**Topics, M**

**Macdonald Polynomials**

@ __References__: Gadde et al CMP(13)-a1211 [and gauge theories];
Blondeau-Fournier et al a1211 [Macdonald superpolynomials];
Morozov & Morozov JHEP(20)-a1907 [generalized].

> __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})
*e* ∧ *e* + (1/*l*) d_{ω}*e* ,

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 *F*_{corr}
= *R* ± (1/*l*^{ 2}) *e*
∧ *e*, and *l*^{ 2} = 3/Λ, then

*S*_{corr}
= # ∫ tr(*F*_{corr}
∧ *F*_{corr}) ;

Notice that at the action level, the Λ = 0 case gives just *R* ∧ *R*,
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 CQG(17)-a1703 [covariant Hamiltonian formulation].

**Machine Learning**
> s.a. neural networks.

* __Idea__: An application
of artificial intelligence (AI) in which systems learn automatically
from experience and improve without being explicitly programmed.

@ __General references__:
Vanchurin ML:ST-a2004 [general theory];
Hosni & Vulpiani a2005
[reflections on complexity, data and models];
Roberts a2104 [and physics].

@ __Quantum machine learning__:
Poland et al a2004 [limits, generalized No Free Lunch theorem];
Blance & Spannowsky JHEP(21)-a2010 [for particle physics].

@ __And quantum theory__: Wang et al SciB(19)-a1901;
Das Sarma et al PT(19)mar; Grimmer et al a1910 [local measurements in quantum field theory];
Han & Hartnoll PRX(20) [matrix quantum mechanics];
Carrasquilla AiP(20)-a2003 [quantum matter, many-body physics];
Bharti et al a2003 [quantum foundations];
Bachtis et al PRD(21)-a2102 [quantum field theory].

@ __And other physics__:
Carrasquilla & Melko nPhys(17) [phases of matter];
Singh et al AJUR(19)-a1810 [lensed gravitational waves];
Shiba Funai & Giataganas PRR(20)-a1810 [Ising spin systems];
Bartók et al PRX(18) [predicting material properties];
Wei & Huerta PLB(20)-a1901 [denoising gravitational wave data from binary mergers];
Carleo et al RMP(19)-a1903 [and the physical sciences];
He & Kim a1905 [learning algebraic structures];
Caron et al EPJC(19)-a1905 [constraining the parameters of physical models];
Arjona & Nesseris PRD(20)-a1910 [cosmology, background expansion of the universe];
Betzler & Krippendorf FdP(20)-a2002 [dualities];
Bachtis et al PRR(21)-a2010 [within Hamiltonians in statistical physics];
Bao et al a2103 [using Hilbert series].

@ __Other applications__:
Middleton PT(19)feb [image restoration];
He a2101 [mathematical structures];
> s.a. QCD effects [confinement].

> __Online resources__: see
Wikipedia page.

**MACHOs**
> see types of dark matter.

**Macrorealism**
> see realism.

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

@ __References__: Wang a1912
[evolution of the concepts of "macroscopic" and "microscopic"]

**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** > s.a. Quasiparticles.

* __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];
Berk & Schmidt Phys(20) [strongly coupling to phonons].

**Magnus Effect**
> s.a. turbulence.

* __Idea__: The effect in which
a spinning object curves away from its flight path, because it feels a force
orthogonal to its velocity and rotation axis; 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].

@ __Gravitational version__: Costa et al PRD(18)-a1805 [spinning black holes and other bodies].

> __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 CMP(17)-a1601 [for Yang-Mills theory on the plane, three proofs],
CMP(17)-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.

**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*/4*E* to go from one state to an orthogonal state.

@ __References__: Margolus & Levitin PhyD(98)qp/97.

> __Online resources__:
see Wikipedia page.

**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 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).

**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 d*P*_{c}/d*t*
= ∑_{c'} (*W*_{c'c}
*P*_{c'} −
*W*_{cc'}
*P*_{c}) in its Markov
version, where *W*_{cc'}
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];
Cresser & Facer a1710 [Markovian, coarse-graining approach].

@ __Applications__: Mendes & Farina qp/06 [atomic energy level corrections].

@ __Special types__: Öttinger a1002 [dissipative, non-linear];
Rivas et al NJP(10) [Markovian, model derivation].

@ __Quantum__: Belavkin TMP(97)qp/05 [irreversible];
Manzano AIPa(20)-a1906 [intro to Lindblad master equation];
Donvil & Muratore-Ginanneschi a2102 [unravelling in quantum trajectories with jumps].

@ __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 PRA(17)-a1610 [spin and spin bath].

> __Related topics__:
see brownian motion; Lindblad Equation.

> __Online resources__:
see Wikipedia page.

**Material Cause**
> see causality.

**Materials**
> see condensed-matter physics.

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

**Mathieu Equation, Function**
> s.a. oscillator.

@ __References__: Frenkel & Portugal JPA(01) [algebraic methods].

**Mathieu Groups**
> see finite groups.

**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],
a1709
[resolved Big Bang in Yang-Mills matrix models];
Klinkhamer a1912 [emergence of an expanding universe].

@ __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) (*A*
⊂ *B*) ∧ (*B* ∈ *I*) implies *A* ∈
*I*; (2) (*A*, *B* ∈ *I*) ∧ (|*A*| <
|*B*|) implies that there exists *b* ∈ *B* \ *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; baryogenesis [matter-antimatter asymmetry].

**Matter-Gravity Entanglement Hypothesis**

* __Idea__: A hypothesis by Bernard Kay,
according to which the entropy of a closed quantum gravitational system is equal to
the system's matter-gravity entanglement entropy; The entropy of a closed system as
a real and objective quantity.

@ __References__: Kay Ent(15)-a1504;
Kay FP(18)-a1802-in [rev].

**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 *g*_{ab} 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 *R*_{abcd}
= [2/(*n*−1)(*n*−2)] *R* *g*_{c[a}
*g*_{b]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(14)-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**
> s.a. entropy; molecular physics [molecular gas].

@ __And gravity__: Roupas CQG-a2003 [gravitatiomnal potential].

**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 (but
named by Thompson, who pointed out its unphysical nature) in a thought experiment about
a possible way to violate the second law of thermodynamics; The demon stands by a small
window in a partition between two parts of a gas container and lets high-speed particles
cross the window only in one direction, low-speed particles cross only in the opposite
direction; 1929, Leo Szilard's paper on entropy and information; Brillouin and Gabor,
on information acquisition by the demon; Landauer (& Bennett), information erasure
feeds entropy into the environment.

@ __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];
Kostic a2001 [and its fallacies].

@ __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 PNAS(17)-a1702;
news sn(17)jul [testing the role of information];
Hu et al a2105 [and non-locality tests].

@ __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,

*f*_{MB}(*v*)
= 4π*n*(*m*/2π*kT*)^{3/2}
*v*^{2}
exp{−*mv*^{2}/*kT*},
*f*_{MB}(*p*)
= (1/2π*mkT*)^{3/2}
exp{−*p*^{2}/2*mkT*},

where *T* is defined by *U* = (3/2) *NkT*; It 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 / Theory**
> 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];
Hartenstein & Hubert BJPS(18)-a1809 [ontological status of fields].

**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 LMP(17)-a1508.

**Mayer-Vietoris Sequence / Theorem**
> s.a. ER = EPR Conjecture.

* __Idea__: It can be regarded as
a generalization of the finite-set formula card(*A* ∪ *B*)
= card *A* + card *B* − card(*A* ∩ *B*).

**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 d*β*/d*t* = 0,
and Schwarzschild-de Sitter for d^{2}*β*
/ d*t*^{2} = 0, or d*H*/d*t* = 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 CQG(17)-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 EPJC(17)-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];
Jain et al a1802.

@ __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;
Rouffort a1809 [validity];
Rao & Chavda a2012 [wave function for interacting bosons];
> s.a. Hartree Equation.

@

@

>

>

>

**Mean Free Path**
> see scattering.

**Meander**
> see molecular physics [polymers].

**Meaning**
> see mathematics [conceptual aspects].

@ __In mathematics and physics__: Polkinghorne 11;
Sánchez a1604-FQXi;
Rovelli a1611 [meaningful information],
Krzanowski a2004.

**Measurement**
> s.a. experiments in physics; types
of quantum measurements and effects; units.

@ __General references__: Resende a2102 [abstract theory];
Theurel a2104
[in classical Hamiltonian mechanics, and disturbance].

@ __Metrology__: Crease PT(09)dec [Peirce and the first absolute measurement standard];
news sn(19)may [history, milestones].

@ __In quantum theory__: Harrison a2104 [general theory];
> s.a. measurement in quantum physics.

@ __Quantum metrology__: Giovannetti et al nPhot(11)-a1102 [reducing the statistical error by repeating the measurement];
Kohlrus et al PRA(19)-a1811 [Earth satellite];
Du & Mann a2012
[relativistic quantum metrology of temperature in (A)dS using Fisher information].

@ __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.

**Melting**
> see phase transitions.

**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];
Jokela et al PRD(19)-a1903 [Yang-Mills theory].

> __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.

**MERA (Multi-scale Entanglement Renormalization Ansatz)**
> see entanglement phenomenology.

**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 PRB(17)-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];
news sn(18)jan [ultrathin 2D metals];
news cosmos(19)jan [nickel-based porous metal as strong as titanium but four to five times lighter].

> __Online resources__:
see Wikipedia page.

**Metaparticles**
> see particle models.

**Metaphysics**

@ __References__: Beenakker in(07)phy [Hempel's dilemma, computational point of view];
Arroyo & Arenhart a2012 [relationship with physics].

**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];
Alshal et al a1808
[2D, Kelvin and Sommerfeld image methods].

@ __Other settings__: Clifton CQG(14)-a1405
[for cosmological models with discrete masses].

**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; It is motivated by expected changes in gravity at high energies.

@ __General references__:
Gronwald IJMPD(97) [rev];
Tapia & Ujevic CQG(98)gq/06;
Scipioni gq/99;
Mignani & Scipioni GRG(01)gq/00;
Nester et al gq/00-MG9 [energy-momentum];
Heinicke et al PRD(05)gq [and Einstein-aether 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];
Afonso et al PRD(19)-a1810 [and general relativity];
Chen & Chiou a2008 [Riemann curvature and Cartan torsion].

@ __Formal aspects__: Godina et al JGP(01)gq/00 [and Nester-Witten 2-form];
Kleyn a0803;
Gaset & Román-Roy JGM(19)-a1804 [multisymplectic approach];
> s.a. torsion.

@ __With torsion and non-metricity__:
Bahamonde & Gigante a2006;
Ariwahjoedi et al a2012 [3+1 formulation].

@ __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];
Jiménez & Delhom EPJC(19)-a1901 [ghosts].

@ __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"];
García-Parrado & Minguzzi a2006 [vanishing shear hypermomentum and amplified symmetry].

@ __Solutions__: Socorro et al PLA(98)gq [multipoles];
Hehl & Macías IJMPD(99)gq [rev];
Baekler & Hehl IJMPD(06)gq [Kerr-de Sitter black holes];
Hohmann Symm(20)-a1912 [spherically symmetric];
> 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];
Latorre et al PLB(18)-a1709 [4-fermion contact interactions];
Bahamonde & Valcarcel a2103
[constraints from classical gravitational tests].

@ __Quantized__: Kalmykov CQG(97);
Mielke & Rincón Maggiolo GRG(03) [BRST];
Ghosh PDU(19)-a1910;
Percacci IJGMP-a2003-conf [rev];
> 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 montecarlo 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.

**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.

**Millicharged Matter**
> see charge; types of dark matter.

**Millikan's Oil Drop Experiment**
> see physics experiments.

**Milne Universe**
> see minkowski space.

**Mimetic Gravity**
> s.a. massive gravity; 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];
Hammer & Vikman a1512;
Sebastiani et al AHEP(17)-a1612 [rev, cosmology and astrophysics];
Izaurieta et al a2007 [and Einstein-Cartan-Kibble-Sciama gravity];
de Cesare & Husain PRD(20)-a2009 [canonical formalism].

@ __Modified versions__: Momeni et al IJGMP(14)-a1407;
Gorji et al JCAP(18)-a1709,
Zheng JHEP(21)-a1810 [higher-derivative];
Bodendorfer et al CQG(18)-a1806 [with limiting curvature];
Ganz et al PRD(19)-a1812 [mimetic scalar gravity, Hamiltonian analysis];
Sheykhi JHEP(21)-a2009 [2+1 dimensions].

@ __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(17)-a1708 [MOND-like acceleration law];
Dutta et al JCAP(18)-a1711 [cosmology];
Brahma et al PLB(18)-a1803 [singularity resolution];
De Cesare Univ(19)-a1904 [reconstruction procedure];
Khalifeh et al PDU(20)-a1907 [mimetic dark matter].

**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 quantum 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];
Maziashvili & Silagadze a1812 [modified position and momentum operators].

@ __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];
> s.a. conservation laws [continuity equation].

> __Related topics__:
see deformation quantization;
deformed uncertainty relations.

**Minimal Surface**
> see extrinsic curvature.

**Minimally Modified Gravity**
> see modifications of general relativity.

**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 Tensor**
> see energy-momentum tensor.

**Minkowski Sum of Polygons**
> see euclidean geometry.

**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
\(\mathbb R\) × S^{1}, 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*(*φ*^{n }*A*
∩ *B*) → *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* ~ *R*_{0}
exp{−*t*/*τ*_{c}},
where *τ*_{c} is the mixing
or correlator decay time, and *τ*_{c} ~
1/*h*_{0},
with *h*_{0}
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.

@ __Types of systems__: Chernov & Zhang Nonlin(05)mp/04 [slow mixing systems];
Kuksin a1901 [perturbed non-linear parabolic pde].

@ __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 [ug intro];
Tent & Ziegler 12;
Marker 16 [infinitary];
Kirby 19 [ug intro].

> __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];
Hosni & Vulpiani a2007 [modeling and data science].

> __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];
Moffat a2001 [cosmology, galaxies and galaxy clusters].

@ __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];
Jamali et al JCAP(18)-a1707;
Negrelli et al a2004 [tests];
Moffat JCAP-a2006.

@ __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];
Israel & Moffat Gal(18)-a1606 [Train Wreck Cluster Abell 520 and Bullet Cluster 1E0657-558];
De Martino & De Laurentis PLB(17)-a1705 [weak-field approximation and universality];
Negrelli et al PRD(18)-a1810 [Milky Way rotation curve];
> s.a. galaxies.

@ __Black holes__:
Moffat EPJC(15)-a1412;
López & Romero ApSS(17)-a1611 [and jets];
Pradhan a2007 [vs naked singularities].

@ __Other phenomenology__: Moffat et al a1204,
Rahvar & Moffat MNRAS(18)-a1807 [lensing];
Roshan & Abbassi PRD(14)-a1407 [Jeans instability analysis];
Moffat PLB(16)-a1603 [binary systems and gravitational waves];
Green et al PLB(18)-a1710 [and the event GW170817/GRB170817A].

**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.

**Mole**

$ __Def__:
An amount of matter containing *N*_{A}
= 6.02 214 076 × 10^{23} particles (as
of 2018, this is now a definition).

**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.

**Moment of Inertia**
> s.a. Elasticity; fluid; rotation.

* __Idea__: The moment of inertia
for an object characterized by a mass density *ρ*, with d*m*
= *ρ*(**x**) d*V*, is the symmetric tensor

*I*_{ij}:=
∫ d*m* (*r*^{2}
δ_{ij} − *x*^{ 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 and elementary differentiation techniques].

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

**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* ×
*X* → *X*, 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.

**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. specific areas.

**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.

**Mosaic**
> a name sometimes used for a tiling.

**Mössbauer Effect**
> see radiation.

**Mössbauer Rotor Experiment**

* __Idea__: The transverse Doppler shift in a rotating system.

@ __References__: Corda a1805-GRF [interpretation, and general relativity].

**Motion**
> s.a. Dynamics; Kinematics.

@ __References__:
Rynasiewicz PhSc(00)mar [absolute vs relative];
Nolte 18 [history].

> __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]; Rella Sigma(21)-2009 [motivic Feynman integrals].

> __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; Wigner-Weyl-Moyal Formalism.

**Moyal Equation**
> see wigner functions.

**Moyal Plane**
> see non-commutative geometry.

**Mpemba Effect**
> see water.

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

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

**Multifractal Theories / Spacetime **
> see fractals in physics.

**Multigravity**
> see bimetric and multimetric theories of gravity.

**Multimessenger Astronomy / Astrophysics**
> see astronomy.

**Multinomial Coefficient**
> s.a. partitions.

* __Idea__: The multinomial
coefficient *C*(*n*; *n*_{1},
*n*_{2}, ..., *n*_{k})
is the number of distinct ways in which a set of *n* elements can be partitioned
into subsets of cardinalities *n*_{1},
*n*_{2}, ..., *n _{k}*;
Obviously those numbers must satisfy

*C*(*n*; *n*_{1},
*n*_{2}, ...,
*n*_{k})
= *n*!/(*n*_{1}!
*n*_{2}! ...
*n*_{k}!)
= (*n*_{1}, *n*_{2},
..., *n*_{k})! ,

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

(*x*_{1} + *x*_{2} + ...
+ *x*_{k})^{n}
= ∑_{partitions of n} *C*(*n*;
*n*_{1}, *n*_{2},
..., *n*_{k})
*x*_{1}^{n1}
*x*_{2}^{n2} ...
*x*_{ k}^{nk} .

* __Properties__:
It is obviously a generalization of the binomial coefficients, *C*(*n*;
*p*, *n*−*p*) = \(n \choose p\).

> __Online resources__:
see MathWorld page.

**Multiplication**
> see Arithmetic.

**Multiplication Structure on a Manifold**
> see manifold.

**Multiply Connected Space**
> see connectedness.

**Multipole Moments**
> s.a. spherical harmonics.

**Multiscale Physics**
> s.a. montecarlo 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];
Schieber & Hütter PT(20)mar [in non-equilibrium systems].

> __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; quantum field theory techniques.

**Multisymplectic Structure**
> see types of symplectic structures.

**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.

**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*(*V* ∪ *W*) .

* __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];
Katsinis & Pastras a1907 [area law at finite temperature].

> __Related topics__:
see entropy [mutual Rényi information];
quantum information; spin models;
uncertainty relations.

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

send feedback and suggestions to bombelli at olemiss.edu – modified 31 may 2021