Dispersion |

**In General** > s.a. causality; Refraction.

* __Idea__: Dispersion is 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*)
= *v*_{0} / *v*_{p}
with *v*_{p} = d*ω*/d*k*,
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*(**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__:
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 *v*_{p} = *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].

> __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* = *p*^{2}/2*m*,
so *ω* = \(\hbar\)*k*^{2}/2*m*
and *v*_{p} = *p*/*m*.

**In Quantum Theory** > s.a. photon.

* __In QED__: Quantum field
theory effects can change the relation *g*_{ab}* k*^{a}* k*^{b}
= 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

*g*_{ab} *k*^{a} *k*^{b} = *f*_{1} *T*_{ab} *k*^{a} *k*^{b}
+ *f*_{2} *C*_{acbd}*
k*^{a}* k*^{b} *ε*^{c} *ε*^{d}
,

where *ε*^{a} 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,

*E*^{2} = *p*^{2} + *ξ*_{photon} (*E*^{3}/*m*_{P}) ,

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].

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send feedback and suggestions to bombelli at olemiss.edu – modified 14 jun 2017