Topics, H
H Theorem >
s.a. pilot wave [subquantum].
* Idea: A statement of
the second law of thermodynamics; Says that the quantity H defined
by H:= 
i pi ln pi,
where
pi is the probability for
the system to be in the state i, or H:= –S/Vk,
never decreases; In other words, for a fixed volume
the entropy never decreases.
@ References: in Reif 65; Santamato & Lavenda JMP(82)
[stochastic, for diffusion processes]; Succi et al RMP(02)
[and hydrodynamic simulations]; Silva cm/06 [relativistic].
@ Quantum: Farquhar & Landsberg PRS(57).
H-Space > s.a. types
of topologies.
* In general relativity: A complex 4D space used to analyze asymptotically flat
spacetimes.
@ References: Kozameh & Newman CQG(05)gq [and
physical spacetime].
Haag's Theorem [> s.a. quantum
field theory.]
@ References: in Streater & Wightman 80, p165; Lupher IJTP(05)
[history and versions]; Shirokov mp/07 [and particle dressing].
Haar Measure > see measure.
Hadamard Matrices > see entropy.
Hadamard's Conjecture > s.a.
huygens' principle.
* Idea: For wave equations
with non-constant coefficients, it is still true that in odd space dimensions
traveling waves from general localized sources
are
sharp,
while in even space dimensions they are not, like for the ordinary wave equation.
Hadamard's Elementary Function [> s.a. green
functions.]
* For a scalar field: G(1)(x, x'):=
0|
{
(x), (x')}
|0
= G+(x, x')
+ G–(x, x').
* For a spinor field: S(1)(x, x'):=
0| [
(x), (x')]
|0 = –(i 
a
a+m) G(1)(x, x').
* Properties: It satisfies the homogeneous field equation.
Hadrons > s.a. [particle types]; QCD; QCD phenomenology.
Hahn-Banach Theorem
* Idea: A linear functional defined on a subspace of a vector space V,
which is dominated by a sublinear function on V, has a linear extension which
is also dominated by the sublinear function.
@ References: in Zeidler 95; in Casti 00.
Half-Flat Metric > see self-dual.
Hall Effect
* History: Initially
MOSFETs were used to produce effective 2D systems; Now other ways are used.
@ Quantized: Prange & Girvin ed 90; Von Klitzing RMP(86)
[Nobel lecture]; Elvang & Polchinski ht/02 [on R4];
Avron et al pw(03)aug
[and topology]; Nair & Randjbar-Daemi
NPB(04)
[on S3]; Leitner cm/05 [in
2+1 QED].
@ Fractional: Chakraborty & Pietiläinen 88; Eisenstein & Stormer
Sci(90)jun;
Murthy & Shankar RMP(03)
[Hamiltonian theory].
@ Other variations: news pn(97)dec
[photon Hall effect]; Strohm et al PRL(05)
[phonon Hall effect].
Halmos Symbol
*
Notation: A square symbol that looks like that for a d'Alambertian, or spacetime
laplacian.
Hamilton Principal Function
* Idea: The action calculated
on a given solution of the equations of motion.
Hamilton Vector
@ References: Muñoz & Pavic EJP(06)
[for relativistic particle in Coulomb potential].
Hamiltonian Dynamics > s.a. hamiltonian systems.
Hamiltonian Structure, Vector Field > see symplectic geometry.
Hanbury Brown–Twiss Effect
* Idea: A "bunching" effect
for light, first shown in 1956 by
physicists Robert Hanbury Brown and Richard Twiss, who saw that the intensities
of light from
two different points on a random light source such as a star (Sirius) are correlated;
This is possible because photons are bosons (however,
this correlation vanishes when a coherent light source such as a laser is viewed);
Since then, the corresponding "antibunching" effect has been noted
for fermions, which are fermions and cannot occupy the same state.
@ References: news pw(05)sep
[quantum gas analog]; news pw(07)jan
[for two isotopes of He]; Nelson & Shimpi PLA(07)
[for parabosons].
Handles
$ Def: An n-dimensional
k-handle is the topological manifold Dk 
Dn–k ;
It is often used as attached to another n-manifold.
Hankel Functions > see bessel functions.
Hannay's Angle > see phase.
Hardy Spaces / Algebras
$ Hardy algebra: A weak
*-closed subalgebra of Linfty(X,
dm),
for some finite positive measure space (X, S, dm).
@ References: Barbey & König 77; Folland & Stein 82.
Hardy's Theorem > see realism; relativistic quantum mechanics.
Harmonic Analysis, Coordinates, Functions > see harmonic functions.
Harmonic Oscillator > see oscillator.
Hartle-Hawking Vacuum > see quantum field theory in curved backgrounds.
Hartle-Hawking Wavefunction > see boundary conditions in quantum cosmology.
Hartman Effect > see quantum-mechanical effects.
Hartree-Fock Approximation
@
References: Kleinert AP(98)
[systematic improvement]; Bardos et al qp/03,
JSP(04)qp/03 [time-dependent,
accuracy]; Trebbia et al PRL(06)
[evidence for breakdown in Bose gas].
Hasse Diagram
* Idea: The graph of
the immediate predecessor relation for a poset,
with the convention that if x < y, x is drawn lower; A useful
way of representing posets without superfluous info.
Hausdorff Dimension > see dimension.
Hausdorff Distance > see distance between metric spaces.
Hausdorff (Separated) Topological Space > see types of topologies.
Hawking Effect, Radiation > see black hole radiation.
Hawking-Lichnerowicz Theorem > s.a.
Lichnerowicz Theorem.
* Idea: A Generalization
of the Lichnerowicz theorem for black hole solutions.
Heat Bath, Engine, Equation, Flow, Kernel > see heat.
Heat Capacity > see specific heat.
Heat Theorem
* Idea: The statement
that the
heat added to a system divided by its temperature is an exact differential–of
the entropy.
Heaven Spaces, Heavenly Equation > see complex structure [techniques in general relativity].
Hecke Algebra
@ Double: Cherednik m.QA/04 [intro].
Heckmann-Schücking Solution > see bianchi I models.
Hegerfeldt Theorem > see locality [localization].
Height of a Group > see Solvable Group.
Heine-Borel Theorem > see compactness.
Heisenberg Algebra, Group > see group types; deformation quantization; Solvable Group; uncertainty principle.
Heisenberg Chain / Model > see spin models.
Heisenberg Principle > see uncertainty.
Heisenberg Representation of Quantum Mechanics > see representations of quantum mechanics.
Helical Symmetry > see electromagnetism [waves]; Scalar Gravity; wave equation.
Helicity > see spinors in field theory.
Helmholtz Equation > see wave equation.
Helmholtz Free Energy > see Free Energy.
Helmholtz Resonator
* Idea: A bottle or
cavity with a neck and an opening at the end (like a regular bottle); If
one blows across the opening, the air in the neck acts like a mass on a spring
(the air inside the cavity) and sound is produced.
Hénon Map, Hénon-Heiles > see chaotic systems.
Hermite Polynomial
@ Generalized: Dattoli & Torre JMP(95)
[and phase space formalisms in classical and quantum mechanics]; Jing & Yang mp/02 [deformed].
@
In superspace: Desrosiers et al NPB(03)ht,
JPA(04)ht/03;
De Bie & Sommen JPA(07)-a0707.
Hermitian Form > see Bilinear Form.
Hermitian Operator > see operator theory.
Heron's Formula > see simplex.
Hessian
$ Def: For a function
of many variables f : Rn → R,
the matrix Hij(x):= 2f / xix j.
Heun Equation / Functions
@ Solutions: Maier MC(07)m.CA/04 [the
192 solutions]; Gurappa & Panigrahi JPA(04)mp [polynomial
solutions]; Valent mp/05-in.
Heyting Algebra
@ And quantum mechanics: in Markopoulou NPPS(00)ht/99-in.
Hidden Variable Theories > s.a. pilot wave.
Hierarchical Methods / Theme in Physics > see paradigms in physics.
Hierarchical Models in Cosmology > see cosmological models; dark matter; galaxies.
Hierarchy Problem in Particle Physics
* Idea: A fine-tuning
problem for the standard model; The fact that the masses in the electroweak
sector of the standard model (× 100 GeV) are very small wrt the scale
naturally appearing in the theory, set by the (renormalized) 
in
the Higgs potential, which is divergent and can be saved by new physics, possibly
only
at 1016 GeV (GUT scales)! The only quark
whose mass is of the right order is the t; In another form,
the great disparity between the strengths of the gravitational force
and the
other forces; The fact that the Planck length and time are so small
compared
to atomic scales, while the Planck energy is so large.
* Proposed solutions:
(1) Dirac's work on the large number hypothesis, leading to the prediction
of the time-variation of the gravitational constant;
(2) Mechanism involving grand unification and susy (non-renormalization theorem);
(3) Large extra dimensions, or Kaluza-Klein-type without compactification (> see brane
world), in which gravity is much weaker than the others because it leaks
into the extra dimensions.
@ References: Dirac Nat(37)feb;
Gross PT(89)jun;
Goldman & Nieto MPLA(05)
[proposal].
Higgledy-Piggledy > see physics.
Higgs Mechanism / Field > s.a. electroweak
theory [Higgs
boson]; kaluza-klein [alternative].
* Idea: A mechanism
by which one can give a mass to the otherwise massless Goldstone bosons arising
from the spontaneous breaking of some group; Analogous to the formation of
Cooper pairs in a superconductor, giving mass to the photon (not a fundamental
field; disappears above T0).
* Gravitational version:
Spacetime arises due to the dynamical breaking of diffeomorphism invariance
in the early universe, with the connection
and metric as Goldstone/Higgs bosons.
@ General references: Higgs PL(64), PRL(64);
Klein & Lee PRL(64);
Englert & Brout PRL(64);
Higgs
PR(66);
Sardanashvily IJGMP(06)
[geometry]; Gassner & Lesch IJTP-ap/06
[bounds on time variation of expectation value]; Smeenk PhSc(07)
[gauge-invariant content]; Lyre a0806 [criticism].
@ And experiment: He et al PRL(07),
news pw(07)mar [hints at Fermilab?].
@ Gravitational version: Percacci NPB(91)-a0712;
Kakushadze & Langfelder MPLA(00)ht;
Consoli hp/01 [Newtonian
gravity from Higgs field], hp/02 [general
relativity from Higgs condensate]; Kirsch PRD(05)ht;
Leclerc
AP(06)
[gauge theory of gravity]; Boulanger & Kirsch PRD(06)ht/06;
Arkani-Hamed et al JHEP(07)
[low-energy dynamics]; 't Hooft a0708 [unitarity,
and QCD]; Kakushadze IJMPA(08)-a0709 [and
massive gravity], IJGMP(08)
[massless limit]; Oda a0709 [with
Polyakov-type scalar field action]; > [s.a. Induced
Gravity].
@ Other variations: Flachi & Toms PLB(00)
[in Randall-Sundrum]; Petriello NPB(01)ht [non-commutative];
Slavnov TMP(06)
[and extra dimensions]; Nakanishi a0704 [unphysical,
and Lorentz invariance violation]; Porto & Zee a0712 [a
private Higgs field for
each fermion]; Wanng a0801.
Higher-Derivative Theories of Gravity > same as higher-order theories.
Higher-Dimensional Theories of Gravity
Higher-Order Theories of Gravity > s.a. higher-order quantum gravity.
Hilbert Action > see actions for general relativity.
Hilbert Matrix
$ Def: The matrix Hij
= (i + j – 1)–1.
Hilbert Problem
$ Def: Given a connected
region S 
C,
with boundary L =
L0 
L1 ... Lp,
where L0
encloses all the other L's and they are
all disjoint (S is a finite or infinite region with holes), and
two nonvanishing
functions G(t) and g(t) satisfying the
Hilbert condition on L, find a sectionally holomorphic function f of
finite degree at infinity,
with
the boundary condition that f +(t)
= G(t) f –(t)
+ g(t).
* Example: If g(t) = 0, we have the homogeneous Hilbert
problem.
Hilbert Space > s.a. operator theory.
Hilbert-Krein Structure > see Supermanifolds.
Hill System
@ Chaos: Chicone et al HPA(99)gq [perturbation
of Kepler problem].
Hirzebruch Signature > see 4D manifolds.
Hirzebruch Signature Theorem > see Index Theorem.
History of Physics > s.a. by areas; XX century; quantum theory; relativity; or under Chronology.
Hochschild Cohomology > see types of cohomology.
Hodge Dual of a Form > see differential forms.
Hodge Theorem > see decomposition.
Hölder Condition > s.a. analysis [continuity].
$ Def: A function f(t),
t 
L R,
satisfies the Hölder
condition H(r), r > 0, if for some A > 0 and
for all t, t' L,
| f(t') – f(t) | 
A |t' – t|r .
* More variables:
The condition generalizes to, for example, | f(u',v') – f(u,v)
| A |u' –
u|r + B |v' – v|s,
for some A, B > 0
and for all (u,v), (u',v') L.
* Remark: A and B are
the Hölder constants, usually of no interest,
r and s the Hölder indices/exponents, which quantify
the "degree of non-differentiability" of the function.
* Relationships: Clearly,
H(r) implies continuity; It is a generalization of the Lipschitz condition,
the case r = 1.
@ References: in Muskhelishvili 77.
Hölder Inequality
$ Def: If p and q are two numbers such that
1/p + 1/q = 1,
with p, q 
1,
then for all f and g
fg 1 f p g q .
* Relationships: This is a generalization of the Schwarz Inequality.
Hole > s.a. black holes; Dirac
Sea.
* Idea: An empty state
near the top of an energy band that (as Peierls showed) behaves like a positive
charge.
* Applications: Essential
notion in solid state electronics.
Hole Argument (Einstein) > s.a.
Covariance; observables;
spacetime [and substantialism].
* Idea: An argument illustrating
the confusing role of spacetime diffeomorphisms and gauge transformations,
in which
by making a diffeomorphism in a subset of spacetime, one reaches the apparent
conclusion
that general relativity is not deterministic, or the better conclusion that
manifold "points" have
no physical significance, due to the general covariance of the field equations
(has been used against the substantialist position).
* History: The "Lochbetrachtung"
was formulated by Albert Einstein
in 1913 in his search for a relativistic theory
of gravitation, and long deemed to be based on a trivial error
of Einstein until 1980 when John Stachel recognized its highly
non-trivial character (talk on Einstein's Search for General
Covariance, 1912-1915, at the 1980 GRG meeting in Jena); Since then the
argument has been discussed by many physicists and philosophers
of science.
@ References: Rynasiewicz PhSc(96)
[no syntactic solution]; Brans GRG(99)
[logic, gauge, and spacetime model];
Macdonald AJP(01);
Bain PhSc(03) [and Einstein algebras]; Norton in(04);
Stachel & Iftime gq/05;
Rickles SHPMP(05),
comment Pooley SHPMP(06),
reply SHPMP(06)
[and lqg]; Iftime & Stachel GRG(06)gq/05 [covariant
theories]; Lusanna & Pauri SHPMP(06)gq [dissolution];
Iftime gq/06-in
[coordinate-free formulation], gq/06/JMP;
in Brading & Ryckman SHPMP(08)
[and Hilbert's axiomatic approach].
Holevo Capacity > s.a. quantum
states and systems.
* Idea: The maximum Holevo
information at the output
of a quantum channel, which quantifies its capacity
for communication of classical information.
Holism > s.a. philosophy
of physics.
* Idea: A physical theory
is holistic if it is not possible to infer the global
properties of a system purely by local measurements;
Thought to manifest itself primarily in quantum entanglement, but also in other
aspects
of quantum theory and gauge theory.
@ References: Esfeld 01; Healey SHPMP(04)
[and gauge theory]; Bartels et al SHPMP(04)
[intro]; Seevinck SHPMP(04)qp [definition,
and quantum mechanics].
Holography > see in field theory; optical technology; physics.
Holomorphic Function > see analytic function.
Holors
* Idea: A generalization
of tensors.
@ References: Moon & Spencer 86.
Homeomorphism
$ Idea: A mapping between
topological spaces preserving all of the topological structure.
$ Def: A bijection f : X → Y between
two topological spaces, which is continuous together with its inverse.
Homeomorphism Problem
$ Idea: The fundamental
problem of topology, which consists in finding a general way to decide whether
two given topological spaces are homeomorphic; It was proved unsolvable by
A Markov in 1958 (there cannot exist any algorithm that can determine whether
two simplicial complexes of dimension greater than 3 are homeomorphic).
@ References: Markov in(58); Gao T&A(04)
[countable spaces].
Homeotopy Group
$ Idea: The group 
0(Diff M)
of isotopically inequivalent diffeomorphisms of M.
@ References: in Friedman & Witt in(88).
Homoclinic Bifurcations, Orbits > see descriptions of chaos.
Homogeneous Point Process > see statistical geometry.
Homogeneous Space in Mathematics > s.a. Covering
Number; Klein Geometry.
$ Def 1: A topological
group which is the coset space G/H of some Lie group G wrt
a closed subgroup H.
$ Def 2: A topological
space S on which a group G acts effectively and transitively.
* Example: C2 for
the action of SL(2,C).
* Structure: They
have a natural metric, inherited from that on G.
@ References: Sabinin 04 [mirror geometry].
Homogeneous Spacetime > s.a.
bianchi models; galaxy
distribution; Isotropy; matter
in cosmology [problem].
$ Idea: A
spacetime is (spatially) homogeneous if there is a 1-parameter family of
hypersurfaces
t foliating
the spacetime, such that for any t and p,
q in t there
is an isometry taking p to q.
@ Of matter: Rodewald AJP(90)
[and entropy-disorder].
@ Of spacetime: Van den Bergh CQG(89)
[kinematical and observational]; Lemos & Ribeiro a0805-A&A [spatial and observational].
@
Curvature homogeneous pseudo-Riemannian: Gilkey & Nikcevic
CQG(04)m.DG/03,
CQG(04)m.DG,
IJGMP(05)m.DG;
Dunn & Gilkey m.DG/03 [not
locally homogeneous]; Dunn et al m.DG/04 [signature
(2,2), complete].
Homological Algebra
@ References: Northcott 60 [intro]; Jans 64 [intro]; Hilton & Stammbach
71; Strooker 78; Rotman 79.
Homology > s.a. types of homology theories.
Homology Manifold
$ Idea: A generalization
of the concept of manifold.
Homomorphism > s.a. category; Cokernel;
group theory.
$ Idea: A structure-preserving
mapping between two algebraic objects (e.g., groups); The image is a substructure
of the range.
* Properties: It
has an inverse (which is unique) iff it is an isomorphism.
Homothecy (Group, Transformation, Killing vector field) > see conformal structures.
Homothetical Curvature
$ Def: The tensor
a gbc.
Homotopy > s.a. fundamental group.
Hooke's Law
@ References: Glass & Winicour JPA(73)
[geometric generalization].
Hoop Conjecture > see gravitational collapse.
Hoop Group > see loops.
Hopf Algebra > s.a. generalized
coherent states; Noether
Theorem; quantum
groups;
renormalization; self-dual
gravity.
* In quantum field theory:
A Connes and D Kreimer discovered a Hopf algebra structure on the Feynman
graphs
of
scalar field theory.
@ General references: Duchamp et al a0802 [intro].
@ And differential equations, dynamical systems: Cariñena et
al m.CA/07-IJGMP.
@ And non-commutative geometry: Connes & Kreimer CMP(98)
[and renormalization]; Varilly ht/01-ln;
Aschieri ht/07-ln;
> s.a. quantum spacetime.
@ In quantum field theory: Connes & Kreimer LMP(99)ht, LMP(01)
[Feynman graphs and renormalization]; Kastler mp/01-in
[Connes-Moscovici-Kreimer]; Sardanashvily qp/02 [and
Fock representation]; Weinzierl EPJC(04)ht/03-in;
Chryssomalakos
ht/04-in
[applications]; Kreimer & Yeats ht/06 [and
short-distance structure]; Van Suijlekom LMP(06)
[and renormalization group in
QED]; Brouder ht/06;
Prokhorenko a0705 [for
non-abelian gauge theories].
Hopf Bifurcation Theorem
$ Idea: It concerns
the splitting of equilibrium solutions in a family of vector fields, like
the different positions of a marble in a slow vs fast rotating ball (at the
bottom vs on the side); applies to chaotic and fractal systems.
@ References: Marsden & McCracken 76.
Hopf Conjecture
$ Def: A compact, even-dimensional
manifold which admits a Riemannian metric of positive sectional curvatures
must have positive Euler number.
* Status: 1976 [@ Geroch
GRG(76)]
Known to be true for homogeneous manifolds, and for arbitrary
manifolds
in dimensions 2 and 4; The latter result has two, apparently
entirely different, proofs, one using Synge's theorem and the other the Gauss-Bonnet
formula, but neither can be generalized directly to dimensions
6
or greater; The Hopf conjecture in these higher dimensions is open.
Hopf Fibration
* Idea: The fibration
of S3 as a twisted S1-bundle
over S2, or S7 as
a twisted S3-bundle over
S4.
@ In physics: Urbantke JGP(03) [overview].
Hopf Invariant
Hopf-Rinow Theorem
$ Def: For a connected
Riemannian manifold M, the following are equivalent, (1) The distance
function d(x,y) is Cauchy complete; (2) M is
geodesically complete; (3) Any x, y can be joined by a
minimizing geodesic.
Horismos [> s.a. spacetime
subsets.]
$ Def: The future horismos
of a point p in spacetime is E+(p):= J+(p)
\ I+(p).
* Relationships: It is
contained in (but does not in general coincide with) the set of points lying
on future-directed null geodesics from p; It also does not necessarily
coincide with J+(p) = I+(p).
Horizons > s.a. event horizon.
Horizontal Tensor Field > see tensor fields.
Hp Spaces
* History: Studied first
by Hardy in 1915, they have been extended to Rn,
Cn and other topological spaces.
$ Def: If f is
holomorphic on 
:= {z C,
|z| 1}, then f Hp iff
f H_p:=
supr < 1 (
02Pi
|f(r exp{i
})|p d/2 )1/p .
* Special case: Hinfty is
the ring of bounded holomorphic functions on ,
with the sup norm.
@ Text: Koosis 80.
Hubbard Model
@ References: Schupp PLA(97)
[quantum symmetry]; Hou et al NPB(00)
[SU(3)]; Deguchi et al PRP(00),
Lieb & Wu cm/02-in
[1D].
Hubble's Constant > see cosmological expansion.
Hudson's Theorem > see wigner function.
Hughes-Drever Experiment > see torsion.
Hurewicz Isomorphism Theorem
* Result: It concerns
the relationship between homotopy and homology groups for a topological space
X; If q(X)
= 0 for all q < n, with n 2,
then Hq(X)
= 0 as well for all q < n, and Hn(X)
= n(X);
Not true for n = 1, but for that case what is true is that 1(X)/[1(X), 1(X)]
= H1(X), or H1(X)
is the abelianization of 1(X).
@ References: Eilenberg AM(44); Eilenberg & MacLane AM(45).
Hurwitz Zeta Function > see Zeta Function.
Husain-Kuchar Model > see types of gauge theories; BF theory.
Husimi Phase Space Distributions / Functions >
s.a. locality in quantum mechanics; quantum mechanics
in phase space.
* Idea: One of the types
of distribution functions used to describe quantum theory in phase space.
@ References: Davidovic & Lalovic JPA(93),
et al JPA(94);
Novaes & de Aguiar PRA(05)qp/04 [spin
systems]; Fan & Guo qp/06 [e in
constant B field]; Toscano et al a0705-PRS
[intermediate Husimi-Wigner representation]; > s.a. types
of
coherent
states.
Hydrodynamic Decomposition > see decomposition.
Hydrodynamics > see fluids [including smooth particle hydrodynamics].
Hydrogen Atom > see [elements]; canonical quantum mechanics; deformation quantization; interactions; quantum systems; wigner functions.
Hyperbola > see conical sections.
Hyperbolic Functions
* Def:
The functions sinh 
:= 
(exp{} – exp{–}),
cosh := (exp{}
+
exp{–}).
* Basic
properties: cosh2 – sinh2 =
1, d(cosh )/d =
sinh , d(sinh )/d =
cosh .
* Relationship with trigonometric functions: Given by
sin i = i sinh ,
cos i =
cosh , tan i =
i tanh .
Hyperbolic Geometry > see geometry.
Hyperbolic Numbers > same as Perplex Numbers? see trigonometry [hyperbolic].
Hypercharge
$ Def: By definition Y:
= B + S, where...
* Remark: It is a good
quantum number for strong interactions, not for weak interactions.
Hypercolor > see composite models [rishons].
Hypercomplex Number > see spinors.
Hyperfluid > see fluid dynamics.
Hypergeometric Equation / Function
@ General references: Schrödinger PRIA(41)phy/99 [factorization];
in Abramowitz & Stegun
ed-65; Gelfand & Graev LMP(99)
[GG function approach]; Thorsley & Chidichimo
JMP(01),
Chidichimo & Thorsley JMP(01)
[asymptotic expansion of 2F1(a,b;c,x)].
@ Confluent hypergeometric functions: Saad & Hall JPA(03) [integrals].
@ Generalized: Ruijsenaars CMP(99), CMP(03), CMP(03); Tarasov & Varchenko LMP(05)
[identities].
Hypergraph > see types of graphs.
Hypergravity > s.a. BRST [quantization].
* Idea: A first version
was the spin-(5/2) gravity analog of sugra, which has been shown to be inconsistent;
The name has later been used for a different type of theory.
@ References: Aragone & Deser CQG(84), CQG(85); Sijacki gq/99-in.
Hyperkähler Structure > s.a.
symplectic structure.
$ Def: A manifold M with
3 complex structures J1,
J2, J3,
satisfying Ji Jj =
ijk Jk,
and ...
@ General references: in Yano 65; Salamon IM(82);
in Yano & Kon 84.
@ Related topics: Hitchin et al
CMP(87), CMP(87)
[in physics]; Hashimoto et al JMP(97)ht/96 [4D
manifolds]; Grantcharov & Poon
CMP(00)
[with torsion]; Dunajski & Mason CMP(00)
[and twistors].
Hypermomentum > see fluid.
Hyperons > see Baryons, neutron stars.
Hyperphoton
* Idea: The quantum
of the fifth force field postulated after a reanalysis of the Eötvös
experiment by Fischbach et al; Produces a repulsive force with a range 
100
m: m 10–9 eV;
As of 1986, there is little theoretical motivation for it.
@ References: Aronson, Cheng, Fischbach & Haxton PRL(86).
Hyperspace > see embedding.
Hyperspin
@ References: Urbantke IJTP(89).
Hypersurface
$ Def: An embedded
submanifold of codimension 1.
* Null hypersurface: One
which has a non-zero normal tangent vector l a at
each point; That vector must
be null, and there can only be one, up to a local rescaling l a → f l a;
It also is tangent to null geodesics (not necessarily affinely parametrized),
since l a a l b =
l b on
the surface; Under a rescaling of l a,
the "surface gravity" transforms
as → f + l aa f.
@ Null hypersurfaces: Nurowski & Robinson CQG(00)gq [invariants
of null surfaces];
Jezierski in(04)gq [null,
rev]; > s.a. horizons [geometry].
@ Spacelike hypersurfaces: Harris CQG(88)
[closed and complete, in Minkowski]; Izumiya & Takahashi JGP(07)
[in constant curvature, like de Sitter]; Hu et al DG&A(07)
[spacelike, constant scalar
curvature,
in de Sitter].
@ Conformally flat: Garat & Price PRD(00), Valiente Kroon PRL(04)
[not in Kerr].
@ Related topics:
Harris CQG(87), CQG(88)
[complete, in Lorentzian manifolds]; Maia IJMPD(99)
[dynamics in general relativity].
> Related topics: see embedding; First
Fundamental Form; foliations;
Gauss-Codazzi Equations.
> Types of hypersurfaces:
see Cauchy Surface; extrinsic [extremal];
horizons;
spacetime subsets.
Hypersurface-Orthogonal Vector Field > see vector field.
Hysteresis
* Remark: Typically
happens for every property one measures for a system undergoing a first-order
phase
transition.
@ References: Brokate & Sprekels 96 [and phase transitions]; Rudowicz & Sung AJP(03)
[ferromagnets, misconceptions].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008