In General > s.a. causality; Refraction.
* Idea: Dispersion in general is the phenomenon by which something is distributed over a wide area in space; In optics (chromatic) dispersion is the separation of white light into colors when going through certain devices, due to the fact that the phase velocity (and therefore the index of refraction) of a wave in a medium depends on its frequency.
* Dispersion relation: The dependence of the index of refraction of a wave in a medium on frequency; A non-trivial dependence n(ω), or a non-constant n(ω) = v0 / vp with vp = dω/dk, is equivalent to a non-linear relation ω = f(k); Therefore, in general a dispersion relation is a relationship between the components of a wave vector $$k^a$$ that in particle-like terms is described by the shape of the "mass-shell" $$E = E({\bf p})$$.
* Remark: The expression "dispersion relation" is often used to denote integral relations of the Kramers-Kronig type – see below.
* Ordinary dispersion: The refractive index of a material changes with increasing wavelength; This stretches out the pulse and reduces the group velocity–the speed at which the peak of the pulse travels.
* Anomalous dispersion: It occurs in materials that absorb radiation in a certain range of wavelengths; On either side of this absorption band, n changes sharply with wavelength; In these regions, the components of radiation at the tail of the pulse interfere destructively, and the peak of the wave is effectively pushed forward.
@ References: Labuda & Labuda EPJH(14) [history of mathematical methods, Titchmarsh's theorem]; Zwicky a1610-ln [and analyticity].

In Classical Theories > s.a. phenomenology of gravity; types of wave equations; wave phenomena.
* Electromagnetic waves: A flat vacuum is non-dispersive, since vp = c if we neglect quantum field theory effects; In a medium, several effects can lead to dispersion.
@ General references: Hagedorn 64; Gratton & Perazzo AJP(07)feb [from dimensional analysis].
@ Electromagnetic waves: Lucarini et al RNC(03) [optics]; Marino et al AP(07)phy [in a regular lattice of oscillators]; Itin PLA(10) [anisotropic media].
@ Other types of waves: Harko & Cheng ApJ(04)ap [multidimensional cosmology]; Amore & Raya Chaos(06)mp/05 [non-linear Klein-Gordon equation]; Likar & Razpet AJP(13)apr [surface waves on water]; Baggioli et al a1904 [gapped momentum dispersion relations].
> Related topics: see dirac equation; FLRW spacetimes.

In Quantum Mechanics > s.a. schrödinger equation [solutions].
* Idea: Free particle propagation is dispersive, since p = $$\hbar$$k and E = $$\hbar$$ω, related by E = p2/2m, so ω = $$\hbar$$k2/2m and vp = p/m.

In Quantum Theory > s.a. photon.
* In QED: Quantum field theory effects can change the relation gab ka kb = 0 into a dispersive one; An effective coupling to the background matter stress-energy and the Weyl tensor, or a temperature effect, replaces it by one of the form

gab ka kb = f1 Tab ka kb + f2 Cacbd ka kb εc εd ,

where εa is the polarization vector.
@ References: Gharibyan PLB(05)he/03 [possible observation of photon dispersion in vacuum]; Pomeau a1409 [in QED, estimate and suggested test].

Quantum-Gravity-Motivated Modifications > s.a. modified lorentz symmetry.
* Idea: Several models have been proposed in which quantum-gravity effects modify the usual dispersion relations, such as DSR models, and ones in which Lorentz invariance is broken and one has, for example for photons,

E2 = p2 + ξphoton (E3/mP) ,

or DSR models.
@ General references: Rätzel et al PRD(11)-a1010 [restrictions on possible forms].
@ Phenomenology: Shore CP(03)gq [QED in curved spacetime]; Buhmann & Welsch PQE(07)qp/06 [macroscopic QED]; Biswas & Faruk a1706 [thermodynamics of a photon gas].
> Related topics: see Chandrasekhar Limit; graviton; matter phenomenology and photon phenomenology in quantum gravity; phenomenology of uncertainty relations; Spectral Dimension.

Kramers-Kronig Relations
* Idea: Integral relations between the real and imaginary parts of the index of refraction n(ω) of waves in a medium, related to the condition that the propagation of the waves be causal – They relate a dispersive process to an absorption one; > s.a. equivalence principle.
@ General references: Wigner ed-64; in Arfken 85; in Weinberg 95; in Alastuey et al 16.
@ Related topics: Kowar a0901 [and causality]; Bohren EJP(10) [history and meaning]; Akyurtlu & Kussow PRA(10) [and negative index of refraction]; Kinsler EJP(11) [and spacetime causality]; Llosa & Salvat a1905 [beyond the optical approximation].