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 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; 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, Newman GRG(18) [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]; Sbisà RBEF(20)-a2010 [on van Hove's 1952 article]; Maiezza & Vasquez a2011 [and renormalization].
> 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; Brum & 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 LMP(17)-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 γaa+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 S3]; 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; Foa Torres Phys(20) [in a 3D electronic system].
@ 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]; Oancea et al a1904 [gravitational spin Hall effect, rev].

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).

Hamilton-Jacobi Theory

Hamilton-Randers Systems > see origin of quantum mechanics.

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]; Qureshi & Ushba Quanta(17)-a1509 [for massive particle wave packets]; Aspect proc(19)-a2005 [with photons and atoms].

$ Def: An n-dimensional k-handle is the topological manifold Dk × Dnk ; It is often used as attached to another n-manifold.

Hankel Functions > see bessel functions.

Hannay's Angle > see phase.

Hard Spheres > see classical systems; chaotic systems.

Hardy Spaces / Algebras
$ Hardy algebra: A weak *-closed subalgebra of L2(X, dm), for some finite positive measure space (X, S, dm).
@ 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: Hardy PRL(92); 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]; Engelbert & Angelo FP(20)-a1910 [and quantum irrealism].
@ Consequences and applications: Mansfield PRA(17)-a1608.
> Related topics: see relativistic quantum mechanics; types of quantum measurements [weak].

Harmonic Analysis, Coordinates, Functions > see harmonic functions.

Harmonic Maps

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]; Klatzow et al PRL(19) + Anders Phys(19) [power increase induced by quantum coherence].
> 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 (bulk motion) 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; 2019, Quantum fluctuations of the vacuum have been shown to also transport heat; > s.a. Conductivity; Convection.
* 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: Hovhannisyan & Allahverdyan JSM(10)-a1007 [enhanced heat transfer, model].
@ Radiation: Miller et al PRL(15) + Messina Phy(15) [near-field radiative heat transfer].
@ Special systems: Hirschfeld & Scalapino Phys(10)aug [iron-based superconductor]; news sci(14)jul [perfect crystals that are poor heat conductors]; Cepellotti & Marzari PRX(16) + McGaughey Phy(16) [in electrical insulators, relaxons]; news sn(19)dec, pt(19)dec [vacuum fluctuations, mediated by the Casimir effect]; > s.a. Josephson Effect [backwards heat flow]; Lennard-Jones Fluid.
> Online resources: see Wikipedia page.

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 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 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 Picture / 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]; Amore & Fernández a2004.
> 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 a2 f(x), and when one looks for a solution u of the same frequency,

2u + k2u = −f(x) ,   with   k:= ω2/a2 .

@ 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 Theorem > 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) = k2 [\(1\over2\)(x2 + y2) + (x2y − \(1\over3\)y3)/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.

Herglotz Lagrangians > see lagrangian dynamics.

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.

$ 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 or naturalness 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 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 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; Kane PT(19)mar [importance].
@ 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 PRL(18)-a1802 [non-linearly realized discrete symmetries]; Patrascu a2002 [approach based on the renormalisation group and the ER-EPR interpretation of entanglement].

Higgledy-Piggledy > see physics.

Higgs Mechanism / Field / Boson

* Idea: The fast growth with the Higgs multiplicity of the cross section found in multi-Higgs boson production in scalar theories with spontaneous symmetry breaking at sufficiently large energies; It has been argued that this "Higgsplosion" solves the Higgs hierarchy and fine-tuning problems.
@ Problems with the idea: Monin a1808 [scenario impossible].

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 Gauge Theories > see types of gauge theory.

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 attributed 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 Hij = (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 L0 encloses all the other Lis 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.

* 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), tL ⊂ \(\mathbb R\), 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
$ 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 (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; Weatherall SHPMP-a1812 [development]; Johns a1907; Pooley a2009 [as an argument against spacetime substantialism].
@ 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]; D'Ariano a2102-in [in quantum 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.


* Idea: A generalization of tensors.
@ References: Moon & Spencer 86.

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

$ 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
$ Def: In its general form, the linearized relationship between the stress and strain tensors for an elastic object, \(\sigma_{ij} = \sum_{kl}C_{ijkl}\,\epsilon_{kl}\).
@ 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 < 4πGM 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; Hod EPJC-a2012.
@ 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]; Hod EPJC(18)-a1903 [in charged curved spacetimes]; Hod EPJC-a2101 [inverse hoop conjecture].

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]; > s.a. algebraic quantum field theory; quantum field theory formalism.
@ Variations, generalizations: Basti et al a1701 [q-deformed Hopf coalgebras and Hopf Algebras]; Zhou a2006 [for abstract finite sets].
> 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 S3 as a twisted S1-bundle over S2, or S7 as a twisted S3-bundle over S4.
@ 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 pq; 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 elf-tuning are called Fab Four; Einstein-dilaton-Gauss-Bonnet.
* Phenomenology: The theory is severely constrained after the multimessenger observations of GW170817].
@ General references: 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(17)-a1705 [local well-posedness of the initial value problem]; Kobayashi RPP(19)-a1901 [rev]; Helpin & Volkov a1906 [Palatini approach]; Kovacs & Reall a2003 [strongly hyperbolic]; Lu & Pang a2003 [as d → 4 limit of Einstein-Gauss-Bonnet gravity].
@ 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]; Duniya et al JCAP(20)-a1902 [beyond Horndeski theories on ultra-large scales]; Helpin & Volkov IJMPA-a1911-proc [metric-affine version]; > s.a. higher-order lagrangian theories.
@ Solar-system phenomenology: Hohmann proc(16)-a1508 [and parameterized post-Newtonian limit]; Bhattacharya & Chakraborty PRD(17)-a1607 [constraints]; Mukherjee & Chakraborty PRD(18)-a1712 [and Gravity Probe B].
@ 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]; > s.a. black-hole thermodynamics.
@ 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]; Kase & Tsujikawa IJMPD(19)-a1809 [dark energy, rev]; Frusciante et al PRD(18)-a1810 [and dark energy]; Shi et al PRD(20)-a1911 [and dark matter]; Linder a2104 [darkenergy].

Hp Spaces
* History: Studied first by Hardy in 1915 and named after him by Riesz, they have been extended to \(\mathbb R\)n, \(\mathbb C\)n and other topological spaces.
$ Def: If f is holomorphic on Δ:= {z = r exp{iθ} ∈ \(\mathbb C\), |z| ≤1}, then f ∈ Hp iff

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

* 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 Hq(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.

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.

Huygens' Principle

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(coshα))/dα = sinhα, d(sinhα)/dα = coshα.
* Relationship with trigonometric functions: Given by sin iα = i sinhα, cos iα = coshα, tan iα = i tanhα.

Hyperbolic Geometry > see riemannian geometry.

Hyperbolic Numbers > same as Perplex Numbers? see trigonometry [hyperbolic].

$ 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 2F1(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]; Rahman JHEP(19)-a1905 [uniqueness result for generalized Aragone-Deser hypergravity].

Hyperkähler Structure > s.a. symplectic structure.
$ Def: A manifold M with 3 complex structures J1, J2, J3, satisfying Ji Jj = \(\epsilon_{ijk}^~\, J_k^~\), 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.

* Idea: An extension of the concept of number, which includes infinite quantities.
@ References: Burgin 12.

Hyperons > see Baryons, neutron stars.

* 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.

@ References: Urbantke IJTP(89).

$ 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 af 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 probability; 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].

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