Topics, B

Baby Universe > s.a. minisuperspace quantum cosmology; multiverse.
@ References: Dijkgraaf et al PRD(06)ht/05, IJMPD(06) [in string theory].

Bach Equation > see Conformal Gravity.

Bach Tensor > s.a. huygens principle.
$ Def: For a Lorentzian 3-manifold with metric q_ab,

B_abc := D_[a (R_b]c – R q_b]c) .

* Applications: Used to study the conformal symmetries of q_ab (vanishes iff conformally flat).
@ References: in Ashtekar & Magnon CQG(84); Glass CQG(01)gq [and conserved current]; Álvarez et al ht/02 [AdS-cft].

Back-Reaction > see self-force; semiclassical gravity.

Background Independence > see Covariance.

Bäcklund Transformation > see solitons.

Bag Model > s.a. casimir; Gravitational Bag [different kind]; QCD.
* Idea: A phenomenological model for hadrons, originated at MIT, in which quark confinement is simulated by enclosing them in a bag.
@ References: Chodos et al PRD(74) [baryon structure]; Colanero & Chu JPA(02) [spherical, solution]; Lavenda ht/06 [thermodynamic problem].

Baire Category Theorem > see distance.

Baker Map > s.a. chaotic systems.
* Idea: A discrete chaotic system.
@ Quantum: in Schack PRA(98)qp/97 [on quantum computer]; Rubin & Salwen AP(98)qp; Schack PRA(98) [and quantum computers]; Soklakov & Schack PRE(00)qp/99; Inoue et al qp/01, JMP(02)qp/01 [semiclassical]; Tracy & Scott JPA(02) [classical limit]; Lozinski et al PRE(02)qp [irreversible]; Degli Esposti et al CMP(06)mp/04 [variance and ergodicity]; Nonnenmacher & Anantharaman AHP(07)mp/05 [entropy of semiclassical measures]; Scherer et al PRD(06) [coarse-grained evolution].

Ball > see sphere.

Banach Space > s.a. Degree Theory; Orlicz Space; tensors [tensor product].
$ Def: A complete normed vector space.
* Types: A Banach space E is reflexive iff the canonical injection E → E** is onto.
* Examples: Any finite-dim'l or Hilbert space; L^p for 1 < p < ; C([0,1], R) with the Sup norm, not reflexive.
* Fréchet derivative: The differential Df of a mapping f: X → Y between (possibly infinite-dimensional) Banach spaces, defined by f(x+h) – f(x) = Df(x) + R(h), where R(h) = o(h); Example: The operator giving the linearized version of a non-linear pde.
@ References: Banach 32; Cirelli 72; Lindenstrauss & Tzafriri 77 [standard].

Barbour-Bertotti Model > see parametrized theories.

Bargmann Invariant > see phase.

Bargmann-Segal Representation / Transform > see representations of quantum theory.

Barker's Theory of Gravitation
@ References: Yepes & Domínguez-Tenreiro PRD(86) [cosmological models].

Barnett Effect > see magnetism.

Barometric Formula > see gas.

Barrett-Crane Model > see spin foams.

Barycentric Coordinates > see simplex.

Baryogenesis > see early universe.

Baryons > see hadrons.

Base for a Topology
* Idea: A collection of open sets generating the topology; every other one is the union of some subcollection of it.

Base Space > see bundle.

Basis for an R-Module > see module.

Batalin-Tyutin Quantization > see quantization of first-class constraints; quantum particles.

Bateman's Dual System > see oscillator.

Baum-Connes Conjecture > see deformation quantization.

Bayes Theory > see probability theory.

BCS Theory > s.a. superconductivity.
@ References: Hainzl et al CMP(08) [for general pair interactions].

Beable
* Idea: Within quantum mechanics, theories of "beables" are, e.g, the theories of de Broglie, Bohm, Bell, Vink, and also "modal" theories.
@ In quantum mechanics: Vink PRA(93); Finkelstein PLA(96)qp/95 [measurement]; Clifton qp/97-in [algebraic]; Elze JPA(08) [symmetry of beables]; > s.a. hidden variables.

Beal Conjecture > see conjectures.

Bel, Bel-Robinson Tensor

Bell's Inequality > s.a. foundations of quantum mechanics.

Bell-Szekeres Spacetime > see gravitational wave solutions.

Berezinian
* Idea: The superdeterminant of a supermetric.
@ References: Khudaverdian & Voronov mp/05 [formula].

Bergmann Manifold
@ References: Holm IJTP(90) [connections].

Bergmann-Wagoner Theory > see scalar-tensor.

Bernoulli Equation
$ Def: The equation p + v² + gy = constant.

Bernoulli Inequality > see inequalities.

Bernoulli Numbers
$ Def: The numbers B_n that appear in the power series expansion

x / (e^x – 1) = _n=0^infty B_n xⁿ/n!

* History: First studied by Faulhaber, but made popular by James Bernoulli.
> Online resources: CRC page.

Bernoulli Shift > see classical systems.

Berry's Phase > see geometric phase.

Berry-Hannay Model > see quantum systems.

Bertotti-Robinson Spacetime > see Robinson-Bertotti.

Bertrand's Theorem > see classical systems.

Bessel Functions

Bessel Transforms
@ References: Oberhettinger 72.

Beta Function > s.a. renormalization group [in quantum field theory].
$ In mathematics: The function

B(x,y):= 2 ₀^infty dt t^2x–1 / (t² + 1)^x+y ,   with Re x > 0 and Re y > 0 .

related to the Gamma function by B(x,y) = (x) (y) / (x+y).

Bethe Ansatz
* Idea: An ansatz to obtain the energy eigenstates of the one-dimensional version of Heisenberg's model of interacting, localized spins.
@ References: Batchelor PT(07)jan [history].

Bethe-Peierls Approximation > see ising model.

Bethe-Salpeter Equation > s.a. Salpeter Equation [same?].
* Idea: Relativistic generalization of the Schrödinger equation, used to describe bound states.
@ References: Nakanishi ed-PTPS(88)#95; Karmanov & Carbonell EPJA(06)ht/05 [solution method].

Betti Numbers > s.a. euler classes.
* Idea: Topological invariants representing roughly the number of independent p-dimensional boundaryless surfaces which are not boundaries themselves; b₀ is the number of connected components and b_k effectively counts the number of k-dimensional holes; More specifically, the number b_k is the dimension of the k-th de Rham cohomology group.
$ Def: The k-th Betti number of a manifold M, b_k(M) or R_k(M), is the rank of the free part of the homology group H_k(M).
* Special cases: If there is no torsion subgroup, b_k = dim H_k(M); If M is closed, B_p = B_4–p, B₁ = B₄ = 1; If M is simply connected, B₁ = B₃ = 0.
@ References: Yano & Bochner 53; Garvín & Lechuga T&A(03) [elliptic space, NP-hard]; Robins mp/06/PRE [for Poisson-centered spheres of given radius].

BF Theory

BFV Formalism > see quantization of first-order constraints.

Bhabha Scattering
@ References: Bonciani & Ferroglia NPPS(06), Becher & Melnikov JHEP(07) [2-loop QED corrections].

Bi-Differential Calculus > see symmetry.

Bi-Hamiltonian Structure > see integrable systems; quantum systems and states; symmetry.

Bi-Local Fields > see types of field theories.

Bialgebras > see algebra.

Bianchi Classification > s.a. bianchi I; bianchi IX; bianchi models.

Bianchi Identities > see curvature.

Bicompact Space
* Idea: A compact Hausdorff space; A bicompact space is T₄.

Bicomplex
@ References: Dimakis & Müller-Hoissen IJMPB(00)ht, JPA(00)nlin.SI [and integrable models], JPA(01) [and Bäcklund transformations].

Biconformal Spaces > see formulations of quantum mechanics.

Biconformal Vector Fields > see conformal structures.

Bidifferential Calculi > s.a. symmetries.
@ References: Chavchanidze mp/01 [and non-Noether symmetries].

Bieberbach Conjecture > see conjectures.

Bieberbach Manifolds
@ References: Pfäffle JGP(00) [Dirac spectrum].

Bifurcation Theory > s.a. Stability Theory.
* History: 1879, Originated with Poincaré; 1933, Developed by Andronov.
* Types: Pitchfork, Hopf bifurcations.
@ General references: Iooss & Joseph 80; Chow & Hale 82 [standard treatise]; Ruelle 89; Gaeta PRP(90); Crawford RMP(91).
@ Example: Johnson AJP(98) [unicycle].

Big Rip > see cosmology.

Big Trip
* Idea: A cosmological process thought to occur in the future by which the entire universe would be engulfed inside a gigantic wormhole and might travel through it along space and time.
@ References: González-Díaz PLB(06)ht [viability]; Faraoni PLB(07)gq [unfounded claims].

Bigravity > see under bimetric theory of gravity.

Bilinear Form
$ Def: A map B: V × V → R or C, linear in both arguments, with V a vector space.
$ Hermitian form: A bilinear form with B(x, y) = B(y, x)*.
* Relationships: Any quadratic function f : V → R determines a bilinear form by B(u,v):= [f(u+v) – f(u) – f(v)].
$ Strongly non-degenerate: A bilinear form (B: V × V → R, considered as) B: V → V*, with V a vector space, is (strongly) nondegenerate if it is an isomorphism (1-1 and onto).
$ Weakly non-degenerate: B is weakly nondegenerate if it is only injective or 1-1; This means B(X, · ) = 0 iff X = 0.

Billiard > see classical systems; spectral geometry and quantum systems.

Bilocal Fields, Bilocality > see foundations of quantum mechanics; types of quantum field theories.

Bimetric Theory of Gravity

Binary Operation on a Set > see set.

Binary System > see dynamics of gravitating bodies; Two-Body Problem.

Binding Energy > see matter phenomenology in gravity.

Bing Topology

Binomial Coefficients
$ Def: The number of ways to choose n different objects (unordered) out of k: ⁿC_k {n \choose k}:= n! / k! (n–k)! .
* Properties: {n \choose m} + {n \choose m+1} = {n+1 \choose m+1} ;

_{k = 0}ⁿ (ⁿC_k) = 2ⁿ ,   _{k = 0}ⁿ k (ⁿC_k) = n 2^n–1 ,   _{k = 0}ⁿ k² (ⁿC_k) = n (n + 1) 2^n–2 ,   and   _{k = 0}ⁿ (–1)^k (ⁿC_k) = 0 .

* Remark: They get their name from the fact that (a + b)ⁿ = _{k = 0}ⁿ (ⁿC_k) a^n–k b^k .
@ References: Zhang DM(06) [generalization of Calkin's identity]; Sun DM(08) [sums, and applications].

Binomial Distribution
$ Def: Given that a property occurs with probability p per trial, the probability that it occurs exactly n times out of N trials is

P_bin(n, N, p) = {N \choose n} pⁿ (1–p)^N–n .

* Cumulative binomial distribution: The probability of it occurring j n times, P_cb(n, N, p) = _j P_bin(j, N, p) for j = n, ..., N.
@ References: Kowalski JMP(00) [generalized].

BIon
* Idea: A finite energy solution of a non-linear field theory (> see, e.g., Born-Infeld) with distributional sources (a soliton has no sources).
@ References: Gibbons CQG(99)ht/98 [from branes]; Tamaki & Torii PRD(00)gq [Einstein-BI-dilaton], PRD(01)gq [string-inspired].

Birefringence > see polarization.

Birkhoff's Theorem > s.a. spherical symmetry in general relativity.
* Idea: The only vacuum, spherically symmetric solution of Einstein's equation is static (and it is the Schwarzschild metric).
* Generalizations: In general relativity, it can be generalized to electrovacuum solutions, giving, as unique spherically symmetric solution, Reissner-Nordström (one cannot, however, generalize it to axisymmetric solutions); Generalizes to other theories, but is violated in braneworld models, such as Randall-Sundrum.
@ General references: Jebsen AMAF(21), translation GRG(05); in Birkhoff 23; in Hawking & Ellis 73; Bondi & Rindler GRG(97) [addendum re meaningful time coordinates]; Schmidt G&C(97)gq; Abbassi gq/98, gq/01 [more solutions??]; Severa gq/02 [geometry]; Johansen & Ravndal GRG(06)phy/05 [history, J T Jebsen]; Deser GRG(05) [re Jebsen]; Deser & Franklin AJP(07) [and t-independence in general relativity].
@ Higher-order gravity: Zegers JMP(05)gq [Lovelock]; Deser & Franklin CQG(05)gq [with second-order field equations].
@ With cosmological constant: Rindler PLA(98) [reformulation, Bertotti-Kasner as extra solution]; Ayón-Beato et al PRD(04)ht [2+1, < 0].
@ For various theories: Venkateswarlu & Reddy ASS(89) [scalar-tensor]; Brodbeck & Straumann JMP(93) [Einstein-Yang-Mills]; Bronnikov & Melnikov GRG(95) [higher dimensions]; Schmidt G&C(97)gq [including other signatures]; Cavaglià ht/98-in [quantum dilaton gravity]; Cavaglià G&C(99)gq [topologically massive gravity]; Keresztes & Gergely a0712 [5D].
@ Consequences of non-validity: Dai et al a0709 [DGP model].
@ Other generalizations: Szenthe JGP(07).

Bistochastic Matrix > see matrices.

Bivector > s.a. types of field theories [bivector fields].
* Idea: An object of the form u_[av_b], representing the u-v plane; The magnitude can be given by u^a v^b u_[a v_b].
* In a Lorentzian metric: The sign of the "magnitude" is related to whether the plane is spacelike, timelike, or null.

Black-Body Radiation > see thermal radiation.

Black Holes > s.a. black hole solutions, thermodynamics; quantum black holes; [< part of gravitation and cosmology].

Black Tides > see black hole phenomenology.

Blandford-Znajek Effect > see black hole phenomenology.

Blazar > see black hole phenomenology; astrophysics; gamma-ray astronomy.

Bloch Ball / Sphere > see quantum systems.

Bloch Theory
* Application: Analyze spectral properties of differential operators which are invariant under an abelian group.
* Bloch's theorem: Electron wave functions in the presence of a periodic potential (such as the electric potential of a crystalline lattice of atoms) are of the form exp{ i kx} u(x), where u(x) is periodic with the same period.
* Bloch oscillations: A phenomenon that occurs when particles subject to a periodic potential are exposed to an additional static force, say, an electric (or gravitational) force in a single direction; The electrons then do not all move in the direction of the force, but instead oscillate back and forth in place; > s.a. tests of newtonian gravity.
@ References: Gruber JMP(01)mp/00 [non-commutative generalization]; Bouda & Meziane IJTP(06)qp/07 [Hamilton-Jacobi formulation].

Block Universe > see time.

BMS Group > see asymptotic flatness at null infinity.

Bochner Theorem
* Idea: A result on measures in constructive quantum field theory.
@ References: in Gel'fand & Vilenkin 64; in Yamasaki 85.

Bode's Law > see Titius-Bode Law.

Bogoliubov Quasiparticle > see Quasiparticles.

Bogoliubov Transformation

Bogomolny Equation
* Idea: The equation B = D one gets in Yang-Mills-Higgs theories, minimizing the energy with the constraint ² = C², whose solution gives a class of monopoles.
* And self-duality: If one considers as the 5th component of A, the equation becomes the self-dual equation F_ab = _abcd F^cd, whose solutions are static self-dual monopoles, characterized by an integer m; For m = 1, we have the Prasad-Sommerfied solutions.
@ References: Bogomolny SJNP(76); Coleman, Parke, Neveu & Sommerfield PRD(77).

Bogomol'ny Inequality > s.a. positive energy theorems.
* Idea: A lower bound on the mass of a monopole solution in a gauge theory in terms of its electric and magnetic charges.
@ For Kaluza-Klein monopoles: Lee & Sorkin CMP(88).

Bohm Metrics
* Idea: Infinite sequences of inhomogeneous Einstein metrics on spheres and products of spheres of dimension 5 d 9.
@ References: Gibbons et al PRD(03).

Bohr Compactification of R > s.a. functions [almost periodic].
* Idea: A compact group obtained as the dual of the real line endowed with the discrete topology; Can be used as the configuration space for a non-standard, polymer representation for the quantum theory of a system on the real line.
@ General references: Halvorson SHPMP(04)qp/01; in Bratteli & Robinson 02.
@ And quantum mechanics of point particles: Ashtekar et al CQG(03)gq/02; > s.a Polymer Representation of Quantum Theory.
@ And quantum gravity / quantum cosmology: Husain & Winkler PRD(04)gq/03; Velhinho CQG(07)-a0704; > s.a. minisuperspace.

Bohr Magneton
* Idea: The constant ₀ = e/2mc.

Bohr Model of the Atom > see history of quantum physics.

Bohr-Rosenfeld > see quantum measurement [quantum field theory].

Boltzmann Brains
@ References: Overbye NYT(08)jan.

Boltzmann Constant > see constants.

Boltzmann (Transport) Equation > s.a. fluids [Navier-Stokes]; stochastic process.
* Approximations involved: (i) Dilute gas, there are only binary collisions; (ii) Ignore the walls of the container; (iii) Ignore the effect of the external force on the collision cross section; (iv) Molecular chaos, the velocity of a molecule is uncorrelated to its position.
* Chapman-Enskog method: A successive approximations method used to find some solutions of the Boltzmann equation.
@ General references: in Huang 63*; in Gorban & Karlin cm/03 [rev].
@ Solutions and techniques: in Huang 63 [Chapman-Enskog]; Kandrup ap/97 [collisionless, t-independent]; Cercignani JSP(05) [global, weak]; Yu JSP(06) [Green's function]; Yang & Zhao JMP(06) [energy method]; Bardos et al JSP(06) [in half space].
@ Quantum: & Joichi, Matsumoto, Yoshimura; Singh & Srednicki PRD(00)hp/99; Yamamoto IJMPA(03)ap [fermions in curved spacetime]; Chen CMP(06) [as limit of random Schrödinger equation]; Breuer & Vacchini a0707 [Monte Carlo simulation].
@ Relativistic: Lucquiaud JMP(78); Horwitz FP(95); Calogero JMP(04)mp [Newtonian limit]; Noutchegueme et al GRG(05)gq, & Dongo CQG(06)gq/05 [in Bianchi I]; Takou & Noutchegueme gq/05 [in flat FRW].

Bolzano-Weierstrass Theorem > see compactness.

Bondi Mass > see asymptotic flatness at null infinity.

Bondi-Sachs Metric > see gravitational wave solutions.

Bonnor / Bonnor-Swaminarayan Solutions > s.a. c-metric.
* Idea: Boost-rotation symmetric spacetimes describing pairs of accelerating particles, possibly connected to struts.
@ References: Podolsky & Griffiths GRG(01)gq/00 [null limits]; Garecki CQG(05)gq/04 [energy-momentum].

Bonnor-Ward Spacetimes
@ References: Rosa & Letelier PLA(07) [closed timelike geodesics, stability].

Boolean Algebra > s.a. logic.
$ Def: A ring R of subsets of a space X, with X in R.

Boost > see kinematics of special relativity.

Bootstrap Theory
* Idea: An approach to understanding elementary particles in hadronic physics that was very popular in the 1960s (one of the main proponents was Geoffrey Chew), as an alternative to quantum field theory.
@ References: Chew SA(64)apr; Redhead FP(05) [overview, assessment].

Bordism

Borel Fixed Point Theorem > see fixed point theorems.

Borel Measure, Sigma-Field > s.a. ring.
* Idea: A positive measure on Borel sets of a locally compact Hausdorff topological space.
$ Borel sets: The Borel -field of a topological space X is the one generated by the open (or the closed) sets in X; An element of it is a Borel set.

Born Rule
* Idea: The statement that = ||², or in general that the probability of obtaining a certain eigenvalue of an observable in a measurement is the square modulus of the corresponding coefficient in the expansion of the state in eigenvectors.
@ References: Born ZP(26); Zurek PRL(03), Schlosshauer & Fine FP(05)qp/03 ["envariance" derivation]; Valentini & Westman PRS(05)qp/04 [dynamical origin, in pilot wave]; Mould qp/05 [replace by probability current alone]; Brumer & Gong PRA(06)qp [in quantum and classical mechanics]; Buniy et al PLB(06)ht [and state space discreteness]; Landsman a0804 [mathematical clarificaton of role].

Born-Infeld Theory

Born-Oppenheimer Approximation > s.a. semiclassical quantum gravity.
* Idea: An approximation used in the quantum theory of molecules, in which ones divides the degrees of freedom into "heavy" ones, the nuclei, and "light"ones, the electrons.

Bose-Einstein Condensation > s.a. atomic; effective field theories; light; sound; temperature; vacuum [fluctuation].
* Idea: A phase transition of a gas of bosons consisting in the amalgamation of many bosonic atoms so cold and dense (chilled to nearly 0 K) that they act as a single quantum state, essentially a single "superparticle"; This occurs when T is so low that the atoms' de Broglie wavelength becomes comparable to the distance between them; Similar to Cooper pairs in superconductors.
* History, theory: Started with S N Bose's letter to Einstein in 1924, after his paper was rejected by Philosophical Magazine; Einstein extended the ideas to massive particles in 1925; Viewed with skepticism (how can you have condensation in an ideal gas, without forces; No applications, ...) until 1938, when F London proposed it to explain He superfluidity, discovered in 1928.
* History, experiment: First produced in 1995, with 5 million Rb atoms, directly observable; The 2001 Nobel prize for physics was given for Bose-Einstein condensation in dilute gases of alkali atoms; 1998, For H atoms (at T 40 K, with about 100 million atoms); 2003, Observed in Ytterbium, which differs from most of the elements that have previously been condensed because it has two valence electrons rather than one, and can be prepared in a non-magnetic state; 2005, Observed in Chromium, which has a very large magnetic dipole moent.
* Properties: Enormous indices of refraction.
@ Books: Pitaevskii & Stringari 03 [r PT(04)oct]; Annett 04 [intro].
@ General references: Scharf AJP(93); Cornell & Wieman SA(98)mar; Ketterle PT(99)dec [experiments]; Burnett et al PT(99)dec [theory]; Collins SA(00)dec; Yukalov PLA(06) [self-consistent theory]; Schlein a0704-in [dynamics].
@ Specific types of gases: news pn(95)jul, pn(95)aug, pn(98)nov, pn(99)jun, Bradley et al PRL(95) [atoms]; Wynar et al Sci(00)feb + pn(00)feb [Rb₂ molecules]; Hall AJP(03)RL [trapped dilute gases]; Takasu et al PRL(03) + pw(03)jul [in Yb]; news pw(05)mar [chromium]; Grether et al PRL(07) [relativistic ideal Bose gas].
@ Related topics: Reichel SA(05)feb [and microchips]; Dorlas et al mp/05 [and long cycles]; Lye et al PRL(05) [in a random potential]; Schützhold PRL(06) [accurate phonon detection]; Damski & Zurek PRL(07) [spin-1, quantum phase transition]; news pw(08)may [use to measure small forces].
> Gravity-related topics: see black hole analogs; lorentz symmetry breaking; quantum field theory effects; matter in quantum garvity; types of dark matter.

Bosons > s.a. particle statistics.
@ Related effects: Gogolin et al PRL(08) [solution to three-body problem, including Efimov trimers]; > s.a. atomic physics, Chirality.

Boson Star > see astronomical objects.

Bosonization > s.a. particle statistics; types of field theories.
@ References: Liguori & Mintchev NPB(98)ht/97 [on the half-line]; Ilinskaia & Ilinski cm/98-in; Dhar & Mandal PRD(06)ht [non-relativistic fermions on S¹]; Kanakoglou & Daskaloyannis mp/07-in [and parastatistics].

Bott's Periodicity Theorem
* Idea: A theorem on periodicities in _q(G) for Lie groups (> see homotopy).
@ References: in Milnor 73 [last chapter].

Boulware Vacuum > see quantum field theory in curved backgrounds.

Bounce > see relativistic cosmological models.

Bound State > see atomic physics; types of quantum states.

Boundary > s.a. boundaries in field theory; spacetime boundary.
$ In homology: A q-chain c such that c = (c'), for some (q+1)-chain c'.
$ In topology: The boundary of a subset S X is its closure minus its interior, S ^· or S:= \ovr S – int(S).

Boundary Conditions > see boundaries in field theory; electromagnetism; quantum cosmology.

Boundary Value Problems > see Dirichlet, Neumann and Robin problem; laplace equation.

Bounded Convergence Theorem
* Idea: Under appropriate circumstances, the integral of a limit function, is the limit of the integrals.

Bounded Operator > see operator theory.

Bounded Variation > see functions.

Boyer-Lindquist Coordinates > see kerr metric.

BPS Solutions > see black hole solutions, and black holes in modified theories.

Bradyon
* Idea: A slower-than-light particle (> aot a tachyon).

Braided Geometry > see geometry.

Braids
* Idea: A braid is a set of n strings stretching between two parallel planes.
* Application: They have been used as tools in the calculation of knot and link invariants; > s.a. 3D general relativity; composite models.
@ Braid group: Cappuccio & Guadagnini PLB(90) [statistics]; > s.a. group types.
@ Invariants: Berger LMP(01); Nechaev & Voituriez NPB(05) [3-strand, Brownian].

Brane World > s.a. brane cosmology.

Brans-Dicke Theory

Bravais Lattice > see crystals.

Bremsstrahlung > see radiation.

Brick Wall Model > see black hole entropy.

Brieskorn Sphere > see 3-manifolds.

Brillouin Zone > see cell complex.

Brillouin-Wigner Perturbation Expansion
@ References: see Ziman 69, pp 55–56.

Brinkman's Theorem > see types of spacetimes [Einstein spaces].

Brouwer Theorem > s.a. fixed-point theorems.
$ Def: The unit sphere Sⁿ in Eⁿ is not a retract of the closed unit ball Bⁿ which it bounds.

Brown Dwarfs > s.a. extrasolar planets.
* Idea: Substellar objects that can be seen through the release of some of their internal gravitational energy; Their masses are between 0.001 and 0.1 M_Sun.
* Consequences: They are a candidate for dark matter (Bahcall 1984).
* Observation methods: Look for methane lines in spectrum.
* Examples: Jupiter; VB8B, first outside the solar system, M 10–50 M_Jup, T = 1360 K, R = 0.09 R_Sun); Gliese 229B, first unambiguous sighting, d 30 ly, M 20–50 M_Jup, T < 1000 K [@ Nakajima et al Nat(95)nov].
@ I: PW(90)oct, 34–38; Martin et al AS(97), Basri SA(00)apr [discovery]; Mohanty & Jayawardhana SA(06)jan.
@ General references: Jameson & Hodgkin CP(97); Oppenheimer et al ap/98-in; Chabrier ap/99-in [physics].

Brownian Motion

BRST Transformations

Brudno Theorem
* Idea: A result in classical information theory, relating entropy rate and algorithmic complexity per symbol.
@ References: Benatti et al CMP(06) [quantum version, connecting von Neumann entropy rate and quantum Kolmogorov complexity].

BTZ Spacetime > see 3D black holes.

Bubbles > see fluids; matter [soap].

Buchdahl Inequality > see astrophysics.

Buffon's Needle > see statistical geometry.

Bunch-Davies Vacuum > see quantum field theory in curved backgrounds.

Bundle [including Bundle Gerbes, Bundle Map]

Bures Metric > see mixed quantum states; riemannian geometry; types of distances.

Burgers Equation [@ wikipedia.]
* Idea: A partial differential equation used in fluid dynamics.
@ References: Kirsch & Simon mp/01 [forced, approach to equilibrium]; Hamanaka & Toda JPA(03) [non-commutative]; Bec & Khanin PRP(07) [turbulence].

Burnside Ring of a Group > see ring.

Butcher Group > related to Hopf Algebras.

Butcher Series > see scalar field theory.

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