**Topics, J**

** j-Function** > see finite groups.

**Jack Polynomials**

* __Remark__: Related to the eigenfunctions
of a well-known exactly solvable quantum many-body system of the Calogero-Sutherland type.

@ __Generalizations__: Desrosiers et al CMP(03)ht/02,
AiM(07)mp/05 [in superspace];
Langmann mp/05-proc [and review];
Lapointe & Mathieu Sigma(15)-a1503 [double Jack polynomials].

**Jackiw-Pi Theory**

* __Idea__: A 3D non-Abelian gauge model for massive vector fields.

@ __References__: Dayi MPLA(98) [Hamiltonian formulation];
Del Cima JPA(11) [symmetries];
Nikoofard & Abreu AdP(16)-a1601 [BV quantization].

**Jackiw-Rebbi Model**

* __Idea__: A (1+1)-dimensional model
that Jackiw and Rebbi considered to show that half-integral fermion numbers are possible
due to the presence of an isolated self charge conjugate zero mode.

@ __References__: Charmchi & Gousheh PRD(14)-a1402 [complete spectral analysis],
NPB(14)-a1404 [massive].

**Jackiw-Teitelboim Theory** > see 2D gravity; 2D quantum gravity.

**Jacobi Algebra / Bracket / Manifold / Structure**
> see poisson structure.

**Jacobi Conjecture**

@ __References__: Abdesselam AHP(03)m.CO/02 [perturbative quantum field theory approach].

**Jacobi Elliptic Functions** > see Elliptic Functions.

**Jacobi Equation, Field** > see geodesics.

**Jacobi Metric, Hamiltonian**
> s.a. variational principles in physics [Jacobi principle].

**Jacobi Principal Function**

@ __In quantum mechanics__: Ferraro JPA(99)qp/96.

**Jamiolkowski Criterion** > see operator theory.

**Jamming** > a phase transition in granular metamaterials.

**Janis-Newman Algorithm** > see under Newman-Janis Algorithm.

**Java, Javascript** > see programming languages.

**Jarzynski Equality**

* __Idea__: A statement relating
transient irreversible processes to infinite-time free energy differences.

@ __References__: Gittes AJP(18)jan-a1704 [two examples of applications];
Bartolotta & Deffner PRX-a1710 [for driven quantum field theories].

**Jaynes-Cummings Model** > see atomic physics.

**Jeans Instability, Model, Swindle** > s.a. Instability.

* __Idea__: An instability in astrophysics,
which causes the gravitational collapse of interstellar gas clouds and star formation;
The name "Jeans swindle" alludes to the fact that Jean's original model and analysis contained a flawed assumption.

@ __References__: Kiessling AAM(03)ap/99;
Ershkovich a1108 [the Jeans model is self-consistent];
Falco et al MNRAS(13)-a1210 [formal justification];
Arbuzova et al PLB(14)-a1406 [Jeans instability in classical and modified gravity];
> s.a. Modified Gravity.

> __Online resources__:
see Wikipedia page.

**Jebsen-Birkhoff Theorem** > see under Birkhoff's Theorem.

**Jericho Effect** > see locality [quantum localization].

**Jerk**

$ __Def__: The time rate
of change of acceleration, d^{3}*x*/d*t*^{3}.

* __And chaos__: Jerk is
the lowest order effect in particle motion that can give rise to chaos.

@ __References__: Schot AJP(78)nov;
Sandin TPT(90)jan;
Leen AJP(94)may [letter];
Gottlieb AJP(96)may [question],
von Baeyer ThSc(98)jan [and chaos].

> __Online resources__:
see John Baez's page [including names for further derivatives: snap, etc].

**Jet (in astrophysics and cosmology)**

@ __References__: Gralla et al PRD(16)-a1504
[force-free solutions modeling rotating stars and black holes immersed in the magnetic field of a thin disk];
Pian a1512-proc [recent progress];
Parfrey et al PRL(19)
+ Penna Phy(19) [first-principles simulations].

> __Related topics__: see matter and radiation
around black-holes; acceleration and bianchi-I
spacetimes [cosmic jets]; star formation and evolution.

**Jet (in differential geometry)**

$ __Def__: The *n*-th
jet of a function *g* defined on a manifold *M* is the function
itself together with all its partial derivatives up to the *n*-th order,
or *j*^{(n)}*g*
= (*g*, ∂_{i} *g*,
∂_{i}∂_{j} *g*,
..., up to *n*-th derivatives).

* __Idea__: Whereas the *n*-th
derivatives of a function by themselves are not a geometrical object, the *n*-th
jet is geometrical; given its values in some coordinate system, one can calculate
its values in any other.

@ __Jet bundle__: in Mather AM(69);
Saunders 89;
Sardanashvily mp/02-ln [in classical and quantum field theory];
Tulczyjew mp/06-proc [modification of Ehresmann's jet theory];
Sardanashvily a0908-ln [and fiber bundles and Lagrangian theory];
> s.a. field theory.

**Jet (in particle physics)** > see QCD phenomenology.

**Johnson-Mehl Model** > see random tilings.

**Jones Polynomial** > see knot invariants.

**Jordan Algebra** > s.a. formulations
of quantum mechanics [Jordan geometry]; lie algebras [invariants].

* __Idea__: An algebra in which
the product satisfies [*a*, *b*, *a*^{2}] = 0
for all *a*, *b*, with [*a*,* b*, *c*]:= (*ab*)*c*
− *a*(*bc*) (the "associator"); This property is weaker than associativity.

@ __General references__: Raptis mp/01 [Jordan-Lie superalgebras];
Rios mp/05 [exceptional, spectrum];
Bremner a1008,
& Peresi a1008 [quasi-Jordan algebras];
Iordanescu a1106 [in mathematics and physics];
Hegazi & Abdelwahab LA&A(16)-a1401 [5D nilpotent, classification];
Carotenuto et al a1803 [differential calculus];
> s.a. Jordan-Lie-Banach Algebra.

@ __ And quantum mechanics__: Townsend proc(85)-a1612 [rev];
Niestegge IJTP(04)-a1001,
FP(09) [and quantum observables];
Wilce a1110 [from operational or physical principles];
> s.a. entangled quantum systems.

**Jordan Curve**

* __Idea__: A simple closed
curve, i.e., a non-self-intersecting continuous loop in the plane.

* __Jordan curve theorem__:
Every simple closed curve divides the plane into exactly two components,
an "interior" and an "exterior" region.

> __Online resources__:
see Wikipedia page.

**Jordan Frame** > see scalar-tensor theories of gravitation.

**Jordan Normal Form of a Matrix** > see matrices.

**Jordan Operator Algebras** > see operator theory.

**Jordan Theory** > see higher-dimensional gravity;
kaluza-klein theory; scalar-tensor theories.

**Jordan Triple** > see spin.

**Jordan-Lie Superalgebra** > see algebra.

**Jordan-Lie-Banach Algebra**
> see Algebraic Quantum Theory.

**Jordan-Wigner Transformations**

* __Idea__: A mapping between
spin chains and fermionic systems in one dimension.

@ __References__: Backens et al a1810 [extension to arbitrary tree structures].

**Josephson Effect** > s.a. superconductivity.

* __Idea__: The tunneling effect proposed
in 1962 by Brian Josephson of Cooper pairs between two superconductors separated by a thin
insulator, maintaining phase coherence between the two superconductors; The difference
*δ* between the phases on the two sides is related to the supercurrent *I*
through the barrier by sin *δ* = *I*/*I*_{0},
where *I*_{0} is the critical current, the maximum current
that the junction can sustain; This was quickly verified in the lab and Josephson was awarded
the 1973 Nobel Prize for Physics

* __Applications__: A device based on this effect
is now called a Josephson junction and it has become an important technology in its own right,
for example in superconducting quantum interference devices (SQUIDs).

* __Thermal Josephson effect__: Heat transport
across a gap between two superconductors; Predicted in 1965 and measured in the lab for the first time.

@ __References__: in Tinkham 75;
Anandan & Pati PLA(97) [geometry];
news pw(07)oct [in atomic gas];
news pw(12)apr [magnetic version];
Williams et al PRL(12)
+ Moore Phy(12)
[in hybrid superconductor-topological insulator devices, and Majorana fermions].

@ __Thermal Josephson effect__: Giazotto & Martínez-Pérez Nat(12)dec
+ news pw(13)jan [measurement, and heat flow from colder to hotter].

**Jost Functions**

@ __References__: Damanik & Simon IMRN-m.SP/05 [for Jacobi matrices].

**Joule Expansion**

* __Idea__: The
free adiabatic expansion of a gas against a vacuum.

@ __References__: Camalet PRL(08) [from the principles of quantum mechanics].

**Joule's Law**

* __Joule's first law__: The amount
of heat generated by an electric current *I* flowing through a conductor of
resistance *R* for a time *t* is \(Q = I^2R\,t\).

* __Joule's second law__: The internal
energy of an ideal gas does not change if volume and pressure change, but does change
if the temperature changes.

> __Online resources__:
see Wikipedia page.

**Joule-Thomson Effect / Process**

* __Idea__: The temperature change
of a gas or liquid when it is forced through a valve or porous plug while kept
insulated, so that no heat is exchanged with the environment (throttling process);
The effect was discovered in 1852 by Lord Kelvin (William Thomson), following earlier
work by Joule on Joule expansion; it is applied in refrigerators, air conditioners,
heat pumps, and liquefiers.

> __Online resources__:
see Wikipedia page.

**Julia Set** > see fractals.

**Junction Conditions**
> see boundaries in field theory;
metric matching.

**JUNO (Jiangmen Underground Neutrino Observatory)** > see neutrinos.

**JWKB Approximation** (for Jeffreys, Brillouin, Wentzel and Kramers)
> see under WKB Approximation.

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send feedback and suggestions to bombelli at olemiss.edu – modified 12 feb 2019