Topics, M

M-Theory

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 = + (1/) e that combines an actual connection valued in the Lie algebra of a group G with a cotetrad e; Used for gravity with a cosmological constant; For > 0 the group G = SO(4,1), while for < 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: 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); Has also been considered as a perturbed BF theory [& Freidel, Smolin & Starodubtsev]; Can be generalized to other Cartan geometries with dim(G/H) = 4.
* Important quantities: The connection A = + (1/) e has values in the vector space so(3,1) + R3,1. Its curvature can be written as

F[A] = R (1/2) e e + (1/) domegae ,

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 (1/2) e e, and 2 = 3/, then

Scorr = # tr(Fcorr Fcorr) ;

Notice that at the action level, the = 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 > 0]; Wise gq/06 [and Cartan geometry]; Anabalón JHEP(08)-a0805 [and gauged Wess-Zumino-Witten term].

Mach's Principle

Macroscopic Systems > see quantum-classical relationship.

Madelung Constant
@ References: Tyagi cm/04 [fast converging series representation].

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

Magnetic Mass > see duality.

Magnetic Part of the Weyl Curvature > see weyl tensor.

Magnetism (including magnetohydrodynamics)

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) sigma-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 a0802

Magnus Effect > s.a. turbulence.
@ Optical: Bliokh & Bliokh PLA(04) [and Berry phase].

Majorana Equation > see arbitrary spin field theories.

Majorana Spinors > see 4-spinors.

Majumdar-Papapetrou Solutions > see under Papapetrou-Majumdar.

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, C) identities, like the spinor identity (tr A) (tr B) – tr AB – tr AB–1 = 0, for all A, B in SL(2, C).

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

Many-Body Physics > see classical systems; composite quantum systems; gravitating matter.

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].

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

Margolus-Levitin Theorem > see quantum effects, quantum information.

Markov Chain, Process > see stochastic processes.

Martingale > s.a. diffusion.
@ References: Revuz & Yor 91; MathWorld page.

Maslov Index > s.a. geodesics.
@ References: Pletyukhov & Brack JPA(03) [canonically invariant calculation]; de Gosson & de Gosson JPA(03) [Hamiltonian periodic orbits].

Mass

Mass Inflation
* Idea: The apparent increase of the mass of a black hole for a traveler moving toward it.
@ References: Oda gq/97 [for Reissner-Nordström black hole, in quantum gravity].

Massless Particles > see field theory.

Master Equation > s.a. brownian motion; Lindblad Equation.
* Idea: An equation describing a classical stochastic process, of the form Pc = c' (Wc'c Pc' – Wcc' Pc) in its Markov version, where the W's are transition rates.
@ General references: Joos qp/05-in [from strong decoherence]; Sun PRL(06) [path summation formulation]; Kryszewski & Czechowska-Kryszk a0801 [pedagogical].
@ Applications: Mendes & Farina qp/06 [atomic energy level corrections].
@ Special types: Belavkin TMP(97)qp/05 [quantum, irreversible].
@ Non-Markovian: Maniscalco PRA(07) [spin-1/2, with exponential memory]; Krovi et al PRA(07)-a0707 [qubit + Ising spin bath].

Matching of Metrics

Mathematica > see programming languages; brst; 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.

Matrix > s.a. characteristic polynomial.

Matrix Mechanics
@ References: Effros mp/04 [history].

Matrix Models / Theories in Physics > s.a. entropy.
* 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.
@ 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].

Matroid Theory
* Idea: A branch of combinatorics started by Whitney (1935); Also called (combinatorial) pregeometry.
$ 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 in I) implies A in I; (2) (A, B in I) (|A| < |B|) implies that there exists b in B \ A, such that A {b} is in I.
* 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].
@ And physics: Nieto & Marín IJMPA(03) [gravity].

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

Maupertuis Principle > see hamiltonian dynamics.

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

Maximum Entropy Principle > see entropy.

Maximum Power / Force Principle > see force.

Maximum Tension Principle > see matter phenomenology in gravity.

Maximal Surface > see extrinsic [extremal surface].

Maxwell's Demon > s.a. computation; heat; [thermodynamics].
* Idea: A creature used by Maxwell in a thought experiment about a possible way to violate the second law of thermodynamics.
@ References: Bennett SA(87)nov; Leff & Rex AJP(90)RL; Maddox Nat(90)may; letters Nat(90)347, 24; Von Baeyer 99; Serreli Nat(07)feb + pw(07)jan [implementation by molecular information ratchet]; Maruyama et al a0707 [and information].
> Online resources: 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-Boltzmann Distribution
* Idea: The distribution of speeds/momenta in a dilute hard sphere gas in a box with hard walls at equilibrium,

fMB(v) = 4n(m/2kT)3/2 v2 exp{–mv2/kT},   fMB(p) = (1/2mkT)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.
@ References: in Srednicki cm/94-in; Cubero et al a0705 [in special relativity].

Mayer-Vietoris Sequence
* Idea: 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 > see schwarzschild solution; [relativistic cosmology].

Mean Curvature > see riemann tensor.

Mean-Field Method > s.a. Defects; ising model; QCD phenomenology; spin models [spin glasses].
@ References: Caracciolo et al AP(98) [statistical theory]; Pluchino et al PhyA(05) [Montecarlo study]; Ponomarenko et al JPA(06) [finite quantum system, canonical ensemble]; Kiessling in(08)-a0711 [and thermodynamical equilibrium]; Yapage & Nagaoka JPA(08) [Ising model, information-theoretic approach].

Mean Free Path > see scattering.

Meander > see molecular physics [polymers].

Measure Theory

Measurement > s.a. experiments in physics; measurement in quantum physics, types and effects; units.
@ In classical physics: Ridgeway a0707 [measurements in infinite lattices].

Mechanical Similarity > see conformal and scale symmetry.

Meissner Effect > see superconductivity.

Mellin Transform
@ References: Oberhettinger 74.

Melnikov Integral / Method > see description of chaos.

Melvin Solution
* Idea: A solution of Einstein's equation representing a spacetime with a strong 'homogeneous' electric field.
@ References: Melvin PL(64); Havrdová & Krtous GRG(07)gq/06 [as limit of C-metric].

Membrane

Meromorphic Function
* Idea: A complex function which only has poles as singularities.
@ References: Yang & Yi 04 [uniqueness theory].

Meron > see solutions of gauge theories.

Mersenne Primes > see number theory.

Mesons > see hadrons.

Metamaterials

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

Metaplectic Group > see group types.

Metric Space

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

Metric-Affine Gravity Theories > s.a. formulations of general relativity; gravity; 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 a0711 [relationship with Riemann-Cartan].
@ Higher-order: Cotsakis et al JMP(99)gq/97; Sotiriou & Liberati AP(07)gq/06, gq/06-in.
@ Solutions: Socorro et al PLA(98)gq [multipoles]; Hehl & Macías IJMPD(99)gq [rev]; Baekler & Hehl IJMPD(06)gq [Kerr-dS black holes]; > s.a. reissner-nordström, FRW 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 ap/05-in [cosmology]; Cheng et al PRD(05)gq [radiation transport].
@ Quantized: Kalmykov CQG(97); Mielke & Rincón Maggiolo GRG(03) [BRST]; > s.a. phenomenology.
> Related topics: see metric matching; spherical general relativity.

Metrizable Manifold > see manifold types.

Metropolis Algorithm > see computational physics.

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; Its results led to the abandonment of the ether concept.
@ General references: Michelson & Morley AJS(1887); Shankland et al RMP(55) [status]; Holton Isis(69).
@ Modern version: Müller et al PRL(03); Consoli phy/05; Müller et al a0706 + news pw(07)jun [10–16 limits on violations]; > s.a. tests of lorentz invariance.
@ Interpretation: Lämmerzahl & Haugan PLA(01)gq; Consoli & Costanzo ap/03 [preferred frame].

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

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

Microscopes

Microsuperspace > an even more restricted type of Minisuperspace.

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

Midisuperspace > see canonical quantum gravity.

Millikan's Oil Drop Experiment > see physics experiments.

Milne Universe > see minkowski space.

Mind

Minimal Surface > see extrinsic curvature.

Minimum Length > see deformation quantization; modified lorentz symmetry and uncertainty relations; quantum gravity phenomenology.

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

Minkowski Inequality > see inequalities.

Minkowski Spacetime

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 > see Homogeneous Space; lie algebra; lie group.

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

Misner Space
* Idea: A 2D space with topology R S1, in which the light cones progressively tilt as one moves forward in time, and has closed timelike curves after a certain point.

Mixed State in Quantum Physics

Mixing System > s.a. chaos; ergodic system; group action; quantum chaos.
$ Def: A dynamical system (X, , ) such that, for all measurable A and B, (n A B) → (A) (B) as n → ; In other words, for all functions f and g, as T → the correlator (... is phase space average)

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

* Comments: It implies ergodicity; The notion is not invariant under t-reparametrizations [@ in Motter PRL(03)gq].
* Examples: The baker's transformation; the Lorenz system.
* Decay time: As a rule, R R0 exp{–t/c}, where c is the mixing or correlator decay time, and c 1/h0, with h0 the sum of the positive Lyapunov exponents (hard to prove).
@ References: in Zaslavsky et al 91; Antonious & Tasaki IJQG(93) [spectral decompositions]; Kandrup MNRAS(98)ap; Chernov & Zhang mp/04 [slow mixing systems]; Richter PRA(07) [quantum speedup of mixing].

Mixmaster Universe > see bianchi IX.

Möbius Transformation > see Complex Number.

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) [@ Wikipedia page].
@ References: Prest 88; Chang & Keisler 90; Poizat 00 [r BAMS(02)].

Models in Physics > s.a. physics; theory.
@ References: Giere PhSc(04) [representing reality]; Emch SHPMP(07) [and theory building]; Pincock PhSc(07) [mathematical idealizations].

Modified Gravity (MOG)
@ References: Moffat & Toth a0712 [solutions from action principle], a0805 [light bending and lensing]; > s.a. galaxies.

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

Modular Forms > s.a. number theory.

Modular Group > see diffeomorphisms.

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-in; > s.a. connections; yang-mills gauge theory.

Modus Ponens, Tollens > see logic.

Molecular Physics

Moment of Inertia > s.a. Elasticity; fluid.
* Idea: The symmetric tensor

Iij:= dm (r2 ij – x i x j) ,

closely related to the quadrupole moment of the mass distribution (> see multipoles).
@ References: Lawton & Noakes JMP(01) [computation]; Díaz et al EJP(06) [for solids of revolution].

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

Momentum > s.a. conservation laws; hamiltonian dynamics and systems.
* History: According to one story, the name and symbol p came from Newton, who actually called it pimentum.
* Idea: The conserved quantity related to spatial translation invariance of a theory.
* In classical mechanics: A particle with velocity v has momentum p = mv.
* In special relativity: A particle with 4-velocity ua has momentum pa = mua.
* For a wave: A wave with wave vector k has momentum p = k, or p = h/.
* In quantum mechanics: A momentum operator conjugate to a configuration variable is one with the right commutation relations; If classically the momentum is associated to a vector field ua on configuration space , a quantum momentum operator can be defined by û(x):= i (u + div u) (x), where the divergence is calculated using the volume element on wrt which the operator must be self-adjoint.
* In field theory: The momentum density of matter Tab as seen by an observer ta is – tbT ab; > s.a. canonical general relativity [various formulations].
@ General references: Sibelius FP(90) [mechanical and wave-theoretical aspects]; Gillespie AJP(95) [why "p"?]; Sonego & Pin EJP(05), EJP(05) [in special relativity]; Crenshaw PLA(05) [electromagnetic, and Fresnel relations]; Roche EJP(06) [general definition].
@ In quantum mechanics: Jordan AJP(75); Paz EJP(01)qp/00, Mosley mp/03 [radial]; Roy et al a0706 [in general coordinates]; Shikano & Hosoya JMP(08) [on a half-line].
@ In quantum field theory: de Haan ht/06 [e mechanical momentum in QED].
> Physics in momentum space: see Fermi Surface; wigner function.

Momentum Map > s.a symplectic manifold.
@ Generalization: in Ortega & Ratiu RPMP(06) [cylinder-valued].

MOND > see under modified Newtonian mechanics.

Monge Metric on the Sphere > see sphere.

Monge-Ampère Equation > see symplectic structures.

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, ), X a set, 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.

Monopoles

Monotonic Sequence > see sequences.

Montecarlo Method > see computational physics.

Monstrous Moonshine > see finite groups.

Moon > see earth and its moon.

Mordell Conjecture > see conjectures.

Morita Equivalence > see non-commutative gauge theory.

Morphism > see category.

Morse Theory

Mosaic > a name sometimes used for a tiling.

Mössbauer Effect > see radiation.

Motion > s.a. geodesics; orbits of gravitating objects; Test Body.
@ References: Rynasiewicz PhSc(00) [absolute vs relative].

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.
@ And quantum field theory: Connes & Marcolli JGP(06) [perturbative renormalization and motivic Galois theory]; Bloch et al CMP(06).

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

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

Multigravity > see theories of gravity.

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) x1n_1 x2n_2 ... xkn_k .

* Properties: It is obviously a generalization of the binomial coefficients, C(n; p, n–p) = {n \choose p}.
@ References: MathWorld page.

Multiplication Structure on a Manifold > see manifold.

Multiply Connected Space > see connectedness.

Multipole Moments > s.a. spherical harmonics.

Multisymplectic Structure > see symplectic structures.

Multiverse

Muon > see particle types.

Murphy's Law
@ References: Matthews SA(97)apr.

Music

Mutual Information > see information.

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