass].
* Idea: Crystalline materials that conduct electricity; Their behavior (at least at small temperatures) is normally described by modelling the electrons in them as Fermi liquids, based on the (partially unoccupied) band theory, but there are metals whose properties deviate from the Fermi-liquid phenomenology, possibly as a result of electron-electron interaction and perhaps disorder; Studies of Hg clusters shows that the transition to metallic behavior occurs in the 20–70 atom range.
* Semimetal: A material with a very small overlap between the bottom of the conduction band and the top of the valence band (no band gap).
@ General references: Brack SA(97)dec [metal clusters]; Grimvall et al RMP(12) [lattice instabilities in metallic elements].
@ Metal-insulator transition: Collier et al Sci(97)sep; Jian et al a1703 [interacting Majorana fermion model].
@ Special types and related topics: Kaul Phy(12) [orthogonal metals]; news pw(13)oct [shape-memory metal]; news Phy(16) [evidence for type-II Weyl semimetals].
> Online resources: see Wikipedia page.

Metamaterials

Metaphysics
@ References: Beenakker in(07)phy [Hempel's dilemma, computational point of view].

Metaplectic Group > see group types.

Method of Images > s.a. schrödinger equation.
* Idea: A fictitious source for a field used to solve boundary-value problems; It is placed behind the boundary and, together with the physical sources present in the problem, produces the same field on the boundary as the prescribed boundary conditions in a new, simpler problem; The method was developed for electric charges by Sommerfeld.
@ Images charges in electricity: Roulet & Saint Jean AJP(00)apr; Bonanno et al EJP(11) [method of images in magnetostatics]; de Melo e Souza et al IJMPCS(12)-a1201 [Sommerfeld's method in the calculation of van der Waals forces], AJP(13)may [van der Waals force between an atom and a conducting surface].
@ Other settings: Clifton CQG(14)-a1405 [for cosmological models with discrete masses].

Metric Space

Metric Tensor > s.a. metric matching and perturbations; types of metrics.

Metric-Affine Gravity Theories > s.a. formulations of general relativity; gravity theories; teleparallel gravity; unified theories.
* Idea: Spacetime is a real, oriented 4-manifold equipped with a metric and an affine connection; The simplest possibility is just the Palatini formulation of general relativity, but when the metric and the connection are considered as independent, many more possibilities arise; Non-metricity and torsion can appear as field strengths, in addition to curvature; Motivated by expected changes in gravity at high energies.
@ Mathematical aspects: Kleyn a0803; > s.a. torsion.
@ General references: Gronwald IJMPD(97) [rev]; Tapia & Ujevic CQG(98)gq/06; Scipioni gq/99; Godina et al JGP(01)gq/00 [and Nester-Witten 2-form]; Mignani & Scipioni GRG(01)gq/00; Nester et al gq/00-MG9 [energy-momentum]; Heinicke et al PRD(05)gq [and Einstein-ether theory]; Cacciatori et al JGP(06)ht/05 [3D, Chern-Simons form]; Sobreiro & Vasquez Otoya BJP(10)-a0711 [relationship with Riemann-Cartan structure]; Kiriushcheva & Kuzmin EPJC(10)-a0912 [Hamiltonian, and gauge invariance]; Vitagliano et al AP(11)-a1008 [dynamics]; Vitagliano CQG(14)-a1308 [role of non-metricity].
@ With matter: Karpelson gq/01 [matter as curvature and torsion]; Capozziello & Vignolo IJGMP(11)-a1003 [Klein-Gordon field, Cauchy problem]; Fatibene et al IJGMP(10) [matter coupled directly to the connection].
@ Higher-order: Cotsakis et al JMP(99)gq/97; Sotiriou & Liberati AP(07)gq/06, JPCS(07)gq/06; Magnano a1601-conf [Legendre transformation method].
@ Other variations: Haghani et al JCAP(12)-a1202 [with Weitzenböck condition]; Vazirian et al AHEP-a1310 [Weyl-invariant extension]; Bambi et al PRD(15)-a1507 [3D]; Golovnev et al PRD(16)-a1509 [effectively non-local metric-affine gravity]; Pasic & Barakovic AHEP(15)-a1509 [with Yang-Mills action for the affine connection, "Yang-Mielke theory"].
@ Solutions: Socorro et al PLA(98)gq [multipoles]; Hehl & Macías IJMPD(99)gq [rev]; Baekler & Hehl IJMPD(06)gq [Kerr-de Sitter black holes]; > s.a. reissner-nordström solutions; FLRW spacetimes.
@ Phenomenology: King & Vassiliev CQG(01)gq/00 [torsion waves, neutrinos]; Solanki et al PRD(04) [constraints from solar observations]; Kleyn gq/04 [tidal forces]; Puetzfeld eConf(04)ap/05 [cosmology]; Cheng et al PRD(05)gq [radiation transport].
@ Quantized: Kalmykov CQG(97); Mielke & Rincón Maggiolo GRG(03) [BRST]; > s.a. approaches to canonical quantum gravity; phenomenology of gravity.
> Related topics: see conservation laws; metric matching; Palatini Formulation; Riemann-Cartan Geometry; spherical general relativity; torsion in physics.

Metrizable Manifold > see manifold types.

Metrology > see Measurement.

Metropolis Algorithm > see Monte Carlo Method.

Michel's Theorem > see symmetry breaking.

Michelson-Morley Experiment > s.a. Ether.
* Idea: An interferometer experiment that tested the universality of the speed of light by comparing light beams moving in different directions and observed in reference frames with different velocities; It observed no variation in the measured speed of light, and the results eventually (not right away!) led to the abandonment of the ether concept.
@ General references: Michelson & Morley AJS(1887); Shankland et al RMP(55) [status]; Holton Isis(69); van Dongen AHES(09)-a0908 [and Einstein].
@ Modern version: Müller et al PRL(03); Consoli phy/05; Müller et al PRL(07)-a0706 + news pw(07)jun [10–16 limits on violations]; > s.a. tests of lorentz invariance [including tests with electrons].
@ Interpretation: Lämmerzahl & Haugan PLA(01)gq; Consoli & Costanzo ap/03 [preferred frame]; Perez PRP-a1004 [new aether theory]; Langangen et al a1102 [quantum-optics framework].

Mickelsson-Faddeev Algebra
@ References: Larsson mp/05 [lack of unitary representations].

Microcanonical Ensemble > see modified thermodynamics; states in statistical mechanics; quantum statistical mechanics.

Microcausality > see causality in quantum field theory.

MicroSCOPE (Micro-Satellite à trainée Compensée pour l'Observation du Principe d'Équivalence) > see tests of the equivalence principle.

Microscopes

Microstate > s.a. Boltzmann Principle; Macrostate; Tropical Mathematics; types of entropy.
* Idea: A (pure, classical) microstate is a full specification of the phase-space variables for a physical system at some time t, e.g., \(({\bf r}_i,\,{\bf p}_i),\ i = 1, ..., N\) for the particles in a gas.
> For black holes: see 2D and 3D black holes; origin of black-hole entropy; quantum black holes.
> Other gravity-related: see gravitational thermodynamics.
> Online resources: see Wikipedia page.

Microsuperspace > an even more restricted type of Minisuperspace.

Microwave Radiation > s.a. CMB; contents of the universe; observational cosmology.

Midisuperspace > see models in canonical general relativity; models in canonical quantum gravity.

Millennium Problems > see mathematics.

Millikan's Oil Drop Experiment > see physics experiments.

Milne Universe > see minkowski space.

Mimetic Gravity > s.a. unimodular gravity.
* Idea: A conformal extension of Einstein's theory of general relativity proposed by Chamseddine & Mukhanov; The conformal factor of the metric becomes dynamical even in the absence of matter, and the field equations differ from the Einstein equation by the appearance of an extra mode of the gravitational field which mimics the behavior of an imperfect-fluid-like cold dark matter.
@ General references: Chamseddine & Mukhanov JHEP(13)-a1308; Golovnev PLB(14); Malaeb PRD(15)-a1404 [Hamiltonian formulation]; Deruelle & Rua JCAP(14)-a1407 [and disformal transformations]; Momeni et al IJGMP(14)-a1407 [modified]; Hammer & Vikman a1512; Sebastiani et al AHEP(17)-a1612 [rev, cosmology and astrophysics].
@ Solutions, phenomenology: Momeni et al EPJC(16)-a1505 [cylindrical, cosmic strings]; Myrzakulov et al CQG(16)-a1510 [static spherically symmetric]; Astashenok & Odintsov PRD(16)-a1512 [neutron stars and quark stars]; Vagnozzi CQG-a1708 [MOND-like acceleration law].

Mind > s.a. philosophy [mind-body problem].

Minicharged Particles > see particle physics [beyond the standard model].

MiniBooNE Experiment > see neutrinos; neutrino oscillations.

Minimal Length
* Motivation: Ultraviolet cutoff and granularity of spacetime from quantum gravity, and related qantum uncertainties in spacetime measurements.
* Consequences: Modified particle dispersion relations; Modified commutation relations and uncertainty relations; Possible Lorentz-symmetry breaking.
@ General references: Chang et al IJMPA(16)-a1602-proc [and gauge invariance].
@ Origin: Moniruzzaman & Faruque a1602 [from atomic and nuclear physics]; > s.a. quantum-gravity phenomenology; quantum spacetime.
@ And specific systems, phenomenology: Rossi et al PRD(16)-a1606 [probe from measurements on a harmonic oscillator]; Louzada & Belich IJMPA(17)-a1706 [corrections to the Stark effect].
> Related topics: see deformation quantization; deformed uncertainty relations.

Minimal Surface > see extrinsic curvature.

Minimization > see Simulated Annealing; statistics in physics.

Minimum Length > same as Minimal Length.

Minisuperspace > s.a. lagrangian systems [symmetric variations].

Minkowski Inequality > see inequalities.

Minkowski Spacetime

Minkowski Tensor > see energy-momentum tensor.

Minkowski Sum of Polygons > see euclidean geometry.

Mirrors

Mirror Manifold
@ References: Greene et al CMP(95) [higher dimensions].

Mirror Matter > see matter; universe contents.

Mirror Symmetry > s.a. Homogeneous Space; lie algebra; lie group.
@ References: Alim a1207-ln [and topological string theory]; Quigley ch(15)-a1412 [introduction, and conformal field theory].

Misner Metric
* Idea: A spatial metric representing two black holes.

Misner Space
* Idea: A simplified 2D model of the 4D Taub-NUT space, that reproduces some of its pathological behaviors; It is a 2D space with topology \(\mathbb R\) × S1, in which the light cones progressively tilt as one moves forward in time, and has closed timelike curves after a certain point.
@ References: Margalef-Bentabol & Villaseñor GRG(14)-a1406 [topology and g-boundary].

Mixed State in Quantum Physics

Mixing System > s.a. chaos; ergodic system; group action; quantum chaos; statistical mechanics [approach to equilibrium].
$ Def: A dynamical system (X, μ, φ) such that, for all measurable A and B, μ(φn AB) → μ(A) μ(B) as n → ∞; In other words, for all functions f and g, as T → ∞ the correlator (\(\langle\)...\(\rangle\) is phase space average)

R(f, g; T):= \(\langle\) f(Tz) g(z)\(\rangle\) – \(\langle\) f(z)\(\rangle\) \(\langle\)g(z)\(\rangle\) → 0 .

* Comments: It implies ergodicity; The notion is not invariant under t-reparametrizations [@ in Motter PRL(03)gq]; > A similar concept is Topological Transitivity.
* Examples: The baker's transformation; The Lorenz system.
* Decay time: As a rule, R ~ R0 exp{–t/τc}, where τc is the mixing or correlator decay time, and τc ~ 1/h0, with h0 the sum of the positive Lyapunov exponents (hard to prove); > s.a. graphs in physics [quantum walk].
@ General references: in Zaslavsky et al 91; Antoniou & Tasaki IJQC(93) [spectral decompositions]; Kandrup MNRAS(98)ap; Chernov & Zhang Nonlin(05)mp/04 [slow mixing systems].
@ Quantum systems: Richter PRA(07) [quantum speedup of mixing]; Zhang et al PRE(16)-a1601 [and ergodicity].
> Online resources: see Wikipedia pages on mathematical notion and physics notion.

Mixmaster Universe > see bianchi IX models.

Möbius Transformation / Group > see Complex Numbers; examples of lie groups [SL(2, \(\mathbb C\))].

Modal Interpretation > see interpretations of quantum theory.

Modal Quantum Theory
* Idea: A "toy model" of quantum theory in which amplitudes are elements of a general field; The theory predicts not the probabilities of a measurement result, but only whether or not a result is possible.
@ References: Schumacher & Westmoreland FP(12), a1204-ch [rev].

Model Theory > s.a. logic.
* Idea: The study of mathematical structures in terms of their first-order definability; Some branches are classical model theory, model theory applied to groups and fields, geometric model theory, and computable model theory (which can also be viewed as an independent subfield of logic).
@ References: Prest 88; Chang & Keisler 90; Poizat 00 [r BAMS(02)]; Prestel & Delzell 11 [IIb intro]; Tent & Ziegler 12; Marker 16 [infinitary].
> Online resources: see Wikipedia page.

Models in Physics > s.a. physics; Simulations of Physical Systems; theory.
* Idea: In a hierarchical view of a physical theory, with the higher levels being paradigms, models are the lower, more specific levels.
* Idealizations: Models for physical situations involve idealizations, simplifying assumptions that we know to be literally false; > s.a. Approximations.
@ References: Giere PhSc(04)dec [representing reality]; Emch SHPMP(07) [and theory building]; Pincock PhSc(07)dec [mathematical idealizations]; Davey PhSc(11)jan [contextualism and beliefs from idealizations]; issue SHPSA(11)#2; Bueno et al SHPMP(12) [phenomenological and partial models, and superconductivity]; Borrelli SHPMP(12) [model-building "beyond the Standard Model" and the composite Higgs particle].
> Related topics: see Multiscale Physics.

Modified Gravity (Hu-Sawicki Scenario) > s.a. cosmological models; phenomenology of higher-order theories; theories of cosmological acceleration.
@ References: Boubekeur et al a1407 [current status, viability].

Modified Gravity (MOG) > s.a. TeVeS.
* Idea: A Scalar-Tensor-Vector modified theory of gravity (STVG) proposed by Moffat and collaborators; In it, gravity is described, in addition to the metric tensor, by two scalar fields G(x) and μ(x), and one vector field \(\phi_{\alpha}(x)\).
@ General references: Moffat & Toth CQG(09)-a0712 [solutions from action principle], MNRAS(09)-a0805 [light bending and lensing], AIP(10)-a0908 [predictions]; Moffat a1410 [constraints, light deflection and lensing].
@ And cosmology: Toth a1011-conf; Moffat & Toth Gal(13)-a1104; Khurshudyan et al IJTP(14)-a1403 [time variation of the gravitational and cosmological constants]; Moffat a1409 [structure growth and the cmb]; Roshan EPJC(15)-a1508 [exact cosmological solutions]; Moffat a1510 [evidence from cmb and structure growth data]; Jamali & Roshan EPJC(16)-a1608 [phase-space analysis].
@ And galactic dynamics: Haghi & Rahvar IJTP(10)-a1002 [constraints using the Magellanic System]; Moffat & Toth a1005 [and the Bullet Cluster]; Moffat & Toth a1103 [Tully-Fisher relationship for gas-rich galaxies, MOG and MOND]; Mishra & Singh PRD(13)-a1108 [galaxy rotation curves, quadrupole gravitational polarization]; Suzuki a1202 [rotation curves and estimation of ξ parameter]; De Martino & De Laurentis PLB-a1705 [weak-field approximation and universality]; > s.a. galaxies.
@ Other phenomenology: Moffat et al a1204 [lensing]; Roshan & Abbassi PRD(14)-a1407 [Jeans instability analysis]; Moffat EPJC(15)-a1412 [black holes]; Moffat PLB(16)-a1603 [binary systems and gravitational waves]; López & Romero a1611 [black holes and jets].

Modular Arithmetic, Equation, Function > see Arithmetic; elementary algebra.

Modular Forms > s.a. number theory; quantum field theory formalism.

Modular Group > see diffeomorphisms.

Modulation Spaces > see deformation quantization.

Module over an Operad
@ Introduction: Markl ag/97 [and examples].

Module over a Ring R > s.a. types of modules.

Moduli Space > s.a. 2D manifolds [Riemann surface].
* Idea: The space of invariant parameters which characterize an object in a category, i.e., the set of equivalence classes of structures; In general, they are singular spaces and can be described as stratified manifolds.
* In physics: Used in gauge theory for connections modulo gauge transformations (applied to instantons, monopoles, duality), and in string theory for 2D conformal metrics (conformal field theory, M-theory).
@ In physics: Nelson PRP(87) [string theory]; Hitchin in(90) [geometry and topology]; Tsou ht/00-proc; > s.a. connections; yang-mills gauge theory.

Modus Ponens, Tollens > see logic.

MOG > see Modified Gravity.

Mogami (Pseudo)Manifolds > see Triangulations.

Molecular Chaos > s.a. H Theorem.
* Idea: The assumption that the velocities of colliding particles in a gas are uncorrelated, and independent of position.
> Online resources: see Wikipedia page.

Molecular Physics

Moment of Inertia > s.a. Elasticity; fluid; rotation.
* Idea: The momet of inertia for an object characterized by a mass density ρ, with dm = ρ(x) dV, is the symmetric tensor

Iij:= dm (r2 δijx i x j) ,

closely related to the quadrupole moment (or 2nd moments) of the mass distribution (> see multipoles).
* Eigenvectors and eigenvalues: The eigenvectors of the moment of inertia tensor of an object define the principal axes of inertia for that object, and the corresponding eigenvalues are the moments of inertia for rotations around those axes.
@ References: Lawton & Noakes JMP(01) [computation]; Díaz et al EJP(06) [for solids of revolution]; Hong & Hong TPT(13)mar [for uniform disks and spheres, without integration]; Rizcallah TPT(15) [using dimensional analysis end elementary differentiation techniques].

Moments of a Distribution > see multipoles [physics notion, for fields] and probability [mathematics notion].

Momentum

Momentum Map > s.a symplectic manifold.
* Idea: A concept introduced in 1965 by Kostant and Souriau which allows one to describe the conserved quantities associated to the symmetries of a given dynamical system, and has been crucial for the development of the theory of reduction (of a dynamical system to a smaller one by dividing out the symmetries).
@ References: Weinstein m.SG/02-conf [momentum maps]; in Ortega & Ratiu RPMP(06) [generalization, cylinder-valued]; Esposito PhD-a1203 [and the theory of reduction]; Esposito & Nest JGP-a1208 [uniqueness].
> Online resources: see PlanetMath page; Wikipedia page.

MOND (Modified Newtonian Dynamics) > s.a. MOND and astrophysics / cosmology.

Monge Metric on the Sphere > see sphere.

Monge-Ampère Equation > s.a. symplectic structures.
* Idea: A special type of non-linear second-order partial differential equation.
> Online resources: see Wikipedia page.

Monodromy, Monodromy Matrix
* Idea: Monodromy is just a name for what you get when you integrate something around a loop (nearly the same as the concept of holonomy); For example, when studying the stability of an orbiting object, you can see what effect a small displacement would have after one whole period by linearizing its equation of motion and solving the resulting linear equation; To solve this you need to do an integral over one period of its orbit; The final displacement will depend on the initial displacement in a linear way, so the answer is neatly encapsulated in something called the "monodromy matrix" [from this page].

Monoid > s.a. Semigroup.
$ Def: A pair (X, \(\circ\)), X a set, \(\circ\) a composition X × XX, associative and with an identity.
* Idea: A structure which is almost a group, but with no inverses; The same as a semigroup with identity.
* Types: It is cancellative if a + c = b + c implies a = b.

Monomorphism > see category.

Monopoles

Monotonic Sequence > see sequences.

Monster Group > see finite groups.

Monsters > s.a. gravitational thermodynamics.
* Idea: Objects with finite ADM mass and surface area, but potentially unbounded entropy.

Monstrous Moonshine > see finite groups.

Monte Carlo Method > s.a. computational physics; integration; markov process; Simulated Annealing.
* Idea: A statistical method used to calculate quantities that are too difficult to compute analytically, in which one generates random events in a computer; Versions are the random walk (Metropolis) and the Hamiltonian ones.
* Markov Chain Monte Carlo method: Different random configurations of a system are generated by small variations as in a Markov chain (for example, a random walk), and are then given a probability of being accepted; Two versions are the Metropolis algorithm and Hamiltonian Monte Carlo.
* Metropolis algorithm: A version of the MCMC method which applies to a thermal system, for which the probability of acceptance depends on the temperature; The algorithm fails in systems on the verge of a phase transition.
@ Texts and reviews: Binder & Heermann 92; Jadach phy/99 [guide]; Newman & Barkema 99, Krauth 06 [in statistical physics].
@ General references: Kosztin et al AJP(96)may [diffusion method for minima]; Binder RPP(97) [in statistical physics]; Doye & Wales PRL(98) [optimization and thermodynamics], CPC(00)phy/99 [self-adapting simplicial grid]; Landau et al AJP(04)oct [Wang-Landau sampling in statistical mechanics]; Kendall et al 05; Hajian PRD(07)ap/06 [Hamiltonian version, and cosmology]; Ambegaokar & Troyer AJP(10)feb [error estimation]; During & Kurchan EPL(10)-a1004 [statistical mechanics of Monte Carlo sampling].
@ Markov Chain Monte Carlo method: Ottosen a1206 [rev]; Alexandru et al PRL(16)-a1605 [real-time dynamics on the lattice using the Schwinger-Keldysh formalism]; Betancourt a1706 [history].
@ Metropolis algorithm: Bhanot RPP(88); Berg PRL(03) [for rugged dynamical variables]; > s.a. path integrals.
@ Other algorithms: Suwa & Todo PRL(10)-a1007 [without detailed balance]; Jansen et al JPCS(13)-a1211, CPC(14)-a1302 [quasi-Monte Carlo method, and lattice field theories]; Herdeiro & Doyon PRE(16)-a1605 [method for critical systems in infinite volume]; > s.a. Glauber Dynamics.
@ Quantum Monte Carlo: Suzuki ed-93 [condensed matter]; Rombouts et al PRL(06) [new updating scheme]; Anderson 07; Pollet et al JCP(07) [optimality]; Temme et al Nat(11)mar-a0911 [sampling from Gibbs distribution]; Destainville et al PRL(10); Fantoni & Moroni JChemP(14)-a1408 [for quantum Gibbs ensemble]; Zen et al PRB(16)-a1605 [improved accuracy and speed]; Gubernatis et al 16 [pedagogical overview]; Becca & Sorella 17 [for correlated systems].
@ For fermions: Corney & Drummond PRL(04)qp, PRB(06)cm/04; Assaraf et al JPA(07).
@ Other systems: Janke PhyA(98) [disordered systems]; Talbot et al JPA(03) [exact results for simple harmonic oscillator]; Lahbabi & Legoll JSP(13) [multiscale systems in time]; Pavlovsky et al a1410-conf [path integral for relativistic quantum systems]; Silva Fernandes & Fartaria AJP(15)sep [gas-liquid coexistence].
> Other systems: see black-hole formation; Chemical Potential; composite systems; diffusion; lattice field theory; Mean-Field Method; observational cosmology; schrödinger equation.
> Online resources: see Wikipedia page.

Montevideo Interpretation > see interpretations of quantum theory.

Moon > see earth and its moon.

Moonshine > see finite groups.

Mordell Conjecture > see conjectures.

Morita Equivalence > see Star Product; non-commutative gauge theory.

Morphism > see category.

Morse Theory

Mosaic > a name sometimes used for a tiling.

Mössbauer Effect > see radiation.

Motion > s.a. Dynamics; Kinematics.
@ References: Rynasiewicz PhSc(00)mar [absolute vs relative].
> Specific types of systems: see gauge theories of gravity; geodesics; orbits of gravitating objects; Test Body.

Motives
* And quantum field theory: The main result is that all quantum field theories share a common universal symmetry realized as a motivic Galois group, whose action is dictated by the divergences and generalizes that of the renormalization group.
@ General references: Rej & Marcolli a0907-proc [introductory survey for physicists]; Marcolli 10.
@ And quantum field theory: Connes & Marcolli JGP(06) [perturbative renormalization and motivic Galois theory]; Bloch et al CMP(06); Ceyhan & Marcolli CMP(12)-a1012 [in configuration space, and renormalization].
> Applications: see Potts Model.
> Online resources: see Wikipedia page.

Mott Insulators > see Hubbard Model; Insulators.

Mott Transition
* Idea: A metal-nonmetal transition in condensed-matter physics.
@ References: Bottesi & Zemba AP(11) [in a 2D lattice spinless fermion model].
> Online resources: see Wikipedia page.

Moufang Loop / Transformations > s.a. lie algebra; noether theorem; types of gauge theories.
* Idea: A non-associative generalization of a (Lie) group; An example are octonions with norm one.
@ References: Vojtechovsky EJC(06) [up to order 64].

Moving Frame on a Manifold > see tetrads; vector fields.

Moyal Algebra / Bracket / Deformation > see algebra; poisson brackets; deformation quantization; supersymmetry.

Moyal Plane > see non-commutative geometry.

MSSM (Minimal Supersymmetric Standard Model) > see supersymmetric theories.

Multi-Fingered Nature of Time > see canonical general relativity; time in gravity.

Multifractal > see fractals in physics.

Multigravity > see bimetric and multimetric theories of gravity.

Multimessenger Astronomy > see astronomy.

Multinomial Coefficient > s.a. partitions.
* Idea: The multinomial coefficient C(n; n1, n2, ..., nk) is the number of distinct ways in which a set of n elements can be partitioned into subsets of cardinalities n1, n2, ..., nk; Obviously those numbers must satisfy n1 + n2 + ... + nk = n; If some of the integers are zero, the value of C is the same as that of the coefficient with the zeroes omitted; It is given by

C(n; n1, n2, ..., nk) = n!/(n1! n2! ... nk!) = (n1, n2, ..., nk)! ,

and gets its name from the fact that it appears as a coefficient in the expansion

(x1 + x2 + ... + xk)n = ∑partitions of n C(n; n1, n2, ..., nk) x1n1 x2n2 ... xknk .

* Properties: It is obviously a generalization of the binomial coefficients, C(n; p, np) = \(n \choose p\).
> Online resources: see MathWorld page.

Multiplication Structure on a Manifold > see manifold.

Multiply Connected Space > see connectedness.

Multipole Moments > s.a. spherical harmonics.

Multiscale Physics > s.a. Monte Carlo Methods [multiscale in time]; renormalization; Scaling.
* Idea: Systems described at different scales by different sets of variables, obeying different laws; Going from a small-scale description to a larger-scale one usually requires a huge reduction of dimensionality of configuration or phase space, and in a given system the correspondence may or may not be understood.
@ General references: Mei & Vernescu 10 [mathematics of homogenization methods]; Weinan E 11 [r PT(12)jun]; Ghosh 11 [using the Voronoi-cell finite element method, for materials with complex microstructures]; Butterfield IF(14)-a1406 [laws at different levels]; Atay et al DNC(16)-a1606 [common framework].
@ Related topics: Rudykh et al PRS(14) [multiscale instabilities in elastomers]; Perryman & PRS(14) [non-obvious thresholds in multi-scale systems in time, and singular perturbation theory].
> Gravitational and astrophysical systems: see fractals in physics [multiscale gravity, spacetime]; galaxy distribution [cosmic web].
> Other related topics: see computational physics [Brownian motion]; descriptions of matter; Models in Physics.

Multisymplectic Structure > see symplectic structures.

Multiverse

Muon > see particle types.

Murphy's Law
* Idea: Anything that can go wrong, will.
* Examples: Buttered toast falling off a table will fall buttered side down (based on gravity); If odd socks can be created, they will be (based on combinatorics).
@ References: Held & Yodzis GRG(81); Matthews EJP(95), SA(97)apr; news tel(01)mar [test].
> Online resources: see Wikipedia page.

Music

Mutual Information > s.a. information.
* Idea: In general, a quantity that quantifies the determinism that exists in a relationship between random variables; A measure of the shared information or correlation between the two regions; For disjoint spacetime regions V and W,

I(V,W):= S(V) + S(W) – S(VW) .

* Properties: It satisfies the inequality I(V,W) ≤ 2 min(S(V), S(W)), and can be thought of as a "point-splitting" regularization of the entropy.
@ References: Casini CQG(07)gq/06; Evans PRS(08) [computationally efficient estimator]; Eisler & Zimboras PRA(14)-a1403 [area-law violation in a non-equilibrium steady state]; Agón et al a1505 [long-distance expansion].
> Related topics: see entropy [mutual Rényi information]; quantum information; spin models; uncertainty relations.


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