**Topics, H**

**H**** Theorem**
> s.a. pilot-wave theory [subquantum].

* __Idea__: A statement
of the second law of thermodynamics; It says that the quantity* H*
defined by *H*:= ∑_{i}
*p*_{i}
ln *p*_{i},
where *p*_{i}
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; In his proof, Boltzmann inadvertently
smuggled in a premise that assumed the very irreversibility he was trying
to prove: 'molecular chaos.'

@ __References__: von Neumann ZP(29)-a1003
[proof in quantum mechanics]; in Reif 65;
Santamato & Lavenda JMP(82) [stochastic, for diffusion processes];
Succi et al RMP(02) [and hydrodynamic simulations];
Silva cm/06 [relativistic];
Brown et al SHPMP(09) [objections and Boltzmann's response];
Boozer EJP(11) [and molecular chaos].

@ __Quantum__: Farquhar & Landsberg PRS(57);
Lesovik et al SRep(16)-a1407.

> __Online resources__:
see Wikipedia page.

**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.

* __Idea__: Any field
unitarily equivalent to a free field must itself be a free field; It shows
that the transformation between interacting and free field operators in a
reasonable quantum field theory cannot be unitary; Or, a system with
continuous degrees of freedom possesses an infinite number of inequivalent
representations of them, so the choice of representation matters.

@ __References__: in Streater & Wightman 64, p165;
Lupher IJTP(05) [history and versions];
mp/07
[and particle dressing]; Weiner CMP(11)
[algebraic version]; Antipin et al PAN(13)-a1202 [in non-commutative quantum field theory];
Antipin et al IJMPA(13)-a1305 [in a non-degenerate indefinite-inner-product space];
Seidewitz FP(17)-a1501
[avoidance in parameterized quantum field theory];
Klaczynski a1602
[renormalisation bypasses Haag's theorem].

> __Online resources__:
see Wikipedia page.

**Haar Measure** > see measure.

**Hadamard Matrices** > see entropy.

**Hadamard States**
> s.a. quantum field theory in curved spacetime.

* __Idea__: States for
quantum field theories which satisfy a constraint on the singular
structure of the associated two-point function; The condition is used as a
criterion to distinguish physically reasonable states for free fields.

@ __References__: Sahlmann & Verch RVMP(01)mp/00;
Moyassari a0705 [2D];
Sanders CMP(10)-a0903;
&Fredenhagen CQG(14)-a1307
[modified Sorkin-Johnston states]; Fewster & Verch CQG(13)-a1307
[new motivation]; Brum PhD-a1407
[explicit construction]; Dappiaggi a1501-proc [construction for spacetimes with null conformal boundaries];
Drago & Gérard a1609 [adiabatic limit].

**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'*):=
\(\langle\)0| {*φ*(*x*), *φ*(*x'*)}
|0\(\rangle\) = *G*^{+}(*x*,*
x'*) + *G*^{−}(*x*, *x'*).

* __For a spinor field__:
*S*^{(1)}(*x*, *x'*):=
\(\langle\)0| [*φ*(*x*), *φ*(*x'*)]
|0\(\rangle\) = −(i *γ*^{a}∇_{a}+*m*)
*G*^{(1)}(*x*,* x'*).

* __Properties__: It
satisfies the homogeneous field equation.

**Hadrons** > s.a. particle
types / QCD; QCD phenomenology.

**Hagedorn Temperature**

* __Idea__: The
"boiling point" of hadronic matter in QCD.

__References__: Rafelski EPJA(15)-a1508.

> __Online resources__:
see Wikipedia page.

**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.

**Hair** > see black-hole hair.

**Half-Flat Metric** > see self-dual
fields and self-dual solutions in general
relativity

**Hall Effect**

* __History__:
Initially MOSFETs were used to produce effective 2D systems; Now other
ways are used; The quantum Hall effect was discovered accidentally.

* __4D version__: Proposed theoretically in 2001, demonstration using a waveguide array with inscribed synthetic dimensions published in 2018.

@ __Quantized__: Prange & Girvin ed 90;
Von Klitzing RMP(86)
[Nobel lecture]; Elvang & Polchinski ht/02
[on \(\mathbb R\)^{4}];
Avron et al pw(03)aug [and topology];
Nair & Randjbar-Daemi NPB(04) [on S^{3}];
Leitner ATMP(08)cm/05 [in 2+1 QED];
Lederer SHPMP-a1406 [philosophical];
Jellal PLA(16)-a1504 [in graphene];
Karabali & Nair PRD(16)-a1604 [effective action, all dimensions];
Tong a1606-ln;
in Chang & Ge 17.

@ __Fractional__: Chakraborty & Pietiläinen 88;
Eisenstein & Stormer Sci(90)jun;
Murthy & Shankar RMP(03) [Hamiltonian theory];
Levin & Fisher PRB(09)
+ Burnell & Sondhi Phy(09)
[in 3D]; Roddaro et al PRL(09)
+ Grayson Phy(09)
[point junction between integer and fractional quantum Hall phases]; Jain
Phy(10) [the 5/2 enigma].

@ __4D version__: Zhang & Hu Sci(01)oct;
Price et al PRL(15); Zilberberg et al Nat(18)jan + news sn(18)jan [experimental demonstration].

@ __Other variations__: news pn(97)dec [photon Hall effect];
Strohm et al PRL(05) [phonon Hall effect];
Nagaosa et al RMP(10) [anomalous].

**Halmos Symbol**

* __Notation__: A square
symbol that looks like that for a d'Alembertian, 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].

**Hamilton's Principle**
> see variational principles in physics.

**Hamilton's Theory of Turns** > see SO(3)
and SU(2).

**Hamiltonian Dynamics**
> s.a. hamiltonian systems.

**Hamiltonian Structure, Vector Field** > see symplectic
manifolds; generalized symplectic structures.

**Hanbury Brown–Twiss Effect**
> s.a. astronomy [intensity interferometers].

* __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 cannot occupy the same state.

@ __References__: Hanbury Brown & Twiss Nat(56);
news pw(05)sep
[quantum gas analog];
news pw(07)jan
[for two isotopes of He];
Nelson & Shimpi PLA(07) [for parabosons];
Westbrook & Boiron a1004-conf [for atomic matter waves];
Perrin et al NPhys(12)-a1012 [Hanbury Brown and Twiss correlations across the BEC threshold];
Silva & Freire HSNS(13)
[and the concept of photon];
Ushba & Qureshi a1509 [for massive particle wave packets].

**Handles**

$ __Def__: An *n*-dimensional
*k*-handle is the topological manifold D^{k}
× D^{n−k} ; It is often
used as attached to another *n*-manifold.

**Hankel Functions** > see bessel functions.

**Hannay's Angle** > see phase.

**Hard Spheres** > see classical ;
chaotic systems.

**Hardy Spaces / Algebras**

$ __Hardy algebra__: A
weak *-closed subalgebra of L^{2}(*X*,
d*m*), for some finite positive measure space (*X*, *S*,
d*m*).

@ __References__: Barbey & König 77;
Folland & Stein 82.

**Hardy's Thought Experiment / Paradox **
> s.a. bell's theorem; Contextuality;
quantum locality; quantum causality;
realism.

* __Idea__: A proof
without Bell-type inequalities that certain non-local correlations violate
local realism.

@ __References__: Wechsler a0812,
Yokota NJP(09)
[implementation proposal with photons];
Sokolovski et al PLA(08)-a0903 [Feynman-path analysis];
Kastner a1006 [in the transactional interpretation];
Xiang ChPB(11)-a1007 [and violation of Bell inequality];
Ahanj PRA(10)-a1007 [and successive spin measurements, hidden-variable analysis];
Fedrizzi et al PRL(11)-a1011 [in time];
Meglicki PLA(11) [avoidance, and weak measurements];
Mansfield & Fritz FP(12)-a1105 [and possibilistic non-locality];
Suarez a1204
[questioning whether it tests non-locality of a single-particle];
Mančinska & Wehner JPA(14)-a1407 [and the CHSH inequality];
Abramsky et al I&C(16)-a1506 [for all *n*-qubit entangled states];
Bolotin JMP(18)-a1801 [in non-classical semantics].

@ __Consequences and applications__: Mansfield a1608.

> __Related topics__:
see relativistic quantum mechanics; types of quantum measurements [weak].

**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 tunneling.

**Hartree Equation** > s.a.
composite quantum systems [many-body systems].

* __Idea__: An
effective evolution equation for mean-field systems.

@ __References__: Knowles & Pickl CMP(10) [bounds on the rate of convergence of
the quantum *N*-body dynamics to the Hartree dynamics].

**Hartree-Fock Approximation / Equation**

* __Idea__: The
Hartree-Fock approximation is the basis of molecular orbital theory; Its
goal is to approximately solve the electronic Schrödinger equation by
describing each electron's motion by a single-particle function (orbital),
approximated by a single Slater determinant made up of one spin orbital
per electron, which does not depend explicitly on the instantaneous
motions of the other electrons.

@ __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];
in Lipparini 08;
Fröhlich & Knowles a0810
[from Fermi gas with Coulomb interaction]; Kraus & Cirac NJP(10)-a1005
[generalized, for fermionic systems in lattices].

> __Online resources__:
see David Sherrill's introduction;
Wikipedia page.

**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 information.

**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.

**Hawking-Malament Theorem** > see causal
structures.

**Heat Bath** > see under Thermal
Bath.

**Heat Capacity** > see specific
heat.

**Heat Engine** > s.a. thermodynamic
systems.

* __Idea__: A system
that uses thermal energy at a high temperature to produce mechanical
energy, which can then be used to do mechanical work, and exhausts the
remaining heat, which cannot be converted to mechanical energy, at a lower
temperature.

* __Quantum heat engines__:
Thermal machines where the working substance is a quantum object (in the
extreme case, the working medium can be a single particle or a few-level
quantum system); The study of QHE has shown a remarkable similarity with
macroscopic thermodynamical results.

@ __General references__: Kieu PRL(04)qp/05
[second law, Maxwell's demon]; news pw(11)jan [smaller than a typical biological cell];
Roßnagel et al PRL(14) [nanoscale, beyond the Carnot limit].

@ __Quantum heat engines__: Scully PRL(02);
Uzdin et al PRX(15) [thermodynamical equivalence of all small-action quantum engines];
interview Phy(17) [Marcus Huber].

> __Online resources__:
see Wikipedia page.

**Heat Equation, Kernel** > see heat;
specific heat.

**Heat Flow, Transfer, Transmission,
Transport** > s.a. heat [history,
heat equation]; non-equilibrium thermodynamics.

* __Mechanisms__:
Macroscopically, they are conduction (diffusion), convection and radiation; Microscopically,
in metals heat is mainly carried by free electrons, whereas in electrically insulating solids
it is transported by atomic vibrations–phonons or, in crystalline electrical insulators,
relaxons; In particular, thermal conductivity results from two processes, intrinsic scattering
between phonons due to atomic vibrations at finite temperature and disruptions to the periodic
lattice such as interfaces and point defects; > s.a. Convection.

* __Thermal conduction__:
Governed by Fourier's law **J** = −*κ* ∇*T*,
with **J** = heat flux, *κ* = coefficient of thermal
conductivity, *T* = temperature.

* __Radiative heat
transfer__: The maximum amount of heat transferred between two objects
is described by black-body theory and the Stefan-Boltzmann law in the far
field; In the near field it is more complicated, because the heat flux can
be many orders of magnitude greater due to the contribution from
evanescent waves that can tunnel between the two bodies; > s.a. thermal radiation.

@ __General references__: Bertola & Cafaro PLA(07) [conduction in Liouvillean form, speed of propagation];
Komatsu et al PRL(08)
[microscopic derivation]; Collet & Eckmann CMP(09) [model,
and Boltzmann equation]; Hovhannisyan & Allahverdyan JSM(10)-a1007
[enhanced heat transfer, model]; Wu & Segal PRA(11)-a1105
[role of quantum correlations]; Miller et al PRL(15)
+ Messina Phy(15)
[near-field radiative heat transfer].

@ __Fourier's law__: Bonetto et al mp/00 [derivation];
Seligman & Weidenmüller JPA(11)-a1011
[in quantum mechanics].

@ __Thermal conductivity coefficient__: Desloge AJP(62)dec [gas].

@ __Special systems__: Hirschfeld & Scalapino Phys(10)aug [iron-based
superconductor]; news pw(13)jan
[thermal Josephson effect and heat flow from colder to hotter];
news sci(14)jul [perfect crystals that are poor heat conductors];
Cepellotti & Marzari PRX(16)
+ McGaughey Phy(16) [in electrical insulators, relaxons];
> s.a. Lennard-Jones Fluid.

> __Online resources__:
see Wikipedia page.

**Heat Theorem
** *

**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 localization.

**Height of a Group** > see Solvable Group.

**Height of a Poset** > see partially
ordered sets.

**Heine-Borel Theorem** > see compactness.

**Heisenberg Algebra, Group**
> see group types; uncertainty principle.

* __Idea__: The group
of observables arising in 1D quantum mechanics with *x* and *p*
(plus the identity) as generators, or its higher-dimensional generalization.

$ __Def__: Abstractly,
a group with three generators *x*, *y* and *z*,
and commutation relations [*x*, *y*] = *z*, [*x*,
*z*] = 0, [*y*, *z*] = 0; If we think of these
respectively as *x*, *p*, and I, then the commutation relations become

[*q*,* p*] = iℏ I
, [*q*, *q*] = 0 , [*p*,
*p*] = 0 , or [*a*,* a**] = 1 .

@ __General references__: Costella qp/95
[[*p*,* q*] ≠ iℏ].

@ __Representations__: Mnatsakanova et al LMP(03)mp/02
[holomorphic functions]; Brodlie PhD(04)qp
[classical and quantum mechanics]; Dereziński LNP(06)mp/05.

@ __Extensions, deformations__: Baskerville & Majid JMP(93)ht/92 [braided]; Masood et al PLB(16)-a1611 [deformations motivated by the generalized uncertainty principle]; > s.a. deformation quantization.

> __Related topics__:
see Commutation Relations; Solvable
Group.

> __Online resources__:
see Wikipedia page.

**Heisenberg Chain / Model** > see spin
models.

**Heisenberg-Euler-Type Electrodynamics** > see modified electrodynamics.

**Heisenberg Principle** > see uncertainty
relations in quantum theory.

**Heisenberg Representation of Quantum Mechanics** > see
representations of quantum mechanics.

**Helical Symmetry** > see electromagnetism[waves];
Scalar Gravity; types
of wave equations.

**Helicity** > see spinors in
field theory; spin-2 fields.

**Hellmann-Feynman Theorem**

* __Idea__: A result
in quantum mechanics relating the derivative of the total energy with
respect to a parameter to the expectation value of the derivative of the
Hamiltonian with respect to that same parameter.

@ __References__: Esteve et al PLA(10)
[generalization]; Xu et al IJTP(12)
[generalization to ensemble averages].

> __Online resources__:
see Wikipedia page.

**Helmholtz Equation** >
s.a. types of wave equations.

* __Idea__:
The partial differential equation obtained from the wave equation for a homogeneous medium,
with an oscillating disturbance of frequency *ω* and amplitude *a*^{2}
*f*(*x*), and when one looks for a solution *u* of the same frequency,

∇^{2}*u* + *k*^{2}*u*
= −*f*(*x*) , with *k*:=
*ω*^{2}/*a*^{2} .

@ __References__: Schmalz et al AJP(10)feb [Green's
function]; Amore JMP(10)
[in general domains, ground state and excited states].

> __Online resources__:
see MathWorld page.

**Helmholtz Free Energy** > see Free Energy.

**Helmholtz's Free Mobility Postulate** > see under Free
Mobility Postulate.

**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.

**Helmholtz Therem** > see vector
field decomposition.

**Hempel's Dilemma** > see philosophy
of physics.

**Hénon Map ** > s.a. chaotic systems.

* __Hénon map__: An
unstable map \({\mathbb R}^2 \to {\mathbb
R}\)^{2}, given by (*x*, *y*)
\(\mapsto (1+y-ax^2,\,bx)\); Its behavior depends crucially on the values of *a*
and *b*; For an interesting case, look at *a* = 1.4, *b* = 0.3.

@ __ References__: Benedicks & Carleson AM(91);
Harsoula et al JPA(15)-a1502
[analytical formulae for the chaotic regions].

**Hénon-Heiles System** > s.a.
chaotic systems.

* __Idea__: A 2D
chaotic dynamical system, with potential that can be written as

*V*(*x*, *y*) = *k*^{2}
[\(1\over2\)(*x*^{2} + *y*^{2})
+ (*x*^{2}*y* − \(1\over3\)*y*^{3})/*a*)] ;

For *a* = *k* = 1, *E* \(\in\) [0, 1/10]
the behavior is regular, for *E* \(\in\) [1/10, 1/6] it is chaotic, and for *E* > 1/6 unbounded.

@ __References__: Hénon & Heiles AJ(64);
Fordy PhyD(91);
Vernov TMP(03)mp/02
[solutions]; Ballesteros & Blasco AP(10)-a1011
[2D integrable systems and perturbations]; > s.a. toda lattice.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Hermite Polynomial**

@ __General references__: Moya-Cessa a0809
[new expression].

@ __Properties and related results__: Wang a0901
[integrals of products].

@ __Generalized__: Dattoli & Torre JMP(95)
[and phase-space formalisms in classical and quantum mechanics]; Jing
& Yang mp/02
[deformed]; Maheshwari et al a1411 [tensor].

@ __In superspace__: Desrosiers et al NPB(03)ht,
JPA(04)ht/03;
De Bie & Sommen JPA(07)-a0707.

> __Online resources__:
see Wikipedia page.

**Hermitian Form** > see Bilinear Form.

**Hermitian Operator** > see operator theory.

**Heron's Formula** > see simplex.

**Hessian**

$ __Def__: For a
function of many variables \(f : {\mathbb R}^n \to {\mathbb R}\), the matrix
\(H_{ij}(x):= \partial^2 f / \partial x^i\partial x^j\).

**Heun Equation / Functions**

* __Idea__: The local
Heun function is the solution of Heun's equation, a second-order linear
ordinary differential equation.

@ __References__: Maier MC(07)m.CA/04 [the 192 solutions];
Gurappa & Panigrahi JPA(04)mp [polynomial solutions];
Valent mp/05-conf;
Hortaçsu a1101-proc [in physics];
Fiziev a1405 [novel representation];
Birkandan & Hortaçsu RPMP(17)-a1605
[applications in quantum field theory].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Heyting Algebra**

@ __And quantum mechanics__: in Markopoulou NPPS(00)ht/99.

**Hidden Sector in Particle Physics** > see particle
physics; dark matter.

**Hidden Variable Theories**
> s.a. pilot-wave interpretation.

**Hierarchical Methods / Theme in Physics** > see paradigms in physics.

**Hierarchical Models in Cosmology** > see cosmological
models; dark matter;
galaxies.

**Hierarchy Problem in Particle Physics**
> s.a. neutrino [neutrino mass hierarchy problem].

* __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
with respect to the scale naturally appearing in the theory, set by the
(renormalized) *μ* in the Higgs potential, which is divergent and
can be saved by new physics, possibly only at 10^{16}
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 supersymmetry (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; (4) Warped extra dimensions; (5) Technicolor.

@ __General references__: Dirac Nat(37)feb;
Gross PT(89)jun;
Tkach MPLA(09)-a0808 [and higher-derivative quantum gravity];
Fabbri IJGMP(16)-a1504 [and standard model extension];
Fowlie a1507
[the big- and little-hierarchy problems and Bayesian probability]; blog forbes(15)12.

@ __Proposed solutions__: Goldman & Nieto MPLA(05) [proposal];
Cassel et al PLB(10)
[supersymmetry, and tests]; Graham et al PRL(15)
+ Dine Phy(15)
[solution using relaxion field]; Fabbri IJGMP(16)-a1504 [generalized Dirac equation and Higgs boson as top-quark condensate];
Arvanitaki et al JHEP(17)-a1609 [and Weinberg's anthropic solution to the cosmological constant problem];
Hook a1802 [non-linearly realized discrete symmetries].

**Higgledy-Piggledy** > see physics.

**Higgs Mechanism / Field / Boson**

**Higher-Derivative Theories of Gravity ** > same as higher-order theories.

**Higher-Dimensional Theories of Gravity**

**Higher-Order Lagrangian Systems**

**Higher-Order Theories of Gravity**
> s.a. types of theories,
phenomenology
and higher-order quantum gravity.

**Higher-Spin Gravity** > see theories of gravitation.

**Hilbert Action** > see actions for general relativity.

**Hilbert's Grand Hotel**

* __Idea__: A story of
an imaginary hotel with infinitely many rooms that illustrates the bizarre
consequences of assuming an actual infinity of objects or events; Invented
in 1947 by George Gamow, who jokingly attributing it to Hilbert; Since the
1970s it has been used in arguments, ranging from cosmology to philosophy
and theology.

@ __References__: Kragh a1403.

**Hilbert Matrix**

$ __Def__: The matrix
*H*_{ij} = (*i* + *j*
− 1)^{−1}.

**Hilbert Problem**

$ __Def__: Given a
connected region \(S \subset {\mathbb C}\), with boundary \(L =
L_0 \cup L_1 \cup\ldots\cup L_p\), where *L*_{0}
encloses all the other *L*_{i}s and they
are all disjoint (*S* is a finite or infinite region with holes), and two
non-vanishing 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's Program**
> see mathematics.

**Hilbert Space**
> s.a. operator theory.

**Hilbert Transform**

@ __References__: Cundin & Barsalou a1105 [and Stieltjes' integral theorem].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Hilbert-Krein Structure** > see Supermanifolds.

**Hilbert-Polya Conjecture** > see Zeta Function.

**Hill System**

@ __Chaos__: Chicone et al HPA(99)gq
[perturbation of Kepler problem].

**Hipparcos Satellite** > see stars.

**Hirzebruch Signature** > see 4D
manifolds.

**Hirzebruch Signature Theorem** > see Index
Theorem.

**History of Physics** >
s.a. by areas; XX-century
physics; quantum theory; relativity;
or under Chronology.

**HJW Theorem** > see mixed
quantum states.

**Hochschild Cohomology** > see types
of cohomology.

**Hodge Dual of a Form** > see differential
forms.

**Hodge Operator**

@ __References__: Castellani et al a1511
[construction based on a Fourier (Berezin)-integral representation].

**Hodge Theorem** > see decomposition.

**Hodograph**

* __Idea__: The
hodograph of a non-relativistic particle motion in Euclidean space is the
curve described by its momentum vector.

@ __References__: Gibbons a1509
[fate in special and general relativity].

**Hofstadter's Butterfly**

* __Idea__: An effect
appearing in the energy levels of electrons exposed to a magnetic field in
a 2D lattice; When plotted as a function of the magnetic field, the
electronic energy spectrum takes on a complex pattern that resembles a
butterfly.

@ __References__: Hofstadter PRB(76);
Petschel & Geisel PRL(93);
news cuny(13)may,
SA(13)sep
[experimental confirmation]; Chin & Mueller Phy(13)
[in an optical lattice of atoms]; Jones-Smith & Wallace IJTP(14)-a1407
[non-Hermitian continuation].

**Hölder Condition** > s.a.
analysis [continuity].

$ __Def__: A function *f*(*t*),
*t* ∈ *L* ⊂ \(\mathbb R\), 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
** $

|| *fg* ||_{1}
≤ || *f* ||_{p} || *g*
||_{q} .

* __Relationships__:
This is a generalization of the Schwarz Inequality.

**Hole (mathematics)** > see graph theory.

**Hole (physics)** > s.a. black
holes; Dirac Sea.

* __Idea__: A
quasiparticle consisting of an empty state near the top of an energy band
that (as Peierls showed) behaves like a positive charge.

* __Applications__: it
is an essential notion in solid state electronics.

@ __References__: Berciu Phy(09)
[hole motion through an ordered insulator].

**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.

@ __General references__: Rynasiewicz PhSc(96)sep
[jstor] [no
syntactic solution]; Brans GRG(99)
[logic, gauge, and spacetime model]; Macdonald AJP(01)feb;
Bain PhSc(03)dec [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 in(08)gq/06
[coordinate-free formulation], gq/06/JMP;
in Brading & Ryckman SHPMP(08) [and Hilbert's axiomatic approach];
Stachel LRR(14) [historical-critical study and contemporary implications];
Weatherall a1412,
comment Roberts a1412 [re mathematical formalism];
Weinstein a1504 [and Einstein's uniformly rotating disk];
Gryb & Thébault PhSc(16)-a1512

@ __Quantum version__: Schmelzer a0902;
Weinstein a1301 [and the PBR theorem].

**Hole-ography** > see ads/cft correspondence.

**Holevo Bound / 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.

@ __References__: Giovannetti et al PRA(12)-a1012 [procedure for asymptotically achieving the Holevo bound];
Zwolak & Zurek sRep-a1303 [and quantum discord];
Bousso PRL(17)-a1611 [universal upper bound on the communication channel capacity].

**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];
Arageorgis SHPMP(13) [holism and non-separability in quantum field theory].

> __Online resources__:
see Wikipedia page;
James Schombert page.

**Holographic Principle** > see holography in field theory.

**Holographic Screen**

* __Idea__: A future
holographic screen is a hypersurface foliated by marginally trapped
surfaces.

* __Examples__: Future
holographic screens can be found in collapsing stars and near a big
crunch; Past holographic screens exist in any expanding universe. Unlike
event horizons, these objects can be identified at finite time and without
reference to an asymptotic boundary.

* __Result__: The area
of a future holographic screen increases monotonically along the
foliation, and similarly for past holographic screens.

@ __References__: Bousso & Engelhardt PRL(15)-a1504 [area law].

**Holography **> see optical technology;
holographic principle and physical theories.

**Holometer** > see holographic principle.

**Holomorphic Function** > see analytic function.

**Holon** > see Luttinger Liquid.

**Holonomic Frame, Vierbein** > see Frames; tetrads.

**Holors**

* __Idea__: A generalization of tensors.

@ __References__: Moon & Spencer 86.

**Holst Action** > see first-order
actions and connection formulation for general relativity.

**Homeomorphism**

$ __Idea__: A mapping
between topological spaces preserving all of the topological structure.

$ __Def__: A bijection
\(f : X \to Y\) between two topological spaces that 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 \(\pi_0({\rm Diff}\ M)\)
of isotopically inequivalent diffeomorphisms of *M*.

@ __References__: in Friedman & Witt in(88).

**HOMFLY Invariant / Polynomial** > see knot invariants.

**Homoclinic Bifurcations, Orbits** > see descriptions of chaos.

**Homogeneity, Spatial**

* __Idea__: Translational invariance.

> __Local concept__:
see Position [position invariance].

> __In cosmology__:
see cosmological principle; galaxy
distribution; large-scale geometry of the
universe; matter distribution
[correlations, fractal].

**Homogeneous
Manifold with Metric** > s.a. 3-manifolds;
bianchi models; geodesic
[homogeneous geodesic]; Isotropy.

$ __Locally homogeneous
manifold__: One in which for every two \(p,\ q \in M\)
there exist neighborhoods *U* of *p* and *V* of *q*
and an isometry \((U, p) \to (V, q)\).

$ __Globally
homogeneous manifold__: One in which the isometry group acts
transitively on all of *M*.

$ __Spatially
homogeneous spacetime__: A spacetime is (spatially) homogeneous if
there is a 1-parameter family of hypersurfaces \(\Sigma_t\) foliating
the spacetime, such that for any *t* and *p*, *q*
in \(\Sigma_t\) there is an isometry taking *p* to *q*.

@ __General references__: Papadopoulos & Grammenos JMP(12)-a1106
[finding all symmetries of an *n*-dimensional locally homogeneous space].

@ __Matter distribution__: Rodewald AJP(90)feb
[and entropy-disorder].

@ __Spacetime__: Van den Bergh CQG(89)
[kinematical and observational]; Lemos & Ribeiro A&A(08)-a0805
[spatial and observational].

@ __Curvature-homogeneous pseudo-Riemannian__: Gilkey & Nikčević
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].

**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* with respect to a closed subgroup *H*.

$ __Def 2__: A
topological space *S* on which a group *G* acts
effectively and transitively.

* __Example__: \(\mathbb C\)^{2} for
the action of SL(2, \(\mathbb C\)).

* __Structure__: They
have a natural metric, inherited from that on *G*.

@ __References__: Sabinin 04
[mirror geometry].

**Homological Algebra**

* __Idea__: The branch
of mathematics which studies homology in a general algebraic setting.

@ __References__: Northcott 60
[intro]; Jans 64 [intro]; Strooker 78;
Hilton & Stammbach 97; Osborne 08
[III, with examples and exercises]; Rotman 08;
Gelfand & Manin 10;
Grandis 12
[and distributive lattices, orthodox semigroups], 13
[strongly non-abelian settings].

> __Online resources__:
see Wikipedia page.

**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 / Homothetic Transformation, Vector Field**
> see conformal structures.

**Homothetical Curvature**

$ __Def__: The tensor \(\nabla_{\!a}g_{bc}\).

**Hong-Ou-Mandel Effect / Interference** > see quantum optics.

**Homotopy** > s.a. fundamental group.

**Hooke's Law**

@ __Generalizations__: Glass & Winicour JPA(73) [geometric];
dell'Isola et al a1008 [isotropic second gradient materials].

**Hoop Conjecture** > s.a. gravitational
collapse; quantum-gravity phenomenology.

* __Idea__: Any chunk
of matter compressed enough in all directions (for example two colliding
particles or a gravitationally collapsing object), becomes a black hole
and develops a horizon; In the spherically symmetric case, this occurs
when the system occupies a sphere whose radius is smaller than its
Schwarzschild radius; A more general precise formulation of the hoop
conjecture in four spacetime dimensions is that the Birkhoff invariant *β*
(the least maximal length of any sweepout or foliation by circles) of an
apparent horizon of energy *E* and area *A* should satisfy
*β* ≤ 4π* E*.

$ __Def__: A black hole
will form iff a mass *M* is compacted to a region with
circumference *C* < 4π*GM* in every direction.

@ __General references__: Thorne in(72);
in Misner et al 73, p868;
Bonnor PLA(83),
PLA(84);
Ponce de León GRG(87)
[counterexample]; Barrabès et al CQG(92);
Chiba & Maeda PRD(94)
[+ Λ]; Gonçalves gq/03-GRF,
PRD(03)gq,
CQG(03)gq [evidence, with isometries];
Nakao et al PLB(03) [brane world];
Senovilla EPL(08)-a0708 [general reformulation];
Ó Murchadha et al PRL(10) [in terms of Brown-York mass];
Cvetič et al CQG(11)-a1104.

@ __Quantum version__: Casadio et al PLB(14)-a1311 [and particle collisions];
Yang RiP(16)-a1512 [natural cutoff for vacuum energy];
Anzà & Chirco PRL(17)-a1703.

@ __Specific types of situations__: Chiba PRD(99)gq [non-axisymmetric];
Yoshino & Nambu PRD(02) [high-energy collisions];
Yoshino PRD(08)-a0712
[collision of two *pp* waves, highly distorted apparent horizon];
Choptuik & Pretorius PRL(10)-a0908 [simulations of ultrarelativistic collisions];
Khuri PRD(09)-a0912 [spherically symmetric];
Mujtaba & Pope PLB(13)-a1211 [black rings];
Müller a1607 [proof for Einstein-Maxwell theory];
Saini & Stojkovic JCAP(18)-a1711 [in expanding spacetimes].

**Hoop Group** > see loops.

**Hopf Algebra** > s.a. generalized
coherent states; noether theorem; quantum
groups; renormalization.

* __Idea__: A bialgebra
equipped with an antiautomorphism satisfying a certain property.

* __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];
Andruskiewitsch & Schneider AM(10)
[finite-dimensional pointed, classification]; Balachandran et al 10;
Underwood 11;
Radford 11.

@ __And differential equations, dynamical systems__: Cariñena et al IJGMP(07)-m.CA/07.

@ __And non-commutative geometry__: Connes & Kreimer CMP(98) [and renormalization];
Várilly ht/01-ln;
Aschieri ht/07-ln;
Tanasa CQG(10)-a0909 [and spin-foam models];
Kovačević & Meljanac JPA(12)-a1110;
Dubois-Violette & Landi CMP-a1201 [Weil algebra of a Hopf algebra];
> s.a. quantum spacetime.

@ __In quantum field theory__: Connes & Kreimer LMP(99)ht,
LMP(01)
[Feynman graphs and renormalization];
Kastler mp/01-proc [Connes-Moscovici-Kreimer];
Sardanashvily qp/02 [and Fock representation];
Weinzierl EPJC(04)ht/03-conf;
Chryssomalakos ht/04-conf [applications];
Kreimer & Yeats NPPS(06)ht [and short-distance structure];
Van Suijlekom LMP(06) [and renormalization group in QED];
Brouder MN(09)ht/06;
Prokhorenko a0705
[for non-abelian gauge theories]; Mestre & Oeckl CM-a0808-proc [and combinatorics of connected graphs];
Duchamp et al JPCS(11)-a1011
[polyzeta functions and Euler's constant *γ*];
Solomon et al PS(10)-a1203 [for quantum statistical mechanics];
Stigner PhD-a1210
[in conformal field theory]; Brouder et al a1502-conf
[Borcherds geometric version of renormalized perturbative quantum field
theory]; Basti et al a1701 [*q*-deformed Hopf coalgebras and Hopf Algebras]; > s.a. algebraic quantum
field theory; quantum field theory formalism.

> __And gravity__:
see modified approaches to quantum gravity; self-dual
gravity.

> __Online resources__:
see Wikipedia page.

**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.

> __Online resources__:
see Wikipedia page.

**Hopf Fibration**

* __Idea__: The
fibration of S^{3} as a twisted S^{1}-bundle
over S^{2}, or S^{7}
as a twisted S^{3}-bundle over S^{4}.

@ __In physics__: Urbantke JGP(03)
[overview];
Lyons a0808
[conventions and unifying framework]; > s.a. liquid
crystals.

> __Online resources__:
see Wikipedia page.

**Hopf Invariant**

**Hopf Sphere Theorem** >
s.a. spheres [pinching problem and related
results on topological spheres].

$ __Def__: Any
compact, simply connected Riemannian manifold with constant curvature 1 is
isometric to the standard sphere.

@ __References__: in Brendle & Schoen BAMS(11).

**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.

**Hořava
(Hořava-Lifshitz) Gravity**

**Horismos** \ s.a. spacetime
subsets.

$ __Def__: The future
horismos of a point *p* in spacetime is *E*^{+}(*p*):=
*J*^{ +}(*p*) \ *I*^{
+}(*p*); Analogously for the past horismos.

* __Horismos relation__:
As a relation, \(q\in E^+(p)\)
is indicated by *p* → *q*; It is not transitive.

* __Characterization__:
The future horismos 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 the boundary of *J*^{
+}(*p*) or *I*^{ +}(*p*).

@ __References__: Minguzzi CQG(09)-a0904
[horismos relation as generator of causal relation in distinguishing
spacetimes].

**Horizon** > s.a. event
horizon; isolated and dynamical
horizon.

**Horizon Problem** > s.a. inflation.

* __Idea__: A problem
in the standard model for cosmology, the fact that the universe appears to
be homogeneous and isotropic on scales at which different points have not
been in causal contact; The most popular solution is provided by
inflationary scenarios.

@ __Proposed solutions__: Romano a0811-wd
[inhomogeneities]; Salesi PRD(12)-a1110
[Lorentz-violating dynamics]; Lolli Savi a1704 [there is no problem].

> __Online resources__:
see NCSA page;
Wikipedia page.

**Horizontal Tensor Field** > see tensor
fields.

**Horndeski Action / Theory **
> s.a. Effective Field Theory;
lovelock gravity; scalar-tensor
gravity.

* __Idea__: The most
general form of the action for a scalar-tensor gravitational theory (or a
vector-tensor one) that leads to second-order field equations in 4D (and
the vector field respects the gauge symmetry), thus evading Ostrogradsky
instabilities; It is a natural extension of the well known scalar-tensor
theories, and is also known as "Generalized Galileons".

* __And general
relativity__: These theories usually rely on non-linear screening
mechanisms to recover general relativity in regions of high density.

* __Special types__:
Theories with a sub-class of Lagrangians that enjoy the very special
property of self-tuning are called Fab Four.

* __Phenomenology__: Severely constrained after the multimessenger observations of GW170817].

@ __Scalar-tensor theory__: Horndeski IJTP(74);
Deffayet et al PRD(11);
Avilez & Skordis PRL(14)-a1303 [cosmological constraints];
Koyama et al PRD(13)-a1305 [effective theory for the Vainshtein mechanism];
Peng PLB(16)-a1511 [off-shell Noether current and conserved charge];
Papallo & Reall PRD-a1705 [local well-posedness of the initial value problem].

@ __And disformal transformations__: Bettoni & Liberati PRD(13)-a1306;
Bettoni a1405-proc.

@ __Vector-tensor theory__: Barrow et al JHEP(13) [and cosmology];
Jiménez et al JCAP(13)-a1308 [stability].

@ __Special types__: Babichev et al CQG(15)-a1507 [extended Fab Four theories];
McManus et al JCAP(16)-a1606 [conditions for Einstein gravity limit];
Linder JCAP(18)-a1801 [no-slip subclass].

@ __Extended / generalized theories__: Gleyzes et al PRL(15)-a1404,
Lin et al JCAP(14)-a1408,
Gleyzes et al JCAP(15)-a1408 [new class of extended theories without Ostrogradski instabilities];
Ohashi et al JHEP-a1505 [bi-scalar extension];
Sakstein et al JCAP(16)-a1603 [Beyond Horndeski theories, constraints from galaxy clusters];
Langlois a1707-proc [higher-order theories];
> s.a. higher-order lagrangian theories.

@ __Solar-system phenomenology__: Hohmann proc(16)-a1508 [and parameterized post-Newtonian limit];
Bhattacharya & Chakraborty PRD(17)-a1607 [constraints].

@ __Astrophysical phenomenology__: Koutsoumbas et al PRD(17)-a1512 [collapse];
Silva et al IJMPD(16)-a1602,
Maselli et al PRD(16)-a1603 [black holes and neutron stars];
Ezquiaga & Zumalacárregui a1710 [constraints from GW170817].

@ __And cosmology__: Martín-Moruno et al JCAP(15)-a1502;
De Felice et al JCAP(15)-a1503;
Bellini et al JCAP(16)-a1509,
Salvatelli et al JCAP(16)-a1602 [constraints];
Rinaldi PDU(17)-a1608 [mimicking dark matter].

**H _{p} Spaces**

*

$

\[\Vert f\Vert_p := \sup_{r<1}\Big({1\over2\pi}\int_0^{2\pi}{\rm d}\theta\,\big|\,f(z)\big|^{\,p}\Big)^{1/p} < \infty\;.\]

* __Special case__:
*H*_{∞} is the ring of bounded
holomorphic functions on Δ, with the sup norm.

@ __Text__: Koosis 80.

> __Online resources__:
see Wikipedia page.

**HR (Hertzsprung-Russell) Diagram** >
see star clusters; star
properties; star types; history
of astronomy [Russell's diagram].

**Hubbard Model**

* __Idea__: A model
of interacting particles in a lattice, used to describe the transition
between conducting and insulating behavior, and high-temperature
superconductors; As in the Ising model, the Hubbard model puts electrons
on a simple lattice, but in this case the electrons are allowed to hop
from site to site; The model also insists on a quantum-mechanical
treatment of the interactions between electrons; These two features make
the Hubbard model a much harder nut to crack.

* __Rules__: Each
site can be in one of four occupation states (no electrons, one up
electron, one down electron, a pair of electrons with opposite spins); An
electron can hop to any neighboring site, provided the move is allowed by
the exclusion principle.

* __History__:
Formulated by John Hubbard in the 1960s, it has since become a "standard
model" of condensed-matter physics and materials research; 2009, Given the
difficulties encountred in solving or simulating the model numerically,
several groups have built macroscopic replicas of the Hubbard lattice out
of light waves and trapped atoms, thus creating a physical analog of an
abstract model that in turn represents another physical system.

@ __General references__: Montorsi ed-92 [reprints];
Schupp PLA(97) [quantum symmetry];
Hou et al NPB(00) [SU(3)];
Wojtkiewicz JSP(09) [upper and lower bounds on partition function];
Peets et al PRL(09)
+ news UPI(09)aug [limitations];
Hayes AS(09)nov [I];
de Leeuw & Regelskis a1509 [algebraic approach];
Wecker et al PRA(15) [quantum simulation].

@ __1D__: Deguchi et al PRP(00);
Lieb & Wu PhyA(03)cm/02-proc;
Essler et al 05;
Wang & Liu PLA(09) [ground state properties].

@ __2D__: Giuliani & Mastropietro CMP(10) [on the honeycomb lattice];
Claveau et al EJP(14) [square lattice, mean-field solution];
Cocchi et al PRL(16)
+ Gemelke Phy(16) [equation of state].

@ __Related topics__: Büchner PRL(10)
+ Phy
[for ultracold atoms]; Assaad & Herbut PRX(13)
[new continuous quantum phase transition and correlation between magnetic order and electrical insulation in Mott insulators];
Murmann et al PRL(15)
[using ultracold neutral atoms to simulate the Fermi-Hubbard model];
> non-extensive statistics [Hubbard dimers].

> __Online resources__:
see Wikipedia page.

**Hubble's Constant / Law **>
see cosmological expansion.

**Hubble Diagram** > 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 \(\pi_q(X) = 0\) for all *q* < *n*,
with *n* ≥ 2, then H_{q}(*X*)
= 0 as well for all *q* < *n*, and \({\rm H}_n(X) = \pi_n(X)\);
This is not true for *n* = 1, but for that case what is true is that
\(\pi_1(X)/[\pi_1(X), \pi_1(X)] = {\rm H}_1(X)\), or H\(_1(X)\) is the
abelianization of \(\pi_1(X)\).

@ __References__: Eilenberg AM(44);
Eilenberg & MacLane AM(45).

**Hurwitz Zeta Function** > see Zeta Function.

**Husain-Kuchař Model** > see types
of gauge theories; BF theory.

**Husimi Phase Space Distributions / Functions**
> s.a. locality in quantum mechanics; quantum
mechanics in phase space; Star Product [Husimi product].

* __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 [electron in a constant magnetic field];
Toscano et al PRS(08)-a0705 [intermediate Husimi-Wigner representation];
Calixto et al PRA(12)-a1409 [and model quantum phase transitions];
> s.a. types of coherent states.

> __Online resources__:
see Wikipedia page.

**Hydrodynamic Decomposition** > see decomposition.

**Hydrodynamics** > see fluids / also computational
physics; hydrodynamic formulation of quantum theory.

@ __References__: Marchetti et al RMP(13) [continuum models for soft active matter].

**Hydrogen Atom** > see hydrogen
/ canonical quantum mechanics; deformation
quantization; interactions; quantum
systems; wigner functions.

**Hyperbola** > see conical
sections.

**Hyperbolicity of Differential Equations ** > see partial
differential equations; canonical formulation
of general relativity; numerical general relativity.

**Hyperbolicity in Other Areas** > see graph
invariants and types.

**Hyperbolic Functions**

* __Def__: The
functions \(\sinh\alpha:= {1\over2}\)(e\(^\alpha\)
− e\(^{-\alpha}\)), \(\cosh\alpha:= {1\over2}\)(e\(^\alpha\) + e\(^{-\alpha}\)).

* __Basic properties__:
\(\cosh^2\alpha - \sinh^2\alpha = 1\), d(cosh*α*))/d*α*
= sinh*α*, d(sinh*α*)/d*α* = cosh*α*.

* __Relationship with
trigonometric functions__: Given by sin i*α* = i sinh*α*,
cos i*α* = cosh*α*, tan i*α* = i tanh*α*.

**Hyperbolic Geometry** > see riemannian
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 Algebra / Number**
> s.a. algebra; spinors.

* __Idea__: An
element of an algebra over a field where the field is the real numbers or
the complex numbers; Examples are the number systems called quaternions,
tessarines, coquaternions, biquaternions, and octonions.

@ __References__: Hertig et al a1406
[and applications to quantum theory]; Sepunaru a1501
[in quantum theory].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**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 _{2}*F*_{1}(*a*,*b*;*c*,*x*)];
Ancarani & Gasaneo JPA(10) [derivatives].

@ __Confluent hypergeometric functions__: Saad & Hall JPA(03) [integrals].

@ __Generalized__: Ruijsenaars CMP(99),
CMP(03),
CMP(03);
Tarasov & Varchenko LMP(05) [identities];
> s.a. integrals.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Hypergraph** > see graph theory
/ types of quantum states [hypergraph states].

**Hypergravity** > s.a. BRST [quantization].

* __Idea__: A first
version was the spin-(5/2) gravity analog of supergravity, 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 in(99)gq;
Fuentealba et al JHEP(15)-a1508 [3D, asymptotic structure].

**Hyperkähler Structure** >
s.a. symplectic structure.

$ __Def__: A manifold *M*
with 3 complex structures *J*_{1},
*J*_{2}, *J*_{3},
satisfying *J*_{i} *J*_{j} = \(\epsilon_{ijk}^~\, J_k^~\), 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];
Gaeta & Rodríguez JMP(14)-a1512 [canonical transformations].

**Hyperlogarithms** > see Feynman Diagrams.

**Hypermomentum** > see fluid.

**Hypernumber**

* __Idea__: An
extension of the concept of number, which includes infinite quantities.

@ __References__: Burgin 12.

**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:
\(m \sim 10^{-9}\) eV; As of 1986, there is little theoretical motivation for it.

@ __References__: Aronson, Cheng, Fischbach & Haxton PRL(86).

**Hyperradiance** > see Superradiance.

**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 \(l^a\, \nabla_{\!a}\, l^b = \kappa\, l^b\)
on the surface; Under a rescaling of *l*^{ a},
the "surface gravity" transforms as \(\kappa \mapsto f\, \kappa + l^a\nabla_{\!a}\, f\).

@ __Null hypersurfaces__: Nurowski & Robinson CQG(00)gq [invariants of null surfaces];
Jezierski in(04)gq [null, rev];
Navarro et al JGP(13) [in Minkowski space];
> s.a. horizons [geometry].

@ __Spacelike hypersurfaces__: Harris CQG(88) [closed and complete, in Minkowski];
Izumiya & Takahashi JGP(07) [in a constant-curvature space, like de Sitter];
Hu et al DG&A(07)
[spacelike, constant scalar curvature, in de Sitter spacetime];
Tibrewala CQG(15)-a1403 [spherically symmetric spacetimes, non-uniqueness of representation of generators of deformations].

@ __Conformally flat__: Garat & Price PRD(00),
Valiente Kroon PRL(04)
[not in Kerr spacetime].

@ __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 curvature [extremal hypersurface];
horizons; spacetime subsets.

**Hypersurface-Orthogonal Vector Field** > see vector field.

**Hypothesis Testing** > see statistics.

**Hysteresis** > s.a. meta-materials.

* __Remark__:
It 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)oct [ferromagnets, misconceptions].

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

send feedback and suggestions to bombelli at olemiss.edu – modified 11 jun 2018